ICT use in the teaching of mathematics: Implications for professional development of pre-service teachers in Ghana Abstract Included in the contemporary mathematics curricula in Ghana is the expectation that mathematics teachers will integrate technology in their teaching. However, importance has not been placed on preparing teachers to use ICT in their instruction. This paper reports on a study conducted to explore the feasibility of ICT use in mathematics teaching at senior high school levels in Ghana. Interviews and survey data were used for data collection. Preliminary results showed that mathematics teachers in Ghana do not integrate ICT in their mathematics instruction. Among the major perceived barriers identified were: Lack of knowledge about ways to integrate ICT in lesson and Lack of training opportunities for ICT integration knowledge acquisition. To overcome some of these barriers, opportunities of a professional development arrangement for pre-service mathematics teachers were explored. Findings from the study revealed specific features of a professional development scenario that matters for ICT integration in mathematics teaching in the context of Ghana. In Ghana, mathematics is a compulsory subject at all levels in pre-university education. Due to its importance the government is committed to ensuring the provision of high quality mathematics education. Various attempts have been made in the past to improve the achievement of mathematics in schools. The most recent is seen in the New Educational Reforms (Anamuah-Mensah National Education Review Committee Report 2002) of which implementation started in September, 2007 . The new curriculum in Mathematics at the Senior High School (SHS) places emphases on skill acquisition, creativity and the arts of enquiry and problem solving. It aims at developing in the student the ability and willingness to perform investigations using various mathematical ideas and operations. As part of the reforms the curriculum places a lot of emphasis on Information and Communication Technology (ICT) as a tool for teaching mathematics (MOESS 2007) . It is therefore, designed to meet expected standards of mathematics in many parts of the world. In spite of government efforts, mathematics has not undergone much change in terms of how it is presented. These reflect consistently in low achievement levels in mathematics among students at the high school levels. Results from the Trends in International Mathematics and Science Study (TIMSS) in 2003 and 2007 at the junior high school level (grade 8 equivalents) are instances of poor mathematics achievement in the country. In the aforementioned study, Ghana's eighth graders were ranked 43rd among 44th and 46th among 47 countries that participated in the study in 2003 and 2007 respectively (Mullis et al. 2004 (Mullis et al. , 2008 . The situation is not too different in SHS's. For many years the failure rate in mathematics has been dramatically high in SHS's. The low scores of students' over the years in the Senior Secondary School Certificate Examination attest to this (Ottevanger et al. 2007 ). In Ghana not many studies have been conducted to explain such poor students' performance in mathematics. Ampiah et al. (2004) reported that both pre-service and in-service programmes in mathematics predominantly reflect teacher-centred approaches to learning. Curriculum documents in this context suggest that teachers should start every lesson with a practical problem to help students acquire the habit of analytical thinking and the ability to apply knowledge in solving practical problems (MOE 2000) and also make use of the calculator and the computer for problem solving and investigations of real life situations (MOESS 2007) , but this orientation to teaching and learning requires more than recommendations contained in syllabuses. More particularly the report on Developing Science, Mathematic and ICT (SMICT) education in Sub-Saharan Africa suggested changes to the teacher's instructional role from presenter of knowledge and the use of drill-oriented methods to participatory teaching and learning (Ottevanger et al. 2007 ). On a much broader note, research conducted in other Sub-Saharan Africa highlights some of the factors responsible for poor students' achievement in mathematics: poorly-resourced schools; large classes; a curriculum hardly relevant to the daily lives of students; a lack of qualified teachers; and inadequate teacher education programmes (Ottevanger et al. 2007 ).The government of Ghana recognizes the need for teacher support for mathematics teachers in various ways. He considers ICT literacy as an engine for accelerated development outlined in the Ghana Information and Communication Technology for Accelerated development (Ghana ICT4AD Policy document 2003) . Ghana introduced ICT into the school curriculum in September 2007 following the recommendations of the ICT4AD document and the AnamuahMensah National Education Review Committee Report (2002) . Both documents highlight the importance of integrating ICT into the curriculum at all levels. As a result, the government and other institutions have invested huge sums of money in procurements of computers and establishment of computer labs in most SHS's, but it is still unclear whether these computers are being used effectively by teachers in their instruction. Thus the question of whether mathematics teachers need any further support to be able to integrate effectively the use of ICT in their daily teaching routines remains unanswered. The overall goal of the present study was to explore the feasibility of ICT use in mathematics classrooms in Ghana as part of an ongoing research project to design a professional development programme for pre-service teachers. The relevance of this study was to (1) provide an understanding of the context of mathematics teachers in the SHS's in Ghana regarding ICT integration in mathematics lessons and (2) determine the features of an ICT intervention that fits the realities in the SHS's that can prepare pre-service teachers to effectively design and implement ICT in teaching mathematics. The study was guided by the following questions:. 1. What are the barriers of ICT use in teaching mathematics in SHS's in Ghana? 2. What are the needs of pre-service and in-service mathematics teachers in teaching mathematics with ICT in SHS's in Ghana? 3. What are the opportunities of ICT use in the teaching of mathematics in SHS's in Ghana? Despite the positive policy statements regarding the need for technology in Ghanaian Senior High Schools, and in the mathematics curriculum in particular, the implementation of technology in educational practice will be a major effort. In this study we explored the barriers, learning needs and opportunities for preparing mathematics teachers to integrate ICT in their instructions. We collected data from in-service teachers who were teaching in schools which had a relative good technology infrastructure. This choice was made because we expect that these schools (and their teachers) will be the first ones to implement the ambitions of the Ghanaian government regarding technology integration in the Senior High School. This is also confirmed by the school principals that were interviewed as part of the study. Also we collected data from pre-service teachers. To realize the implementation of technology, pre-service teachers need to be well prepared to use technology in their future teaching. For this reason we invited pre-service teachers from one of the two main teacher education institutions to participate in the study. Although findings from this study can not be generalized to Ghanaian in-and pre-service mathematics teachers, we believe that they provide information about conditions and opportunities to realize the first steps in the process of implementation of technology in Senior High Schools. Fullan (2007) indicated that school-based professional community can offer support and motivation to teachers as they work to overcome the tight resources, isolation, time constraints and other obstacles they commonly encounter in today's schools. He maintains that within a strong professional community, for example, teachers can work collectively to set and enforce standards of instruction and learning. The results showed a number of barriers identified to be reasons why mathematics teachers in this study did not integrate ICT in their instruction. Among others, lack of ICT knowledge in integration: lack of knowledge about ways to integrate ICT in lesson and lack of training opportunities for ICT integration knowledge acquisition (Ottevanger et al. 2007; Snoeyink and Ertmer 2002; Jones 2004) were the major perceived barriers identified by both group of teachers. Particularly, the pre-service teachers in this study reported fairly lower attitudes about knowledge of technology use in instruction than the in-service teachers. In-service training programmes organised for in-service teachers as was reported by school heads could explain this difference. Lack of ICT infrastructure (Snoeyink and Ertmer 2002; Jones 2004) were also reported by the participant to be real challenges faced in the mathematics classrooms in Ghana. The study showed that schools lacked common mathematical software (such as derive, graphic calculus, geometer's sketchpad etc) used in teaching mathematics. It was particularly surprising to find that no classroom was identified to be web-based although few computer laboratories were supported by internet connectivity. Low ICT access levels as reported by the teachers are possible barriers for their inability to integrate ICT in instruction. Similar study conducted in Ghana (Agyei and Voogt, submitted) reported low levels of ICT integration of these teachers as a result of their low ICT competencies and access levels. Another barrier to ICT integration as reported by the teachers was the teaching strategy used in SHS's. The most frequently used strategy for teaching as reported was the chalk and talk approach (Ottevanger et al. 2007; Ampiah et al. 2004) ; in which teachers did most of the talking and intellectual work, while students were passive receptacles of the information provided. Both in-and pre-service teachers barely differed in opinions on this subject. These teachers also have been taught in the same manner and for most of them effectively integrating ICT in their instruction is a complex innovation (AACTE 2008; Voogt 2008 ) which requires them to change their routines of teaching. This was reiterated by the preservice teachers who reported that most instructors at the teacher education programme were mainly dependent on lecture-based instruction. The programme also did not include courses where teachers were taught how to integrate ICT in their lessons (Adamy and Boulmetis 2006; Brown and Warschauer 2006) .This means that the pre-service teachers' experience to integrate ICT in teaching is limited making the programme fall short of the practical approach. This leads to the big question whether the presently trained pre-service teachers are sufficiently prepared for new teaching methods which are flexible and involve appropriate use of technology. The teachers also indicated that although schools are generally interested in ICT use, regular school practices did not promote ICT use in classrooms. Most in-service teachers reported that schools did not offer them sufficient time to manage and familiarise themselves with ICT. They maintained that as well as lack of time, schools did not provide support network for them to take up the challenges of using ICT in teaching. An additional barrier that could have contributed to these teachers use of ICT was the curriculum factor. Although the curriculum requires mathematics teachers to use ICT in instruction, some teachers (especially in-service ones) alleged that the current status of the curriculum presented serious threats to possibilities of teaching to integrate ICT in the classroom. As a result most of them reinforce the phenomenon of "teaching to the test" where they rush to cover all the topics mechanically in order to finish on time for examinations rather than striving for in depth student learning. Thus they contended that, to allow for meaningful teaching by integrating ICT, the concern of curriculum overload should be addressed.. In spite of the barriers, the study also showed several opportunities that existed for the teachers to be trained to be able to integrate ICT in their teaching. Curriculum (MOESS 2007; MOE 2000) and Policy documents (Ghana ICT4AD Policy 2003) highlight the importance of integrating ICT to enhance teaching and learning. The teacher education programme and SHS's are open to any such ICT innovation. Both in-and pre-service teachers expressed the need for mathematics teachers to integrate ICT in their lessons. More importantly they showed a lot of enthusiasm to be part of any professional development programme related to integrating ICT in teaching and learning mathematics. It was encouraging to find that contemporary mathematics teachers appeared generally supportive and confident in wanting to use computers in their classrooms. Their overwhelming high perceptions (more especially the inservice teachers) to integrate ICT in an innovative way to improve teaching in their future lessons confirmed this. This is in line with similar study (Authors; in preparation) which reported positive attitudes of these teachers towards ICT integration which is a necessary condition for teachers' willingness to participate in an ICT related programme and a predictor of future classroom integration (Authors; in preparation). Although the study reported overall low ICT infrastructure and accessibility in the schools, it could be argued that with computer numbers in schools having increased, the access difficulties identified are likely to stem from a situation in which demand outstrips supply rather than simply a shortage of hardware; although this may be the case in some SHS's in other parts of the country. This augurs well for the future with respect to the design of a professional development programme and anticipated levels of computer competence and its use among new, younger, mathematics teachers as they enter the profession.. The findings reported here highlight areas that require further attention to enable teachers use ICT in mathematics teaching. In particular, a professional development scenario that will assist pre-service and in-service teachers develop skills on ways to integrate ICT in their teaching processes was one of the significant issues identified by the researchers. Such a programme need not differ in content but in format for both groups of teachers. Bearing in mind the complexity of the problems most mathematics classroom in Ghana face in terms of ICT infrastructure and lack of application software, an environment with a more generalised application that offer a technology readily available and user friendly among mathematics classroom with the potential for supporting students' higher-order thinking in mathematics (such as spreadsheet) is proposed for use in professional development programmes. This will ensure that teachers will be able to use existing hardware and software in creative and situation specific ways to design ICT resources to accomplish their teaching goals. Thus this study is in line with SMICT study (Ottevanger et al. 2007 ) which discusses the lessons for improving teaching and learning mathematics in secondary education in Africa. Among their recommendations, effective use of ICT needs to be optimized through extensive programmes of teacher support to improve mathematics and science teaching in Sub-Saharan Africa.

B I O O O O O O O B I O O O O B O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O B O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O B O B O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O B O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O B O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O B O O O O O O O O O O O O O O O O O O O B O O O O O O B I O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O


Placement-Proximity-Based Voltage Island Grouping Under Performance Requirement Abstract-High power consumption not only leads to short battery life for hand-held devices but also causes on-chip thermal and reliability problems in general. As power consumption is proportional to the square of supply voltage, reducing supply voltage can significantly reduce power consumption. Multi-supply voltage (MSV) has previously been introduced to provide finer grain power and performance tradeoff. In this paper, we propose a methodology on top of a set of algorithms to exploit nontrivial voltage island boundaries for optimal power versus design-cost tradeoff under performance requirement. Our algorithms are efficient, robust, and error-bounded and can be flexibly tuned to optimize for various design objectives (e.g., minimal power within a given number of voltage islands, or minimal fragmentation in voltage islands within a given power bound) depending on the design requirement. Our experiment on real industry designs shows a tenfold improvement of our method over current logical-boundary-based industry approach. Reducing power consumption is essential in current chip designs. MSV has previously been used to reduce the overall power consumption without degrading the entire circuit performance, by applying high voltage on timing-critical cells and low voltage on noncritical cells. However, the resulting fragmented power network causes extra design cost, and grouping cells into a practical number voltage islands is, therefore, desired. Logical boundaries are largely used today in such groupings but they are almost always far from optimal for the voltage islands. In this paper, we initiated the study of the problem of balancing power consumption and power-network fragmentation. In particular, we investigated the problem of partitioning a placement region into a small number of voltage islands, while keeping the overall voltage supply low. We formulated this problem as a VPP and presented an efficient TS-Alg for solving it. We have implemented and applied our TS-Alg on a wide selection of real industry designs, and it has been shown to be fast, scalable, and produce superior results compared with the logical-boundary-based approach commonly used today.

O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O


A three-dimensional multidimensional gas-kinetic scheme for the Navier-Stokes equations under gravitational fields This paper extends the gas-kinetic scheme for one-dimensional inviscid shallow water equations (J. Comput. Phys. 178 (2002), pp. 533-562) to multidimensional gas dynamic equations under gravitational fields. Four important issues in the construction of a well-balanced scheme for gas dynamic equations are addressed. First, the inclusion of the gravitational source term into the flux function is necessary. Second, to achieve second-order accuracy of a well-balanced scheme, the Chapman-Enskog expansion of the Boltzmann equation with the inclusion of the external force term is used. Third, to avoid artificial heating in an isolated system under a gravitational field, the source term treatment inside each cell has to be evaluated consistently with the flux evaluation at the cell interface. Fourth, the multidimensional approach with the inclusion of tangential gradients in two-dimensional and three-dimensional cases becomes important in order to maintain the accuracy of the scheme. Many numerical examples are used to validate the above issues, which include the comparison between the solutions from the current scheme and the Strang splitting method. The methodology developed in this paper can also be applied to other systems, such as semi-conductor device simulations under electric fields. Since most astrophysical problems are related to the hydrodynamical evolution in a gravitational field, the correct implementation of the gravitational force in an astrophysical hydrodynamical code is essential, especially in systems with long time integrations, such as modeling star and galaxy formation. Even though many hydrodynamical codes have been successfully applied to astrophysical problems, including the Piecewise parabolic method (PPM) and total variation diminishing (TVD) codes [5, 12] , most have considered only short time evolutions with strong shock or expansion waves. With the slowness of galaxy evolution, many codes have difficulties due to the improper treatment of the gravitational force effect, the so-called source term in the Euler or the Navier-Stokes equations. After integration of millions of time steps, many codes cannot settle down to determine an isothermal steady state solution for a gas in a time-independent external gravitational field. The solution will either oscillate around the equilibrium solution, or simply deviate from equilibrium due to artificial heating, which triggers numerical gravitation-thermal instability, i.e., the collapse of the gas core. There have been many attempts to construct such a well-balanced gas dynamic code that preserves the hydrostatic solution accurately [8, 20, 2] .. In recent years, the study of flow equations with source terms has attracted much attention in the computational fluid dynamics community [18] . Flow equations with source terms in one-dimensional (1D) case can be written as. where W is the vector of conservative flow variables with the corresponding fluxes F (W ). Here, S is the source, such as the gravitational force. If we integrate the above equation with respect to dx in a spatial element, j, from x j−1/2 to x j+1/2 , and dt in a time interval from t n to t n+1 , where ∆x = x j+1/2 − x j−1/2 is the cell size and W j is the averaged value of W in cell j, i.e.,. The above equation is an integral form of Eq. (1) and is exact and equivalent to Eq. (1) . In an earlier paper on the shallow water equations [16] , we emphasized the importance of including the source term effect into the flux function at a cell interface in order to develop a well-balanced scheme. However, the exact equilibrium solution, such as u = 0 and h + B = constant, for the shallow water equations, is relatively simple and special. The simplicity can be realized through the relation dh/dx = −dB/dx = constant inside each computational cell once the bottom topology is approximated as a linear function inside each cell. Therefore, the MUSCL-type scheme with a single slope inside each cell can be precisely used to reconstruct the initial well-balanced solution. However, these techniques developed for the shallow water equations cannot be directly extended to gas dynamics equations. For example, the hydrostatic equilibrium solution for the gas dynamic equations may become ρ ∼ e −αx , which is an exponential function. This exact equilibrium solution cannot even be precisely reconstructed as an initial condition in the MUSCL-type scheme with a single limited slope inside each numerical cell. Even though, it seems impossible to develop a well-balanced scheme for the gas dynamic equations with an error up to the order of machine zero, a proper numerical treatment is still essential to get accurate solutions. For a gas flow under an external time-independent gravitational field, there exists an isothermal solution, the so-called hydrostatic or well-balanced equilibrium solution, with a constant temperature and zero fluid velocity. This solution is an intrinsic solution due to the balance between the flux gradient and source term, i.e., F (W ) x = S. However, based on the macroscopic equations, it is unclear how the gas settles down to an isothermal solution. Usually, additional assumptions, such as a constant temperature, are needed before deriving such an equilibrium result. Numerically, it is challenging to obtain such a solution. If Eq. (1) is solved by simple operator splitting method, such as W t + F x = 0 for the flux at a cell interface and W t = S to account for the source term inside each cell, as observed frequently, many schemes simply deviate from the exact solution after millions of time step integrations. The systematic errors will accumulate and eventually crash the calculation. This happens to be the so-called numerical heating phenomenon, which was reported in [13] . Due to the accumulation of numerical errors, the internal energy in an isolated gas system with adiabatic boundary conditions under a constant gravitational field will increase forever. Figure 1 shows such a solution, where the internal energy goes up after millions of time steps. The non-conservation of the total energy in the system is not surprising because the gas dynamic equation (1) itself is not written in a conservative form. The problem in the gas dynamic equations is that the numerical error accumulates in one direction. On the contrary, there is no such problem in the shallow water equations even if the numerical scheme is not a well-balanced one.. Based on the Boltzmann equation, a flow under a gravitational field in a hydrostatic state can be described as. where f is the gas distribution function, c is the particle velocity, and φ is the external gravitational potential. The general solution of the above differential equation is. where F is an arbitrary function of. . This equilibrium solution clearly shows that the gas will have the same temperature, because the temperature becomes a constant multiplier for the function φ + 1 2 c 2 . For example, the function F in hydrostatic equilibrium will become exp(−( 1 2 c 2 + φ)/kT ), where kT is a constant temperature. The above analysis shows the usefulness of using a kinetic equation instead of macroscopic equations to describe a gas flow under a gravitational field. It also implicitly demonstrates that the flux functions, which are the moments of f , have to take into account the source term effect, i.e., φ, in the solution of f .. In the past several years, a gas-kinetic BGK Navier-Stokes (BGK-NS) scheme has been developed for compressible Navier-Stokes equations [15, 9, 10] . It achieves success in computing viscous flow solutions, such as the hypersonic viscous and heat conducting flows [14] . Due to the simplicity of particle transport, a multi-dimensional scheme can be constructed by incorporating a multidimensional particle propagation mechanism through the cell interface, where the particle transport in both normal and tangential directions of a cell interface can be taken into account. However, constructing a well-balanced BGK-NS method for gas dynamic equations under a gravitational field is not straightforward. As shown above, with a hydrostatic state, the density inside each numerical cell is distributed exponentially in space, which cannot be precisely reconstructed by a simple limited slope in a standard MUSCL-type scheme. In other words, even starting from the initial condition, some numerical error is already introduced. Another important issue that needs to be addressed is the Chapman-Enskog expansion of the Boltzmann equation, where the external force effect has to be added to account for the balance between the dissipative and external forcing terms. This is important to capturing the viscous flow solution, especially with a scheme that should work under both continuous and discontinuous flow conditions. To achieve a well-balanced scheme, the gravitational force effect on the particle acceleration and on the numerical flux at the cell interface needs to be explicitly taken into account as well. In order to avoid the numerical heating (cooling) effect, a consistent treatment between the source term inside each cell and the flux at the cell interface is required. The BGK scheme presented in this paper is a well-balanced scheme up to second-order accuracy. On the contrary to the shallow water solutions, the hydrostatic equilibrium state for the gas dynamic equations, such as zero flow velocity, cannot be precisely kept by the current scheme up to the machine zero, i.e., 10 −15 . How to extend a well-balanced strategy from the shallow water equations to the gas dynamic equations remains an open problem. This paper is organized as follows. Section 2 gives the details of the multi-dimensional BGK-NS scheme under a gravitational field. Section 3 presents the results from many test cases, which confirm the importance of some strategies in the development of a well-balanced scheme. At the same time, the numerical heating phenomenon is presented. The functions from the nonlinear limiter and the multi-dimensional transport mechanism on the well-balanced scheme are described. The last section is the conclusion. The gas-kinetic BGK scheme for the Navier-Stokes equations is extended to include external gravitational forces. Our study allows us to draw the following conclusions. (1) The gravitational force effect should be included in the evaluation of the numerical fluxes across a cell interface. It has the first-order effect in time [16] , which cannot be neglected in a second-order scheme. In a gas kinetic scheme, the external force modifies the time evolution of both the initial gas distribution function and the local equilibrium state through the particle acceleration. Without such an implementation, it is impossible for the system to settle down to a highly accurate hydrostatic equilibrium state. (2) Numerical heating is mainly caused by the inconsistent treatment of the gravitational source terms inside each cell in the energy source term scheme. After a long time evolution, the numerical error will accumulate and dramatically affect the final solution of an isolated or insulated gravitational system. For a system that exchanges energy with its surrounding environment or that has a short evolution time, such as the star explosion case, this kind of numerical heating may not appear to be significant. (3) The numerical simulation of the onset of two-dimensional laminar convection shows that the current BGK scheme can accurately determine the critical Rayleigh numbers. However, the nonlinear limiter is usually harmful to the smooth flow solution. High-order interpolation may alleviate this problem. (4) The results of finite amplitude laminar convection with a central interpolation agree well with Graham's results. The van Leer limiter introduces a bit of additional dissipation. By including the tangential gradient terms, i.e., the so-called multidimensional approach, the accuracy of the scheme is improved. The kinetic scheme presented in this paper is not only limited to hydrodynamics under gravitational fields; it can be applied to other systems in which the external force term can be an electric field force, such as the Boltzmann-type equations for semiconductor devices.. Developing a well-balanced scheme for gas dynamic equations under gravitational fields is still an open problem. It is much more difficult than developing a scheme for the shallow water equations. We hope that this paper will help to shift attention from shallow water equations to gas dynamic equations. At the current stage, it seems that no currently available scheme can maintain the velocity of the hydrostatic equilibrium state up to the machine zero, such as on the order of 10 −16 . At the same time, the scheme has a shock capturing ability. Even with consideration of all the techniques presented in this paper, the current method has only second-order accuracy in maintaining the hydrostatic solution. Much work is left for the future.. In the gas-kinetic scheme, to obtain the derivatives of a Maxwellian, the solution of the following equations need to be evaluated,. where b and M are known. The matrix M is defined as. where f eq is the Maxwellian. The details of the matrix M are given by [6] .. b Results from current gas-kinetic BGK scheme.. c Courant number.

O O O B I O O B I O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O


A mobile tourist assistance and recommendation system based on complex networks In recent years, the tourism assistance community is gradually shifting its emphasis to digital, interactive systems. Our project aims for digital tourism assistance by combining mobile guiding and route recommendation based on community networks. We use complex networks to tackle problems, such as deriving optimal routes in conjunction with knowledge deduced from our social tourist network, and giving a respectable prediction of which meta-and multimedia data about touristic objects are needed while the tourist is on tour. This article describes ongoing work on the recommendation system and mobile semantic replication strategies. This paper describes a strategy for combining mobile tourist assistance and recommendation with tourist community management on top of complex networks. As a result, lists of touristic objects are created, which serve on the one hand as guiding recommendations for tourists and on the other hand as predictions of data that need to be made available on mobile clients with limited internet connection and storage capabilities. As usual in ongoing work, the next step will be an evaluation of the system within a real usage environment together with our industrial partner.

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The stable set of associated prime ideals of a polymatroidal ideal Abstract. The associated prime ideals of powers of polymatroidal ideals are studied, including the stable set of associated prime ideals of this class of ideals. It is shown that polymatroidal ideals have the persistence property and for transversal polymatroids and polymatroidal ideals of Veronese type the index of stability and the stable set of associated ideals is determined explicitly. Let I be an ideal in a Noetherian ring R. It is customary to denote by Ass(I) the set of associated prime ideals of R/I. Brodmann [3] showed that Ass(I k ) = Ass(I k+1 ) for all k ≫ 0. One calls the smallest number k 0 for which this happens the index of stability and Ass(I k 0 ) is called the stable set of associated prime ideals of I. It is denoted by Ass ∞ (I). Several natural questions arise in the context of Brodmann's theorem.. (1) Is there an upper bound for the index of stability depending only on R? (2) What can be said about the set Ass ∞ (I)? Can Ass ∞ (I) be computed in case that R is a polynomial ring and I is a graded ideal? (3) Is it true that Ass(I) ⊂ Ass(I 2 ) ⊂ . . . ⊂ Ass(I k ) ⊂ . . .? All these questions are widely open, even for monomial ideals, though in several interesting special cases, including edge ideals and vertex cover ideals, these questions have been answered quite comprehensively, see [5] , [9] and [17] . A nice survey on what is known about the stability of associated prime ideals of powers of edge ideals is given in [18] . Question (3) doesn't have a positive answer in general, see [12] and [17] for counterexamples. The ideals which provide these counterexamples are monomial ideals, but not squarefree. An ideal I for which (3) holds true is said to satisfy the persistence property. It is an open question whether all squarefree monomial ideals satisfy the persistence property.. Suppose now that (R, m) is local or a standard graded K-algebra with graded maximal ideal m. We say that an ideal I ⊂ R has non-increasing depth functions, if for all prime ideals P in the support V (I) of R/I, one has that depth R P /I k R P is a non-increasing function of k, and I is said to have strictly decreasing depth functions, if the depth functions of all its localizations are strictly decreasing until they reach their limit value. In the case that I is a graded ideal, respectively a monomial ideal, we require the defining property of non-increasing (strictly decreasing) depth functions only for localizations with respect to prime ideals P ∈ V * (I), where V * (I) denotes the set of graded, respectively monomial prime ideals containing I.. It is easily seen that for an ideal which has non-increasing depth functions the persistence property holds, see Proposition 1.1. Moreover, if an ideal has strictly decreasing depth functions, then its index of stability is bounded by dim R − 1. We do not know of any example of a squarefree monomial ideal which does not have non-increasing depth functions. On the other hand it is shown in [12, Theorem 4.1] that, given any non-decreasing function f : N → N, there exists a monomial ideal in a polynomial ring S (with sufficiently many variables) such that depth S/I k = f (k) for all k. This shows that among the monomial ideals, non-increasing depth functions can be expected in general only for squarefree monomial ideals.. There is at least one case known to us in which Ass ∞ (I) can be computed efficiently. Namely, if I is a monomial ideal in a polynomial ring S = K[x 1 , . . . , x n ] whose Rees algebra R(I) is Cohen-Macaulay. By a result of Huneke [16] it follows that the associated graded ring of I is Cohen-Macaulay, and this implies that lim k depth S/I k = n − ℓ(I), where ℓ(I) denotes the analytic spread of I, that is, the Krull dimension of the fiber ring R(I)/mR(I), see Eisenbud and Huneke [8, Proposition 3.3] . This theorem allows us to identify the elements of Ass ∞ (I) in terms of the exponent matrix associated with the unique minimal monomial set G(I) of generators of I. All this is explained in detail in Section 1. There we also define the invariants dstab(I) and astab(I). The first of them is the smallest integer k with the property that depth I k = depth I ℓ for all ℓ ≥ k, while the second is the smallest integer with Ass(I k ) = Ass(I ℓ ) for all ℓ ≥ k. One may ask whether there is any relation between these numbers. At the end of Section 1 we show that either one may be smaller than the other. However we show in Proposition 1.1 that astab(I) is bounded below and above by local data of dstab provided it has non-increasing depth functions, which is for example the case if all powers of I have a linear resolution, see Proposition 1.2. These facts are used to compute the index of stability for Stanley-Reisner ideal of the natural triangulation of the projective plane.. In Section 2 the strategies discussed in Section 1 are applied to study the associated prime ideals of powers of polymatroidal ideals. Two general properties of polymatroidal ideals are crucial: 1. all powers of polymatroidal ideals have a linear resolution, as shown in [7, Theorem 5.3 ], 2. localizations of polymatroidal ideals at monomial prime ideals are again polymatroidal, see Corollary 2.2. These two facts combined with Proposition 1.1 immediately yield that polymatroidal ideals have the persistence property, see Proposition 2.3. We recall in Theorem 2.4 the result of Villarreal [19, Proposition 3.11] , which says that the Rees ring of a polymatroidal ideal is normal, and consequently Cohen-Macaulay. Applying then the HunekeEisenbud result, the limit depth of a polymatroidal ideal can be expressed by its analytic spread. From this one easily deduces an algorithm, described at the end of the section, to compute Ass ∞ (I) for any polymatroidal ideal I. All data required to compute Ass ∞ (I) are given by the exponent matrix of the minimal set of monomial generators of I.. In the remaining two sections we consider special classes of polymatroidal ideals where the questions concerning associated prime ideals of powers of ideals have complete answers. The ideals considered in Section 3 are the polymatroidal ideals of transversal polymatroids. Algebraically speaking, ideals of this type are simply arbitrary (finite) products of monomial prime ideals. In [7, Lemma 3 .2] a primary decomposition of products of ideals generated by linear forms is given. However this decomposition is not at all irredundant and it is not easy to obtain an irredundant decomposition from that given in [7, Lemma 3.2] .. Our first result (Lemma 3.1) asserts that the presentation of a transversal polymatroidal ideal as product of monomial prime ideals is unique. The key result of Section 3 is Theorem 3.3 where it is shown that the graded maximal ideal m is associated to the transversal polymatroidal ideal I = P F 1 · · · P Fr if and only if r i=1 F i = [n] and the intersection graph G I is connected. Here G I is the graph with vertex set {1, . . . , r} and for which {i, j} is an edge of G I if and only if F i ∩ F j = ∅. By using this result we conclude in Corollary 3.6 that Ass(I) = Ass ∞ (I) for any transversal polymatroidal ideal. Furthermore we show in Theorem 3.7 that Ass(I) is determined by the trees of the graph G I . As nice consequences of all this we classify in Corollary 3.9 all subsets S = {F 1 , . . . , F r } of 2. [n] for which there exists a transversal polymatroidal ideal I with Ass(I) = {P F 1 , . . . , P Fr }, and in Corollary 3.10 we give an irredundant primary decomposition of all powers I k of I. We conclude this section with two results concerning the depth of S/I. In Theorem 3.12 it is shown that depth S/I is essentially determined by the number of components of G I , and Corollary 3.14 says that dstab(I) = 1. Thus for any transversal polymatroidal ideal, dstab(I) = astab(I) = 1.. The situation for ideals I d;a 1 ,...,an of Veronese type, which is the class of polymatroidal ideals considered in Section 4, is completely different. Here Ass ∞ (I) = V * (I), as shown in Proposition 4.3, and the invariant astab(I) can be any number between 1 and n − 1 determined by an explicit formula given in terms of the numbers d and a 1 , . . . , a n , see Corollary 4.6. Moreover it is shown in Corollary 4.7 that astab(I) = dstab(I) and lim k→∞ depth S/I k and ℓ(I) are computed for any Veronese type ideal.. The common feature to transversal polymatroidal ideals and to ideals of Veronese type is that astab(I) = dstab(I). It would be interesting to know whether this equality holds for any other polymatroidal ideal. As we have seen in Section 1, arbitrary monomial ideals, even when they are squarefree, do not satisfy this equality.. 1. Generalities about the depth and the associated primes of powers of an ideal. Let (R, m) denote a Noetherian local ring or standard graded K-algebra with graded maximal ideal m, and I ⊂ R an ideal. In the graded case we assume that I is graded ideal.. We are going to relate the index of stability and the persistence property of I to the property of I to have non-increasing depth functions. We say that P ∈ V (I) is a persistent prime ideal of I, if whenever P ∈ Ass(I k ) for some exponent k, then 3 P ∈ Ass(I k+1 ). If this happens to be so for k, then of course we have P ∈ Ass(I ℓ ) for all ℓ ≥ k. The ideal I is said to have the persistence property if all prime ideals P ∈ k Ass(I k ) are persistent prime ideals. By a famous theorem of Brodmann [2] it is known that depth R/I k is constant for all k ≫ 0. We call the smallest number k 0 such that depth R/I k = depth R/I k 0 for all k ≥ k 0 , the index of depth stability of I, and denote this number by dstab(I). Brodmann also showed [3] that there exists an integer k 1 such that Ass(. The smallest such number is called the index of stability of I. We denote this number by astab(I).. At the end of this section we show by examples that the invariants dstab(I) and astab(I) are unrelated. In other words, either one of these numbers may be smaller than the other one or they may also be equal. However we have (c) max P ∈Ass ∞ (I) {dstab(IR P )} ≤ astab(I). In addition, if I has non-increasing depth functions, then astab(I) ≤ max P ∈V (I) {dstab(IR P )}.. One has that P ∈ Ass(I k ) if and only if P R P ∈ Ass R P (I k R P ). By part (a) this is implies that P R P ∈ Ass R P (I ℓ R P ) for all ℓ ≥ k. Thus P ∈ Ass(I ℓ ) for all ℓ ≥ k.. (c) Let r = astab(I). Then, whenever P ∈ Ass ∞ (I), we have P ∈ Ass(I ℓ ) for all ℓ ≥ r. This implies that depth R P /I ℓ R P = 0 for all ℓ ≥ r. Hence dstab(IR P ) ≤ r, which yields the first inequality. Now let s = max P ∈V (I) {dstab(IR P )}, and suppose that r > s. Then there exists P ∈ Ass ∞ (I) such that P ∈ Ass(I r ), but P ∈ Ass(I s ). Indeed, otherwise we would have that depth R P /I s R P = 0 for all P ∈ Ass ∞ (I). Since I has non-increasing depth functions it would follow that astab(I) ≤ s < r, a contradiction.. It follows that depth R P /I s R P > depth R P /I r R P = 0, in contradiction to the definition of s. . Proof. Let f ∈ I be a homogeneous polynomial of degree d. Then f I k is generated in degree (k + 1)d and f I k ⊂ I k+1 . The short exact sequence. induces the long exact sequence Let I ⊂ S be a monomial ideal. Throughout this paper S stands for the polynomial ring K[x 1 , . . . , x n ] where K is a field. We denote by G(I) the unique minimal set of monomial generators of I. In the case that. denote by an abuse of notation the ideal G(I)T again by I. Observe that by using this notation it follows that Ass S (I) = Ass T (I).. Let u = i∈L x i be a squarefree monomial in S. Then. and where I L ⊂ S ′ is the ideal which is obtained from I by applying the K-algebra homomorphism S → S ′ with x i → 1 for all i ∈ L.. Let P = (x i 1 , . . . , x ir ) be a monomial prime ideal, and I ⊂ S any monomial ideal. We denote by I(P ) the monomial ideal in the polynomial ring. In the later proofs we need the following simple facts. Lemma 1.3. Let I ⊂ S be a monomial ideal. Then (a) P ∈ Ass(I) if and only if depth S(P )/I(P ) = 0; (b) Ass(I L ) = {P ∈ Ass(I) :. Proof. (a) has been observed in [9, Lemma 2.11].. (b) As before let. . Then T = S u where u = i∈L x i . Thus by using the basic rules concerning the behaviour of associated prime ideals with respect to localization and polynomial ring extension we obtain. On the other hand,. Since the assignment P → P T establishes a bijection between the set Ass S ′ (I L ) and {P T : P ∈ Ass S (I L )}, the desired conclusion follows.. If I is a monomial ideal, we say that I has non-increasing depth functions if depth S(P )/I(P ) k is a non-increasing function of k for all P ∈ V * (I).. Since the associated prime ideals of a monomial ideal are monomial prime ideals, it follows (in analogy to Proposition 1.1(b)) that a monomial ideal has the persistence property if I has non-increasing depth functions as defined for monomial ideals.. For monomial ideals, the corresponding statement of Proposition 1.1(c) reads as follows: Proposition 1.4. Let I ⊂ S be a monomial ideal which has non-increasing depth functions. Then. {dstab(I(P ))}.. In particular, if Ass ∞ (I) = V * (I), one has astab(I) = max P ∈V * (I) {dstab(I(P ))}.. As a consequence, in the case of a monomial ideal which has non-increasing depth functions we need to compute the depth stability only for a finite number of monomial prime ideals in order to obtain bounds for its index of stability. The following example demonstrates this strategy.. Let I be the Stanley-Reisner ideal that corresponds to the natural triangulation of the projective plane. Then. The following table displays in the jth row and the kth column the depth of S(P )/I(P ) k where P ∈ V * (I) is of height j.. I 1 2 3 4 3 0 0 0 0 4 1 1 1 1 5 2 2 0 0 6 3 0 0 0 The ideal I is of height 3, so depth S(P )/I(P ) k = dim S(P )/I(P ) k = 0 for all k and all P ∈ V * (I) of height 3. For P ∈ V * (I) of height 4 the ideal I(P ) is a monomial ideal complete intersection of height 3. Therefore depth S(P )/I(P ) k = 1 for all k ≥ 1. This explains the first two rows of the table. If P ∈ V * (I) is of height 5, then I(P ) is the edge ideal of a 5-cycle. It follows from [5, Lemma 3.1] that Ass ∞ (I(P )) = Ass(I(P )) ∪ {m} and astab(I(P )) = 3, which explains the third row of the table. In particular, we have that depth S(P )/I(P ) k = 0 for all P ∈ V * (I) of height 5 and for all k ≥ 3. Finally, by using CoCoA [6] we find that depth S/I = 3, depth S/I 2 = 0 and depth S/I 3 = 0. Borna [1, Corollary 3.3] has shown that I k has a linear resolution for k ≥ 3 and when char(K) = 0. Applying Proposition 1.2, we see that depth S/I k = 0 for all k ≥ 3. It follows also that I is an ideal with nonincreasing depth functions and consequently I satisfies the persistence property, by Proposition 1.1(b). By applying the second inequality of Proposition 1.4 we obtain that astab I ≤ 3. By using Singular [10] , we find that all prime ideals of height 5 are in Ass(I 3 ). Since I satisfies persistence property we get that all prime ideals of height 5 are in Ass ∞ (I). It follows then that max P ∈Ass ∞ (I) {dstab(IS P )} ≥ 3. Finally, by applying again Proposition 1.4 we obtain that astab(I) = 3. As a byproduct of computing the astab(I) we obtain that Ass ∞ (I) = Ass(I 3 ). Calculations with 6 Singular show that Ass(I 3 ) consists of all prime ideals of height 3, 5 and 6 which belong to V * (I), altogether 17. Moreover we see that in this example, dstab(I) < astab(I).. As a second example we consider the ideal I = (xyz, ytu, xzv, tuv, xtv) ⊂ S = K [x, y, z, t, u, v] . Then I = F ∈F (∆) P F where ∆ is the simplicial complex with facets F = {{z, t}, {x, t}, {x, u}, {y, v}, {z, u, v}} and where P F is the monomial prime ideal whose generators correspond to the vertices of F . The simplicial complex ∆ has no special odd cycles in the sense of [14] . Thus as a consequence of [14, Theorem 2.2.] it follows that the vertex cover algebra of ∆ is standard graded which implies that Ass(I) = Ass ∞ (I). Thus astab(I) = 1. On the other hand one can check with CoCoA that depth S/I = depth S/I 2 = 3 and depth S/I 3 = 2 . Thus astab(I) < 3 ≤ dstab(I).. As a last topic of this section we want to recall a few facts about the limit depth of an ideal. As we mentioned already, the function f (k) = depth R/I k is constant for k ≫ 0. We call lim k→∞ depth R/I k the limit depth of I, see [12] . This limit depth can be computed under certain conditions that we are going to describe now.. Recall that the analytic spread of an ideal I is the Krull dimension of the fiber ring R(I)/mR(I). It is known by Brodmann [2] that. Thus in particular, if the analytic spread of I is equal to n, then lim k→∞ depth R/I k = 0. Eisenbud and Huneke [8, Proposition 3.3] showed that equality holds in Brodmann's inequality if the associated graded ring gr I (R) is Cohen-Macaulay, which by Huneke [16] is the case if R and R(I) are Cohen-Macaulay.. In the case that I is a monomial ideal generated in a single degree, the analytic spread of I is the rank of the integer matrix whose rows correspond to the monomial generators of I.

O O O O B I I O O O O O O B I I O O O B I O O O O O O O B I I O O O O O O O O O B I O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O 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O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O B I I O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O 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Searching in metric spaces with user-defined and approximate distances Novel database applications, such as multimedia, data mining, e-commerce, and many others, make intensive use of similarity queries in order to retrieve the objects that better fit a user request. Since the effectiveness of such queries improves when the user is allowed to personalize the similarity criterion according to which database objects are evaluated and ranked, the development of access methods able to efficiently support user-defined similarity queries becomes a basic requirement. In this article we introduce the first index structure, called the QIC-M-tree, that can process userdefined queries in generic metric spaces, that is, where the only information about indexed objects is their relative distances. The QIC-M-tree is a metric access method that can deal with several distinct distances at a time: (1) a query (user-defined) distance, (2) an index distance (used to build the tree), and (3) a comparison (approximate) distance (used to quickly discard from the search uninteresting parts of the tree). We develop an analytical cost model that accurately characterizes the performance of the QIC-M-tree and validate such model through extensive experimentation on real metric data sets. In particular, our analysis is able to predict the best evaluation strategy (i.e., which distances to use) under a variety of configurations, by properly taking into account relevant factors such as the distribution of distances, the cost of computing distances, and the actual index structure. We also prove that the overall saving in CPU search costs when using an approximate distance can be estimated by using information on the data set only (thus such measure is independent of the underlying access method) and show that performance results are closely related to a novel "indexing" error measure. Similarity queries are a fundamental search paradigm for modern dataintensive applications, such as multimedia databases, data mining, pattern recognition, molecular biology, and e-commerce. In its essence, the problem is to find in a given collection of objects those which better fit (i.e., which are more similar to) a given query specification. In order to improve the effectiveness of such queries, it is nowadays well recognized that the notion of similarity, which ultimately determines how database objects are evaluated and ranked, should be made user-dependent, so as to give the users enough flexibility to state their preferences [Chomicki 2002] . As an example, a user looking for a secondhand car on a trading site could be more interested in the car's price than in its speed, whereas the opposite could hold for another user. Although on simple object domains preferences could be explicitly expressed by the user (e.g., by moving sliders on a graphical interface), in more complex cases, such as those arising in multimedia and data mining systems, the fine tuning of the (many) parameters is best left to the system, which takes the burden of automatically "learning" them by monitoring the user's activity [Ç etintemel et al. 2000] or by exploiting some feedback on results of previous queries [Ortega-Binderberger et al. 2002] .. The problem of how to support (user-defined) similarity queries on large databases has recently attracted a lot of research interests. As representatives of the different approaches, Börzsönyi et al. [2001] and Kießling and Köstler [2002] present solutions aiming to smoothly integrate preferences into SQL queries, Ortega-Binderberger et al. [2002] explore how mechanisms of query refinement can be implemented on top of object-relational database systems, Bartolini et al. [2001] show how the feedback cycle needed to converge to optimal query settings can be "bypassed" by reusing past feedback information, and Hristidis et al. [2001] concentrate on query processing issues arising when preferences are expressed as a weighted linear combination of (numerical) attributes' values. Because of the large data volumes that arise in the considered scenarios, and because in complex objects domains even assessing the degree of similarity between a data object and the user query can be a time-consuming task, a fundamental issue is the development of index-based strategies able to avoid the overhead of scanning the entire data set in order to find the best matches for the query at hand. To this end, several solutions (which are summarized in Section 2.1) have been developed for the specific case when objects are represented by means of numerical features (color histograms [Faloutsos et al. 1994 ] and texture descriptors [Manjunath and Ma 1996] for still images, Fourier and wavelet coefficients for time series [Agrawal et al. 1993; Yi and Faloutsos 2000] , etc.) and such feature vectors are indexed with a multidimensional Spatial Access Method (SAM), such as the X-tree [Berchtold et al. 1996] and the SR-tree [Katayama and Satoh 1997] (see Gaede and Gunther [1998] for a survey on SAMs). In this approach user preferences are taken into account by suitably adapting the geometry of the query region in the feature space (leading, say, to complex-to-evaluate ellipsoid queries) and/or by querying the SAM with an approximate geometry that guarantees to return a superset of the result (thus possibly including so-called false drops).. In the general case, however, objects are not necessarily represented as vectors, rather as points of a metric space. In this case it is only required that the distance function used to measure the dissimilarity of the objects is a metric (thus no geometric operations based on objects' "positions" can now be used) and the triangle inequality property of the metric is the key method used to • P. Ciaccia and M. Patella prune irrelevant parts of the search space. Based on this simple principle, several Metric Access Methods (MAMs) have been developed so far to efficiently solve similarity queries on generic metric spaces (see Chávez et al. [2001] for a recent survey).. Although MAMs have an intrinsic broader applicability than SAMs (since vector spaces are a proper subset of metric spaces), it is a common belief that MAMs are bound to only use the distance function with which they are built Korn et al. 1998; Chakrabarti and Mehrotra 1999; Sakurai et al. 2001] . Based on this opinion, neither the use of MAMs with user-defined distance functions nor the use of approximate distances for speeding-up query evaluation with MAMs have been considered before. As a consequence, MAMs are nowadays used only for query distances that are known in advance and they can become CPU-bound when the distances are costly to evaluate, as happens in complex metric spaces (e.g., when comparing long strings in genomic databases and with ellipsoid queries in highdimensional vector spaces [Chávez et al. 2001] ).. In this article we start by showing that the limitation on the applicability of metric index structures is only apparent. The keys for adding flexibility to MAMs are the so-called Lower-Bounding property, which is also commonly used with SAMs, and "distance scaling," which allow MAMs to be applied in a wider variety of situations. We then provide an extension of the M-tree [Ciaccia et al. 1997 ] (which we take as the reference MAM throughout the article) so as to process queries using distance functions based on user preferences. In particular, any distance function that is lower-bounded by the distance used to build the tree can be used to correctly query the index without any false dismissal, that is, without losing any relevant object.. Turning to performance issues, we introduce the concept of the comparison (approximate) distance function, to quickly prune the nodes of the tree that do not contain relevant objects. The so-extended M-tree, also called the QIC-Mtree, can therefore deal with several distinct distances at a time:. 1. the query (user-defined) distance (according to which the actual result must be computed), 2. the index distance (used to build the tree), and 3. the comparison distance (used to quickly discard uninteresting index paths).. It is worth pointing out that this enlarged scenario also includes the possibility of having nonmetric query and/or comparison distance functions, a fact which indeed further broadens the scope of applicability of the QIC-M-tree. In order to estimate performance of the QIC-M-tree when solving similarity (range and k-nearest neighbors) queries, we present a cost model to analytically predict search costs given statistics about the data set and the index at hand. Specifically, the only information exploited by the cost model is the distance distribution, that is, the distribution of pairwise objects' distances. Experiments with real metric data sets clearly demonstrate the high accuracy of the cost model. Besides being applicable to predict query costs, the cost model . routing object of node N r [N ] covering radius of node N n h number of nodes at level h of the QIC-M-tree n = H h=1 n h total number of nodes of the QIC-M-tree r h average covering radius for nodes at level h of the QIC-M-tree can also be used to help with some relevant query processing issues, such as the following.. 1. When several comparison distances are available, which one is the most effective for a given query? 2. On the other hand, is the use of a comparison distance always effective?. Our results are not peculiar to the QIC-M-tree but can also be applied to other types of (properly extended) MAMs. In particular, the saving that can be obtained by using a comparison distance is independent of the underlying access method, and can be derived from statistics on the data set only.. The article is organized as follows. In Section 2 we give background information on MAMs and on how user-defined similarity queries are processed by SAMs. Section 3 introduces the basic concepts underlying our approach. In Section 4 we present the algorithms for correctly solving range and k-nearest neighbors queries with the QIC-M-tree. Section 5 introduces a cost model for analytically predicting index performance and Section 6 validates the cost model through extensive experiments over real metric data sets. Section 7 presents a discussion on related work and Section 8 concludes. Relevant symbols used throughout the article are given in Table I. • P. Ciaccia and M. Patella To the best of our knowledge, this is the first work to investigate the use of metric access methods to answer similarity queries based on user-defined distance functions. Indeed, the complaint that MAMs are bound to use a given distance function has been raised several times (e.g., see Seidl and Kriegel [1998] , , Korn et al. [1998] , Chakrabarti and Mehrotra [1999] , and Sakurai et al. [2001] ). For the same reason, the issue of supporting adaptable similarity criteria for searching in generic metric spaces has not been investigated before. Thus, being unable to compare our approach with direct competitors, and having already discussed in Section 2.1 how SAMs deal with user-defined similarity queries, in the following we briefly discuss other approaches and techniques that are peculiar to generic metric spaces and have been somewhat influential to our work.. Three major trends for speeding up query evaluation in metric spaces can be observed. The first one, well represented by the so-called GEMINI approach [Zaniolo et al. 1997, Chapter 12] , is to extract from each object a feature vector and to replace the original object distance with a distance in the resulting feature (vector) space. This allows SAMs to be used and, provided the feature distance lower-bounds the object distance, the F&R processing algorithm to be applied. The problem of finding a distance in a feature space that lower-bounds the object distance has, however, to be solved using domain-specific considerations (see, e.g., Faloutsos et al. [1994] for the case of color images). Alternatively, and without the need of domain knowledge, one could try to map objects into points of a (low-dimensional) vector space so that relative object distances are preserved as much as possible in the resulting target space. FastMap [Faloutsos and Lin 1995] is a well-known representative of this approach. In Hjaltason and Samet [2000] it is, however, demonstrated that mappings achieved through FastMap do not satisfy the lower-bounding property unless the original space is (isometric to) the Euclidean space. Correctness of search algorithms has to be given up therefore. Other mappings are proposed in Hjaltason and Samet [2000] that satisfy the lower-bounding property, but they require high computation costs, and therefore cannot be used to improve search performance. Finally, several techniques have been proposed to reduce the number of distance computations at query time without resorting to a vector space mapping [Chávez et al. 2001; Figueira Santos Filho et al. 2001] . Such techniques use optimizations that are very similar to those used for the M-tree, in that precomputed distances between indexed objects and a set of suitable objects, called pivots, 12 are used together with the triangle inequality to exclude objects from the result. In this section we provide the necessary background on how Spatial Access Methods have been used so far to process user-defined similarity queries and summarize the basic principles of Metric Access Methods. In what follows we consider that the data set O = {O 1 , . . . , O M } of the objects of interest is a subset of a domain U and that similarity queries are expressed by specifying a query object Q ∈ U and a query (user-defined) distance function d Q : U × U → + . Similarity queries are of two basic types: the range query

O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O


The

IFC-based

path

planning

for

indoor

spaces

Path

planning

is

a

fundamental

problem,

especially

for

various

AEC

applications,

such

as

architectural

design,

indoor

and

outdoor

navigation,

and

emergency

evacuation.

However,

the

conventional

approaches

mainly

operate

path

planning

on

drawings

or

building

layouts

by

simply

considering

geometric

information,

while

losing

abundant

semantic

information

of

building

components.

To

address

this

issue,

this

paper

introduces

a

new

method

to

cope

with

path

planning

for

indoor

space

through

an

IFC

(Industry

Foundation

Classes)

file

as

input.

As

a

major

data

exchange

standard

for

Building

Information

Modeling

(BIM),

the

IFC

standard

is

capable

of

restoring

both

geometric

information

and

rich

semantic

information

of

building

components

to

support

lifecycle

data

sharing.

The

method

consists

of

three

main

steps:

extracting

both

geometric

and

semantic

information

of

building

components

defined

within

the

IFC

file,

discretizing

and

mapping

the

extracted

information

into

a

planar

grid,

and

finally

finding

the

shortest

path

based

on

the

mapping

for

path

planning

using

Fast

Marching

Method.

The

paper

aims

to

process

different

kinds

of

building

components

and

their

corresponding

properties

to

obtain

rich

semantic

information

that

can

enhance

applications

of

path

planning.

In

addition,

the

IFC-based

distributed

data

sharing

and

management

is

implemented

for

path

planning.

The

paper

also

presents

some

experiments

to

demonstrate

the

accuracy,

efficiency

and

adaptability

of

the

method.

Video

demonstration

is

available

from

Path

planning

generally

refers

to

finding

the

shortest

path

connecting

two

points,

while

avoiding

collision

with

obstacles.

It

is

a

fundamental

problem

in

robotics,

automation,

computer-aided

design

and

computer

graphics.

Especially,

path

planning

may

arise

in

various

applications

of

construction

automation,

such

as

architectural

design

indoor

and

outdoor

navigation

emergency

evacuation

and

path

planning

of

construction

sites

A

common

way

to

solve

the

problem

of

path

planning

in

construction

automation

has

been

approached

in

a

number

of

relevant

literature,

which

primarily

operates

on

drawings

or

building

layouts,

possibly

with

few

attached

attributes

for

obstacles.

Although

several

digital

building

models

in

the

form

of

CAD

have

been

used

for

path

planning,

they

usually

contain

only

geometric

information

while

losing

abundant

semantic

information

of

building

components

(e.g.

types

and

attributes

of

building

components

and

their

relationships).

Therefore,

it

becomes

important

to

develop

a

reliable

method

that

can

enhance

applications

of

path

planning

by

combining

both

geometric

and

semantic

information

of

building

components..

BIM

(Building

Information

Modeling)

technology

has

been

receiving

a

growing

amount

of

attention

in

the

AEC

(Architecture,

Engineering

and

Construction)

industries

Compared

to

the

traditional

CAD

technology,

BIM

is

capable

of

restoring

both

geometric

and

rich

semantic

information

of

building

components,

as

well

as

their

relationships,

to

support

lifecycle

data

sharing.

As

a

major

data

exchange

standard

for

BIM,

the

IFC

(Industry

Foundation

Classes)

standard

led

by

the

buildingSMART,

formerly

known

as

International

Alliance

for

Interoperability

(IAI),

plays

a

crucial

role

in

the

process

The

IFC

data

model,

which

contains

geometric

and

rich

semantic

information

of

building

components,

is

intended

to

facilitate

interoperability

in

the

AEC

industries.

It

is

a

neutral

and

open

specification

that

is

not

controlled

by

a

single

vendor

or

group

of

vendors.

Today,

the

IFC

standard

has

been

supported

by

most

BIM

software

vendors.

A

list

of

software

applications/

utilities,

that

provide

IFC

import

and/or

export

functionality,

is

available

at

the

buildingSMART

website

A

number

of

recent

research

articles

have

concerned

extracting

and

managing

semantic

information

on

building

components

produced

in

the

form

of

IFC

for

various

applications,

such

as

automatic

rule-based

checking

evaluation

of

design

solutions

construction

cost

estimating

construction

management

comparison

and

metrics

between

IFC

files

This

paper

aims

to

study

the

particular

problem

of

path

planning

for

indoor

spaces,

all

relying

on

the

IFC

building

models

as

input.

The

system

developed

in

the

paper

automatically

extracts

and

manages

geometric

and

semantic

information

on

the

building

space

and

its

inside

components

from

the

IFC

file,

which

enhances

applications

of

path

planning.

The

paper

introduces

a

novel

IFC-based

path

planning

method

for

indoor

spaces.

In

contrast

with

the

conventional

path

planning

approaches,

that

mainly

operate

on

drawings

or

building

layouts

by

considering

the

geometric

information

while

losing

abundant

semantic

information

of

building

components,

the

paper

uses

the

IFC

file

as

input

for

path

planning.

The

IFC

standard

is

a

major

data

exchange

standard

for

BIM

and

it

contains

both

geometric

information

and

rich

semantic

information

of

building

components,

to

support

lifecycle

data

sharing.

The

system

developed

in

this

paper

automatically

extracts

and

manages

geometric

and

semantic

information

on

building

components

and

spaces

from

the

IFC

file,

which

enhances

applications

of

path

planning.

Meanwhile,

using

the

IFC

files

as

input

makes

better

extensibility

of

the

system.

The

experimental

results

demonstrate

that

the

presented

method

is

able

to

cope

with

the

properties

of

building

components

and

is

more

accurate

and

efficient

than

some

previous

work

used

for

path

planning,

such

as

the

A

and

the

ant

colony

algorithm.

In

addition,

the

paper

also

demonstrates

the

adaptability

of

the

presented

algorithm

in

handling

some

special

applications,

e.g.

the

status

of

doors,

the

risk

level

of

equipment,

the

functional

spaces,

and

extension

to

multi-storey

building,

all

relying

on

semantic

information

extracted

from

the

IFC

models.

In

addition,

a

library

BIM

model

in

Xuzhou

city

at

China

is

selected

as

a

case

study.

Finally,

the

IFC-based

distributed

data

sharing

and

management

is

implemented

for

path

planning..

The

current

algorithm

presented

in

the

paper

still

has

some

limitations.

One

is

the

execution

efficiency

and

precision

of

the

algorithm

that

partly

depends

on

the

size

of

the

grid,

where

the

grid

is

used

to

restore

the

discretized

building

environment.

If

an

IFC

model

with

large

and

complex

environment

is

imported

for

path

planning,

one

possible

way

to

improve

the

precision

of

the

result

is

to

increase

the

size

of

the

grid.

When

the

number

of

the

grid

nodes

is

infinite,

a

continuous

orientation

of

the

path

can

be

theoretically

obtained.

However,

a

larger

grid

will

increase

the

storage

and

computing

cost

on

path

planning

computation.

In

addition,

the

increase

in

the

precision

is

still

limited

by

the

size

of

the

smallest

obstacle.

The

automatic

determination

of

discretization

remains

a

future

research

problem..

For

the

extraction

of

semantic

information,

the

additional

properties

of

building

components

should

be

previously

defined

by

the

IFC

Property

Sets.

For

some

IFC

files

without

these

additional

properties

defined

or

with

different

names,

the

current

implementation

cannot

automatically

process

the

semantic

information,

where

only

the

geometric

information

will

be

dealt

with.

In

addition,

another

limitation

of

our

algorithm

is

in

the

case

of

multi-storey

building.

As

the

number

of

stairs

and

elevators

increases,

the

path

planning

in

such

environment

will

be

much

more

complicated.

All

the

above

issues

will

be

studied

in

the

future

work.

Many

path

planning

methods

have

been

developed

in

the

fields

of

computer

science

and

robotics

in

the

past

years

Today

some

studies

have

begun

to

exploit

the

path

planning

methods

in

various

AEC

applications

due

to

the

significant

improvement

in

computer

power.

One

direct

application

is

indoor

and

outdoor

navigation

(e.g.

for

the

blind/vehicle/pedestrian)

in

which

a

fast

and

efficient

path

planning

algorithm

often

generates

the

collision-free

paths

for

providing

the

navigational

assistance.

Another

application

is

emergency

evacuation

simulation

(e.g.

for

surveying

fire

safety

design)

in

the

public

buildings

where

the

planed

paths

provide

the

possible

evacuation

alternatives

in

different

hazard

occurrence

processes.

In

addition,

path

planning

can

help

the

architectural

designer

for

implementing

automatic

rule-based

checking

of

building

designs

for

example,

the

shortest

circulation

path

length

between

a

public

space

and

fire

exits

should

be

less

than

a

minimum

value

in

many

fire-code

checking..

Path

planning

is

also

a

key

element

in

many

construction

automation

applications

For

instance,

Kang

and

Miranda

developed

three

path

planning

algorithms

for

implementing

the

automated

robotic

crane

erection

processes

in

construction.

Sivakumar

et

al.

provided

a

path

planning

algorithm

to

assist

a

cooperative

lifting

plan.

Soltani

et

al.

presented

a

novel

application

of

path

planning

in

construction

sites

based

on

multicriteria

evaluation

of

transportation,

safety

and

visibility

measures.

This

application

can

assist

the

site

planners

to

reduce

hazards

and

select

the

best

available

route

for

site

operatives

and

vehicles..

Most

of

the

above

existing

efforts

require

drawings

or

building

layouts

as

input,

possibly

with

few

attached

attributes

for

obstacles,

in

path

planning.

However,

they

usually

contain

only

geometric

information

while

losing

abundant

semantic

information.

A

survey

of

many

available

path

planning

methods

in

various

AEC

applications

is

beyond

the

scope

of

this

paper.

Instead,

the

following

subsection

briefly

reviews

the

most

related

works

associated

with

BIM

models.

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