Near-Optimal Compression for the Planar Graph Metric

Abboud, Amir; Gawrychowski, Paweł; Mozes, Shay; Weimann, Oren

doi:10.1137/1.9781611975031.35

Cited by 14 publications

(38 citation statements)

References 68 publications

(125 reference statements)

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“…This approach is inspired by the approximate diameter algorithm of Weimann and Yuster [WY16], and the metric compression problem by Abboud at el. [AGMW18]. We start by defining the following problem, a special case of Abboud at el., which will underlie the combinatorial basis for our diameter computation.…”

Section: Our Resultsmentioning

confidence: 99%

Section: The Metric Compression Problemmentioning

confidence: 99%

“…This problem can observed as a special case of the metric compression problem studied by Abboud et al [AGMW18]. In particular, [AGMW18] considered an arbitrary subset S ⊆ V with the objective to compress the S × S distances (rather than the S × T distances). Our formulation is motivated by diameter computation, where the set S corresponds to the cycle separator of the graph and T = V, we then wish to compress the two sides across the cycle separator to speed up the computation of the diameter.…”

Section: The Metric Compression Problemmentioning

confidence: 99%

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Planar diameter via metric compression

Parter

2019

Proceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing

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We develop a new approach for distributed distance computation in planar graphs that is based on a variant of the metric compression problem recently introduced by Abboud et al. [SODA'18]. In our variant of the Planar Graph Metric Compression Problem, one is given an n-vertex planar graph G = (V, E), a set of S ⊆ V source terminals lying on a single face, and a subset of target terminals T ⊆ V. The goal is to compactly encode the S × T distances.One of our key technical contributions is in providing a compression scheme that encodes all S × T distances using O(|S| · poly(D) + |T|) bits 1 , for unweighted graphs with diameter D. This significantly improves the state of the art of O(|S| · 2 D + |T| · D) bits. We also consider an approximate version of the problem for weighted graphs, where the goal is to encode (1 + ) approximation of the S × T distances, for a given input parameter ∈ (0, 1]. Here, our compression scheme uses O(poly(|S|/ ) + |T|) bits. In addition, we describe how these compression schemes can be computed in near-linear time. At the heart of this compact compression scheme lies a VC-dimension type argument on planar graphs, using the well-known Sauer's lemma.This efficient compression scheme leads to several improvements and simplifications in the setting of diameter computation, most notably in the distributed setting:• There is an O(D 5 )-round randomized distributed algorithm for computing the diameter in planar graphs, w.h.p.• There is an O(D 3 ) + poly(log n/ ) · D 2 -round randomized distributed algorithm for computing an (1 + ) approximation of the diameter in weighted graphs with polynomially bounded weights, w.h.p.No sublinear round algorithms were known for these problems before. These distributed constructions are based on a new recursive graph decomposition that preserves the (unweighted) diameter of each of the subgraphs up to a logarithmic term. Using this decomposition, we also get an exact SSSP tree computation within O(D 2 ) rounds. 1 As standard, O is used to hide poly log n factors. Peleg in their seminal work [GKP93, KP95], real world networks usually do have small diameter. In addition, global graph problems admit a trivial Ω(D) lower bound in the distributed setting. Thus, a separator with O(D) vertices is small w.r.t to the total round complexity. Distributed Planar Graphs via Low-Congestion Shortcuts. The area of distributed planar algorithm was initiated by Ghaffari and Haeupler [GH16a], who introduced the notion of lowcongestion shortcuts. Roughly speaking, low-congestion shortcuts augment vertex disjoint subgraphs of potentially large diameter, with edges from the original graph in order to considerably reduce their diameter. Using this machinery, [GH16a] has provided improved algorithms for MST and minimum-cut. Low-congestion shortcuts and their algorithmic applications have been studied extensively since then [HIZ16a, HIZ16b, HLZ18, Li18, HHW18]. Recently, Ghaffari and Parter [GP17]presented a distributed construction of shortest path separator in nearly optimal time. W...

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Section: Our Resultsmentioning

confidence: 99%

Section: The Metric Compression Problemmentioning

confidence: 99%

Section: The Metric Compression Problemmentioning

confidence: 99%

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Planar diameter via metric compression

Parter

2019

Proceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing

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“…Our representation of M is as follows, and fairly standard [2,16,35]. We define a (k − 1) × (k − 1) matrix P , satisfying…”

Section: Fr-dijkstramentioning

confidence: 99%

An Almost Optimal Edit Distance Oracle

Charalampopoulos,

Gawrychowski,

Mozes

et al. 2021

Preprint

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We consider the problem of preprocessing two strings S and T , of lengths m and n, respectively, in order to be able to efficiently answer the following queries: Given positions i, j in S and positions a, b in T , return the optimal alignment of S[i . . j] and T [a . . b]. Let N = mn. We present an oracle with preprocessing time N 1+o(1) and space N 1+o(1) that answers queries in log 2+o(1) N time. In other words, we show that we can query the alignment of every two substrings in almost the same time it takes to compute just the alignment of S and T . Our oracle uses ideas from our distance oracle for planar graphs [STOC 2019] and exploits the special structure of the alignment graph. Conditioned on popular hardness conjectures, this result is optimal up to subpolynomial factors. Our results apply to both edit distance and longest common subsequence (LCS).The best previously known oracle with construction time and size O(N ) has slow Ω( √ N ) query time [Sakai, TCS 2019], and the one with size N 1+o(1) and query time log 2+o(1) N (using a planar graph distance oracle) has slow Ω(N 3/2) construction time [Long & Pettie, SODA 2021]. We improve both approaches by roughly a √ N factor.

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“…They did not calculate the exact constant, but later experimental comparison by Fischer [16] showed that, even after some tweaking, in the worst case it is around 8. In a later paper, Alstrup et al [9] showed an NCA labeling scheme with labels of length 2.772 log n+O(1) 1 and proved that any such scheme needs labels of length at least 1.008 log n − O (1). The latter non-trivially improves an immediate lower bound of log n + Ω(log log n) obtained from ancestry.…”

Section: Introductionmentioning

confidence: 98%

Labeling Schemes for Nearest Common Ancestors through Minor-Universal Trees

Gawrychowski

Kühn

Łopuszański

et al. 2018

Proceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms

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Preprocessing a tree for finding the nearest common ancestor of two nodes is a basic tool with multiple applications. Quite a few linear-space constant-time solutions are known and the problem seems to be well-understood. This is however not so clear if we want to design a labeling scheme. In this model, the structure should be distributed: every node receives a distinct binary string, called its label, so that given the labels of two nodes (and no further information about the topology of the tree) we can compute the label of their nearest common ancestor. The goal is to make the labels as short as possible. Alstrup, Gavoille, Kaplan, and Rauhe [Theor. Comput. Syst. 37(3): 441-456 2004] showed that O(log n)-bit labels are enough, with a somewhat large constant. More recently, Alstrup, Halvorsen, and Larsen [SODA 2014] refined this to only 2.772 log n, and provided a lower bound of 1.008 log n.We connect the question of designing a labeling scheme for nearest common ancestors to the existence of a tree, called a minor-universal tree, that contains every tree on n nodes as a topological minor. Even though it is not clear if a labeling scheme must be based on such a notion, we argue that all already existing schemes can be reformulated as such. Further, we show that this notion allows us to easily obtain clean and good bounds on the length of the labels. As the main upper bound, we show that 2.318 log nbit labels are enough. Surprisingly, the notion of a minoruniversal tree for binary trees on n nodes has been already used in a different context by Hrubes et al. [CCC 2010], and Young, Chu, and Wong [J. ACM 46(3):416-435, 1999] introduced a very closely related (but not equivalent) notion of a universal tree. On the lower bound side, we show that any minor-universal tree for trees on n nodes must contain at least Ω(n 2.174 ) nodes. This highlights a natural limitation for all approaches based on defining a minoruniversal tree. We complement the existential results with a generic transformation that allows us, for any labeling scheme for nearest common ancestors based on a minoruniversal tree, to decrease the query time to constant, while increasing the length of the labels only by lower order terms.

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Near-Optimal Compression for the Planar Graph Metric

Cited by 14 publications

References 68 publications

Planar diameter via metric compression

Planar diameter via metric compression

An Almost Optimal Edit Distance Oracle

Labeling Schemes for Nearest Common Ancestors through Minor-Universal Trees

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