An Ejection Chain Algorithm for the Quadratic Assignment Problem Abstract -In this study we present a new tabu search algorithm for the quadratic assignment problem (QAP) that utilizes an embedded neighborhood construction called an ejection chain.Our ejection chain approach provides a combinatorial leverage effect, where the size of the neighborhood grows multiplicatively while the effort of finding a best move in the neighborhood grows only additively. Our results illustrate that significant improvement in solution quality is obtained in comparison to the traditional swap neighborhood. We also develop two multi-start tabu search algorithms utilizing the ejection chain approach in order to demonstrate the power of embedding this neighborhood construction within a more sophisticated heuristic framework. Comparisons to the best large neighborhood approaches from the literature are presented. The quadratic assignment problem (QAP) is a classical combinatorial optimization problem that has garnered much attention due to both its large number of applications and its solution complexity. Originally used to model a location problem in the 1950's, the QAP is computationally very difficult to solve which makes it an ideal candidate for testing new algorithmic approaches. While facility location problems remain the most popular application area for the quadratic assignment problem, many other applications for this problem exist including scheduling problems, statistical data analysis, information retrieval, as well as problems in transportation. The attractiveness of the QAP is also due to the fact that many other combinatorial optimization problems can be formulated as a QAP, including the traveling salesman problem, the maximum clique problem and the graph partitioning problem. (See Cela (1998) for a survey of both classical and practical applications.). In the context of facility location problems, the QAP can be stated as follows. Let is an associated matrix of flows to be transmitted (or shipped) between pairs of facilities ,. The objective is to find a minimum cost assignment of facilities to locations considering both the flow of materials between facilities and the distance between locations.. In mathematical terms, each assignment can be defined as a permutation π of the underlying index set = … {1, , }, N n i.e. π The objective of the QAP is to find a permutation vector π ∈ Π n that minimizes the total assignment cost, where Π n is the set of all possible permutations of N. Such a formulation can be generically described as This study examined the use of ejection chains for the QAP. The results indicate the use of these embedded neighborhood structures result in higher solution quality than obtained by using the traditional 2-exchange neighborhood. We also demonstrate the power of coupling this neighborhood definition with more sophisticated adaptive memory guidance. Our resulting ejection chain approaches are shown to be competitive with recently proposed path enumeration techniques embodied in very large search neighborhood (VLSN) methods.. Future studies could examine integrating the ejection chain algorithms with the VLSN methods and with higher level adaptive memory techniques such as path relinking. Our computational testing showed the parameters chosen for the tabu search framework and the multi-start variants produced the best results from the several combinations examined, but more thorough analysis of these parameters could also provide better quality results, particularly by means of dynamic parameter manipulation. The addition of the diversification method in the EC3 algorithm disclosed the importance of strategic diversification to find enhanced solutions, and we anticipate that additional attention to diversification strategies may also yield gains for future algorithms.

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


Accuracy estimate and optimization techniques for SimRank computation The measure of similarity between objects is a very useful tool in many areas of computer science, including information retrieval. SimRank is a simple and intuitive measure of this kind, based on graph-theoretic model. SimRank is typically computed iteratively, in the spirit of PageRank. However, existing work on SimRank lacks accuracy estimation of iterative computation and has discouraging time complexity. In this paper we present a technique to estimate the accuracy of computing SimRank iteratively. This technique provides a way to find out the number of iterations required to achieve a desired accuracy when computing SimRank. We also present optimization techniques that improve the computational complexity of the iterative algorithm from O(n 4 ) to O(n 3 ) in the worst case. We also introduce a threshold sieving heuristic and its accuracy estimation that further improves the efficiency of the method.As a practical illustration of our techniques we computed SimRank scores on a subset of English Wikipedia corpus, consisting of the complete set of articles and category links. The requirement for measuring similarity between objects arises in many fields of computer science; examples include recommender systems, "related pages" queries of web search engines, document classification and clustering. Large amounts and fast growth of information require machine aid to humans for finding, classifying and analyzing requested information. Such a range of challenges includes automatically detecting objects similar to a given object and ranking them Permission to copy without fee all or part of this material is granted provided that the copies are not made or distributed for direct commercial advantage, the VLDB copyright notice and the title of the publication and its date appear, and notice is given that copying is by permission of the Very Large Data Base Endowment. To copy otherwise, or to republish, to post on servers or to redistribute to lists, requires a fee and/or special permission from the publisher, ACM. VLDB '08, August 24-30, 2008, Auckland, New Zealand Copyright 2008 VLDB Endowment, ACM 000-0-00000-000-0/00/00. in accordance with their similarity scores. While humans make judgment on object similarity intuitively, based on their previous experience, the task of systematically computing object similarity by a machine remains nontrivial. For practical applicability, an effective similarity measure should both reflect human intuition on objects similarity and provide reasonable computational complexity.. For the existing similarity measures, two broad categories can be outlined: (i) content-or text-based similarity measures that treat each object as a bag of items or as a vector of word weights, and (ii) link-based ones that consider objectto-object relations expressed in terms of links. In the recent research [12] , the extensive evaluation of different similarity measures was performed, and link-based measures produced systematically better correlation with human judgments compared to text-based measures. It is worth mentioning that the success of the Google search engine began with its ability to rank search results in accordance with human expectations; the latter feature was essentially based on a purely link-based ranking algorithm called PageRank [1] . From this perspective, it is reasonable to assume that an effective similarity measure would have a comparable impact for computer science techniques as PageRank ranking algorithm had for web search.. Considering the outlined state of the art, the similarity measure SimRank [7] can be considered as one of the promising ones, due to the following reasons. SimRank is a link-based similarity measure, and builds on the approach of previously existing link-based measures. SimRank is based on both a clear human intuition and a solid theoretical background. Similarly to PageRank, SimRank is defined recursively with respect to "random surfer" model and is computed iteratively. Unlike the similarity measures that require human-built hierarchies, SimRank is applicable to any domain with object-to-object relationships, including the Web.. Nevertheless, existing work on SimRank lacks two important issues. Firstly, although SimRank iterative similarity scores are known to converge [7] , a real-life computation naturally involves performing a finite number of iterations.. However, a potential difference between SimRank iterative similarity scores and theoretical ones remains an open question. The symmetric question is finding out the precise number of iterations sufficient to guarantee a desired accuracy.. Secondly, optimization issue of SimRank computation is not the primary focus of the original SimRank proposal [7] . To the best of our knowledge, there exists the only research work concerning SimRank optimization [3] . That work is initially oriented on SimRank probabilistic computation, details are considered in the Related Work section. As for SimRank iterative computation, optimization has not been addressed in scientific literature yet, and the time complexity of the straightforward SimRank computation is an obstacle for using SimRank on practical data corpora.. This paper presents a solution to both issues, both mathematically proven and practically approved by experimental results. In summary, the main contributions of this paper are the following:. • A precise accuracy estimate is presented for SimRank scores computed iteratively, with respect to the theoretical ones. This allows one to find out the number of iterations required for achieving the desired accuracy.. • Optimization techniques for SimRank computation are presented that improve SimRank computational complexity from O(n 4 ) to O(n 3 ). A threshold sieving heuristic is introduced and its accuracy estimation is given that further improves the efficiency of the method.. • SimRank computational viability for relatively large object corpora in the presence of the suggested optimization techniques is verified experimentally by computing SimRank similarity scores for a subset of English Wikipedia corpus, consisting of the complete set of articles and category links.. The rest of the paper is organized as follows. In the next section, SimRank overview is given and the necessary notations and formulae are introduced. In Sect. 3, accuracy estimate for the SimRank iterative computation model is established. In Sect. 4, SimRank optimization techniques are suggested and summarized in the general algorithm; the computational complexity of the proposed algorithm is given. Sect. 5 gives the overview of the related work. Experimental results are presented in Sect. 6. Future work is discussed in Sect. 7. Although optimization techniques presented in this paper provide a considerable improvement to SimRank computational complexity, our profiling experiments revealed that a significant proportion of computation time is occupied by graph access operations I(v) and O(v). Although SimRank computational complexity is guaranteed to remain at most cubic in the number of nodes, graph access operations may become a performance bottleneck in the growing size of input graphs. Our future work thus involves developing fur-ther optimization techniques for speeding up graph access operations. Our current vision to achieving scalability in the growing size of input graph is splitting the graph into several (generally, intersecting) subgraphs in such a way that 1. each subgraph could be passed to its own parallel computation instance as if the complete graph without changing the result of similarity scores computation for node pairs processed by that instance; and 2. each subgraph is small enough to fit into main memory for maximizing the speed of access operations.. Our future plans include developing a systematic procedure for splitting a general graph into subgraphs that satisfy the above listed requirements. In particular, we believe that scale-free graphs theory [8] can be exploited for developing such a procedure for scale-free graphs, which constitute an underlying model for many existing practical domains. The paper addresses the issues missing for similarity measure SimRank, namely, accuracy estimate and optimization techniques, for facilitating SimRank wider application.. A precise accuracy estimate for SimRank iterative computation is established. The estimate reveals that SimRank computation parameters suggested in the original SimRank proposal implied a relatively low accuracy, and the choice for different parameter values is suggested. The accuracy estimate allows a-priori finding out the correct number of iterations required for achieving a desired accuracy. The number of iterations turns out to be independent of input graph characteristics, the fact to benefit scalability.. Optimization techniques are suggested and integrated into the general algorithm to provide a systematic improvement for SimRank computational complexity.. Experimental results show a 50 times speedup achieved by the optimization techniques for a graph with 10K nodes, and relative improvement in computation time further increases for larger graphs. The experience in computing SimRank scores over the English Wikipedia corpus exhibits practical viability of the approach for relatively large data corpora.. We believe that the results presented in the paper would facilitate a wider application of SimRank to computer science techniques, as this similarity measure definitely deserves. Due to practical importance of measuring object-to-object similarity, different approaches to defining similarity measures were suggested in scientific literature, e.g. the ones based on domain hierarchies [5] , information theory [9] , network flow computation [11] . With respect to the focus of this paper, a detailed discussion is given to related work that either correlate with SimRank or present similarity measures with complexity analysis claimed applicable for large data corpora.. Xi et al. suggested a similarity-calculating algorithm called SimFusion that aims at "combining relationships from multiple heterogeneous data sources" [17] . The basic intuition behind SimFusion approach somewhat resembles the one for SimRank: "the similarity between two data objects can be reinforced by the similarity of related data objects from the same and different spaces" [17] . SimFusion provides the following extensions with respect to SimRank: (i) support for different kinds of intra-nodes relations, e.g. outgoing links, content commonality; (ii) support for different weights associated with different kinds of relations; (iii) support for several information spaces.. Iterative similarity computation formula for SimFusion has much in common with the one for SimRank. Indeed, with Lurm being a row-stochastic matrix that combines all the relationships between nodes, SimFusion reinforcement assumption is reified as follows:. where Susm is a "unified similarity matrix" that represents similarity values between node pairs [17] . Let us denote a row in Lurm that corresponds to node a as Lurm(a), and a matrix element in S k urm that corresponds to a pair of nodes (a, b) as S k usm (a, b). If we then consider node relations of the kind "has an incoming link" and treat all incoming links as of an equal priority, then Lurm(a) contains 1 |I(a)| in column Ii(a), i = 1, n and zeroes in all remaining columns. In accordance with (25), SimFusion iterative similarity value between nodes a and b thus takes the form:. which is the same as for SimRank iterative formula minus the decay factor C. Unlike SimFusion, SimRank similarity score for any node with itself always equals to 1, and these initial similarity scores are iteratively propagated to the other node pairs. In SimFusion, initial similarity scores are redistributed in a flow fashion through node relations, and thus node similarity with itself may not be equal to 1.. If a node has no incoming links, it has zero SimRank similarity score with every other node except for itself. In SimFusion, a node with no relationship to the other nodes in data space has each element in the corresponding row of relationship matrix set to 1/n for preventing similarity sinks. This treatment has the same effect as introducing a source of rank in PageRank [13] .. In spite of the noted differences between SimRank and SimFusion, equation (26) shows that the overall iterative formulae for SimRank and SimFusion have much resemblance to each other. Consequently, some of the optimization techniques presented in this paper should apply to SimFusion computation as well, e.g. essential node pairs selection in terms of sparse matrices. We also believe that there should exist an analogous accuracy estimate for SimFusion as the one revealed for SimRank.. Computational complexity for SimFusion is O(Kn 3 ) in the worst case [17] . Cubic computational complexity in the number of nodes for SimFusion directly follows from matrix representation (25); as for SimRank, the double summation in the iterative formula (3) misleaded SimRank computational complexity, which was previously considered O(Kn 4 ) in the worst case [7, 17] . The idea of partial sums suggested in this paper fixes this inconsistency between SimRank and SimFusion, for now both are known to have cubic computational complexity in the number of nodes.. Fogaras and Rácz [3] suggested a scalable framework for SimRank computation based on Monte Carlo method. The main idea of their approach is to generate reversed random walks for each node in a graph, calculate the first meeting time τ a,b for a pair of random walks started in nodes a and b, and estimate s(a, b) by C τ a,b . In their work, Fogaras and Rácz suggest several excellent ideas, in particular, fingerprint trees, random permutations on graph nodes for effectively generating coupled random walks, parallelization possibilities for SimRank computation under their framework. The probabilistic approach they took allowed them significantly reduce the computational complexity and create a framework for similarity computation scalable enough for performing SimRank precomputation phase for a graph with 79M nodes. Fogaras and Rácz provide a solid theoretical basis for approximations they make and a (probabilistic) error estimate for their Monte Carlo similarity function. Valuable experimental results are obtained for path length and the decay factor value; surprisingly however, Monte Carlo similarity was not compared with iterative SimRank similarity from the scores quality perspective. The differences in our approaches are that Fogaras and Rácz initially base their framework on Monte Carlo method, and thus their computation is inherently probabilistic, whereas our work is focused on iteratively computing the exact similarity scores.. For improving time and space requirements for SimRank computation, Jeh and Widom [7] suggested pruning the logical graph G 2 . Their proposal is to "set the similarity between two nodes far apart to be 0, and consider node-pairs only for nodes which are near each other" [7] . This technique is based on the assumption that "it is very likely that the neighborhood (say, nodes within a radius of 2 or 3) of a typical node will be a very small percentage (< 1%) of the entire domain" [7] . Firstly, this assumption does not hold for scale-free graphs, as these have a very small average distance (or diameter) between nodes [2] . As our early experimental studies over practical scale-free graphs showed, even the neighbourhood within the radius of 2 contains a considerable proportion of graph nodes.. Secondly, pruning graph G 2 in the suggested way is an approximation; Jeh and Widom provide no theoretical argument about the error of approximating [3] , but admit that "the quality of the approximation needs to be verified" [7] . Finally, removing node pairs not near each other from consideration undermines the basic SimRank design principle of being a generalization to a conventional immediate inneighbours analysis [15] . As verified by Fogaras and Rácz in their experiments, "the multi-step neighborhoods of pages 2 contain valuable similarity information" [3] .. For comparison, optimization techniques suggested in this paper are free from the above mentioned drawbacks. Selecting essential node pairs does not affect iterative similarity scores. Threshold sieving has a controllable effect over iterative scores precision that can be restored by performing an additional iteration. The techniques do not decrease the radius of a node neighbourhood considered for similarity scores computation.. Maguitman et al. [12] introduce an information-theoretic measure of semantic similarity that exploits human-generated topical directories metadata and relies on "both hierarchical and non-hierarchical structure of an ontology" [12] . Maguitman et al. suggest a flexible mechanism for extending the previously existing tree-based semantic similarity measures to a graph-based semantic similarity; however, their results are based on the assumption that a graph necessarily has a "hierarchical (tree) component" T .. Maguitman et al. claim computational complexity O(n 3 ) for their semantic similarity measure for n topics [12] ; however, that is computational complexity for just a single matrices product A B used in their reasoning. Precisely, semantic similarity measure requires the closure matrix T + computed; the latter is defined as T + = limr→∞ T (r) , with. where h is "the maximum depth of the tree T " [12] . Generally, h depends on n, and the worst case computational complexity for T + is actually O(n 4 ), not O(n 3 ). Maguitman et al. required significant computational and storage resources for computing similarity scores for their semantic similarity measure for a data corpus consisting of 0.5M topics containing totally 1.23M pages [12] . For comparison, our claim is that SimRank optimization techniques suggested in this paper allow computing SimRank similarity scores on a desktop machine in reasonable time even for a larger data corpora, as our experimental results illustrate.. Using once computed semantic similarity scores as a baseline, Maguitman et al. introduce and compare several approximation measures. Notably, link-based similarity measures systematically produced better correlation with semantic similarity measure and correspondingly with human judgments compared to text-based measures [12] . This important result can in particular serve as an approval for SimRank being a purely link-based similarity measure.. Lin et al. [10] suggest a similarity measure based on PageRank scores propagation through link paths. With r standing for the propagation radius and d for an average node degree, finding similar nodes to a given node with respect to that measure has computational complexity of O(d 2r ), which we believe being too ineffective for on-line computation assumed in [10] .. Geerts et al. [6] introduce the concept of a database graph for expressing relationships between partial tuples in a relational database and explore methods for ranking partial tuples. Defining similarity measures in a database graph is pointed out as an interesting question for future work. We believe that SimRank can be a candidate measure for this domain, since a database graph contains enough information for computing SimRank similarity scores for partial database tuples, and the suggested optimization techniques provide viable computational complexity.

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


Node density independent localization This paper presents an enhanced version of a novel radio interferometric positioning technique for node localization in wireless sensor networks that provides both high accuracy and long range simultaneously. The ranging method utilizes two transmitters emitting radio signals at almost the same frequencies. The relative location is estimated by measuring the relative phase offset of the generated interference signal at two receivers. Here, we analyze how the selection of carrier frequencies affects the precision and maximum range. Furthermore, we describe how the interplay of RF multipath and ground reflections degrades the ranging accuracy.To address these problems, we introduce a technique that continuously refines the range estimates as it converges to the localization solution. Finally, we present the results of a field experiment where our prototype achieved 4 cm average localization accuracy for a quasi-random deployment of 16 COTS motes covering the area of two football fields. The maximum range measured was 170 m, four times the observed communication range. Consequently, node deployment density is no longer constrained by the localization technique, but rather by the communication range. Categories and SubjectThe DARPA/IXO NEST program has supported the research described in this paper. We have recently proposed a novel approach to sensor node localization, the Radio Interferometric Positioning System (RIPS) [4] . RIPS creates a low-frequency interference signal by one pair of nodes transmitting simultaneously at close frequencies. The relative phase offset at a pair of receiver nodes is used to determine a distance measure between the transmitting and receiving nodes. Unlike tradiPermission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. To copy otherwise, to republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. tional ranging approaches, which determine the pairwise distance between two sensor nodes, RIPS measures dABCD, a distance aggregate called the "q-range" involving four nodes: two transmitters A, B and two receivers C, D. We reported a localization accuracy of 3 cm in a 16-node setup covering an area of 324 m 2 . We estimated the maximum range of RIPS on Mica2 motes to be 160 m, but this was not experimentally verified. We have successfully demonstrated in multiple field experiments that the accuracy of RIPS is as good as the best ultrasonic techniques while allowing two orders of magnitude smaller node density. At the same time, RIPS works at less than one tenth of the node density of current outdoor acoustic methods and achieves two orders of magnitude better precision.. The range that can be accurately measured with RIPS is about 4 times the communication range. That means that any deployment that forms a reasonable connected communication graph can be localized. In other words, the necessary deployment density is determined by the communication (or sensing) needs of the application and not the localization. In this sense, RIPS is indeed a node-density independent localization method.. These results correspond to minor to moderate RF multipath environments such as rural areas or open spaces within an urban area. Our current work focuses on extending the approach to cluttered urban environments. Range-based approaches to sensor node localization provide higher accuracy than range-free methods. Ranging in WSNs is typically based on acoustic or Radio Signal Strength (RSS) measurements. Unless a powerful central beacon is used as a sound source, the acoustic range is limited to 10 meters or so. The reliable range of RSS in urban deployment is even less. The largest-scale acoustic localization setup found in the literature used 45 nodes in an offset grid of approximately 9 m cell size [3] . The average accuracy achieved was 2.4 m, while the deployment density was larger than 10 4 nodes/km 2 . Ultrasonic methods provide centimeter-scale accuracy, but due to their limited range typical setups have > 10 5 nodes/km 2 density [9, 5] . The accuracy of RSS experiments is measured in meters, with a required density of 10 5 nodes/km 2 [9, 1] . Recently a novel approach, called Spotlight [6] , was proposed for sensor node localization. A powerful laser beacon with precisely known position and orientation is used to scan the sensor field. Nodes detect the light and record the time of detection. Correlating the detection times with the orientation of the laser, the node locations can be computed using a 3D map of the area. While the accuracy and range of Spotlight is comparable to that of RIPS, the need for an expensive central beacon, light sensors and a map of the area makes its overhead prohibitive for many deployments.. Interferometry is a widely used technique in both radio and optical astronomy to determine the precise angular position of celestial bodies as well as objects on the ground. For example, very-long-baseline interferometry, where the antennas can be thousands of miles apart, is used to track the motion of the tectonic plates on earth with millimeter accuracy [8] . However, as these sophisticated techniques require very expensive equipment, it is just the underlying idea of using radio interference for positioning that we have borrowed and applied to the sensor network domain.

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


Adoption of electronic health records in the presence of privacy concerns: the elaboration likelihood model and individual persuasion Within the emerging context of the digitization of health care, electronic health records (EHRs) constitute a significant technological advance in the way medical information is stored, communicated, and processed by the multiple parties involved in health care delivery. However, in spite of the anticipated value potential of this technology, there is widespread concern that consumer privacy issues may impede its diffusion. In this study, we pose the question: Can individuals be persuaded to change their attitudes and opt-in behavioral intentions toward EHRs, and allow their medical information to be These results as well as other theoretical and practical implications are discussed. All that may come to my knowledge in the exercise of my profession or in daily commerce with men, which ought not to be spread abroad, I will keep secret and will never reveal.. -The Hippocratic Oath, 4 th Century BC In today's increasingly digital and networked society, as the volume of personal information captured in electronic databases continues to grow at exponential rates (Edmonds et al. 2004) , the concept of privacy has been elevated to the forefront of public discourse. Widely publicized compromises of personal information 2 are fueling a heated debate on how much information about oneself should be made available to others, and the extent to which this information can be used by various entities such as the government and private corporations. Indeed, recent research underscores the significance of privacy concerns when the Internet is used as a medium for transferring information (Dinev and Hart 2006; Hui et al. 2007; Malhotra et al. 2004 ), when information is gathered and used in an organizational context (Smith et al. 1996; Stewart and Segars 2002) , and how individuals respond to threats to electronic privacy (Son and Kim 2008) .. The focus of this paper is on an important and arguably controversial technological innovation: electronic health records (EHRs) that capture patient information in digital format and potentially make this information available in an identified way to those who have permission to access it and to others, in a de-identified, aggregated format. From a legal perspective, the Health Insurance Portability and Accountability Act of 1996 (HIPAA) states that the information in an EHR about a patient is actually owned by the practitioner collecting the information and/or the insurance payor covering the patient, yet the patient has the unconditional right to be informed of the data-handling practices of medical practitioners and providers. 3 It is only recently that hospitals, health systems, and insurance payors have begun to offer patients access to their personal EHR (sometimes termed a PHR or personal health record) through secure Internet connections, but oftentimes this access is limited to a single institution.. Although EHRs offer the potential to radically transform the health care system, no study has examined a key component of the adoption equation: What happens if health systems and providers adopt EHR systems, but patients refuse to allow their medical information to be digitized because of privacy concerns (Kauffman 2006) ? National surveys indicate that the public is particularly anxious about privacy in the context of health related issues: a recent report by the California HealthCare Foundation found that 67 percent of the national respondents felt "somewhat" or "very concerned" about the privacy of their personal medical records (Bishop et al. 2005) . U.S. Senators Frist and Clinton (2004) reinforced this point when they observed, "[patients] need…information, including access to their own health records….At the same time, we must ensure the privacy of the systems, or they will undermine the trust they are designed to create.". In the past decade, recognizing that the health care sector is in need of radical transformation, national and international momentum around the application of information technology in health care has grown considerably. In the United States, significant investments in EHR systems are being propelled by support from the highest levels of the federal government (Bush 2004) as the EHR is viewed by many as the foundation for a safer and more efficient healthcare system. In an attempt to connect every doctor and hospital in the country, the National Health Service in the United Kingdom is bearing the cost of providing electronic health records to every citizen (Becker 2004) . A general assumption underlying this momentum is that EHRs will have a positive impact on persistent problems with the delivery of health care such as medical errors and high administrative costs (Bates and Gawande 2003; Becker 2004) . However, there are many examples of information technologies that are promising but fail to diffuse widely because of resistance to use from key stakeholders. To the extent that the value potential of these technologies will not be realized in the face of such resistance, from a public policy perspective, this is a matter of significant concern.. Under the assumption that the adoption of EHRs is desirable, in this study we pose the question: Can individuals be persuaded to change their attitudes and adoption decisions toward electronic health records in the presence of significant privacy concerns associated with use? That is, if people are provided with positively framed messages about the value of EHRs, will they allow their doctors to "digitize" their medical information such that it could be made available to others in an electronic format? The individual patient or consumer is the central focus of our study; we do not investigate the adoption decision of the hospital or providers. There is an extensive and robust literature examining the behavioral aspects of technology adoption and usage, drawing upon multiple theoretical perspectives such as the technology acceptance model, theory of reasoned action, and diffusion of innovation (e.g., Agarwal and Prasad 1998; Davis 1989; Fishbein and Ajzen 1975; Rogers 1995; Venkatesh et al. 2003) . With a few exceptions (e.g., Bhattacherjee and Sanford 2006) , much of this research assumes implicitly that the respondent has developed a well-formed attitude toward the target technology, and there is typically limited discussion of the fact that the individual could be persuaded to change this attitude. While some work has examined pre and post-adoption behavior (Karahanna et al. 1999) , this research does not tap directly into how to persuade a person to change his or her opinion or into the influence process itself (Sussman and Siegal 2003) .. In the psychology literature, the elaboration likelihood model (ELM) provides a theoretical perspective on how attitudes evolve and change over time. To examine the effects of privacy concerns on the modification of attitudes, we integrate concern for information privacy (CFIP) into the ELM. Drawing upon the attitude persuasion literature, we suggest that individuals can be persuaded to support the use of EHRs, even in the presence of significant privacy concerns, if appropriate messages about the value and safety of EHR systems are imparted to them. We report findings from an experiment in which subjects are assigned to two different manipulations (positively framed and neutrally framed arguments) to assess the impact of CFIP on the relationship between these variables, attitude, and likelihood of adoption. We use structural equations modeling to empirically test our predictions with a sample of 366 subjects.. The research reported here makes several contributions to both research and practice. From a theoretical perspective, it extends the ELM to include an important moderating construct affecting persuasion and intentions in the context of the digitization of information which has not been examined in prior literature, viz., CFIP. The moderating effect of CFIP theorized, and empirically examined here, may be useful in other contexts in which personal information is controlled or processed. Second, we propose that attitudes toward the use of certain technologies are malleable in response to some forms of persuasion, and test this assertion empirically. As noted, with few exceptions, prior research has implicitly suggested that the attitudes of potential adopters of technology are relatively immutable. In contrast, extending the work of Bhattacherjee and Sanford (2006) , we posit that people can be persuaded even before they use a technology if value-based arguments resonate with them. 4 This finding is likely to be applicable to any information technology that might result in significant informational disparities between adopters and non-adopters. Third, this paper focuses on EHRs, which are a new and important technology with consequential impacts for the way in which health care is managed by consumers and providers. Given the sensitive and personal nature of one's health information, we also argue that the investigation of privacy of healthcare data offers unique insights into behaviors not found when examining other types of data privacy and security. Finally, the findings from this study offer pragmatic insights that can be used to drive public policy decisions related to public perceptions and attitudes toward the use of EHRs, including the crafting of national messages and education. The ELM specifies a set of theoretical mechanisms that yield attitude change that subsequently leads to behavioral intentions to engage in specific acts. Central to this theory is the notion of persuasion. In the context of digital health information, it is widely acknowledged that a critical barrier to widespread diffusion is the individual's concern about privacy. Will these privacy concerns hinder the adoption of EHR systems or can people be persuaded to accept the technology if proper messages are conveyed? We investigate this research question using the ELM as a theoretical framework. Our choice of a privacy-focused lens to investigate attitudes toward and adoption of electronic health records was motivated by three primary considerations. First, opinion-poll data would lead one to believe that privacy concerns will negatively affect attitudes toward EHRs to such a degree as to render any national efforts unachievable. We sought to demonstrate that through proper messaging and education, attitudes can be changed, even in the presence of great privacy concern. Second, there has been tremendous media attention focused on the privacy of information in a variety of different domains other than health such as financial data.. Typically the discourse has highlighted negative consequences such as breaches of security, fraud, or theft of information. This study is a first attempt at investigating whether these concerns are unfounded or if people weigh the costs and benefits of potentially compromising some degree of privacy for the possibility of getting better results (Dinev and Hart 2006) . This so called "privacy calculus" (Culnan 1993; Laufer and Wolfe 1977) is a cognitive process that people undergo as a means of assessing future ramifications from choices made today (e.g., if one chooses to opt-out of an EHR, could this result in negative consequences in the future?). The final reason we explored privacy is the belief that privacy concerns are the single greatest threat to successful rollouts of EHRs. Considerable work remains with respect to the investigation of the variability in beliefs related to privacy concerns and, in particular, whether public opinion will impact the use of information technology that has been developed to store and maintain personal information.. On the basis of published research, it is also abundantly clear that there is limited knowledge related to the role that patients play in the health information technology arena, especially as it relates to patient involvement in the delivery, monitoring, and dissemination of information related to their health care. This study sought to fill some of these gaps in current knowledge. Thus, it can serve as a foundation not only for making decisions related to EHR design, adoption, and implementation, but also as a basis for future research.. acceptable fit, no modifications were made to the model parameters. All variables were checked for multicollinearity and in all instances the test statistics were within acceptable ranges (tolerance greater than 0.2 (Menard 1995) and VIF less than 10 (Myers 1990) ).. Two questions are used to perform a manipulation check: respondents' assessment of the trustworthiness and reliability of the sources of information. Analysis of variance with trustworthiness and reliability as dependent variables and argument frame as a fixed factor indicate that respondents perceived significantly more trust and reliability when AF was positive, confirming that messages had been read and understood as desired (Trust: M positve = 5.33, M neutral = 4.51, p < 0.001; Reliable: M positive =5.42, M neutral = 4.40, p < 0.001). Overall, results provide empirical support for the core hypotheses proposed in this research. Argument framing, issue involvement, and concern for information privacy are all important influences on individuals' attitudes toward the use of EHRs both as main effects and as interactions with each other. We further find that attitudes and concern for information privacy influence the likelihood that the individual will opt-in to making his/her health-related data available in a digital artifact. While we were unable to collect actual behavioral data related to the opt-in decision, we suggest that this is an important area for future research. Much work remains with respect to the interrelations between privacy, attitudes and beliefs, and actual behavior. This, coupled with the increasing prevalence of volitional (versus mandated) information systems, offers numerous research opportunities.. As noted earlier, the interaction term AF × II exhibited counter-intuitive behavior. In Model 2, the interaction term was nonsignificant but in model 5, it became significantly negative. One explanation for the nonsignificant result is that the motivated subjects have reason to believe in the value of the use of EHRs and therefore respond favorably to all messages, irrespective of whether they are positively framed. It is also possible that the highly motivated individuals would have been previously exposed to the positively framed arguments and in many cases may be more familiar with the source, thus the endorsement may not have been as striking. However, the post hoc examination yielded other interesting insights. Following the median split analysis, we learned that there was a significant difference in attitude change but only under conditions of positive AF. One explanation for this is that it is easier to persuade highly involved individuals of the value of EHRs than it is to persuade weakly involved individuals. It can be argued that high II individuals want to believe in the value of EHR use, while low II individuals need to be persuaded by more compelling arguments. This may be due to a lack of understanding, or a misunderstanding, of the uses of EHRs by low II respondents. It may also be the case that the uninformed are unnecessarily concerned about functions and features of EHRs, which may or may not be grounded in fact.. A second explanation can be offered by insights that the ELM brings. As noted, elaboration is a function of both ability and motivation and if one or both factors are low, a peripheral route of persuasion may be acting. In this particular case, we found that attitude change was only significant when AF was Note: Attitude change represents the difference between pre-and postmanipulation attitudes about the use of EHRs. The attitudes were measured on three 7-point semantic differential scales anchored at 1 and 7 (bad-good, foolish-wise, unimportant-important). Demographic and descriptive statistics are presented in Table 3 (see also Appendix B). Estimates derived from the SEM analysis are used to test the research hypotheses. The overall fit statistics of the structural model were nearly identical to the initial measurement model, however, the structural model is preferred due to parsimony (i.e., the measurement model includes all paths between variables whereas the struc-tural includes only the hypothesized paths): CFI = .91, AGFI = 0.82, RMSR = 0.06, RMSEA = 0.07.. In the first hypothesis, we proposed a relationship between argument framing and post-manipulation attitude. The path coefficient is positive and significant (β AF = .076, p < .10, see Table 4 , Model 2) suggesting that positively framed arguments yield higher post-manipulation attitudes, supporting hypothesis 1. In hypothesis 2, we argued that greater II will yield greater attitude changes. Our results support this assertion (β II = .123, p < .01, see Table 4 , Model 2). Hypothesis 3 states that II will positively moderate the relationship between AF and post-manipulation attitude. We create the AF × II interaction term by multiplying the variables (Kenny 2004) and the resultant standardized coefficient measures how the effect of AF varies as II varies. In Model 3, the AF × II coefficient is 0.007 and nonsignificant, indicating that the effect of II on attitude change does not change significantly as AF goes from 0 (neutral) to 1 (strong). Thus, H3 is not supported.. A key objective of this study is to evaluate how privacy concerns influence attitude and likelihood of adoption. In hypothesis 4, CFIP is hypothesized to moderate the relationship between AF and attitude. Following the procedure outlined by Kenny (2004) , we create a product term for AF (positive/neutral) and the continuous, aggregated variable CFIP. The relationship between AF × CFIP and attitude is positive and significant (β AFxCFIP = .335, p < .05, Model 4), thus confirming hypothesis 4. In hypothesis 5, CFIP was hypothesized to moderate the relationship between AF × II and attitude. In this three-way interaction analysis, we again followed the procedure outlined by Kenny and created variables for the AF × II interaction and multiplied these by CFIP and calculated the resulting path coefficient. The three-way interaction was positive and highly significant (β AFxIIxCFIP = .629, p < .001), indicating that CFIP does in fact moderate the relationship between AF × II and attitude, thus confirming hypothesis 5. The variance in post-manipulation attitude explained by AF, AF × II, AF × II × CFIP, and the control variables is 87.6 percent, representing a 19.0 percent increase in variance explained when the three-way interaction is included.. In hypothesis 6, we proposed a relationship between postmanipulation attitude and likelihood of adoption of EHRs (opt-in); this relationship is positive and significant (β Att = .50, p < .001; see Table 4 , Models 1-5), confirming hypothesis 6. Finally, in hypothesis 7, we posited a negative relationship between CFIP and likelihood of adoption. The negative relationship between CFIP and LOA is present and highly significant (β CFIP = -.12, p < .05; see Models 1-5); therefore, 11 The left portion of the model is essentially a 2 × 2 factorial design, which has traditionally been tested using an ANOVA. Our intent in using SEM was to assess the entire model simultaneously, including the right-hand portion of the model. Later, in the post hoc analysis, we use ANOVA to further interpret the findings. The χ 2 -statistic is significant at the p<.001 level in all models, when a well-fit model should yield a non-significant result. However, many researchers have noted that the χ 2 -statistic is sensitive to large sample sizes (Froehle and Roth 2004; Hu and Bentler 1999) , and the χ 2 /df test yields values of less than 3 in most models.. *** = p < .001; ** = p < .01; * = p < .05; † = p < .10 hypothesis 7 is also supported (see Table 5 for a summary of results). The model explains a substantial amount of variance in the dependent measures, particularly post-manipulation attitude (i.e., post-manipulation attitude, adjusted R 2 = 87.6 percent; likelihood of adoption, adjusted R 2 = 25.5 percent).

O O B I I O O O O B O O B I I 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 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 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 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 B 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 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 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 B 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 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 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 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 B 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 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 B I I 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 B 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 I B 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 B B O O B 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 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 B 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 O O O 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 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 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 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 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The f-vector of the descent polytope For a positive integer n and a subset S ⊆ [n − 1], the descent polytope DP S is the set of points (x 1 , . For a set S ⊆ [n − 1] = {1, 2, . . . , n − 1}, define the descent polytope DP S to be the set of points (x 1 , . . . , x n ) in R n such that 0 ≤ x i ≤ 1, and. Thus DP S is the order polytope of the ribbon poset Z S = {z 1 , z 2 , . . . , z n } defined by the cover relations z i ·> z i+1 if i ∈ S and z i <· z i+1 if i / ∈ S; see [9] . Descent polytopes occur as a subdivision of the ndimensional unit cube in the recent work [2] of Ehrenborg, Kitaev, and Perry. It is clear that the set S and its complement S = [n − 1] − S yields the same descent polytope up to an affine transformation.. In this paper our primary goal is to compute the f -vector of the descent polytope DP S . Recall that for an n-dimensional polytope, the f -vector is the integer vector (f 0 , f 1 , . . . , f n ), where f i is the number of i-dimensional faces in the polytope. Observe that f n = 1 since the descent polytope is a face of itself. For S ⊆ [n − 1], define the f -polynomial of the descent polytope DP S to be. To simplify notation, we will often write F S instead of F S (t). As we show in Section 2, F S can be expressed as a sum of polynomials taken over all subsets of S.. The volume of the descent polytope DP S is given by β(S)/n!, where β(S) denotes the number of permutations in the symmetric group S n with descent set S. A classical result in combinatorics is that β(S) is maximized when S is an alternating set, that is, S = {1, 3, 5, . . .} ∩ [n − 1], or equivalently S = {2, 4, 6, . . .} ∩ [n − 1]; see [1, 6, 7, 8, 11] . We show that the same result holds for the f -vector of the descent polytope. That is, the number of i-dimensional faces of the descent polytope DP S is maximized when S is the alternating set.. A different way to encode subsets of the set [n − 1] is by a word in letters x and y of length n − 1. Viewing x and y as non-commutative variables and the words as monomials, we consider the non-commutative generating function Φ(x, y) = v F v · v. We determine this generating function, which turns out to be a rational function. Using this rational function we obtain a more concise expression for the f -polynomial of the descent polytope DP S . A second way to encode subsets of [n − 1] is by compositions. In Section 4 we use this encoding to obtain more recurrences to compute the f -polynomial F S .. We end the paper with a few open questions and directions for further research. . We denote F S by F v S ; such notation has an advantage, as v S encodes not only S ⊆ [n − 1] but also the dimension n. Since pairs (n, S ⊆ [n − 1]) are in bijective correspondence with xy-words via S → v S , it is natural to parameterize the f -polynomials of descent polytopes by xy-words and write F v , where v = v S for some S ⊆ [|v|], and |v| denotes the length of the word v.. For a xy-word v = v 1 v 2 · · · v n−1 , define the statistic κ(v) by κ(v) = 2 + |{i : v i = v i+1 }| for v = 1, and κ(1) = 1. A direct observation is the number of facets of the descent polytope is described by κ.. Lemma 2.1. The number of (n − 1)-dimensional faces of the n-dimensional descent polytope DP v is given by. Proof. There are n − 1 supporting hyperplanes of the form x i = x i+1 that each intersect the polytope in a facet. The hyperplane x i = 1 intersects the polytope in a facet if one of the following three cases holds: v i−1 v i = xy; i = 1 and v 1 = y; or i = n and v n = x. A similar statement holds for the hyperplane x i = 0. The lemma follows by adding these three statements.. For a xy-. The following theorem provides a way to compute the f -polynomial F v . Theorem 2.2. Let v be a xy-word of length n − 1. Then the f -polynomial of the descent polytope DP v is given by. Proof. For a face F of a polytope, let F I denote the relative interior of F. Then the polytope is the disjoint union of F I taken over all faces F, including the polytope itself.. Recall that the descent polytope DP v consists of all points (x 1 , . . . , x n ) ∈ R n belonging simultaneously to the half spaces. , and x i ≥ x i+1 (v i = y). A face F of DP v can be uniquely identified by specifying which of these half spaces contain F on their boundary hyperplanes, as long as the intersection of the whole polytope and the specified boundary hyperplanes is non-empty. Forming the specification just for the half spaces of the form x i ≤ x i+1 or x i ≥ x i+1 restricts the location of F I in R n to the region defined by the relations. , where the symbol ≶ denotes strict inequality:. Then T is the set of indexes j for which F does not lie entirely on the boundary hyperplane x j = x j+1 and thus the relative interior F I is contained in the interior of the corresponding half space. Let R(T ) denote the intersection of the region defined by (2.1) and the hypercube [0, 1] n . Each point (x 1 , . . . , x n ) of DP v belongs to exactly one such region R(T ), namely, the one for T = {j | x j = x j+1 }. Thus we have the disjoint union. Let us show that the term corresponding to T = ∅ in the expression in the statement of the theorem is the contribution to F v of the faces F of DP v for which F I is contained in the region R(T ). In other words, we claim that for T = ∅ we have. To select a particular face F from the set of all faces with the property F I ⊆ R(T ), we need to complete the specification started above, that is, we must specify which of the hyperplanes x i = 0, 1 contain F, and we must make sure that the intersection of the set of the specified hyperplanes and R(T ) is non-empty. In terms of defining relations (2.1), this task is equivalent to setting the common value of some of the "blocks" of coordinates (x 1 , . . . , x j 1 ), (x j 1 +1 , . . . , x j 2 ), . . . , (x j k +1 , . . . , x n ) to 0 or 1. Since the relations must remain satisfiable by at least one point in [0, 1] n , only the blocks preceded in (2.1) by > (or nothing) and succeeded by < (or nothing) can be set to 0. Similarly, only the blocks preceded by < (or nothing) and succeeded by > (or nothing) can be set to 1. Thus each block can be set to at most one of 0 and 1. The letters of the xy-word v T = v j 1 · · · v j k encode the inequality signs in (2.1) (x stands for <, and y stands for >), so the number of blocks that can be set to 0 or 1 is the total number of occurrences of x followed by y, or y followed by x, in v T , plus 2, as we also need to count the first and the last blocks. In other words, the number of such blocks is κ(v T ).. Observe that the dimension of the face of DP v obtained by this specification procedure equals the number of blocks that have not been set to 0 or 1: the common values of the coordinates in those blocks form the "degrees of freedom" that constitute the dimension. Let us call such blocks free. The number of faces F with F I ⊆ R(T ) for which the specification procedure results in m free blocks is. the number of ways to choose |T | + 1 − m blocks that are not free out of κ(v T ) possibilities. Hence we have. Finally, for T = ∅, we have R(T ) = {0 ≤ x 1 = · · · = x n ≤ 1}, which is just the line segment joining the two vertices (0, . . . , 0) and (1, . . . , 1) of DP v . Thus the contribution of R(T ) to F v is. Adding this equation to the sum of (2.2) taken over the non-empty T proves the theorem. A more general invariant to study of the descent polytopes is the flag f -vector. The flag f -vector is efficiently encoded by the cd-index. Is there a way to describe the cd-index of the descent polytope in terms of the xy-word u?. Setting t = 1 in the polynomial F v (t) we obtain the number of faces of the descent polytope DP v . Especially, for the alternating word z n we obtain the sequence {F z n−1 (1)} n≥1 = 3, 7, 19, 51, . . .. This sequence has a different combinatorial interpretation, as it matches the sequence A052948 in the Online Encyclopedia of Integer Sequences [5] defined as the number of paths from (0, 0) to (n + 1, 0) with allowed steps (1, 1), (1, 0) and (1, −1) contained within the region −2 ≤ y ≤ 2. The generating function 1 − 2x 2 1 − 3x + 2x 3 given in [5] indeed results if t = 1 is substituted into (3.5) and the constant 1 is added. Is there a bijective proof? A first step to find such a bijective proof would be to find a statistic on these lattice paths with the same distribution as the dimensions of the faces of the descent polytope DP z n−1 .. For a xy-word v = v 1 v 2 · · · v n let v * denote the reverse of the word, that is, v * = v n · · · v 2 v 1 . Note that the two descent polytopes DP v and DP v * only differ by a linear transformation and hence their f -polynomials agree, that is, F v = F v * . However the expressions for the f -polynomials for F v and F v * in Corollary 3.3 differ. Is there a bijection between the factorizations of v and v * ? The number of factorizations of v is also equal the number of alternating subwords of v; see Corollary 2.3. This fact also asks for a bijective proof.. More inequalities for the descent statistic has been proved in [3, 4] . Can these inequalities be extended to the f -polynomial F v ? For instance, Ira Gessel asked the following question: where does the maximum of the descent set statistic occur when restricting to words v of length n − 1 having exactly k runs of x's and y's. He conjectured and it was proved in [4] that the maximum occurs at the composition (r, r + 1, . . . , r + 1 a , r, . . . , r) where r = ⌊(n − 1)/k⌋ and a = (n − 1) − r · k. Would the f -polynomial be maximized at the same composition?

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Non-Markovian dynamics of cavity losses We provide a microscopic derivation for the non-Markovian master equation for an atomcavity system with cavity losses and show that they can induce population trapping in the atomic excited state, when the environment outside the cavity has a non-flat spectrum. Our results apply to hybrid solid state systems and can turn out to be helpful to find the most appropriate description of leakage in the recent developments of cavity quantum electrodynamics. Cavity quantum electrodynamics (CQED) is one of the most important fields of quantum optics, both from a fundamental and an applicative point of view 1 . Indeed, the interaction of an atom and a single mode of the quantized electromagnetic field inside a high-Q cavity can be exploited to study the properties of highly nonclassical states of the atom-cavity system and to use their non-classical features for quantum information processing 2 . In this context it is fundamental to take into account the unavoidable coupling between the atom-cavity system and the environment external to it. In general, the coupling with an environment gives rise to dissipation and decoherence phenomena 3 , which can strongly affect the dynamics of the quantum system under study. In previous papers 4, 5 we derived a master equation for an atom-cavity system which takes into account from the very beginning a coupling between the atom and the cavity described by the Jaynes-Cummings (JC) model 6 . The approach followed in Refs. 4 and 5 is different from the one usually reported in the literature to describe cavity losses. Indeed the usual model of cavity losses has been introduced in a phenomenological way, i.e., one microscopically derives the master equation for the cavity only, described as a quantum harmonic oscillator coupled to a bosonic bath, and then assumes that the presence of the atom inside the cavity does not affect the structure of the dissipator in the master equation 7 . Through the microscopic model derived in Refs. 4, 5 it is possible to give a complete justification of the phenomenological model, which turns out to be valid when the spectrum of the environment is flat and its temperature is zero. The situation is quite different if the environment has a non-flat spectral density. This is not an uncommon situation nowadays: indeed there are new-generation CQED experiments which are performed on high-Q cavities created inside photonic bandgap (PBG) materials 8 . For this system we proposed a non-Markovian model of cavity losses 9 which predicts a dynamics completely different from the one predicted by the phenomenological model, giving rise to phenomena such as population trapping due to cavity losses. Scope of this paper is to provide some details of the derivation of the non-Markovian master equation for the JC model and to show that the main predictions of the theory, namely population trapping due to cavity losses 9 , does not depend on the shape of the spectral density one chooses, but only on the values of the asymptotic decay rates corresponding to the transitions of interest. We clarify this point by taking into account two different models of environmental spectral densities and showing that, for the same asymptotic values of the decay rates, the predictions differ only in the details of the very short-time dynamics, while the two densities give the same predictions for longer times. The paper is structured as follows. In Sec. 2 we present the non-Markovian model of cavity losses, in Sec. 3 we present the calculation of the time-dependent decay rate for a Lorentzian spectrum and summarize the dynamics of the system in this case. In Sec. 4 the case of a spectrum with a Lorentzian gap is presented along with some conclusive remarks.

O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O 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 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 O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O


Hex-splines: a novel spline family for hexagonal lattices Abstract-This paper proposes a new family of bivariate, nonseparable splines, called hex-splines, especially designed for hexagonal lattices. The starting point of the construction is the indicator function of the Voronoi cell, which is used to define in a natural way the first-order hex-spline. Higher order hex-splines are obtained by successive convolutions. A mathematical analysis of this new bivariate spline family is presented. In particular, we derive a closed form for a hex-spline of arbitrary order. We also discuss important properties, such as their Fourier transform and the fact they form a Riesz basis. We also highlight the approximation order. For conventional rectangular lattices, hex-splines revert to classical separable tensor-product B-splines. Finally, some prototypical applications and experimental results demonstrate the usefulness of hex-splines for handling hexagonally sampled data. In this paper, we proposed and studied a novel family of bivariate, nonseparable splines for hexagonal lattices. These splines are defined in a natural way, starting from the Voronoi cell definition. We studied the mathematical foundations and presented the algorithmic tools required to make full use of the hex-splines. As a summary, we show in Table II the (21) can be demonstrated by using the Fourier transform of their sign-version [see (30) ].. The recursive relationship (23) for the case of crossterms, is also proved using the sign-version. Consider the first-order generating functions using respectively the vectors and , and and . The Fourier transforms correspond to with (54). The crossterm of both generating functions, corresponds to the following Fourier expression, which can be separated using partial fractions and which gives us Applying this equation to a general crossterm results in (23) .. This recursive equation can also be used to compute some derivative generating functions. For example, when computing the derivative in the direction of , the Using this result, the remaining terms of the difference between the right and left hand side of (55) can be "neutralized" using the localization of , which equals zero where , according to the definition of . Next, we write the two-sided generating function in terms of one-sided generating functions using (26) and applying (55) and (56) (57) This corresponds to a symmetrized version of the "intermediate" part of the hex-spline (this can easily checked by applying the construction algorithm of Section III-B to ). Analogously, we can obtain (58) Adding up both parts, i.e., (57) and (58), results into the firstorder hex-spline.

O O O O O O B 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 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