Reachability Preservers: New Extremal Bounds and Approximation Algorithms

Abboud, Amir; Bodwin, Greg

doi:10.1137/1.9781611975031.122

Cited by 13 publications

(38 citation statements)

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“…In [11,29,22] pairwise spanners for general metrics were studied, and in particular terminal spanners. Recently [1] introduced reachability preservers from a given set of sources.…”

Section: Related Workmentioning

confidence: 99%

“…In the current paper we devise a suit of terminal embeddings and metric structures, such as spanners, distance oracles and distance labeling schemes (see Section 2 for definitions), for doubling metrics with distortion 1 + , for an arbitrarily small > 0. In particular, Gupta et al [19] devised an embedding of metrics with doubling constant λ into ∞ with distortion 1 + and dimension log n • λ log 1/ +O (1) . Our terminal embedding of doubling metrics into ∞ has the same distortion, but the dimension is log k • λ log 1/ +O (1) , i.e., the dependency on n is replaced by (essentially) the same dependency on k.…”

Section: Introductionmentioning

confidence: 99%

“…In particular, Gupta et al [19] devised an embedding of metrics with doubling constant λ into ∞ with distortion 1 + and dimension log n • λ log 1/ +O (1) . Our terminal embedding of doubling metrics into ∞ has the same distortion, but the dimension is log k • λ log 1/ +O (1) , i.e., the dependency on n is replaced by (essentially) the same dependency on k.…”

Section: Introductionmentioning

confidence: 99%

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Near Isometric Terminal Embeddings for Doubling Metrics

Elkin

Neiman

2021

Algorithmica

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Given a metric space (X, d), a set of terminals K ⊆ X, and a parameter t ≥ 1, we consider metric structures (e.g., spanners, distance oracles, embedding into normed spaces) that preserve distances for all pairs in K × X up to a factor of t, and have small size (e.g. number of edges for spanners, dimension for embeddings). While such terminal (aka source-wise) metric structures are known to exist in several settings, no terminal spanner or embedding with distortion close to 1, i.e., t = 1 + for some small 0 < < 1, is currently known.Here we devise such terminal metric structures for doubling metrics, and show that essentially any metric structure with distortion 1 + and size s(|X|) has its terminal counterpart, with distortion 1 + O( ) and size s(|K|) + 1. In particular, for any doubling metric on n points, a set of k = o(n) terminals, and constant 0 < < 1, there exists A spanner with stretch 1 + for pairs in K × X, with n + o(n) edges.A labeling scheme with stretch 1 + for pairs in K × X, with label size ≈ log k.An embedding into d ∞ with distortion 1 + for pairs in K × X, where d = O(log k). Moreover, surprisingly, the last two results apply if only K is a doubling metric, while X can be arbitrary.

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“…In [11,29,22] pairwise spanners for general metrics were studied, and in particular terminal spanners. Recently [1] introduced reachability preservers from a given set of sources.…”

Section: Related Workmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Near Isometric Terminal Embeddings for Doubling Metrics

Elkin

Neiman

2021

Algorithmica

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“…Using the recent work due to Abboud and Bodwin [AB18], we next show how to get a polynomial improvement on the number of branching events from Lemma 3.5. This in turn gives a polynomial improvement on the size of reachability-preserving minor from Theorem 3.7.…”

Section: An Improved Bound Of O(k 3 )mentioning

confidence: 99%

“…As discussed above, this comes from the assumption that we have access to the sparsest reachability preserver. It is conceivable that a similar approach that appears in [AB18] could be employed to achieve a better running-time. However, the focus of our paper is on optimizing the size of reachability-preserving minors.…”

Section: : Return Hmentioning

confidence: 99%

Improved Guarantees for Vertex Sparsification in Planar Graphs

Goranci¹,

Henzinger²,

Pan³

2020

SIAM J. Discrete Math.

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Graph Sparsification aims at compressing large graphs into smaller ones while preserving important characteristics of the input graph. In this work we study Vertex Sparsifiers, i.e., sparsifiers whose goal is to reduce the number of vertices. We focus on the following notions:(1) Given a digraph G = (V, E) and terminal vertices K ⊂ V with |K| = k, a (vertex) reachability sparsifier of G is a digraph H = (V H , E H ), K ⊂ V H that preserves all reachability information among terminal pairs. In this work we introduce the notion of reachability-preserving minors (RPMs) , i.e., we require H to be a minor of G. We show any directed graph G admits a RPM H of size O(k 3 ), and if G is planar, then the size of H improves to O(k 2 log k). We complement our upper-bound by showing that there exists an infinite family of grids such that any RPM must have Ω(k 2 ) vertices.(2) Given a weighted undirected graph G = (V, E) and terminal vertices K with |K| = k, an exact (vertex) cut sparsifier of G is a graph H with K ⊂ V H that preserves the value of minimum-cuts separating any bipartition of K. We show that planar graphs with all the k terminals lying on the same face admit exact cut sparsifiers of size O(k 2 ) that are also planar. Our result extends to flow and distance sparsifiers. It improves the previous best-known bound of O(k 2 2 2k ) for cut and flow sparsifiers by an exponential factor, and matches an Ω(k 2 ) lowerbound for this class of graphs. minors in undirected graphs by Krauthgamer et al. [KNZ14]. Theorem 1.1. Given a k-terminal digraph G, there exists a reachability-preserving minor H of G with size O(k 3 ).It might be interesting to compare the above result with the construction of reachability preserver by Abbound and Bodwin [AB18], where the reachability preserver for a pair-set P in a graph G is defined to be a subgraph of G that preserves the reachability of all pairs in P . The size (i.e., the number of edges) of such preservers is shown to be at least Ω(n 2/(d+1) |P | (d−1)/d ), for any integer d ≥ 2, which is in sharp contrast to our upper bound O(|P | 3/2 ) on the size of reachability-preserving minors by taking P to be the pair-set of all terminals.Furthermore, by exploiting a tight integration of our techniques with the compact distance oracles for planar graphs by Thorup [Tho04], we can show the following theorem regarding the size of reachability-preserving minors for planar digraphs.Theorem 1.2. Given a k-terminal planar digraph G, there exists a reachability-preserving minor H of G with size O(k 2 log k).We complement the above result by showing that there exist instances where the above upperbound is tight up to a O(log k) factor. Theorem 1.3. For infinitely many k ∈ N there exists a k-terminal acyclic directed grid G such that any reachability-preserving minor of G must use Ω(k 2 ) non-terminals.Cut, Flow and Distance Sparsifiers. We provide new constructions for quality-1 (exact) cut, flow and distance sparsifiers for k-terminal planar graphs, where all the terminals are assumed to lie on the s...

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A note on distance-preserving graph sparsification

Bodwin

2022

Information Processing Letters

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Reachability Preservers: New Extremal Bounds and Approximation Algorithms

Abstract: In this paper we prove new results about the extremal structure of paths in directed graphs. Say we are given a directed graph G = (V, E) on n nodes, a set of sources S ⊆ V of size |S| = n 1/3 , and a subset P ⊆ S × V of pairs (s, t) where s ∈

Cited by 13 publications

References 46 publications

Near Isometric Terminal Embeddings for Doubling Metrics

Near Isometric Terminal Embeddings for Doubling Metrics

Improved Guarantees for Vertex Sparsification in Planar Graphs

A note on distance-preserving graph sparsification

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