Combinatorial techniques for mixed-size placement While recent literature on circuit layout addresses large-scale standard-cell placement, the authors typically assume that all macros are fixed. Floorplanning techniques are very good at handling macros, but do not scale to hundreds of thousands of placeable objects. Therefore we combine floorplanning techniques with placement techniques to solve the more general placement problem. Our work shows how to place macros consistently with large numbers of small standard cells. Proposed techniques can also be used to guide circuit designers who prefer to place macros by hand.We address the computational difficulty of layout problems involving large macros and numerous small logic cells at the same time. Proposed algorithms are evaluated in the context of wirelength minimization because a computational method that is not scalable in optimizing wirelength is unlikely to be successful for more complex objectives (congestion, delay, power, etc.) We propose several different design flows to place mixed-size placement instances. The first flow relies on an arbitrary black-box standard-cell placer to obtain an initial placement and then removes possible overlaps using a fixed-outline floorplanner. This results in valid placements for macros, which are considered fixed. Remaining standard cells are then placed by another call to the standard-cell placer. In the second flow a standard-cell placer generates an initial placement and a force-directed placer is used in the engineering change order (ECO) mode to generate an overlap-free placement. Empirical evaluation on ibm benchmarks shows that in most cases our proposed flows compare favorably with previously published mixed-size placers, Kraftwerk, and the mixed-size floor-placer proposed at the 2003 Conference on Design, Automation, and Test in Europe (DATE 2003), and are competitive with mPG-MS. During the last few decades, academia and industry have invested considerable effort in research on physical design for very large-scale integration (VLSI) [Sherwani 1999] . Through the integration of multiple optimization techniques, design methods and high-performance computer-aided design (CAD) software for integrated circuits (ICs) were developed. However, the growing size of ICs lead to frequent changes to common design flows. Recently, design reuse was introduced as a way to (i) tame the complexity of circuit design for deep submicron technologies, and (ii) improve time-to-market. This trend is further accelerated with the use of hardware description languages and high-level synthesis. Indeed, several current industrial initiatives provide infrastructure and training for the reuse of intellectual property (IP), and also facilitate business models based on IP reuse.. Reuse of design IP is important for multi-million-gate application-specific integrated circuits (ASICs) and considered an integral part of the system-onchip (SoC) design style, that is critical for graphics cards, communication chips, etc. Design IP blocks may implement algorithms or signal transforms, and may contain "canned" table lookups or embedded RAM.. During physical design, IP design blocks appear as black-box macros, that is, blocks of logic with known function and known geometric and electrical properties, but no structural description of their inner workings. Such macros may or may not have flexible geometries, but in any case are considered parts of design. In classical physical design flows a circuit is first partitioned, then floorplanned, and finally standard-cell placement is performed in each partition. This was necessary primarily because older placers, for example, those based on simulated annealing, did not scale very well. However the scalability of min-cut placers dramatically improved after the multilevel partitioning breakthrough in 1997 [Alpert et al. 1997; Karypis et al. 1997; Caldwell et al. 2000a] . In addition to having near-linear runtime, placers based on recursive bisection perform circuit partitioning and, if the cut-lines are allowed to move, also perform floorplanning. Yet macro-placement is not supported in these placers, mainly because large macros that contain more than several percent of layout area introduce considerable discreteness in the solution space and may be difficult to handle within standard recursive min-cut bisection.. Reusing black-box macros in physical design still remains a challenge and existing commercial tools often require help from human designers. For example, the Cadence QPlace manual [Cadence 2000] mentions that the addition of macros may slow down otherwise fairly efficient placement of standard cells and the results may be inferior to what human designers can achieve. Cadence Silicon Ensemble (SEDSM) recommends the following flow for circuits with macros: -Locations of macros are found by invoking block placement. Macros may have overlaps and may not fit in layout area. -Human designer manually removes any overlaps between macros. -Macros are now considered fixed. -QPlace is called to place standard cells.. Figure 1(a) shows the placement of the ibm02 design (see Section 5), produced with the Cadence SEDSM flow recommended for circuits with a large number of macros. As seen in the figure, there is a large amount of overlaps between macros and the designer is expected to remove these overlaps manually. If the design is given directly to the SEDSM placer, QPlace, a legal placement is produced, but the runtimes and solution quality suffer. However, a new version of SEDSM is currently in beta-testing and implements a different macro-placement flow, achieving better results [Varadrajan and DeLendonck 2002] . Recently acquired by Cadence, Silicon Perspective developed the First Encounter tool which performs system-on-chip physical-prototyping and hierarchical physical design. First Encounter allows full-chip physical prototyping and emphasizes early floorplanning. Figure 1 (b) shows the placement of the same ibm02 design produced by the force-directed placer Kraftwerk [Eisenmann and Johannes 1998 ]. This placement also has a large amount of overlap between macros. Figure 1 (c) shows a Kraftwerk placement for design ibm03. As seen in the figure, there are no significant overlaps between macros in this placement and relatively simple techniques should be able to legalize such a placement. However, Kraftwerk does not always produce such nonoverlapping placements and more sophisticated legalization techniques are required.. In addition to physical design with IP blocks, mixed-size placement techniques are relevant in the context of physical synthesis, where layout starts before the netlist is fully synthesized. While our work does not address synthesis, the proposed techniques may be useful in physical synthesis tools that operate at the chip level.. Previous published work on mixed-size placement can be broadly classified into two approaches: continuous optimization techniques and combinatorial optimization techniques. Continuous optimization techniques like force-directed approaches work well with less constrained designs having relatively large amount of white-space [Eisenmann and Johannes 1998; Mo et al. 2000] . On the other hand, combinatorial techniques are particularly promising on constrained designs with less white-space [Nag and Chaudhary 1999] . Published works [Nag and Chaudhary 1999; Vijayan 1991] have focused on overlap removal for macros only and did not consider mixed-size placement. An entirely different approach was pursued in Chang et al. [2003] . Their placer mPG-MS was based on an earlier tool mPG, which recursively clusters the netlist to build a hierarchy. The top-level netlist of approximately 500 clusters is placed using simulated annealing (SA), and then the placement is gradually refined by unclustering the netlist and improving the placement of smaller clusters by SA. mPG-MS contributes a structure of bins, in which large and small blocks are placed during course placement. The coarse placement is necessarily overlapfree for big objects, but small objects must be further re-placed by a detail placer. A significant effort is expended to check for overlap during refinement and legalize possible violations. A recent work [Choi and Bazargan 2003 ] also dealt with the mixed-size placement problem. The authors proposed a mixed-size placement flow which combines a hierarchical simulated annealing-based floorplanner with partitioning-based placement techniques to handle the mixed-size placement problem. Their method starts with a netlist and a fixed-floorplan area. At each level of partitioning, "large" hard macros are extracted from the netlist and the rest of the standard-cells and small macros are partitioned into a number of "soft" modules using a min-cut partitioner. The mixed hard/soft modules are floorplanned using a slicing floorplanner. This is performed recursively until the modules contain fewer than 30 gates. The method employs area migration techniques to satisfy the fixed-outline constraints. However, this method does not produce completely legal placements, with large overlaps remaining between macros. The reader is referred to the book by Sarrafzadeh et al. [2002] for a detailed background discussion of mixed-size placement.. The main contribution of our work is a methodology to place designs with numerous macros by combining floorplanning and standard-cell techniques. The proposed design flow is as follows:. -An arbitrary black-box (no access to source code required) standard-cell placer generates an initial placement.. • S. N. Adya and I. L. Markov -To remove overlaps between macros, a physical clustering algorithm constructs a fixed-outline floorplanning instance. -A fixed-outline floorplanner [Adya and Markov 2001] generates valid locations of macros. -With macros considered fixed, the black-box standard-cell placer is called again to place small cells.. This design flow provides a somewhat new "killer-application" for the many floorplanning techniques developed in the last 5 years (e.g., Lin and Chang [2001] ). Indeed, we do not insist on using a particular floorplan representation, but rather emphasize floorplanning as a step in large-scale placement with macros.. We also propose a second design flow combining a black box standard-cell placer and a force-directed placer. The proposed design flow is as follows:. -An arbitrary black-box (no access to source code required) standard-cell placer generates an initial placement. -A force-directed placer [Eisenmann and Johannes 1998 ] is used in engineering change order (ECO) mode to remove the overlaps while changing the initial placement minimally.. Depending upon the design requirements and characteristics of the design, either of our flows can be used to produce high-quality placements of mixedsize designs. We notice that existing academic placers Capo [Caldwell et al. 2000a ], Dragon 2000 [Wang and Sarrafzadeh 2000] , Feng Shui [Yildiz and Madden 2001] , and Spade [Dutt 2000 ] cannot process movable macros. In fact, all macros are removed in the placement benchmarks described in Wang and Sarrafzadeh [2000] (produced from the ISPD 98 circuit benchmarks), and all cells are artificially made one by one. Therefore, we derived new placement benchmarks from the original ISPD 98 circuits, preserving macros and the areas of all cells. Having converted the benchmarks into Cadence LEF/DEF format, we compared the performance of our methods to the Cadence commercial placer, QPlace, Kraftwerk, mPG-MS, and the mixed-size placement flow proposed in Choi and Bazargan [2003] .. The remaining part of the article is organized as follows. Section 2 covers previous work relevant to force-directed placement and fixed-outline floorplanning. Two new design flows for macro placement are proposed in Sections 3 and 4. Section 5 presents empirical validation of our work, and future directions are discussed in Section 6. Section 7 concludes our work. Modern SoC designs entail placement instances with numerous design IP blocks. Handling such layout problems has become important, and our work addresses this problem. Floorplanning techniques handle designs with macros, but do not scale to a hundred thousand standard cells. On the other hand, standard-cell placers handle large numbers of small, fixed-height cells, but do not handle macros very well. Therefore, we attempt to combine the strengths of both techniques. We propose two design flows to place macro cells consistently with large numbers of standard cells. The first flow uses a combination of techniques from This flow can be modified to include a human designer who uses the initial placement as a hint when manually placing macros. Alternatively, variants of this flow can better preserve the initial placement.. Our proposed Flow 2 combines a standard cell placer with force-directed techniques and can be summarized as follows:. -Shred the original netlist. Use a standard-cell placer to generate an initial placement. -Use a force-directed placer in ECO mode to remove the overlaps while trying to minimize the change in existing placement.. Our Flow 1 produces completely overlap-free placements with reasonably good wirelengths. Our Flow 2 produces high-quality wirelength placements with potentially some overlaps. However, these overlaps are generally very small and can be removed by simple techniques. Either of our flows can be applied for mixed-size design placement depending upon the requirements and characteristics of the design. Our empirical results for mixed-size placement are significantly better than those produced by the Cadence placer QPlace. Our results also compare favorably to those in Chang et al. [2003] and Choi and Bazargan [2003] . It should be noted, however, that the multilevel techniques in Chang et al. [2003] are very different from those used by other researchers and can, in principle, be combined with ours or even applied to placements produced by our methods.. Our experiments show that the proposed flows scale up to at least a thousand macros in addition to hundreds of thousands of standard cells. However, in Flow 1, floorplanning instances with a thousand blocks is a bottleneck and may be improved further. Our ongoing work focuses on techniques for incremental wirelength computation as well as multilevel techniques for floorplanning that can handle greater numbers of macros. We have not explicitly considered timing and congestion, but the significant improvements in wirelength obtained suggest that those metrics can also improve. Moreover, if an objective function can be quickly computed (e.g., circuit delay without false paths can be computed by static timing analysis in linear time), its optimization can be quickly added to simulated annealing that we use for floorplanning. Alternatively, one could use a previously reported force-directed macro block placer [Mo et al. 2000 ] that handles congestion. Congestion and timing can also be addressed at the second call to a black-box placer, assuming that the placer has relevant functionalities [Wang and Sarrafzadeh 2000; Kahng et al. 2002] . Our focus on half-perimeter wirelength is also due to our belief that any large-scale layout tool that cannot successfully optimize wirelength is not going to successfully optimize more complex objectives. Our work can be considered a first step in this direction. Our proposed flow is implemented in C++ and compiled by g++ 2.95.4 -O3. Runtimes are measured on a 2-GHz PC/Intel system running Linux. We compare our results against QPlace v.5.1.67 from Cadence, whose runtimes are measured on a 500-MHz Sun Blade-100 system running Solaris. We also compare our results against force-directed placer, Kraftwerk [Eisenmann and Johannes 1998 ], and mPG-MS [Chang et al. 2003 ]. Runtimes for Kraftwerk are measured on a 2-GHz PC/Intel system and for mPG-MS are observed on a 750-MHz Sun Blade-1000 system running Solaris. In this section we outline the relevant background for our study. We briefly describe the top-down recursive bisection-based placement framework, a generic force-directed placement and floorplanning algorithm, and a fixed-outline floorplanning algorithm. All these algorithms are used in our proposed mixedsize placement flows.

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


Optimal Filter Banks for Multiple Description Coding: Analysis and Synthesis Abstract| Multiple Description (MD) coding is a source coding technique for information transmission over unreliable networks. In MD coding, the coder generates several di erent descriptions of the same signal and the decoder can produce a useful reconstruction of the source with any received subset of these descriptions. In this paper we study the problem of MD coding of stationary Gaussian sources with memory. First, we compute an approximate MD rate distortion region for these sources, which we prove to be asymptotically tight at high rates. This region generalizes the MD rate distortion region of El Gamal, Cover and Ozarow for memoryless Gaussian sources. Then, we develop an algorithm for the design of optimal two-channel biorthogonal lter banks for MD coding of Gaussian sources. We show that optimal lters are obtained by allocating the redundancy over frequency with a reverse \water-lling" strategy. Finally, we present experimental results which show the e ectiveness of our lter banks in the low complexity, low rate regime. In this work we have addressed the problem of MD coding in a subband framework. We have shown how to design perfect reconstruction lter banks that can minimize the side distortion given a certain amount of redundancy. Two other important contributions of this paper are: (a) the characterization of a region which is asymptotically the MD rate-distortion region for general stationary Gaussian sources, and (b) experimental results showing conditions under which our method outperforms other MD methods.

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Variational collision integrator for polymer chains The numerical simulation of many-particle systems (e.g., in molecular dynamics) often involves constraints of various forms. We present a symplectic integrator for mechanical systems with holonomic (bilateral) and unilateral contact constraints, the latter being in the form of a nonpenetration condition. The scheme is based on a discrete variant of Hamilton's principle in which both the discrete trajectory and the unknown collision time are varied (cf. [Fetecau et al., 2003, SIAM J. Applied Dynamical Systems, 2, pp. 381-416]). As a consequence, the collision event enters the discrete equations of motion as an unknown that has to be computed on-the-fly whenever a collision is imminent. The additional bilateral constraints are e ciently dealt with employing a discrete null space reduction (including a projection and a local reparametrisation step) which considerably reduces the number of unknowns and improves the condition number during each time-step as compared to a standard treatment with Lagrange multipliers. We illustrate the numerical scheme with a simple example from polymer dynamics, a linear chain of beads, and test it against other standard numerical schemes for collision problems. In computer simulations of polymeric solutions, the polymers are often thought of as spherical hard bodies (beads) with no internal degrees of freedom, but covalent and non-bonded interactions between them [11] . The typical interaction potentials (such as the Lennard-Jones potential) are very sti↵, which limits the maximum possible time-step in a simulation. A common approach to increase the time-step in order to speed up the simulation consists in replacing the sti↵ forces by constraints [31] . Pushing this idea to the extreme, one can think of a model in which each molecule is represented as a single bead and the sole interactions are in form of perfectly elastic collisions, modelled by inequality constraints that prevent the beads from penetrating each other (excluded volume e↵ect); this is the basis of the classical event-driven algorithm [1] , in which no numerical discretization at all is needed as the motion of the beads follows straight lines in the absence of collisions. For chain-like polymers, a common model is the pearl-necklace-model in which the beads are linked by bond constraints with no other restrictions on their relative positions [29, 14] . It has proven useful for the simulation of melts, and, though conceptually rather simple, is able to explain a variety of phenomena such as reptation, Rouse dynamics, or glassy states on the basis of Monte-Carlo simulations (see [5] and the references therein). Even though the total energy of a chain is independent of the bond angles, within the limitation that the beads cannot penetrate each other, it is possible to control the sti↵ness of the chain (more precisely: its persistence length) by changing the ratio between the bead size and the bond length. The dynamics of the chain are characterised by smooth motions of the bond-constrained particles with impulsive forces when particles collide where the main di culty for the numerical discretization is the robust treatment of both equality constraints and collisions (see, e.g., [36, 10] ). For a recent overview of coarse-grained polymer models, we refer to [28] .. In this article we present an variational algorithm that handles equality and inequality constraints on the same footing. Our approach closely follows the route described in [13] where a seamless variational integrator for collision problems is introduced; by 'seamless' we mean that both the integration scheme for the free motion and the contact conditions at the points of collision follow from a single variational principle. Here we extend the seamless variational scheme to the case where, besides the non-penetration condition for the beads, additional holonomic constraints are present during the entire motion. Although our method is an ordinary time-stepping scheme, it shares some of the spirit of event-driven methodology in that it combines an implicit integrator for the free motion of the chain that allows for large time-steps with an exact treatment of the collision events. In particular, the collision times are computed on-the-fly by solving a simple quadratic equation. As the system is not sti↵-it does not involve soft-core potentials-the method allows for using relatively large time-steps in the implicit integrator where 'relative' is meant in comparison with a typical molecular dynamics (MD) integrator, e.g., the Verlet method. We stress, however, that it is possible to add an interaction potential; since the scheme that we propose is fully implicit, we do not expect that this will lead to serious time-step limitations or major implementation issues (depending on the implicit solver that is used). But since the main bottleneck for the numerics is in the constraints we focus only on this aspect.. The typical approach for treating hard-core collisions in molecular simulations is by means of combining a traditional MD integrator, typically the Verlet method, with a suitable collision operator that comes into play whenever a collision is detected [32, 27, 33] . Imminent collisions are detected by checking whether particles overlap at the end or during the time interval, while the momentum update at the collision point is done by exploiting the balance of total energy and total linear and angular momentum [18, 7, 17] . The underlying mathematical theory for problems of this type (i.e., non-smooth problems) is provided by the theory of di↵erential inclusions in which the impulsive collision forces are understood as measure-valued generalisations of smooth forces; for a thorough treatment of numerical methods for di↵erential inclusions we refer to [12] or [23] . For smooth systems, variational integrators with constant time-step preserve various properties of the exact dynamics such as symplecticity or momentum maps (e.g., linear or angular momentum) at the discrete level [34, 26] . The key idea is simple: rather than discretising the continuous-time equations of motion, variational integrators directly discretise Hamilton's principle by appropriate quadrature rules; the numerical scheme then follows from the now discrete Euler-Lagrange equations. Variational integrators have proven useful, e.g., in understanding the long-term stability of symplectic integrators using backward error analysis [16] . Moreover, the (discrete) variational principle allows for easy generalisations so as to treat, e.g., infinite-dimensional systems [8] , systems with constraints [19] , or contact problems in continuum mechanics [9] . The standard molecular dynamics algorithm for systems with holonomic constraints, that is in fact a variational integrator, is the SHAKE/RATTLE algorithm [31, 2] which can be considered an augmented version of the Verlet algorithm where the constraints are enforced by suitable Lagrange multipliers. As an alternative to Lagrange multipliers that may cause stability problems when the integration time-steps are small (e.g., close to a collision point) the discrete null space method eliminates the constraint force and the Lagrange multipliers by 2 projecting the forces in the system onto the space of admissible momenta. It does so by taking advantage of the d'Alembert principle that states that the constraint forces are always acting perpendicular to the constraint manifold and therefore have no component in the tangential direction. The discrete null space method has been introduced in conjunction with an energy-momentum conserving time integration scheme in [3, 4] and has been transferred to variational time-stepping schemes in [25] . The new idea in this work is to use the discrete null space methodology also for the treatment of the contact constraints while they are active. As a consequence, the proposed method is variational. Yet the method is not symplectic in any obvious sense (neither are variants based on SHAKE/RATTLE), for the time-stepping is altered in the presence of collisions. Therefore backward error analysis cannot be employed to analyse the long-term stability or quasi-conservation of energy (cf. [7] ), even though our numerical test suggest good long-term stability of the scheme. The paper is organised as follows: Section 2 briefly reviews the variational principle for collision problems. The discrete formulation is given in detail in Section 3 where also the discrete null space method is introduced. As a numerical test of our integrator we simulate a linear chain of beads in Section 4 and compare it against an high-order implicit collocation method and the semi-explicit SHAKE/RATTLE algorithm; the test systems are nontrivial (one is non-integrable), yet simple enough to illustrate the basic features of the method. We summarise the results of the article in Section 5. Starting from a discrete version of Hamilton's principle, we have derived a structure preserving integrator for mechanical systems including holonomic (bilateral) constraints as well as unilateral contact (inequality) constraints. The basic idea of the discrete variational principle for systems involving collisions is to treat both the discrete-time trajectory and unknown collision time as unknowns; the resulting discrete Euler-Lagrange equations plus boundary conditions then lead to a time-stepping algorithm that includes the contact force and the constraint forces corresponding to the holonomic constraints. As we have shown all reaction forces can be eliminated 18 from the discrete equations of motion using a discrete null space reduction (involving a projection and a local reparametrisation step) which considerably reduces the number of unknowns as compared to the standard treatment with Lagrange multipliers. As a consequence, the condition number during iteration stays low. This fact, together with the possibility to determine the contact time by solving an algebraic equation rather than searching it via bisection strategies, leads to much lower computational costs while, at the same time, it increases the accuracy to which the contact time is determined. Although the overall scheme is not symplectic in any obvious sense, we found that it shows excellent long-term stability and quasi-energy-conservation akin to symplectic integrators: the total energy remains close to the correct value, even after many thousands of collisions using a large basic time-step, and even in case of multiple collisions happening at the same time node or following each other in quick succession. We stress that the method is rather flexible and can (and should) be extended to, e.g., non-spherical particle geometries or systems involving a smooth interaction potential. For large particle systems, in which the particle density varies in di↵erent regions, i.e., collisions occur more frequently in some parts of the system than in others, a combination with a multiple time-stepping strategy (a multirate integrator [15, 16] or an asynchronous integrator [24] ) might be worthwhile. A still open problem is the backward error analysis of non-smooth systems with variable time-step that will be addressed in future work.

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Influence of the condensate and inverse cascade on the direct cascade in wave turbulence During direct numerical simulation of the isotropic turbulence of surface gravity waves in the framework of Hamiltonian equations formation of the long wave background or condensate was observed. Exponents of the direct cascade spectra at the different levels of an artificial condensate suppression show a tendency to become closer to the prediction of the wave turbulence theory at lower levels of condensate. A simple qualitative explanation of the mechanism of this phenomenon is proposed. Description of the water waves appeals scientists attention during several centuries. At the same time, first attempts to explain observed spatial and temporal spectra are relatively recent. One of the first and at the same time most famous works is the paper by Phillips [1] which was, probably, the first attempt to give an explanation for power-like spectra of surface gravity waves observed in numerous experiments. In just one decade statistical theory of water waves, based on the the kinetic equation for waves derived by Hasselmann [2] and solutions of this equation obtained from Zakharov's theory of wave (or weak) turbulence [3, 4] . These solutions [5, 4] are stationary Kolmogorov solutions of the kinetic equation corresponding to flux of energy from large to small scales (direct cascade) and flux of wave action (waves "number") from small to large scales (inverse cascade). Now the kinetic equation is a base tool for wave forecasting.. The Hasselmann kinetic equation for waves was derived under some assumptions, which include Gaussian statistics for the wave field and prevalence of resonant interactions [4] . These assumptions are subject to confirmation. Modern numerical methods allow us to perform wave field modeling in the framework of kinetic equations faster than the real processes in nature. At the same time, it is still not possible to create a wave forecasting model based on direct numerical simulation of the primordial dynamic equations. Fortunately, in practical applications we do not need to know velocity and elevation at every point of the surface. Statistics, especially mean wave hight and speed, are what really matters for estimation of operational conditions of oil platforms and cargo ships. Such statistics are exactly the subject of the theory of weak turbulence. As a consequence, the problem of confirmation and correction of the theory of wave turbulence is of great practical importance.. Open sea observations confirm temporal and space spectra predicted by the theory of wave turbulence [6, 7, 8] . A comprehensive review of experiments and comparison with the theory of the weak turbulence can be found in [9, 10, 11] . At the same time even the most recent, state of the art wave tanks experiments give strange results [12, 13] , demonstrating dependence of the spectrum slope on the average steepness or forcing level.. One of the ways from this deadend is direct numerical simulation in the framework of dynamical equations. Such numerical experiments can be separated in two major groups. In swell simulation experiments initial condition is some spectral distribution, there is no pumping. Temporal evolution of such initial condition is simulated and than can be compared with corresponding simulation in the framework of kinetic equation [14, 15, 16, 17, 18, 19] . More interesting and more complex is the simulation of the wave turbulence with pumping. Classical sandbox for such simulations is an isotropic turbulence, which allows us to demonstrate of the features of the wave turbulence and, at the same time, simplifies situation due to additional smoothing of the spectra after angle averaging. Although this situation looks pretty artificial, the isotropic turbulence can be observed in the sea with large amount of broken ice, for example, where waves create isotropic wavefield after multiple reflections. But in such a case floating ice have to be taken into account, which makes situation more complex. Simulation of isotropic wave turbulence already confirmed some prediction of the theory of weak turbulence. Direct cascade spectra were obtained in [20, 21, 22] . Inverse cascade spectra, although in simplified equations, were obtained in [23] . For the first time simultaneous simulation of direct and inverse cascades for gravity waves in the framework of primordial dynamical equations was performed in [24] . Current paper is a continuation of the work [24] . The present article will concentrate on the condensate generation and its influence on the direct cascade spectra of wave turbulence of surface gravity waves. A more detailed and deep insight in the mechanism of the condensate influence is presented. Starting from the results of isotropic turbulence simulation author will explore in details how presence of the condensate changes the slope of direct cascade spetrum. It will be demostrated that the presence of the inverse cascade is also important. At the same time in the pure situation with suppressed condensate and inverse cascade observation shows spectrum slope close to previous numerous results of simulations [20, 21, 22] . A simple qualitative explanation of the mechanism of spectrum distortion is provided. Direct analysis of the simulation results supports the theory. The observed phenomena of condensate formation and its influence on the direct cascade have to be quite common for experiments in experimental wave tanks. Finite size of the surface area gives us the same discrete grid of wavevectors as in the case of periodic boundary conditons. It means that at some large scale four waves nonlinear interaction will be arrested, which provides conditions for condensate formation. Such condensate will distort the direct cascade spectrum, changing its exponent through stimulation of whitecapping, resulting in additional dissipation. It is worth to note that this is threshold phenomenon with respect to average steepness as it was shown in [38] . If such dissipation is strong enough to balance the flux of energy, following [37] we shall get Phillips spectrum, as it was demonstrated in the present paper. Gradually suppressing condensate and inverse cascade one can achieve spectrum corresponding to the stationary solution of the kinetic equation. Simple mechanism of condensate interaction with short waves of direct cascade was proposed. This interaction along with other distortion possibilities (see, for example, [39] ) can be the reason of observed spectra deviation from the weak turbulence predictions. These results can be used for explanation of experimental data obtained in the wavetanks.

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On the mathematical theory of living systems, I: Complexity analysis and representation This paper is the first one of a sequel devoted to the challenging goal of developing a mathematical theory for living systems. We consider systems constituted of a number of living entities, called active particles, which have the ability to express specific strategies and interact with other entities. The author proposes a personal path, starting from the identification of a number of common features of living systems that can be viewed as sources of complexity, focusing specifically on the representation of systems based also on a strategy to reduce their complexity. The overall system is decomposed into functional subsystems whose representation is delivered by a probability distribution over the microscopic state of the active particles belonging to such system. Looking ahead, this paper indicates some guidelines to derive mathematical structures, where interactions involving active particles are nonlinearly additive. The study of complex systems, namely systems of many individuals interacting in a nonlinear manner, has received in recent years a remarkable increase of interest from applied mathematicians [1] . Their interest follows that, already well settled, of physicists and researchers in various other fields as engineering, economy or social sciences.. Focusing on large systems constituted by several living entities, it is very difficult to understand and model them based on the sole description of the dynamics and interactions of a few individual entities localized in space and time. In other words, the modeling of individual dynamics does not lead in a straightforward way to a mathematical description of collective emerging behaviors. Understanding the rôle of nonlinear interactions is indeed one of the greatest challenges in the study of complex systems, considering that they are at the core of the emergence of qualitatively different states, namely, new collective states that are not mere combinations of the states of the individual units belonging to the system. In other words, the dynamics of each entity is not determined by the dynamics of all other entities, but by their action as a whole.. A remarkable conceptual difficulty arising from dealing with living system is the lack of fundamental paradigms (first principles) about equilibrium and conservation rules. Some reasonings on this matter are offered in various papers [2] [3] [4] [5] [6] [7] [8] , and books [9] [10] [11] . This lack of paradigms has prevented that various existing models of biological phenomena, even successful under many respects, could generate a biological-mathematical theory analogous to historical theories born from the encounter between mathematical and physical sciences.. Moreover, causality principles are almost always lost due to the heterogeneously distributed ability of living systems to express a strategy often related to their will to survive. More precisely, it is a complex output related to the afore-mentioned ability to express specific strategies that may change not only due to the presence of other entities, but also as a consequence of environmental conditions. * Tel.: +39 0532 974036.

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Solving equations in the relational algebra Enumerating all solutions of a relational algebra equation is a natural and powerful operation which, when added as a query language primitive to the nested relational algebra, yields a query language for nested relational databases, equivalent to the well-known powerset algebra. We study sparse equations, which are equations with at most polynomially many solutions. We look at their complexity, and compare their expressive power with that of similar notions in the powerset algebra. Suppose we are allowed to see only a view on a database B, computed by a relational algebra expression e. If we still want to find out what B is, we might try to "invert" e (assuming we know this expression), which will only work when we also know the finite domain D of B. Specifically, we can enumerate all databases X over D, and test for each X whether it satisfies the equation e(X) = e(B). One of these solutions will be B of course, so if the set of all solutions is not too big, it might provide us with useful information to start our detective work.. The above simple scenario from database security led us to wonder what can be said in general about the solution of equations in the relational algebra. Generally, if e 1 and e 2 are two algebra expressions over some database schema augmented with some relation variables X 1 , . . . , X p , we can consider the equation e 1 = e 2 . A solution of this equation, given a database B with finite domain D, is a tuple (X 1 , . . . , X p ) of relations over D such that e 1 and e 2 evaluate to the same relation on the augmented database (B, X 1 , . . . , X p ).. Asking whether there exists a solution of a relational algebra equation on a database is almost exactly the same thing as asking whether an existential second-order logic sentence is true on that database. Hence, by Fagin's theorem [Fag74, EF95] , the problems that can be formulated as finding a solution of some relational algebra equation are nothing but the problems in NP.. However, in the present paper, we start from the observation that the set of all solutions of an equation, being a set of tuples of relations, is a nested relation. One can therefore consider the enumeration of all solutions of an equation as a query language primitive, which can be added to the nested relational algebra. We introduce and study this extension of the nested relational algebra, which we call the equation algebra. The equation algebra is extremely powerful: it is equivalent to the well-known powerset algebra for nested relations. Our particular interest, however, is in what can be expressed in the equation algebra by using only equations that have a solution set of polynomial size on each database. We call such equations sparse.. Our interest in sparse equations does not stem from time efficiency considerations. Indeed, it is not obvious how knowing that an equation is sparse would help in actually finding even one solution more quickly. It is neither obvious, however, that it would not help. For example, consider the problem of checking on a given database whether some fixed sparse relational algebra equation has a solution. Using an extension of Fagin's theorem to nested relational databases, we show that this problem can be NP-hard only if every problem in NP can already be decided by a polynomial-time nondeterministic Turing machine that has only polynomially many accepting computations on each input. The latter is one of the many unresolved questions in computational complexity theory [All86] .. Nevertheless, sparse equations are still interesting from a space efficiency standpoint. Indeed, for the natural evaluation strategy for equation algebra expressions to run in polynomial space, it is necessary that all equations occurring in the expression are sparse. Interest in fragments of powerful query languages for which the natural evaluation strategy is polynomial-space is not new to database theory research. For example, Abiteboul and Vianu [AV91] showed that the parity query is not expressible in the polynomialspace fragment of various computationally complete query languages.. Closer to our topic is the work of Suciu and Paredaens [SP97] , who showed that queries such as transitive closure and parity are not expressible in the polynomial-space fragment of the powerset algebra for nested relations. This fragment consists of all powerset algebra expressions where all intermediate results are of polynomial size, on each database. Note that this fragment does make sense as there are expressions that always produce a result of logarithmic size; applying the, exponential, powerset operator to such expressions produces a result of polynomial size.. We also mention Grumbach and Vianu [GV95] , who also studied a sparsity notion in connection with queries over nested relational databases, although they considered sparsity as a property of databases rather than of query language expressions.. Suciu and Paredaens conjectured in general that the polynomial-space fragment of the powerset algebra has no more power than the nested relational algebra without powerset. (This conjecture has been confirmed for monadic database schemas [VdB] .) At first sight, the operator that we add to the nested relational algebra, to enumerate all solutions of an equation, does not seem to be that different from the powerset operator. After all, both operators perform some kind of potentially exponential enumeration.. Yet, as we will point out, the analogue of the Suciu-Paredaens conjecture does not hold for the sparse fragment of the equation algebra. Specifically, using sparse equations only, we can express transitive closure; in fact, we can express any fixpoint query. This complements a result by Abiteboul and Hillebrand [AH95] , who showed that transitive closure becomes expressible in the powerset algebra in polynomial space, provided we use a more clever "pipelined" evaluation strategy. Actually, every fixpoint query is already expressible using equations that are not just sparse, but even unambiguous: they have a unique solution on each database. Unambiguous equations in the relational algebra are known as implicit definitions in first-order logic, and were studied in the context of finite model theory by Kolaitis [Kol90] . Kolaitis already showed that every fixpoint query can be implicitly defined. We offer a straightforward, direct proof.. Another example of the differences between the sparse fragment of the equation algebra and that of the powerset algebra is given by the well-known nesting operator of the nested relational algebra. This operator becomes redundant once we extend the algebra with the solution operator or with the powerset operator. However, the original nesting operator never blows up exponentially. We show that, without using the nesting operator itself, nesting is expressible in the equation algebra using sparse equations only, but that the same is not possible with a polynomial-space powerset algebra expression.. However, there are also similarities between the two fragments. Specifically, we prove an analogue to the Suciu-Paredaens result, to the effect that the parity query is not expressible in the sparse fragment of the equation algebra either. This is our main technical contribution; the proof is in the style of an elegant argument of Liebeck [Lie83] , invoking Bochert's theorem on the order of primitive permutation groups. Coming back to connection with implicit definitions in first-order logic, our result generalizes the known and easy fact that the parity query cannot be defined implicitly [Kol90] , in two directions: from unambiguous to sparse; and from a single equation to an arbitrarily complex expression involving several, possibly nested, equations.. This paper is organized as follows. Section 2 recalls the nested relational data model. Section 3 introduces relational algebra equations. Section 4 introduces the equation algebra. Section 5 introduces sparse equations, as well as the natural evaluation strategy for equation algebra expressions. Section 6 studies the time complexity of sparse equations. Finally, Section 7 presents the comparison with the polynomial-space powerset algebra.

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


Quantum Simulation of Simple Many-Body Dynamics Quantum computers could potentially simulate the dynamics of systems such as polyatomic molecules on a much larger scale than classical computers. We investigate a general quantum computational algorithm that simulates the time evolution of an arbitrary non-relativistic, Coulombic many-body system in three dimensions, considering only spatial degrees of freedom. We use a simple discretized model of Schrödinger evolution and discuss detailed constructions of the operators necessary to realize the scheme of Wiesner and Zalka. The algorithm is simulated numerically for small test cases, and its outputs are found to be in good agreement with analytical solutions. Quantum computers could provide an exponential speedup over classical methods of simulating quantum mechanical systems [1] . Wiesner [2] and Zalka [3] have sketched a basic theory of such simulations that includes discretization techniques and decomposition of time evolution operators. Other approaches in the literature include the quantum lattice gas automaton (QLGA) [11, 12, 13] and various methods of decomposing fermionic Hamiltonians into spin operators via the Jordan-Wigner transformation [8, 14, 15, 16] .. However, work based on the first approach has generally focused on bounds for relevant time or space complexities rather than the explicit form of the requisite operators [4, 5, 17] , as well as on the problem of calculating energy spectra [6, 16] . Conversely, in this note, we expand on Wiesner and Zalka's approach to compute position-dependent wavefunctions. We propose detailed constructions of the operators (quantum gates) for such simulations and numerical tests of the algorithm described.. Given the initial state of some number of particles confined to a finite volume in threedimensional space, we aim to compute the wavefunction of the system after an arbitrary period of time. Our simplified model uses only the non-relativistic Coulomb Hamiltonian, whose terms include kinetic energies and electric potentials, and neglects spin-orbit effects. We have constructed the relevant quantum gates for simulating the non-relativistic Coulomb Hamiltonian for spinless many-body quantum systems, building on Wiesner and Zalka's model. We have suggested implementations of these gates and proven the exactness of the kinetic energy operator in the continuum limit. Lastly, we have simulated the algorithm in MATLAB and compared its outputs to analytical solutions, finding good agreement. More numerical tests are needed and forthcoming. In particular, it would be desirable to simulate two particles acting via the Coulomb force in a potential well in 1, 2, and 3 dimensions, and to compare those numerical tests involving interacting particles to classically obtained benchmarks.. Our preliminary investigation of this algorithm, under highly simplifying assumptions, has two main limitations. First, disregarding fermionic spin statistics (the spin component of the wavefunction) means there is no way to enforce the Pauli exclusion principle, which does not currently permit the recovery of realistic atomic and molecular structure. Second, if one chooses to sample the probability density of the many-body wavefunction at the end of the time evolution by making repeated runs of the algorithm and measuring (Section 2.2), the number of runs necessary to obtain sufficiently good "resolution" is unknown. This latter problem is also encountered in the algorithm of Yepez and Boghosian [13] .. It should be possible to incorporate smaller terms into the Hamiltonian with no change in the form of the momentum operator. Doing so lifts many of the simplifications we have made. For example, the Dirac Hamiltonian corrects for both spin and relativistic effects [18] . Two extra qubits are required to represent positive-and negative-energy states as well as spin up or down. An electron wavefunction would thus reside in a Hilbert space H r ⊗H ⊗2 s where H r is 2 3n -dimensional and H s is two-dimensional. This results in four-component spinor wavefunctions where each component is itself a position-dependent wavefunction. The Dirac Hamiltonian for an electron in an electromagnetic field iŝ. where γ µ are the gamma matrices and π ≡ P + e A/c where P and A are operator-valued vectors of momentum operators and vector potential operators, respectively, in the x, y, and z directions. The momentum operators in each of the three spatial dimensions act only on the qubits encoding the electron's position in that dimension. Namely, P = (P x ,P y ,P z ) withP x =P ⊗ I 2 2n ,P y = I 2 n ⊗P ⊗ I 2 n ,P z = I 2 2n ⊗P , andP as defined in equation (7) . Because the gamma matrices act on the spinor components, we can rewrite these operators in terms of matrices of the correct dimension aŝ H relativistic + spin = γ 0 mc 2 ⊗ I 2 3n + (γ 0 γ µ ⊗ I 2 3n )(I 4 ⊗ π µ c) + I 4 ⊗Û .. Both the position and the velocity of every particle in the system make a nonlocal contribution to the vector potential at every point, so implementing A might therefore require additional qubits for keeping track of such velocities. Finally, these corrections we have described account only for electron spin and not nuclear spin.

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


On codes identifying vertices in the two-dimensional square lattice with diagonals AbstractÐFault diagnosis of multiprocessor systems motivates the following graph-theoretic definition. A subset g of points in an undirected graph q Y i is called an identifying code if the sets fv g consisting of all elements of g within distance one from the vertex v are different. We also require that the sets fv g are all nonempty. We take q to be the infinite square lattice with diagonals and show that the density of the smallest identifying code is at least 2/9 and at most 4/17. A multiprocessor system can be modeled as an undirected graph q Y i where is the set of processors and i is the set of links in the system. Fault diagnosis consists of testing the system and locating faulty processors. For this purpose, a set of processors will be selected and they will be assigned the task of testing their neighbors for malfunctions. Whenever a selected processor detects a fault of any kind among its neighbors or if it malfunctions, an error message is issued that specifies only its origin. What is the minimum number of selected processors needed to identify any faulty processor?. Let us reformulate the problem in graph-theoretic terms: q Y i is an undirected graph (finite or infinite). We denote fv fx P X dxY v IgY the ball of radius one centered at the vertex v P , where dxY v equals the number of edges in a shortest path between v and x. If dxY v I, we say that x covers v (and vice versa). A code g is a nonempty subset of . Its elements are called codewords. The code g is an identifying code if the sets fv g, v P , are all nonempty and different.. In this paper, we consider the case when q is the infinite square grid with diagonals. In other words, the vertex set is Â a n d t h e e d g e s e t i s i ffuY vg X u À v P fHY AEIY AEIY HY IY AEIY ÀIY AEIggX We denote this graph by . If we view the vertices as the squares on an infinite chessboard, then fv denotes the squares that are a king's move away from v together with the square v itself.. The density h of g is defined as h jgjaj j if is finite and, for the infinite graph , the density is defined as. where n is the set of vertices xY y such that jxj n and jyj n. For a given q, we would like to find a code g with minimal density.. In fact, instead of this infinite grid, we can consider the finite graphs nYm with vertex set an Â am and edge set i ffuY vg X u À v P fHY AEIY AEIY HY IY AEIY ÀIY AEIgg, where additions are now modulo n in the first and modulo m in the second coordinate.. Assume that we have shown that the density of any identifying code in nYm has density at least and that this is true for all nYm such that n ! k and m ! k, where k is a constant. This implies that the density of any identifying code g in has density at least . Indeed, let g PnIYPnI consist of all codewords xY y of g such that Àn`x`n and Àn`y`n together with all the Vn points ÀnY , nY , Y Àn, Y n, where Àn n. It is easy to verify that g PnIYPnI is an identifying code in PnIYPnI (coordinates now viewed modulo Pn I). Therefore,. i.e., jg n jaj n j ! À VnaPn I P , from which the claim follows.. If we consider the infinite two-dimensional square lattice (without any diagonals), it is proven in [4] , [3] , and [5] that the smallest identifying code has density at least ISaRQ and at most UaPH. If we change the graph a little and add the main diagonal to each square, then the smallest identifying code has density IaR [8] .. In this paper, we consider the case in which both the diagonals are included as edges. In this case, the easy lower and upper bounds are IaS (from inequality (2)) and IaR. We prove that the lower bound can be improved to PaW and the upper bound to RaIU.. For the problem of identifying codes, we refer to Karpovsky et al. [8] and [9] . The papers [1] and [2] give further results in the Hamming spaces and [6] for the hexagonal mesh. For a closely related problem in which fv g are required to be different only for noncodewords v P n g, see Haynes et al. [7] . Such a set g is called a locating-dominating set. Any identifying code is a locatingdominating set. However, for instance, for the infinite twodimensional square lattice (without diagonals), it was shown by Slater [10] that the smallest possible density equals QaIH. We have considered the problem of finding small identifying codes in the infinite square lattice with diagonals. By construction, it is easy to show that there is an identifying code with density RaIU, whereas the best current lower bound is PaW, so a gap still remains. Compared to the square lattice without diagonals, we can clearly do better because, in that case, every identifying code must have density at least ISaRQ. This is not surprising because, in our case, each vertex has degree eight, whereas, in the square lattice without diagonals, each vertex has only degree four. Note added on 26 May 2000: It has recently been shown by I. Charon, O. Hudry, and A. Lobstein that there exists a 1-identifying code with density 2/9 in the two-dimensional square lattice with diagonals.

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