Quasiperiodic and Lyndon episturmian words Recently the second two authors characterized quasiperiodic Sturmian words, proving that a Sturmian word is non-quasiperiodic if and only if it is an infinite Lyndon word. Here we extend this study to episturmian words (a natural generalization of Sturmian words) by describing all the quasiperiods of an episturmian word, which yields a characterization of quasiperiodic episturmian words in terms of their directive words. Even further, we establish a complete characterization of all episturmian words that are Lyndon words. Our main results show that, unlike the Sturmian case, there is a much wider class of episturmian words that are non-quasiperiodic, besides those that are infinite Lyndon words. Our key tools are morphisms and directive words, in particular normalized directive words, which we introduced in an earlier paper. Also of importance is the use of return words to characterize quasiperiodic episturmian words, since such a method could be useful in other contexts. Sturmian words are a fascinating family of infinite words defined on a 2-letter alphabet which have been extensively studied since the pioneering work of Morse and Hedlund in 1940 (see [36] ). Over the years, these infinite words have been shown to have numerous equivalent definitions and characterizations, and their beautiful properties are related to many fields like Number Theory, Geometry, Dynamical Systems, and Combinatorics on Words (see [1, 31, 38, 6] for recent surveys).. Many recent works have been devoted to generalizations of Sturmian words to arbitrary finite alphabets. An especially interesting generalization is the family of episturmian words, introduced by Droubay, Justin, and Pirillo in 2001 [12] (see also [23, 25] for example). Episturmian words include not only the Sturmian words, but also the well-known Arnoux-Rauzy sequences (e.g., see [5, 24, 38, 46] ). More precisely, the family of episturmian words is composed of the ArnouxRauzy sequences, images of the Arnoux-Rauzy sequences by episturmian morphisms, and certain periodic infinite words. In the binary case, Arnoux-Rauzy sequences are exactly the Sturmian words whereas episturmian words include all recurrent balanced words, that is, periodic balanced words and Sturmian words (see [18, 37, 44] for recent results relating episturmian words to the balanced property). See also [17] for a recent survey on episturmian theory.. Episturmian morphisms play a central role in the study of episturmian words (Section 2.3 recalls the definition of these morphisms). Introduced first as a generalization of Sturmian morphisms, Justin and Pirillo [23] showed that they are exactly the morphisms that preserve the aperiodic episturmian words. They also proved that any episturmian word is the image of another episturmian word by some so-called pure episturmian morphism. Even more, any episturmian word can be infinitely decomposed over the set of pure episturmian morphisms. This last property allows an episturmian word to be defined by one of its morphic decompositions or, equivalently, by a certain directive word, which is an infinite sequence of rules for decomposing the given episturmian word by morphisms. In consequence, many properties of episturmian words can be deduced from properties of episturmian morphisms. This approach is used for instance in [7, 16, 27, 43, 44, 46] and of course in the papers of Justin et al.. Morphic decompositions of Sturmian words have been used in [29] to characterize quasiperiodic Sturmian words. Quasiperiodicity of finite words was first introduced by Apostolico and Ehrenfeucht [3] in the following way: "a word w is quasiperiodic if there exists a second word u = w such that every position of w falls within some occurrence of u in w". The word u is then called a quasiperiod of w. In the last fifteen years, quasiperiodicity and covering of finite words has been extensively studied (see [2, 21] for some surveys). In [33] , Marcus extended this notion to infinite words and opened some questions, particularly concerning quasiperiodicity of Sturmian words. After a brief answer to some of these questions in [27] , the Sturmian case was fully studied in [29] where it was proved that a Sturmian word is non-quasiperiodic if and only if it is an infinite Lyndon word. Here we extend this study to episturmian words.. In Sections 2-3, we recall useful results on episturmian words and their directive words. A particularly important tool is the normalization of words directing the same episturmian word, which we recently introduced in [28, 19] . This idea allows an episturmian word to be defined uniquely by its so-called normalized directive word, defined by some factor avoidance. It can be seen as a generalization of a previous result by Berthé, Holton, and Zamboni [7] , which was used in [29] to show that the directive word of a non-quasiperiodic Sturmian word can take only two possible (similar) forms. For non-binary episturmian words, even those defined on a ternary alphabet, this simplicity does not hold since a combinatorial explosion of the number of cases occurs. In particular there exist ternary non-quasiperiodic episturmian words that have infinitely many directive words. As such, the method we use to characterize quasiperiodic episturmian words greatly differs from the one in the Sturmian case.. In Section 4, we characterize quasiperiodic episturmian words. To prove it, we introduce a new way to tackle quasiperiodicity by stating an equivalent definition that is related to the notion of return words. From this, we show that any standard episturmian word (or epistandard word) is quasiperiodic; in particular, sufficiently long palindromic prefixes of an epistandard word are quasiperiods of it. We then extend this result by describing the quasiperiods of any (quasiperiodic) episturmian word (see Theorem 4.19) . This yields a characterization of quasiperiodic episturmian words in terms of their directive words (Theorem 4.28) . Note that the set of quasiperiods of an episturmian word was previously described only for the Fibonacci word [27] . (In [29] this set was not described for quasiperiodic Sturmian words). At the end of Section 4, using the normalization aspect, we give a second characterization of quasiperiodic episturmian words which itself provides an effective way to decide whether or not a given episturmian word is quasiperiodic (see Theorem 4.29) .. Section 5 is concerned with the study of the action of episturmian morphisms in relation to quasiperiodicity. This study leads to non-trivial extensions of results in [29] . Using this approach, we provide a completely different proof of our main characterization of quasiperiodic episturmian words. We also characterize episturmian morphisms that map any word onto a quasiperiodic one (see Section 5.4). This result naturally allows us to consider quasiperiodicity of words defined using episturmian morphisms.. Lastly, in Section 6, we characterize episturmian Lyndon words in terms of their directive words. This result shows that, unlike the Sturmian case, there exist non-quasiperiodic episturmian words that are not infinite Lyndon words.. For this, we recall that a factor u of a finite or infinite word w ∈ A ∞ is right (resp. left) special if ua, ub (resp. au, bu) are factors of w for some letters a, b ∈ A, a = b.. An infinite word t ∈ A ω is episturmian if its set of factors is closed under reversal and t has at most one right (or equivalently left) special factor of each length. Moreover, an episturmian word is standard if all of its left special factors are prefixes of it.. In the initiating paper [12] , episturmian words were defined as an extension of standard episturmian words, which were themselves first introduced and studied as a generalization of standard Sturmian words using palindromic closure (see Theorem 4.5 later). Specifically, an infinite word was said to be episturmian if it has exactly the same set of factors as some standard episturmian word [12] . This definition is equivalent to the aforementioned one by Theorem 5 in [12] . Moreover, it was proved in [12] that episturmian words are uniformly recurrent. Hence ultimately periodic episturmian words are (purely) periodic.. Note. Hereafter, we refer to a standard episturmian word as an epistandard word, for simplicity.. To study episturmian words, Justin and Pirillo [23] introduced episturmian morphisms. In particular they proved that these morphisms, which we recall below, are precisely the morphisms that preserve the set of aperiodic episturmian words. In [35] , Monteil proved that any Sturmian subshift contains a multi-scale quasiperiodic word, i.e., an infinite word having infinitely many quasiperiods. A shorter proof of this fact was provided in [29] . This can be easily extended to episturmian words. Certainly, by Fact 3.4, any episturmian subshift contains at most two epistandard words (one in the aperiodic case and two in the periodic case) and any epistandard word has infinitely many quasiperiods (by Theorem 4.10).. Actually the characterization of quasiperiodic Sturmian words in [29] shows that in any Sturmian subshift there are only two non-quasiperiodic Sturmian words and all other (Sturmian) words in the subshift have infinitely many quasiperiods. It is easy to see that the same result does not hold for episturmian words defined over an alphabet containing more than two letters. For instance, any episturmian word having a spinned directive word in {abc, abc} ω is non-quasiperiodic: all of these non-quasiperiodic episturmian words belong to the subshift of the Tribonacci word r, directed by (abc) ω . Moreover, one can verify (using Theorem 3.2) that the quasiperiodic episturmian word t directed by (abc) n (abc)(abc) ω for some n ≥ 1 (which is in the subshift of r) has exactly n + 1 directive words:. (abc) i abc(ābc) n−i (abc) ω , 0 ≤ i ≤ n.

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The complexity of weighted and unweighted #CSP We give some reductions among problems in (nonnegative) weighted #CSP which restrict the class of functions that needs to be considered in computational complexity studies. Our reductions can be applied to both exact and approximate computation. In particular, we show that the recent dichotomy for unweighted #CSP can be extended to rational-weighted #CSP. The counting complexity of the weighted constraint satisfaction problem, for both exact and approximate computation, has been an active research area for several years. See, for example, [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] . The objective is to give a precise categorisation of the computational complexity of problems in a given class. Easily the most significant development in this stream of research was a recent result of Bulatov [1] . This establishes a dichotomy for exact counting in the whole of (unweighted) #CSP. The dichotomy is between problems in FP and problems which are #P-complete. Dyer and Richerby [16] have given an easier proof of this theorem, and have shown it to be decidable [17] .. In this paper, we study equivalences among problems in weighted #CSP. These equivalences can greatly simplify the classes of problems which need to be considered in studies of computational complexity. A particular consequence of these results is that the dichotomy for unweighted #CSP can be extended to nonnegative rational-weighted #CSP. In the results we present here, the weights will usually lie in some subset of the nonnegative algebraic numbers, since the proofs do not appear to extend to negative weights [18] or complex weights [5] . Neither do we consider general real numbers, since we want our results to apply to standard models of computation and their complexity classes. An extension to a suitable model of real number computation may be possible, though statements about complexity would need to be modified appropriately.. The plan of the paper is as follows. In Section 1.1 we define the weighted constraint satisfaction problem and establish some notation. In Section 1.2, we define a notion of reducibility, which we call weighted reduction, that is used in all our proofs. Its advantage is that the same reductions apply to both exact and approximate computation. Section 2 proves the equivalence of unweighted and rational-weighted #CSP. Section 3 shows that a weighted #CSP problem can be assumed to have only one function, while retaining several useful restrictions on instances. Finally, in Section 4, we show that any rational-weighted problem is computationally equivalent to an unweighted problem with only binary constraints. Thus any #CSP problem is equivalent to a canonical digraph-labelling problem. This gives another proof of the equivalence of unweighted and rational-weighted #CSP.

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Optimal Packet Loss Protection of Progressively Compressed 3-D Meshes Abstract-We consider a state-of-the-art system that uses layered source coding and forward error correction with Reed-Solomon codes to efficiently transmit 3-D meshes over lossy packet networks. Given a transmission bit budget, the performance of this system can be optimized by determining how many layers should be sent, how each layer should be packetized, and how many parity bits should be allocated to each layer such that the expected distortion at the receiver is minimum. The previous solution for this optimization problem uses exhaustive search, which is not feasible when the transmission bit budget is large. We propose instead an exact algorithm that solves this optimization problem in linear time and space. We illustrate the advantages of our approach by providing experimental results for the compressed progressive meshes (CPM) mesh compression technique. A N increasing number of applications require fast delivery of three-dimensional (3-D) models over an unreliable channel. For example, Internet users may want to browse a graphic library consisting of many 3-D objects, which are typically modeled as 3-D triangular meshes. To enable high-quality renderings, millions of triangles may be needed for a single object. Thus, in addition to a fast and reliable error control technique, a compression scheme that provides a compact representation of the 3-D mesh is crucial. While 3-D mesh compression is a mature research area, only a few 3-D mesh error control techniques have been proposed. Yan et al. [1] , [2] partition a 3-D mesh into small pieces that are compressed independently to avoid error propagation and use error concealment techniques at the receiver to recover the missing data. A similar idea was proposed by Bischoff and Kobbelt [3] . Chen et al. [4] use a progressive compression scheme that generates a base mesh and a number of refinement layers. The base mesh and a portion of the refinement layers are sent with the transmission control protocol (TCP), while the remaining layers are sent without protection with the user datagram protocol (UDP). In [5] , the same authors propose a different approach based on UDP and forward error correction. Bici et al. [6] wavelet-based progressive geometry compression scheme [7] to compress the 3-D mesh and the forward error correction system of [8] to protect the resulting embedded bitstream. The wavelet bitstream is packetized into a block of packets where unequal error protection is applied across the packets. However, the framework of [8] does not allow progressive transmission. Also it cannot be used with large 3-D models because of practical limits on the packet size and channel codeword length [9] . In this paper, we focus on a system proposed by AlRegib et al. [10] , which also relies entirely on forward error correction (see also [11] and [12] for variations of this system and [13] for an extension that exploits selective retransmission). But unlike the system used in [6] , it allows progressive transmission and can be used with large models. A progressive compression scheme is used to generate a base mesh and a number of refinement layers. Any transmitted layer is protected by applying Reed-Solomon codes across the packets and packetized in a block of packets (BOP) consisting of information symbols and parity symbols. Given a total transmission bit budget, the performance of the system is optimized by finding the number of layers that should be transmitted and the number of parity symbols allocated to each transmitted layer such that the expected distortion at the receiver is minimum. To solve this optimization problem, AlRegib et al. [10] apply exhaustive search whose complexity is reduced by constraining the number of parity symbols assigned to a given layer to be greater than or equal to the number of parity symbols assigned to the next layer. We show that even with this monotonicity constraint, the time complexity of exhaustive search remains too high, making it unfeasible when large models are used. We also show that this constraint can lead to poor performance and propose a dynamic programming algorithm that solves the unconstrained optimization problem in linear time and space.. Our algorithm is inspired from an algorithm of Dumitrescu and Wu [14] , which was developed to optimize a related system [15] . However, the algorithm of Dumitrescu and Wu [14] cannot be used with the system of [10] . Indeed, the packet size is fixed a priori in the system of [15] , while it can vary with the BOP in the system of [10] . Second, the number of information symbols allocated to a BOP is variable in the system of [15] , whereas it is fixed in the system of [10] . Third, the number of transmitted BOPs is fixed in the system of [15] , whereas it must be optimized in [10] . We also point out that our algorithm has linear time and space complexity, while the algorithm of [14] has quadratic time and space complexity.. The paper is organized as follows. In Section II, we discuss the system proposed in [10] . We show in particular that the proposed heuristic algorithm to optimize the bit allocation problem for this system is unfeasible when the 3-D model is large. In Section III, we propose a better packetization technique for this system and give an algorithm that finds an optimal solution to the bit allocation problem in linear time and space. In Section IV, we give experimental results for 3-D models compressed with the CPM coder [16] . We consider both a two-state Markov channel and a simulated Internet link.

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Model checking LTL with regular valuations for pushdown systems Recent works have proposed pushdown systems as a tool for analyzing programs with (recursive) procedures. In particular, the model-checking problem for LTL has been studied. In this paper we examine an extension of this, namely model-checking with regular valuations. The problem is solved via two different techniques, with an eye on efficiency -both techniques can be shown to be essentially optimal. Our methods are applicable to problems in different areas, e.g., data-flow analysis, analysis of systems with checkpoints, etc., and provide a general, unifying and efficient framework for solving these problems. The authors and the University of Edinburgh retain the right to reproduce and publish this paper for non-commercial purposes.Permission is granted for this report to be reproduced by others for non-commercial purposes as long as this copyright notice is reprinted in full in any reproduction. Applications to make other use of the material should be addressed in the first instance to Copyright Permissions, Division of Informatics, The University of Edinburgh, 80 South Bridge, Edinburgh EH1 1HN, Scotland. Abstract Recent works have proposed pushdown systems as a tool for analyzing programs with (recursive) procedures. In particular, the model-checking problem for LTL has been studied. In this paper we examine an extension of this, namely model-checking with regular valuations. The problem is solved via two different techniques, with an eye on ef£ciency -both techniques can be shown to be essentially optimal. Our methods are applicable to problems in different areas, e.g., data-¤ow analysis, analysis of systems with checkpoints, etc., and provide a general, unifying and ef£cient framework for solving these problems. Pushdown systems can be seen as a natural abstraction of programs written in procedural, sequential languages such as C. They generate in£nite-state transition systems whose states are pairs consisting of a control location (which stores global information about the program) and stack content (which keeps the track of activation records, i.e., previously called procedures and their local variables).. Previous research has established applications of pushdown systems for the analysis of Boolean Programs [1, 8] and certain data-¤ow analysis problems [7] . The model-checking problem has been considered for various logics, and quite ef£cient algorithms have emerged for linear time logics [2, 6, 9] .. In this paper we revisit the model-checking problem for LTL and pushdown systems. Generally speaking, the problem is undecidable for arbitrary valuations, i.e., the functions that map the atomic propositions of a formula to the respective sets of pushdown con£gurations that satisfy them. However, it remains decidable for some restricted classes of valuations. In [2, 6, 9] valuations were completely determined by the control location and/or the topmost stack symbol (we call these valuations 'simple' in the following). Here we propose (and solve) the problem for valuations depending on regular predicates over the complete stack content. We argue that this solution provides a general, ef£cient, and unifying framework for problems from different areas (e.g., data-¤ow analysis, analysis of systems with checkpoints, etc.). We proceed as follows: Section 2 contains basic de£nitions. Most technical content is in Section 3 where we formally de£ne simple and regular valuations and propose our solutions to the model-checking problem with regular valuations, based on a reduction to the case of simple valuations. We can thus re-use most of the theory from [6] . While the reduction itself is based on a standard method, we pay special attention to ensure its ef£ciency, modifying the algorithm of [6] to take advantage of speci£c properties of our constructions. We propose two different techniques -one for regular valuations in general and another for a restricted subclass -both of which increase the complexity by only a linear factor (in the size of an automaton for the atomic regular predicates). By contrast, a blunt reduction and analysis would yield up to a quadric ('9 § @ ') blowup. Even though one technique is more powerful than the other at the same asymptotic complexity, we present them both since it is not clear how they might perform in practice.. In Section 4 we consider applicability of our abstract results. The £rst area (Section 4.1) are problems of interprocedural data-¤ow analysis. Here, regular valuations can be used to 'gather' pieces of information which dynamically depend on the history of procedure calls. LTL can express quite complex relationships among those dynamic properties and allows to solve relatively complicated problems using our model-checking algorithm. To give a concrete example, we indicate how to decide whether a given variable A is dead at a given point of a recursive program with dynamic scoping. Another application (Section 4.2) is connected to systems with checkpoints. First, we introduce a formal model for such systems, called pushdown systems with checkpoints. The idea is that the computation is interrupted at certain points and some property of the stack content is checked. Further computational steps depend on the result of this inspection. This part of our work is motivated by the advent of programming languages which can enforce security requirements. Newer versions of Java, for instance, enable programs to perform local security checks in which the methods on the stack are checked for correct permissions. Jensen et al [10] £rst proposed a formal framework for such systems. With their techniques one can prove the validity of control ¤ow based global security properties as well as to detect (and remove) redundant checkpoints. Our methods are more general, however; for instance, we are not restricted to safety properties, and our model can represent data-¤ow as well. Properties of pushdown systems with checkpoints can be expressed in LTL for which we provide an ef£cient model-checking algorithm. In Section 4.3 we present and analyze a model-checking algorithm for CTLB . In the context of £nite-state systems it is well-known that model-checking the more powerful logic CTLB can be reduced to checking LTL [5] . This technique can be transferred to pushdown systems using model-checking with regular valuations.. The complexity of all of the previously developed algorithms is measured in certain parameters of the problem which are usually small, and our complexity bounds are polynomials in those parameters. In general, those parameters can be exponential in the size of a problem instance. Therefore, it is natural to ask whether it is possible to solve some of the studied problems more ef£ciently by other (possibly quite different) techniques. This question is answered (negatively) in Section 5 where we establish EXPTIME lower bounds for those problems (even for rather restricted forms of them). Hence, all of our algorithms are essentially optimal. Complexity measures are discussed in more detail in Remark 3.1 and in Section 5. We draw our conclusions in Section 6. We have presented two different techniques for checking LTL with regular valuations on pushdown systems. Both techniques rely on a reduction to (and slight modi£cation of) the problem for simple valuations discussed in [6] . Both techniques take linear time and space in t f ô ß j ö õ v j ö ÷ ø p t where ô ß j ö õ v j ö ÷ ø is the set of states of an automaton representing the regular predicates used in the formula. Since both take the same asymptotic time it would be interesting to compare their ef£ciency in practice (for cases where both techniques can be used).. The solution can be seamlessly combined with the concept of symbolic pushdown systems in [8] . These are used to achieve a succinct representation of Boolean Programs, i.e., programs with (recursive) procedures in which all variables are boolean.. The ability to represent data is a distinct advantage over the approaches hitherto made in our areas of application, namely data-¤ow analysis [7] and security properties [10] . For the latter, we have indicated that our model is more general. Our approach provides a unifying framework for these applications without losing ef£ciency. Both techniques take linear time in t f ô ß j ö õ v j ö ÷ ø p t whereas the methods used in [7] were cubic (though erroneously reported as linear there, too). In [10] no complexity analysis was conducted.

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


A positive linear discrete-time model of capacity planning and its controllability properties Abstract--One of the most important concepts in production planning is that of the establishment of an overall or aggregate production plan. In this paper, the problem of establishing an aggregate production plan for a manufacturing plant is considered. A new dynamic discrete-time model of capacity planning utilizing concepts arising in positive linear systems (PLS) theory is proposed and its controllability property is analyzed. Controllability is a fundamental property of the system with direct implications not only in dynamic optimization problems (such as those arising in inventory and production control) but also in feedback control problems. Some new open problems regarding controllability of stationary and nonstationary PLS with linear constraints are posed in the paper. An optimal control problem for capacity planning is formulated and discussed. In this paper, we consider the problem of establishing an aggregate production plan for a manufacturing plant. The basic issue is, given a set of production demands stated in some common unit, what levels of resources (such as inventory, regular time production, overtime production, labour, etc.) should be provided in each period? There has been a long history of academic research on aggregate planning, resulting in many mathematical programming (static) models and in a variety of heuristics [1, 2] . However, as the firms attempt to implement manufacturing planning and control systems, they find deficiencies in these models and heuristics. We attempt to overcome some of these drawbacks by proposing a new dynamic model, utilizing concepts arising in positive linear systems (PLS) theory.. The dynamic modeling approach (and, particularly, the optimal control approach) to the theory of the firm is motivated by three issues: (i) the need for policies, (ii) the contribution of deductive analysis, and (iii) the need to incorporate time.. The state of the art of this area is well exposed in the monograph [3] . In this book, the authors discuss a number of continuous-time dynamic models and exploit the Pontryagin Maximum Principle developed for such models to determine the optimal policy. They do not consider positive systems as well as discrete-time models. However, discrete-time models seem to be somewhat more suitable to describe the firm's dynamics. The model we consider in this paper represents not only the system dynamics, but it contains a number of important parameters not included in the dynamic models described in [3] . Applying some recent results concerning PLS and developing further the theory, we analyze controllability properties of the simplified model. Controllability is a property of the system that shows its ability to move in space. It is a fundamental property with direct implications not only in dynamic optimization problems (such as those arising in inventory and production control) but also in feedback control problems. On the basis of the proposed model, we formulate and discuss a discrete-time optimal control problem for capacity planning.. The paper is organized as follows. In Section 2, we present the model. Positivity properties of the model are identified in Section 3. In Section 4, we extend the definitions for reachabllity and controllability of stationary (time-invariant) positive systems to nonstationary (time-variant) positive systems and study the reachability and controllability properties of the (simplified) model for capacity planning proposed in Section 2. An optimal control problem formulated on the basis of the model is considered in Section 5. Finally, in Section 6, the conclusions are stated. In this paper, a new discrete-time dynamic model of capacity planning is developed. The model is motivated not only by the need for policies, but also by the need to incorporate time and open the way for deductive anMysis. Some interesting new characterizations of production systems important for aims of planning and control appear in the model. The model can be built-in in decision support systems. It is somewhat easier for simulation and decision making than the static models.. The model presented in this paper belongs to the class of nonstationary discrete-time positive linear systems. The theory of positive systems has rapidly been developing during the last decade. Criteria for identifying reachability and controllability properties of time-invariant positive linear systems have been obtained quite recently. Reachability and controllability criteria for nonstationary (time-variant) positive systems even without side linear constraints are not known to date. We have studied the controllability properties of the capacity-planning model (without the side linear constraints) and have obtained interesting results. It turns out that the model is controllable in the short term in some not quite realistic situations, but it is weakly controllable in the long term. By introducing a relevant objective function, a related discrete-time optimal control problems is formulated. A general optimal control solver can solve it, but the structure of the formulated discrete-time optimal control problem above allows developing more efficient specialized algorithms. The idea of such an algorithm is discussed in the paper.

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


Non-photorealistic

virtual

environments

We

describe

a

system

for

non-photorealistic

rendering

(NPR)

of

virtual

environments.

In

real

time,

it

synthesizes

imagery

of

architectural

interiors

using

stroke-based

textures.

We

address

the

four

main

challenges

of

such

a

system

-interactivity,

visual

detail,

controlled

stroke

size,

and

frame-to-frame

coherence

-through

image

based

rendering

(IBR)

methods.

In

a

preprocessing

stage,

we

capture

photos

of

a

real

or

synthetic

environment,

map

the

photos

to

a

coarse

model

of

the

environment,

and

run

a

series

of

NPR

filters

to

generate

textures.

At

runtime,

the

system

re-renders

the

NPR

textures

over

the

geometry

of

the

coarse

model,

and

it

adds

dark

lines

that

emphasize

creases

and

silhouettes.

We

provide

a

method

for

constructing

non-photorealistic

textures

from

photographs

that

largely

avoids

seams

in

the

resulting

imagery.

We

also

offer

a

new

construction,

art-maps,

to

control

stroke

size

across

the

images.

Finally,

we

show

a

working

system

that

provides

an

immersive

experience

rendered

in

a

variety

of

NPR

styles.

Virtual

environments

allow

us

to

explore

an

ancient

historical

site,

visit

a

new

home

with

a

real

estate

agent,

or

fly

through

the

twisting

corridors

of

a

space

station

in

pursuit

of

alien

prey.

They

simulate

the

visual

experience

of

immersion

in

a

environment

by

rendering

images

of

a

computer

model

as

seen

from

an

observer

viewpoint

moving

under

interactive

control

by

the

user.

If

the

rendered

images

are

visually

compelling,

and

they

are

refreshed

quickly

enough,

the

user

feels

a

sense

of

presence

in

a

virtual

world,

enabling

applications

in

education,

computer-aided

design,

electronic

commerce,

and

entertainment..

While

research

in

virtual

environments

has

traditionally

striven

for

photorealism,

for

many

applications

there

are

advantages

to

non-photorealistic

rendering

(NPR).

Artistic

expression

can

often

convey

a

specific

mood

(e.g.

cheerful

or

dreary)

difficult

to

imbue

in

a

synthetic,

photorealistic

scene.

Furthermore,

through

abstraction

and

careful

elision

of

detail,

NPR

imagery

can

focus

the

viewer's

attention

on

important

information

while

downplaying

extraneous

or

unimportant

features.

An

NPR

scene

can

also

suggest

additional

semantic

information,

such

as

a

quality

of

"unfinishedness"

that

Figure

A

non-photorealistic

virtual

environment.

may

be

desirable

when,

for

example,

an

architect

shows

a

client

a

partially-completed

design.

Finally,

an

NPR

look

is

often

more

engaging

than

the

prototypical

stark,

pristine

computer

graphics

rendering..

The

goal

of

our

work

is

to

develop

a

system

for

real-time

NPR

virtual

environments

Figure

The

challenges

for

such

a

system

are

four-fold:

interactivity,

visual

detail,

controlled

stroke

size,

and

frame-to-frame

coherence.

First,

virtual

environments

demand

interactive

frame

rates,

whereas

NPR

methods

typically

require

seconds

or

minutes

to

generate

a

single

frame.

Second,

visual

details

and

complex

lighting

effects

(e.g.

indirect

illumination

and

shadows)

provide

helpful

cues

for

comprehension

of

virtual

environments,

and

yet

construction

of

detailed

geometric

models

and

simulation

of

global

illumination

present

challenges

for

a

large

virtual

environment.

Third,

NPR

strokes

must

be

rendered

within

an

appropriate

range

of

sizes;

strokes

that

are

too

small

are

invisible,

while

strokes

that

are

too

large

appear

unnatural.

Finally,

frame-toframe

coherence

among

strokes

is

crucial

for

an

interactive

NPR

system

to

avoid

a

noisy,

flickery

effect

in

the

imagery..

We

address

these

challenges

with

image-based

rendering

(IBR).

In

general,

IBR

yields

visually

complex

scenery

and

efficient

rendering

rates

by

employing

photographs

or

pre-rendered

images

of

the

scene

to

provide

visual

detail.

Not

surprisingly,

by

using

a

hybrid

NPR/IBR

approach

we

are

able

to

reap

the

benefits

of

both

technologies:

an

aesthetic

rendering

of

the

scene,

and

visual

complexity

from

a

simple

model.

More

subtly,

each

technology

addresses

the

major

drawbacks

of

the

other.

IBR

allows

us

to

render

artistic

imagery

with

complex

lighting

effects

and

geometric

detail

at

interactive

frame

rates

while

maintaining

frame-to-frame

coherence.

On

the

flipside,

non-photorealistic

rendering

appeases

many

of

the

artifacts

due

to

under-sampling

in

IBR,

both

by

visually

masking

them

and

by

reducing

the

viewer's

expectation

of

realism..

At

a

high

level,

our

system

proceeds

in

three

steps

as

shown

in

Figure

First,

during

off-line

preprocessing,

we

construct

an

IBR

model

of

a

scene

from

a

set

of

photographs

or

rendered

images.

Second,

during

another

preprocessing

step,

we

filter

samples

of

the

IBR

model

to

give

them

a

non-photorealistic

look.

The

result

is

a

non-photorealistic

image-based

representation

(NPIBR)

for

use

in

interactive

walkthroughs.

Finally,

during

subsequent

on-line

sessions,

the

NPIBR

model

is

resampled

for

novel

viewpoints

to

reconstruct

NPR

images

for

display.

This

approach

addresses

many

of

the

challenges

in

rendering

NPR

images

of

virtual

environments

in

real-time.

First,

by

executing

the

most

expensive

computations

during

off-line

preprocessing,

our

system

achieves

interactive

frame

rates

at

run-time.

Second,

by

capturing

complex

lighting

effects

and

geometric

detail

in

photographic

images,

our

system

produces

images

with

visual

richness

not

attainable

by

previous

NPR

rendering

systems.

Third,

with

appropriate

representation,

prefiltering,

and

resampling

methods,

IBR

allows

us

to

control

NPR

stroke

size

in

the

projected

imagery.

Fourth,

by

utilizing

the

same

NPR

imagery

for

many

similar

camera

viewpoints

rather

than

creating

new

sets

of

strokes

for

each

view,

our

system

acquires

frame-to-frame

coherence.

Moreover,

by

abstracting

NPR

processing

into

a

filtering

operation

on

an

image-based

representation,

our

architecture

supports

a

number

of

NPR

styles

within

a

common

framework.

This

feature

gives

us

aesthetic

flexibility,

as

the

same

IBR

model

can

be

used

to

produce

interactive

walkthroughs

in

different

NPR

styles..

In

this

paper,

we

investigate

issues

in

implementing

this

hybrid

NPR/IBR

approach

for

interactive

NPR

walkthroughs.

The

specific

technical

contributions

of

our

work

are:

a

method

for

constructing

non-photorealistic

textures

from

photographs

that

largely

avoids

seams

in

images

rendered

from

arbitrary

viewpoints,

and

a

multiresolution

representation

for

non-photorealistic

textures

(called

art-maps)

that

works

with

conventional

mip-mapping

hardware

to

render

images

with

controlled

stroke

size.

These

methods

are

incorporated

into

a

working

prototype

system

that

supports

interactive

walkthroughs

of

visually

complex

virtual

environments

rendered

in

many

stroke-based

NPR

styles..

The

remainder

of

this

paper

is

organized

as

follows.

In

Section

we

review

background

information

and

related

work.

Sections

address

the

main

issues

in

constructing,

filtering,

and

resampling

a

hybrid

NPR/IBR

representation.

Section

presents

results

of

experiments

with

our

working

prototype

system,

while

Section

contains

a

brief

conclusion

and

discussion

of

areas

for

future

work.

This

paper

describes

a

system

for

real-time

walkthroughs

of

nonphotorealistic

virtual

environments.

It

tackles

the

four

main

challenges

of

such

a

system

-interactivity,

visual

detail,

controlled

stroke

size,

and

frame-to-frame

coherence

-through

image-based

rendering

of

non-photorealistic

imagery.

The

key

idea

is

that

an

image-based

representation

can

be

constructed

off-line

through

a

sequence

of

image

capture

and

filtering

steps

that

enable

efficient

reconstruction

of

visually

detailed

images

from

arbitrary

viewpoints

in

any

non-photorealistic

style.

The

technical

contributions

of

this

work

include

a

method

for

constructing

NPR

textures

that

avoids

seams

in

novel

images

and

a

multiscale

texture

representation

(art-maps)

that

provides

control

over

the

size

of

strokes

during

interactive

rendering.

This

work

suggests

a

number

of

areas

for

future

investigation:.

Augmenting

the

scene

with

geometry-based

elements.

Real-time

NPR

rendering

of

simple

geometric

objects

in

the

scene

-perhaps

architectural

accents

such

as

a

plant

or

a

chair

rendered

in

the

NPR

styles

of

Gooch

et

al.

or

Kowalski

et

al.

-would

enhance

the

sense

of

immersion

while

not

greatly

slowing

our

system.

View-dependent

rendering.

We

have

observed

that

many

viewdependent

geometric

and

lighting

effects

are

visually

masked

by

non-photorealistic

rendering

(see

Section

Nonetheless,

viewdependent

texture

mapping

(e.g.

offers

an

opportunity

to

capture

these

effects

for

even

better

fidelity

to

the

environment..

Better

stroke

coherence.

As

mentioned

in

Section

runtime

blending

between

neighboring

levels

of

the

mip-map

hierarchy

causes

visual

blending

between

strokes

in

the

art-maps.

It

may

be

possible

to

achieve

better

coherence

between

neighboring

levels

of

the

mip-maps,

most

likely

by

designing

customized

NPR

filters

that

deliberately

assign

strokes

in

multiple

levels

of

the

art-maps

at

once.

The

desired

visual

effect

might

be

that

strokes

grow

and

eventually

split

apart,

rather

than

fading

in,

as

the

user

approaches

a

surface.

The

traditional

strategy

for

immersive

virtual

environments

is

to

render

detailed

sets

of

polygons

with

appropriate

lighting

effects

as

the

camera

moves

through

the

model

With

this

approach,

the

primary

challenge

is

constructing

a

digital

representation

for

a

complex,

visually

rich,

real-world

environment.

Despite

recent

advances

in

interactive

modeling

tools,

laser-based

range-finders,

computer

vision

techniques,

and

global

illumination

algorithms,

it

remains

extremely

difficult

to

construct

compelling

models

with

detailed

geometry,

accurate

material

reflectance

properties,

and

realistic

global

illumination

effects.

Even

with

tools

to

create

an

attractive,

credible

geometric

model,

it

must

still

be

rendered

at

interactive

frame

rates,

limiting

the

number

of

polygons

and

shading

algorithms

that

can

be

used.

With

such

constraints,

the

resulting

imagery

usually

looks

very

plastic

and

polygonal,

despite

setting

user

expectations

for

photorealism..

In

contrast,

image-based

modeling

and

rendering

methods

represent

a

virtual

environment

by

its

radiance

distribution

without

relying

upon

a

model

of

geometry,

lighting,

and

reflectance

properties

An

IBR

system

usually

takes

images

(photographs)

of

a

static

scene

as

input

and

constructs

a

sample-based

representation

of

the

plenoptic

function,

which

can

be

resampled

to

render

photorealistic

images

for

novel

viewpoints.

The

important

advantages

of

this

approach

are

that

photorealistic

images

can

be

generated

without

constructing

a

detailed

model

or

simulating

global

illumination,

and

the

rendering

time

for

novel

images

is

independent

of

a

scene's

geometric

complexity.

The

primary

difficulty

is

storing

and

resampling

a

high-resolution

representation

of

the

plenoptic

function

for

a

complex

virtual

environment

If

the

radiance

distribution

is

under-sampled,

images

generated

during

a

walkthrough

contain

noticeable

aliasing

or

blurring

artifacts,

which

are

disturbing

when

the

user

expects

photorealism..

In

recent

years,

a

few

researchers

have

turned

their

attention

away

from

photorealism

and

towards

developing

non-photorealistic

rendering

techniques

in

a

variety

of

styles

and

simulated

media,

such

as

impressionist

painting

pen

and

ink

technical

illustration

ornamentation

engraving

watercolor

and

the

style

of

Dr.

Seuss

Much

of

this

work

has

focused

on

creating

still

images

either

from

photographs,

from

computer-rendered

reference

images,

or

directly

from

models,

with

varying

degrees

of

user-direction.

One

of

our

goals

is

to

make

our

system

work

in

conjunction

with

any

of

these

technologies

(particularly

those

that

are

more

automated)

to

yield

virtual

environments

in

many

different

styles..

Several

stroke-based

NPR

systems

have

explored

time-changing

imagery,

confronting

the

challenge

of

frame-to-frame

coherence

with

varying

success.

Winkenbach

et

al.

and

later

Curtis

et

al.

observed

that

applying

NPR

techniques

designed

for

still

images

to

time-changing

sequences

yields

flickery,

jittery,

noisy

animations

because

strokes

appear

and

disappear

too

quickly.

Meier

adapted

Haeberli's

"paint

by

numbers"

scheme

in

such

a

way

that

paint

strokes

track

features

in

a

model

to

provide

frame-to-frame

coherence

in

painterly

animation.

Litwinowicz

achieved

a

similar

effect

on

video

sequences

using

optical

flow

methods

to

affix

paint

strokes

to

objects

in

the

scene.

Markosian

found

that

silhouettes

on

rotating

objects

change

slowly

enough

to

give

frame-to-frame

coherence

for

strokes

drawn

on

the

silhouette

edges.

We

exploit

this

property

when

drawing

lines

on

creases

and

silhouettes

at

run-time.

Kowalski

et

al.

extends

these

methods

by

attaching

non-photorealistic

"graftals"

to

the

geometry

of

a

scene,

while

seeking

to

enforce

coherence

among

the

graftals

between

frames.

The

bulk

of

the

coherence

in

our

system

comes

from

reprojection

of

non-photorealistic

imagery,

so

the

strokes

drawn

for

neighboring

frames

are

generally

slowlychanging..

Several

other

researchers,

for

example

Horry

et

al.

Wood

et

al.

and

Buck

et

al.

have

built

hybrid

NPR/IBR

systems

where

hand-drawn

art

is

re-rendered

for

different

views.

In

this

spirit

our

system

could

also

incorporate

hand-drawn

art,

although

the

drawing

task

might

be

arduous

as

a

single

scene

involves

many

reference

images..

In

this

paper,

we

present

a

system

for

real-time,

NPR

virtual

environments.

Rather

than

attempting

to

answer

the

question

"how

would

van

Gogh

or

Chagall

paint

a

movie?"

we

propose

solutions

to

some

technical

issues

facing

an

artist

wishing

to

use

NPR

styles

in

a

virtual

environment

system.

Two

visual

metaphors

represent

the

extremes

in

a

spectrum

of

aesthetics

one

could

choose

for

an

"artistic"

immersive

experience.

On

one

extreme,

we

could

imagine

that

an

artist

painted

over

the

walls

of

the

model.

In

this

case,

the

visual

effect

is

that

as

the

user

navigates

the

environment

the

detailed

stroke

work

is

more

or

less

apparent

depending

on

her

distance

from

the

various

surfaces

she

can

see.

In

the

other

extreme,

we

could

imagine

that

as

the

user

navigates

the

environment

in

real-time,

a

photograph

of

what

is

seen

is

captured,

and

an

artist

instantaneously

paints

a

picture

based

on

the

photograph.

In

this

case,

the

visual

effect

suffers

from

either

flickering

strokes

(lack

of

frame-to-frame

coherence)

or

the

"shower

door

effect"

(the

illusion

that

the

paintings

are

somehow

embedded

in

a

sheet

of

glass

in

front

of

the

viewer).

Our

goal

is

to

find

a

compromise

between

these

two

visual

metaphors:

we

would

like

the

stroke

coherence

to

be

on

the

surfaces

of

the

scene

rather

than

in

the

image

plane,

but

we

would

like

the

stroke

size

to

be

roughly

what

would

have

been

selected

for

the

image

plane

rather

than

what

would

have

been

chosen

for

the

walls.

The

difficult

challenge

is

to

achieve

this

goal

while

rendering

images

at

interactive

rates..

We

investigate

a

hybrid

NPR/IBR

approach.

Broadly

speaking,

the

two

main

issues

we

address

are:

constructing

an

IBR

representation

suitable

for

NPR

imagery,

and

developing

a

IBR

prefiltering

method

to

enable

rendering

of

novel

NPR

images

with

controllable

stroke-size

and

frame-to-frame

coherence

in

a

realtime

walkthrough

system.

These

issues

are

the

topics

of

the

following

two

sections.

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


Rate-distortion analysis for light field coding and streaming A theoretical framework to analyze the rate-distortion performance of a light field coding and streaming system is proposed. This framework takes into account the statistical properties of the light field images, the accuracy of the geometry information used in disparity compensation, and the prediction dependency structure or transform used to exploit correlation among views. Using this framework, the effect that various parameters have on compression efficiency is studied. The framework reveals that the efficiency gains from more accurate geometry, increase as correlation between images increases. The coding gains due to prediction suggested by the framework match those observed from experimental results. This framework is also used to study the performance of light field streaming by deriving a view-trajectory-dependent rate-distortion function. Simulation results show that the streaming results depend both the prediction structure and the viewing trajectory.For instance, independent coding of images gives the best streaming performance for certain view trajectories. These and other trends described by the simulation results agree qualitatively with actual experimental streaming results. From fly-arounds of automobiles to 360. • panoramic views of cities to walkthroughs of houses, 3-D content is increasingly becoming commonplace on the Internet. Current content, however, offers limited mobility around the object or scene, and limited image resolution. To generate high-quality photorealistic renderings of 3-D objects and scenes, computer graphics has traditionally turned to computationally expensive algorithms such as ray-tracing.. Recently, there has been increasing attention on image-based rendering techniques that require only resampling of captured images to render novel views.. Such approaches are especially useful for interactive applications because of their low rendering complexity. A light field [1, 2] is an image-based rendering dataset that allows for photo-realistic rendering quality, as well as much greater freedom in navigating around the scene or object. To achieve this, light fields rely on a large number of captured images.. The large amount of data can make transmitting light fields from a central server to a remote user a challenging problem. The sizes of certain large datasets can be in the tens of Gigabytes [3] . Even over fast network connections, it could take hours to download the raw data for a large light field.. This motivates the need for efficient compression algorithms to reduce the data size.. Numerous algorithms have been proposed to compress light fields. The most 1 now with NetEnrich, Inc., Santa Clara, CA 2 The authors would like to thank Markus Flierl for his valuable comments on this manuscript.. efficient algorithms use a technique called disparity compensation. Disparity compensation uses either implicit geometry information or an explicit geometry model to warp one image to another image. This allows for prediction between images, such as in [4] [5] [6] [7] [8] [9] [10] , encoding several images jointly by warping them to a common reference frame, as in [11, 12, 8, 13] , or, more recently, by combining prediction with lifting, as in [14, 15, 9] . This paper considers a closed-loop predictive light field coder [16, 9] from the first set of techniques.. In this light field coder, an explicit geometry model is used for disparity compensation. For real-world sequences, this geometry model is estimated from image data using computer vision techniques, which introduces inaccuracies.. In addition, lossy encoding is used to compress the description of the geometry model, which leads to additional inaccuracies in the geometry data. Light field images, captured in a hemispherical camera arrangement, are organized into different levels. The first level is a set of key images, where each image is independently compressed without predicting from nearby images. These images are evenly distributed around the hemisphere. Each image in the second level uses the two nearest neighboring images in the first level to form a prediction image with disparity compensation. The residual image resulting from prediction is encoded. The images in the third levels uses the nearest images in the first two levels for prediction, and so on, until all the images in the light field are encoded.. As theoretical and experimental results later in this paper will show, using prediction typically improves compression efficiency. It also, however, creates decoding dependencies between images. While this is acceptable for a download scenario where the entire light field is retrieved and decoded, for other transmission scenarios such as interactive streaming, decoding dependencies may affect the performance of the system. This paper considers an interactive light field streaming scenario where a user remotely accesses a dataset over a best-effort packet network. As the user starts viewing the light field dataset by navigating around the scene or object, the encoded images that are appropriate for rendering the desired view are transmitted [17, 18] .. For interactive performance, there are stringent latency constraints on the system. This means that a view, once selected by the user, must be rendered by a particular deadline. Due to the best-effort nature of the network and possible bandwidth limitations, not all images required for rendering that view may be available by the rendering deadline. In this case, the view is rendered with a subset of the available set of images, resulting in degraded image quality. The decoding dependencies due to prediction between images will affect the set of images that can be decoded, and thus the rendered image quality and the overall streaming performance.. This paper presents a theoretical framework to study the rate-distortion performance of a light field coding and streaming system. Coding and streaming performance is affected by numerous factors including geometry accuracy, the prediction dependency structure, between-view and within-view correlation, the image selection criteria for rendering, the images that are successively transmitted and the user's viewing trajectory. The framework that is presented incorporates these factors into a tractable model.. This work is based on the theoretical analysis of the relationship between motion compensation accuracy and video compression efficiency [19] [20] [21] [22] [23] . It is also closely related to the theoretical analysis of 3-D subband coding of video [24] [25] [26] . Prior theoretical work has analyzed light fields from a sampling perspective [27] [28] [29] , but not a rate-distortion perspective as in this work.. This paper is organized as follows. The theoretical framework for coding the entire light field is described in Section 2, along with simulation and experimental coding results. Section 3 presents the extension to light field streaming, along simulation results and comparison to the results of light field streaming experiments.. 2 Compression Performance for the Entire Light Field A theoretical framework to analyze the rate-distortion performance of a light field coding and streaming system is proposed. In the framework, the encoding and streaming performance is affected by various parameters, such as the correlation within an image and between images, geometry accuracy and the prediction dependency structure. Using the framework, the effects of these parameters can be isolated and studied. Specifically, the framework shows that compression performance is significantly affected by three main factors, namely, geometry accuracy, correlation between images, and the prediction dependency structure. Moreover, the effects of these factors are inter-dependent.. Prediction is only useful when there is both accurate geometry and sufficient correlation between images. The gains due to geometry accuracy and image correlation are not simply additive. The larger the correlation between images, the greater the benefit of more accurate geometry over less accurate geometry. In converse, with low correlation between images, there is little benefit to any level of accuracy in the geometry. In the datasets studied, the theory indicates that most of the benefit of prediction is realized with only two levels of images in the prediction dependency structure. This result is confirmed by actual experimental results.. The theoretical framework for light field coding is extended and used to study the performance of streaming compressed light fields. In order to extend the framework, a view-trajectory-dependent rate-distortion function is derived.. The derivation shows that the distortion is composed to two additive parts: distortion due to coding, and distortion resulting from using only a subset of the required images for rendering. Theoretical simulation results, using actual streaming traces, reveal that the prediction structure that gives the best streaming performance depends heavily upon the desired viewing trajectory.. Independent coding of the light field images is, in fact, the best trajectory for certain view trajectories of certain datasets. This observation is confirmed with actual streaming experimental results. The simulation results, in general,. show similar trends and characteristics to the actual experimental streaming results. There is not an exact match, due the numerous simplifications and assumptions made in the model to make the analysis tractable. The framework, despite these limitations, has been effectively used as part of the procedure for estimating distortion values for rate-distortion optimized packet scheduling.

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


Sifting micro-blogging stream for events of user interest Micro-blogging is a new form of social communication that encourages users to share information about anything they are seeing or doing, the motivation facilitated by the ability to post brief text messages through a variety of devices. Twitter, the most popular micro-blogging tool, is exhibiting rapid growth [3] : up to 11% of online Americans are using Twitter by December 2008, compared to 6% in May 2008 Due to its nature, micro-blogosphere has unique features: (i) It is a source of extremely up-to-date information about what is happening in the world; (ii) It captures the wisdom of millions of people and covers a broad range of domains.These features make micro-blogosphere more than a popular medium of social communication: we believe that it has additionally become a valuable source of extremely upto-date news on virtually any subject of user interest.Making use of micro-blogosphere in this new role we meet the following challenges: (A) Since any given subject is generally mentioned in the micro-blogging stream on the continuous basis, a method is needed for locating periods of news on this subject. (B) Additionally, even for such periods, stream filtering is required for removing noise and for extracting messages that best describe the news.To address these challenges we make and exploit the following observations: (A) For an arbitrary subject, events that catch user interest gain distinguishably more attention than the average mentioning of the subject resulting in message activity bursts for it. (B) Most of the messages in an activity burst describe common event in close variationseither rephrased or "retweeted" between the users.We demonstrate TweetSieve-a system that allows obtaining news on any given subject by sifting the Twitter stream. Our work is related to frequecy-based analysis applied to blogs [1], but higher latency and lower coverage in blogs makes the analysis less effective than in case of micro-blogs.In TweetSieve demo, the user is able to express the subject of her interest by an arbitrary search string. The system shows the period of events occuring for the subject and outputs tweets that best describe each of the events. Figure 1 shows a screenshot of the system for "Semantic search" as a sample subject.The underlying process consists of two steps: Identifying activity bursts. Counting the messages matching the search string in the stream over time, the fre- Selecting messages that best describe news events. For the set of all messages matching the search string in an activity burst, we apply the message-granular variation of our keyphrase extraction algorithm [2] that is specifically suited to efficiently filtering noisy data. The algorithm clusters messages with respect to their similarity to each other and chooses central messages from the most dense clusters. As the similarity measure we use Jaccard coefficient for the "bag of words" representation of messages.The demonstration illustrates the potential of our approach in bringing news acquisition to a new level of promptness and coverage range.

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


Iterative learning from texts and counterexamples using additional information Abstract A variant of iterative learning in the limit (cf. Lange and Zeugmann 1996) is studied when a learner gets negative examples refuting conjectures containing data in excess of the target language and uses additional information of the following four types: (a) memorizing up to n input elements seen so far; (b) up to n feedback memberships queries (testing if an item is a member of the input seen so far); (c) the number of input elements seen so far; (d) the maximal element of the input seen so far. We explore how additional information available to such learners (defined and studied in Jain and Kinber 2007) may help. In particular, we show that adding the maximal element or the number of elements seen so far helps such learners to infer any indexed class of languages class-preservingly (using a descriptive numbering defining the class)-as it is proved in Jain and Kinber (2007), this is not possible without using additional information. We also study how, in the given context, different types of additional information fare against each other, and establish hierarchies of learners memorizing n + 1 versus n input elements seen and n + 1 versus n feedback membership queries. As we have shown, additional information of the types studied in this paper can add interesting new capabilities to iterative learners getting negative examples to conjectures containing data in excess of the target language. Some problems related to comparisons of help provided by additional information remain open, and solving these problems can offer new (and, possibly, unexpected) insight into advantages of using additional information of certain types for the learners in question. Influence of noise on additional information has been discussed in Sect. 7, however, many questions remain open here as well. Similarly to Jain and Kinber (2007) , one might also consider different types of negative examples (refuting conjectures containing extra elements) by iterative learners and explore how these different types of negative examples may interplay with different types of additional information. Yet another interesting area of research is studying iterative learnability with counterexamples and additional information of specific indexed classes of languages (for example, regular languages or patterns)-as we have shown all such classes are learnable class-preservingly using the maximal element or the number of elements seen so far as additional information, and, therefore, one can now study if and when learnability of such classes may be efficient.. A general open problem for iterative learners of any type using additional (bounded) memory is whether a multiset type memory (when a learner can store the same input item several times; for example, the learner may decide to store, say, 10 copies of the next input element) can have an advantage over a set type memory (where every item is stored just once). We suspect that no such advantage is possible-however, we have not been able to find a proof.

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


Distributed MST for constant diameter graphs This paper considers the problem of distributively constructing a minimum-weight spanning tree (MST) for graphs of constant diameter in the bounded-messages model, where each message can contain at most B bits for some parameter B. It is shown that the time required to compute an MST for graphs of diameter 4 or 3 can be as high as f~(~/'n/B) and f~(x~/-n/x~/-B), respectively. The lower bound holds even if the algorithm is allowed to be randomized. On the other hand, it is shown that O(log n) time units suffice to compute an MST deterministically for graphs with diameter 2, when B = O(log n). These results complement a previously known lower bound of f~(~/'n/B) for graphs of diameter ~(log n). In this paper we have proved the somewhat surprising result that the complexity of distributed MST computation changes exponentially when the diameter changes from 2 to 3. The significance of diameter 2 may be explained by the fact that the only bottlenecks in graphs of diameter 2 may be articulation nodes: nodes, however, may relay linear number of bits in a single time step. This property does not hold for graphs of diameter 3, where the bottleneck may be an edge, that can relay only B bits in a single time step.. We do not know whether the lower bounds on the time complexity we have demonstrated are the best possible. We leave this question open.

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


Quantum networks: Anti-core of spin chains The purpose of this paper is to exhibit a quantum network phenomenon-the anti-core-that goes against the classical network concept of congestion core. Classical networks idealized as infinite, Gromov hyperbolic spaces with least-cost path routing (and subject to a technical condition on the Gromov boundary) have a congestion core, defined as a subnetwork that routing paths have a high probability of visiting. Here, we consider quantum networks, more specifically spin chains, define the so-called maximum excitation transfer probability pmax(i, j) between spin i and spin j, and show that the central spin has among all other spins the lowest probability of being excited or transmitting its excitation. The anti-core is singled out by analytical formulas for pmax(i, j), revealing the number theoretic properties of quantum chains. By engineering the chain, we further show that this probability can be made vanishingly small. Probably the most significant result of the Gromov analysis of classical networks [4, 6] is existence of a congestion core. Under a network protocol that sends the packets along least cost paths, the core can be qualitatively defined as a point where most of the geodesics (least cost paths) converge, creating packet drops, high retransmission rates, and other nuisances under the TCP-IP protocol [10] . Existence of the core has been experimentally observed [11] and mathematically proved [8] if the network is Gromov hyperbolic, subject to some highly technical conditions related to the Gromov boundary [2] . A Gromov hyperbolic network can intuitively be defined as a network that "looks like" a negatively curved Riemannian manifold (e.g., a saddle) when viewed from a distance. See, e.g., [3] for a precise definition.. Next to classical networks, one can envision quantum networks: the nodes are spins that can be up |↑ (not excited) or down |↓ (excited) and the links are quantum mechanical couplings of the XX or Heisenberg type. Given some random source-destination pair (i, j), a valid question is whether some spin ω could act as a "core," that is, a spin that could be excited no matter what the source and the destination are. For a linear chain, one would expect such a congestion core in the center as classically any excitation in one half of the chain would have to transit the center of the chain to reach the other half. In this work, we demonstrate that quantum-mechanically the transmission of excitations does not need to occur this way, and in fact the center ω of an odd-length spin chain can act as an "anti-core," excitation of which is avoided.. This "anti-core," or "anti-gravity" center as it was originally called, was first observed in [7] . The anti-core ω was defined as a point of high inertia, i d α (i, ω), α ≥ 1, as opposed to the classical congestion core that has minimum inertia owing to the negative curvature of the underlying space [9, Theorem 3.2.1]. The inertia quantifies how difficult communication to and from the anticore is.. As it has been done along this line of work, a pre-metric d(·, ·) based on the Information Transfer Capacity (ITC) (see Sec. 2) is employed. Unlike standard quantum mechanical distances [13, 15] , [12, pages 412-413] , this "distance" measure aims to quantify not how distant two fixed quantum states are, but how close to a desired target state a quantum state can get under the evolution of a particular Hamiltonian. The initial and target states are typically orthogonal.. In this paper, we provide an analytical justification of the numerically observed anti-core phenomenon in spin chains with XX coupling, starting with finite-length chains, extending the ITC concept to semi-and bi-infinite cases (Sec. 3), and finally proving that d(ω, j) ≥ d(i, j), ∀j = ω in Sec. 4 . We further show that by adding a bias on the central spin its "anti-core" property can be made stronger in the sense that the probability of transmission of the excitation to and from it is infinitesimally small (Sec. 5).. The remaining nagging question is why was it observed in [7] that spin chains appear Gromov-hyperbolic and have an anti-core, while classical networks are Gromov hyperbolic with the opposite core? This will be clarified in Sec. 5 by means of a spin chain example, showing that its Gromov boundary has only one point, while classical networks need to have at least two points in their Gromov boundary for the core to emerge. The early numerical observation [7] that homogeneous odd length N -spin chains have an "anti-gravity" center has been analytically confirmed in the N → ∞ limit by developing closed-form formulas for the asymptotic maximum excitation transfer probability, and by showing that the anti-core has the lowest probability of being excited or of transmitting its excitation. As shown in Sec. 3, the phenomena exhibited in Fig. 1 can be accurately explained by the N = ∞ asymptotic formula. The existence of an anti-core at the center of a linear array of spins shows that excitations in a spin network do not propagate as they would in a classical network. In a classical linear network any excitation in one half of the chain must transit through the center to reach the other half, and the center would thus be expected to be a congestion core. In quantum networks, however, excitations can be transferred from one end of a chain to the other without passing through the center due to the intrinsic entanglement present in the eigenstates of the system. When a single excitation is created in one location what is really created is a wavepacket, which is a superposition of many eigenstates of the system Hamiltonian that subsequently evolve and interfere. High probability of transmission requires constructive interference at any particular node at some time, and the existence of an anti-core shows that, surprisingly, there is least constructive interference for the center of the chain.. As we have also shown, it is possible to engineer chains such that the diameter of the chain goes to infinity even if the physical number of spins is finite. The simplest way to achieve this is by applying a bias to the central spin in an oddlength chain. By increasing the bias we can increase the diameter even for a finite chain and achieve infinite diameter in the limit of infinite bias. This can be explained in terms of the bias moving the central spin further and further away from the other spins and therefore effectively decoupling the chains. However, for any finite bias, no matter how large, an excitation in one half of the chain can tunnel through the obstruction in the center given sufficient time, allowing almost perfect excitation transfer between the end spins of the engineered chain.. Such finite length, infinite diameter chains lend themselves to a coarse Gromov analysis, as Section 5 shows. The specific feature, demonstrated on chains of limited length but conjectured to hold for longer chains, is a Gromov boundary reduced to a singleton. This strongly contrasts with the classical network paradigm of a Gromov boundary with at least two points, creating the congestion core [2] . Although there is early indication that the difference in cardinalities of the Gromov boundaries might be at least part of the explanation of the core versus anti-core discrepancy, more analysis is needed to prove a general fact and is left for further research.

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