We propose to study unweighted graphs of constant distance VC-dimension as a broad generalization of many graph classes for which we can compute the diameter in truly subquadratic-time. In particular for any fixed @math , the class of @math -minor free graphs has distance VC-dimension at most @math . Our first main result is that on graphs of distance VC-dimension at most @math , for any fixed @math we can either compute the diameter or conclude that it is larger than @math in time @math , where @math only depends on @math . Then as a byproduct of our approach, we get the first truly subquadratic-time algorithm for constant diameter computation on all the nowhere dense graph classes. Finally, we show how to remove the dependency on @math for any graph class that excludes a fixed graph @math as a minor. More generally, our techniques apply to any graph with constant distance VC-dimension and polynomial expansion. As a result for all such graphs one obtains a truly subquadratic-time algorithm for computing their diameter. Our approach is based on the work of Chazelle and Welzl who proved the existence of spanning paths with strongly sublinear stabbing number for every hypergraph of constant VC-dimension. We show how to compute such paths efficiently by combining the best known approximation algorithms for the stabbing number problem with a clever use of @math -nets, region decomposition and other partition techniques.
An early example of linear-time solvable special case for diameter computation is the class of interval graphs @cite_0 . For every interval graph @math and for any integer @math , if we first compute an interval representation for @math in linear-time @cite_31 then we can compute by dynamic programming, for every vertex @math , the contiguous segment of all the vertices at a distance @math from @math in @math . It takes almost linear-time and it implies a straightforward quasi linear-time algorithm for diameter computation. More efficient algorithms for diameter computation on interval graphs and related graph classes were proposed in @cite_23 . Nevertheless we will show in what follows that interval orderings are a powerful tool for diameter computation on more general geometric graph classes.
@cite_0 @cite_31 @cite_23
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Determining the diameter of a graph is a fundamental graph operation, yet no efficient (i.e. linear or quadratic time) algorithm is known. In this paper, we examine the diameter problem on chordal graphs and AT-free graphs and show that a very simple (linear time) 2-sweep LexBFS algorithm identifies a vertex of maximum eccentricity unless the given graph has a specified induced subgraph (it was previously known that a single LexBFS algorithm is guaranteed to end at a vertex that is within 1 of the diameter for chordal graphs and AT-free graphs). As a consequence of the forbidden induced subgraph result on chordal graphs, our algorithm is guaranteed to work optimally for directed path graphs (it was previously known that a single LexBFS algorithm is guaranteed to work optimally for interval graphs). We show a new way to round vector solutions of semidefinite programming (SDP) hierarchies into integral solutions, based on a connection between these hierarchies and the spectrum of the input graph. We demonstrate the utility of our method by providing a new SDP-hierarchy based algorithm for constraint satisfaction problems with 2-variable constraints (2-CSP's). More concretely, we show for every 2-CSP instance I a rounding algorithm for r rounds of the Lasserre SDP hierarchy for I that obtains an integral solution that is at most worse than the relaxation's value (normalized to lie in [0,1]), as long as r > k ( ) ( ) ;, where k is the alphabet size of I, @math , and @math denotes the number of eigenvalues larger than @math in the normalized adjacency matrix of the constraint graph of @math . In the case that @math is a instance, the threshold @math is only a polynomial in @math , and is independent of the alphabet size. Also in this case, we can give a non-trivial bound on the number of rounds for instance. In particular our result yields an SDP-hierarchy based algorithm that matches the performance of the recent subexponential algorithm of Arora, Barak and Steurer (FOCS 2010) in the worst case, but runs faster on a natural family of instances, thus further restricting the set of possible hard instances for Khot's Unique Games Conjecture. Our algorithm actually requires less than the @math constraints specified by the @math level of the Lasserre hierarchy, and in some cases @math rounds of our program can be evaluated in time @math . We develop a new technique for constructing sparse graphs that allow us to prove near-linear lower bounds on the round complexity of computing distances in the CONGEST model. Specifically, we show an @math lower bound for computing the diameter in sparse networks, which was previously known only for dense networks [, SODA 2012]. In fact, we can even modify our construction to obtain graphs with constant degree, using a simple but powerful degree-reduction technique which we define. Moreover, our technique allows us to show @math lower bounds for computing @math -approximations of the diameter or the radius, and for computing a @math -approximation of all eccentricities. For radius, we are unaware of any previous lower bounds. For diameter, these greatly improve upon previous lower bounds and are tight up to polylogarithmic factors [, SODA 2012], and for eccentricities the improvement is both in the lower bound and in the approximation factor [Holzer and Wattenhofer, PODC 2012]. Interestingly, our technique also allows showing an almost-linear lower bound for the verification of @math -spanners, for @math . Consider an undirected weighted graph G=(V,E) with |V|=n and |E|=m, where each vertex v is assigned a label from a set L of labels. We show how to construct a compact distance oracle that can answer queries of the form: "what is the distance from v to the closest lambda-labeled node" for a given node v in V and label lambda in L. This problem was introduced by Hermelin, Levy, Weimann and Yuster [ICALP 2011] where they present several results for this problem. In the first result, they show how to construct a vertex-label distance oracle of expected size O(kn^ 1+1 k ) with stretch (4k - 5) and query time O(k). In a second result, they show how to reduce the size of the data structure to O(kn ^ 1 k ) at the expense of a huge stretch, the stretch of this construction grows exponentially in k, (2^k-1). In the third result they present a dynamic vertex-label distance oracle that is capable of handling label changes in a sub-linear time. The stretch of this construction is also exponential in k, (2 3^ k-1 +1). We manage to significantly improve the stretch of their constructions, reducing the dependence on k from exponential to polynomial (4k-5), without requiring any tradeoff regarding any of the other variables. In addition, we introduce the notion of vertex-label spanners: subgraphs that preserve distances between every node v and label lambda. We present an efficient construction for vertex-label spanners with stretch-size tradeoff close to optimal. In this note, we present first linear time algorithms for computing the center and the diameter of several classes of face regular plane graphs: triangulations with inner vertices of degree ≥ 6, quadrangulations with inner vertices of degree ≥ 4 and the subgraphs of the regular hexagonal grid bounded by a simple circuit of this grid. We present a probabilistic algorithm that, given a connected graph G (represented by adjacency lists) of average degree d, with edge weights in the set 1,...,w , and given a parameter @math , estimates in time @math the weight of the minimum spanning tree (MST) of G with a relative error of at most @math . Note that the running time does not depend on the number of vertices in G. We also prove a nearly matching lower bound of @math on the probe and time complexity of any approximation algorithm for MST weight. The essential component of our algorithm is a procedure for estimating in time @math the number of connected components of an unweighted graph to within an additive error of @math . (This becomes @math for @math .) The time bound is shown to be tight up to within the @math factor. Our connected-components algorithm picks @math vertices in the graph and then grows "local spanning trees" whose sizes are specified by a stochastic process. From the local information collected in this way, the algorithm is able to infer, with high confidence, an estimate of the number of connected components. We then show how estimates on the number of components in various subgraphs of G can be used to estimate the weight of its MST. We investigate the complexity of several fundamental polynomial-time solvable problems on graphs and on matrices, when the given instance has low treewidth; in the case of matrices, we consider the treewidth of the graph formed by non-zero entries. In each of the considered cases, the best known algorithms working on general graphs run in polynomial, but far from linear, time. Thus, our goal is to construct algorithms with running time of the form poly(k)·n or poly(k) · n log n, where k is the width of the tree decomposition given on the input. Such procedures would outperform the best known algorithms for the considered problems already for moderate values of the treewidth, like O(n1 c) for some small constant c. Our results include: -- an algorithm for computing the determinant and the rank of an n × n matrix using O(k3 · n) time and arithmetic operations; -- an algorithm for solving a system of linear equations using O(k3 · n) time and arithmetic operations; -- an O(k3 O n log n)-time randomized algorithm for finding the cardinality of a maximum matching in a graph; -- an O(k4 · n log2 n)-time randomized algorithm for constructing a maximum matching in a graph; -- an O(k2 · n log n)-time algorithm for finding a maximum vertex flow in a directed graph. Moreover, we provide an approximation algorithm for treewidth with time complexity suited to the running times as above. Namely, the algorithm, when given a graph G and integer k, runs in time O(k7 · n log n) and either correctly reports that the treewidth of G is larger than k, or constructs a tree decomposition of G of width O(k2). The above results stand in contrast with the recent work of [SODA 2016], which shows that the existence of algorithms with similar running times is unlikely for the problems of finding the diameter and the radius of a graph of low treewidth. We show that the eccentricity of every vertex in an undirected graph on n vertices can be computed in time n exp O(k*log(d)), where k is the treewidth of the graph and d is the diameter. This means that the diameter and the radius of the graph can be computed in the same time. In particular, if the diameter is constant, it can be determined in time n*exp(O(k)). This result matches a recent hardness result by Abboud, Vassilevska Williams, and Wang [SODA 2016] that shows that under the Strong Exponential Time Hypothesis of Impagliazzo, Paturi, and Zane [J. Comp. Syst. Sc., 2001], for any epsilon > 0, no algorithm with running time n^ 2-epsilon *exp(o(k)) can distinguish between graphs with diameter 2 and 3. Thorup and Zwick showed that for any integer k≥ 1, it is possible to preprocess any positively weighted undirected graph G=(V,E), with |E|=m and |V|=n, in O(kmn @math ) expected time and construct a data structure (a (2k–1)-approximate distance oracle) of size O(kn @math ) capable of returning in O(k) time an approximation @math of the distance δ(u,v) from u to v in G that satisfies @math , for any two vertices u,v∈ V. They also presented a much slower O(kmn) time deterministic algorithm for constructing approximate distance oracle with the slightly larger size of O(kn @math log n). We present here a deterministic O(kmn @math ) time algorithm for constructing oracles of size O(kn @math ). Our deterministic algorithm is slower than the randomized one by only a logarithmic factor. Using our derandomization technique we also obtain the first deterministic linear time algorithm for constructing optimal spanners of weighted graphs. We do that by derandomizing the O(km) expected time algorithm of Baswana and Sen (ICALP’03) for constructing (2k–1)-spanners of size O(kn @math ) of weighted undirected graphs without incurring any asymptotic loss in the running time or in the size of the spanners produced. We continue research into a well-studied family of problems that ask if the vertices of a graph can be partitioned into sets A and B, where A is an independent set and B induces a graph from some specified graph class G. We let G be the class of k-degenerate graphs. The problem is known to be polynomial-time solvable if k=0 (bipartite graphs) and NP-complete if k=1 (near-bipartite graphs) even for graphs of diameter 4, as shown by Yang and Yuan, who also proved polynomial-time solvability for graphs of diameter 2. We show that recognizing near-bipartite graphs of diameter 3 is NP-complete resolving their open problem. To answer another open problem, we consider graphs of maximum degree D on n vertices. We show how to find A and B in O(n) time for k=1 and D=3, and in O(n^2) time for k >= 2 and D >= 4. These results also provide an algorithmic version of a result of Catlin [JCTB, 1979] and enable us to complete the complexity classification of another problem: finding a path in the vertex colouring reconfiguration graph between two given k-colourings of a graph of bounded maximum degree. In the recent past, there has been considerable progress in devising algorithms for the all-pairs shortest paths (APSP) problem running in time significantly smaller than the obvious time bound of O(n3). Unfortunately, all the new algorithms are based on fast matrix multiplication algorithms that are notoriously impractical. Our work is motivated by the goal of devising purely combinatorial algorithms that match these improved running times. Our results come close to achieving this goal, in that we present algorithms with a small additive error in the length of the paths obtained. Our algorithms are easy to implement, have the desired property of being combinatorial in nature, and the hidden constants in the running time bound are fairly small. Our main result is an algorithm which solves the APSP problem in unweighted, undirected graphs with an additive error of 2 in time @math . This algorithm returns actual paths and not just the distances. In addition, we give more efficient algorithms with running time @math for the case where we are only required to determine shortest paths between k specified pairs of vertices rather than all pairs of vertices. The starting point for all our results is an @math algorithm for distinguishing between graphs of diameter 2 and 4, and this is later extended to obtaining a ratio 2 3 approximation to the diameter in time @math . Unlike in the case of APSP, our results for approximate diameter computation can be extended to the case of directed graphs with arbitrary positive real weights on the edges. The Near-Bipartiteness problem is that of deciding whether or not the vertices of a graph can be partitioned into sets @math and @math , where @math is an independent set and @math induces a forest. The set @math in such a partition is said to be an independent feedback vertex set. Yang and Yuan proved that Near-Bipartiteness is polynomial-time solvable for graphs of diameter 2 and NP-complete for graphs of diameter 4. We show that Near-Bipartiteness is NP-complete for graphs of diameter 3, resolving their open problem. We also generalise their result for diameter 2 by proving that even the problem of computing a minimum independent feedback vertex is polynomial-time solvable for graphs of diameter 2. Chapter 36 Decomposing Graphs into Regions of Small Diameter* Nathan Linialt Michael .lss A decomposition of a graph G = (V, E) is a partition of the vertex set into subsets (called lhks). The diameter of a decomposition is the least. d such that any two vertices belonging to the same connected component of a block are at distance < d. In this paper we prove (nearly best possible) statements of the form: .4ny n–vertex graph has a decomposition into a small number of blocks each having small diameter. Such decompositions provide a tool for efficiently decentralizing distributed computations. In [AGLP1 it was shown that every graph has a decomposition into at most s(n) blocks of diameter at most s(n) for s(n) = o( loglog d h n). usinga,technique of Awerbuch [A] and Awerbuch and Peleg [AP], we improve this result by showing that every graph has a decomposition of diameter ()(log n) into O(log n) blocks. In addition, we give a randomized distributed algorithm that produces such a decomposition and runs in time 0(log2 n). The construction can be parametrized to provide decompositions that trade-off between the number of blocks and the diameter. We show that this trade-off is nearly best possible for two families of graphs the first consists of skeletons of certain triangulations of a simplex and the second consists of grid graphs with added diagonals. The proofs in both cases rely on basic results in combinatorial topology, Sperner’s lemma for the first class and Tucker’s lemma for the second. *This work was supported in part by NSF contracts DMS87-03541 and CCR-8911388 tDepartment of Computer Science, Hebrew University, Jerusalem, Israel. + ‘Department of Computer Science and Engineering, Mail Code C-014, University of California, San Diego, La Jolla, CA 92093-0114. We present new and improved data structures that answer exact node-to-node distance queries in planar graphs. Such data structures are also known as distance oracles. For any directed planar graph on n nodes with non-negative lengths we obtain the following: * Given a desired space allocation @math , we show how to construct in @math time a data structure of size @math that answers distance queries in @math time per query. As a consequence, we obtain an improvement over the fastest algorithm for k-many distances in planar graphs whenever @math . * We provide a linear-space exact distance oracle for planar graphs with query time @math for any constant eps>0. This is the first such data structure with provable sublinear query time. * For edge lengths at least one, we provide an exact distance oracle of space @math such that for any pair of nodes at distance D the query time is @math . Comparable query performance had been observed experimentally but has never been explained theoretically. Our data structures are based on the following new tool: given a non-self-crossing cycle C with @math nodes, we can preprocess G in @math time to produce a data structure of size @math that can answer the following queries in @math time: for a query node u, output the distance from u to all the nodes of C. This data structure builds on and extends a related data structure of Klein (SODA'05), which reports distances to the boundary of a face, rather than a cycle. The best distance oracles for planar graphs until the current work are due to Cabello (SODA'06), Djidjev (WG'96), and Fakcharoenphol and Rao (FOCS'01). For @math and space @math , we essentially improve the query time from @math to @math . In this paper we consider the problem of determining a balanced ordering of the vertices of a graph; that is, the neighbors of each vertex v are as evenly distributed to the left and fight of v as possible. This problem, which has applications in graph drawing for example, is shown to be NP- hard, and remains NP- hard for bipartite simple graphs with maximum degree six. We then describe and analyze a number of methods for determining a balanced vertex-ordering, obtaining optimal orderings for directed acyclic graphs, trees, and graphs with maximum degree three. For undirected graphs, we obtain a 13 8-approximation algorithm. Finally we consider the problem of determining a balanced vertex-ordering of a bipartite graph with a fixed ordering of one bipartition. When only the imbalances of the fixed vertices count, this problem is shown to be NP-hard. On the other hand, we describe an optimal linear time algorithm when the final imbalances of all vertices count. We obtain a linear time algorithm to compute an optimal vertex-ordering of a bipartite graph with one bipartition of constant size. We give efficient algorithms for volume sampling, i.e., for picking @math -subsets of the rows of any given matrix with probabilities proportional to the squared volumes of the simplices defined by them and the origin (or the squared volumes of the parallelepipeds defined by these subsets of rows). In other words, we can efficiently sample @math -subsets of @math with probabilities proportional to the corresponding @math by @math principal minors of any given @math by @math positive semi definite matrix. This solves an open problem from the monograph on spectral algorithms by Kannan and Vempala (see Section @math of KV , also implicit in BDM, DRVW ). Our first algorithm for volume sampling @math -subsets of rows from an @math -by- @math matrix runs in @math arithmetic operations (where @math is the exponent of matrix multiplication) and a second variant of it for @math -approximate volume sampling runs in @math arithmetic operations, which is almost linear in the size of the input (i.e., the number of entries) for small @math . Our efficient volume sampling algorithms imply the following results for low-rank matrix approximation: (1) Given @math , in @math arithmetic operations we can find @math of its rows such that projecting onto their span gives a @math -approximation to the matrix of rank @math closest to @math under the Frobenius norm. This improves the @math -approximation of Boutsidis, Drineas and Mahoney BDM and matches the lower bound shown in DRVW . The method of conditional expectations gives a algorithm with the same complexity. The running time can be improved to @math at the cost of losing an extra @math in the approximation factor. (2) The same rows and projection as in the previous point give a @math -approximation to the matrix of rank @math closest to @math under the spectral norm. In this paper, we show an almost matching lower bound of @math , even for @math . In the densest subgraph problem, given an edge-weighted undirected graph (G=(V,E,w) ), we are asked to find (S V ) that maximizes the density, i.e., w(S) |S|, where w(S) is the sum of weights of the edges in the subgraph induced by S. This problem has often been employed in a wide variety of graph mining applications. However, the problem has a drawback; it may happen that the obtained subset is too large or too small in comparison with the size desired in the application at hand. In this study, we address the size issue of the densest subgraph problem by generalizing the density of (S V ). Specifically, we introduce the f-density of (S V ), which is defined as w(S) f(|S|), where (f: Z _ 0 R _ 0 ) is a monotonically non-decreasing function. In the f-densest subgraph problem (f-DS), we aim to find (S V ) that maximizes the f-density w(S) f(|S|). Although f-DS does not explicitly specify the size of the output subset of vertices, we can handle the above size issue using a convex concave size function f appropriately. For f-DS with convex function f, we propose a nearly-linear-time algorithm with a provable approximation guarantee. On the other hand, for f-DS with concave function f, we propose an LP-based exact algorithm, a flow-based (O(|V|^3) )-time exact algorithm for unweighted graphs, and a nearly-linear-time approximation algorithm. In a seminal paper, Bar- (1992) considered broadcasting in radio networks whose nodes know only their own label and labels of their neighbors. They claimed a linear lower bound on the time of deterministic broadcasting in such radio networks, by constructing a class of graphs of diameter 3, with the property that every broadcasting algorithm requires linear time on one of these graphs. Due to a subtle error in the argument, this result is incorrect. We construct an algorithm that broadcasts in logarithmic time on all graphs from the work of Bar- Moreover, we show how to broadcast in sublinear time on all n-node graphs of diameter o(log log n). On the other hand, we construct a class of graphs of diameter 4, such that every broadcasting algorithm requires time spl Omega (4 spl radic n) on one of these graphs. In view of the randomized algorithm, running in expected time O(D log n + log sup 2 n) on all n-node graphs of diameter D, our lower bound gives the first correct proof of an exponential gap between determinism and randomization in the time of radio broadcasting. Given an undirected graph @math with @math edges, @math vertices, and non-negative edge weights, and given an integer @math , we show that for some universal constant @math , a @math -approximate distance oracle for @math of size @math can be constructed in @math time and can answer queries in @math time. We also give an oracle which is faster for smaller @math . Our results break the quadratic preprocessing time bound of Baswana and Kavitha for all @math and improve the @math time bound of Thorup and Zwick except for very sparse graphs and small @math . When @math and @math , our oracle is optimal w.r.t. both stretch, size, preprocessing time, and query time, assuming a widely believed girth conjecture by Erd o s. This paper studies the problem of constructing a minimum-weight spanning tree (MST) in a distributed network. This is one of the most important problems in the area of distributed computing. There is a long line of gradually improving protocols for this problem, and the state of the art today is a protocol with running time O(Λ(G)+n⋅log∗n) due to Kutten and Peleg [S. Kutten, D. Peleg, Fast distributed construction of k-dominating sets and applications, J. Algorithms 28 (1998) 40–66; preliminary version appeared in: Proc. of 14th ACM Symp. on Principles of Distributed Computing, Ottawa, Canada, August 1995, pp. 20–27], where Λ(G) denotes the diameter of the graph G. Peleg and Rubinovich [D. Peleg, V. Rubinovich, A near-tight lower bound on the time complexity of distributed MST construction, in: Proc. 40th IEEE Symp. on Foundations of Computer Science, 1999, pp. 253–261] have shown that Ω˜(n) time is required for constructing MST even on graphs of small diameter, and claimed that their result “establishes the asymptotic near-optimality” of the protocol of [S. Kutten, D. Peleg, Fast distributed construction of k-dominating sets and applications, J. Algorithms 28 (1998) 40–66; preliminary version appeared in: Proc. of 14th ACM Symp. on Principles of Distributed Computing, Ottawa, Canada, August 1995, pp. 20–27]. In this paper we refine this claim, and devise a protocol that constructs the MST in Ω˜(μ(G,ω)+n) rounds, where μ(G,ω) is the MST-radius of the graph. The ratio between the diameter and the MST-radius may be as large as Θ(n), and, consequently, on some inputs our protocol is faster than the protocol of [S. Kutten, D. Peleg, Fast distributed construction of k-dominating sets and applications, J. Algorithms 28 (1998) 40–66; preliminary version appeared in: Proc. of 14th ACM Symp. on Principles of Distributed Computing, Ottawa, Canada, August 1995, pp. 20–27] by a factor of Ω˜(n). Also, on every input, the running time of our protocol is never greater than twice the running time of the protocol of [S. Kutten, D. Peleg, Fast distributed construction of k-dominating sets and applications, J. Algorithms 28 (1998) 40–66; preliminary version appeared in: Proc. of 14th ACM Symp. on Principles of Distributed Computing, Ottawa, Canada, August 1995, pp. 20–27]. As part of our protocol for constructing an MST, we develop a protocol for constructing neighborhood covers with a drastically improved running time. The latter result may be of independent interest.