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12
Subcubic Equivalences Between Path, Matrix, and Triangle Problems
"... We say an algorithm on n × n matrices with entries in [−M,M] (or nnode graphs with edge weights from [−M,M]) is truly subcubic if it runs in O(n 3−δ · poly(log M)) time for some δ> 0. We define a notion of subcubic reducibility, and show that many important problems on graphs and matrices solvab ..."
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We say an algorithm on n × n matrices with entries in [−M,M] (or nnode graphs with edge weights from [−M,M]) is truly subcubic if it runs in O(n 3−δ · poly(log M)) time for some δ> 0. We define a notion of subcubic reducibility, and show that many important problems on graphs and matrices solvable in O(n 3) time are equivalent under subcubic reductions. Namely, the following weighted problems either all have truly subcubic algorithms, or none of them do: • The allpairs shortest paths problem on weighted digraphs (APSP). • Detecting if a weighted graph has a triangle of negative total edge weight. • Listing up to n 2.99 negative triangles in an edgeweighted graph. • Finding a minimum weight cycle in a graph of nonnegative edge weights. • The replacement paths problem on weighted digraphs. • Finding the second shortest simple path between two nodes in a weighted digraph. • Checking whether a given matrix defines a metric. • Verifying the correctness of a matrix product over the (min,+)semiring. Therefore, if APSP cannot be solved in n 3−ε time for any ε> 0, then many other problems also
Improved Distance Sensitivity Oracles via Fast SingleSource Replacement Paths
"... Abstract—A distance sensitivity oracle is a data structure which, given two nodes s and t in a directed edgeweighted graph G and an edge e, returns the shortest length of an st path not containing e, a so called replacement path for the triple (s,t,e). Such oracles are used to quickly recover from ..."
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Abstract—A distance sensitivity oracle is a data structure which, given two nodes s and t in a directed edgeweighted graph G and an edge e, returns the shortest length of an st path not containing e, a so called replacement path for the triple (s,t,e). Such oracles are used to quickly recover from edge failures. In this paper we consider the case of integer weights in the interval [−M,M], and present the first distance sensitivity oracle that achieves simultaneously subcubic preprocessing time and sublinear query time. More precisely, for a given parameter α ∈ [0,1], our oracle has preprocessing time Õ(Mnω+1 2 +Mn ω+α(4−ω) ) and query time Õ(n 1−α). Here ω < 2.373 denotes the matrix multiplication exponent. For a comparison, the previous best oracle for small integer weights has Õ(Mnω+1−α) preprocessing time and (superlinear) Õ(n1+α) query time [Weimann,YusterFOCS’10]. The main novelty in our approach is an algorithm to compute all the replacement paths from a given source s, an interesting problem on its own. We can solve the latter singlesource replacement paths problem in Õ(APSP(n,M))) time, where APSP(n,M) < Õ(M 0.681 n 2.575) [ZwickJACM’02] is the runtime for computing allpairs shortest paths in a graph with n vertices and integer edge weights in [−M,M]. For positive weights the runtime of our algorithm reduces to Õ(Mnω). This matches the best known runtime for the simpler replacement paths problem in which both the source s and the target t are fixed [VassilevskaSODA’11]. Keywordsreplacement paths; distance sensitivity oracles; shortest paths. I.
Approximate shortest paths avoiding a failed vertex : optimal data structures for unweighted graphs
"... Abstract. Let G = (V, E) be any undirected graph on V vertices and E edges. A path P between any two vertices u, v ∈ V is said to be tapproximate shortest path if its length is at most t times the length of the shortest path between u and v. We consider the problem of building a compact data struct ..."
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Abstract. Let G = (V, E) be any undirected graph on V vertices and E edges. A path P between any two vertices u, v ∈ V is said to be tapproximate shortest path if its length is at most t times the length of the shortest path between u and v. We consider the problem of building a compact data structure for a given graph G which is capable of answering the following query for any u, v, z ∈ V and t> 1. report tapproximate shortest path between u and v when vertex z fails We present data structures for the single source as well allpairs versions of this problem. Our data structures guarantee optimal query time. Most impressive feature of our data structures is that their size nearly match the size of their best static counterparts. 1.
Replacement Paths via Fast Matrix Multiplication
"... Abstract. Let G = (V, E) be a directed edgeweighted graph and let P be a shortest path from s to t in G. The replacement paths problem asks to compute, for every edge e on P, the shortest stot path that avoids e. Apart from approximation algorithms and algorithms for special graph classes, the na ..."
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Abstract. Let G = (V, E) be a directed edgeweighted graph and let P be a shortest path from s to t in G. The replacement paths problem asks to compute, for every edge e on P, the shortest stot path that avoids e. Apart from approximation algorithms and algorithms for special graph classes, the naive solution to this problem – removing each edge e on P one at a time and computing the shortest stot path each time – is surprisingly the only known solution for directed weighted graphs, even when the weights are integrals. In particular, although the related shortest paths problem has benefited from fast matrix multiplication, the replacement paths problem has not, and still required cubic time. For an nvertex graph with integral edgelengths in {−M,..., M}, we give a randomized Õ(Mn1+ 2 3 ω) = O(Mn 2.584) time algorithm that uses fast matrix multiplication and is subcubic for appropriate values of M. In particular, it runs in Õ(n1+ 2 3 ω) time if the weights are small (positive or negative) integers. We also show how to construct a distance sensitivity oracle in the same time bounds. A query (u, v, e) to this oracle requires subquadratic time and returns the length of the shortest utov path that avoids the edge e. In fact, for any constant number of edge failures, we construct a data structure in subcubic time, that answer queries in subquadratic time. Our results also apply for avoiding vertices rather than edges. 1
Fault tolerant approximate bfs structures
 In SODA
, 2014
"... A faulttolerant structure for a network is required to continue functioning following the failure of some of the network’s edges or vertices. This paper addresses the problem of designing a faulttolerant (α, β) approximate BFS structure (or FTABFS structure for short), namely, a subgraph H of th ..."
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A faulttolerant structure for a network is required to continue functioning following the failure of some of the network’s edges or vertices. This paper addresses the problem of designing a faulttolerant (α, β) approximate BFS structure (or FTABFS structure for short), namely, a subgraph H of the network G such that subsequent to the failure of some subset F of edges or vertices, the surviving part of H still contains an approximate BFS spanning tree for (the surviving part of) G, satisfying dist(s, v,H \ F) ≤ α · dist(s, v,G \ F) + β for every v ∈ V. We first consider multiplicative (α, 0) FTABFS structures resilient to a failure of a single edge and present an algorithm that given an nvertex unweighted undirected graph G and a source s constructs a (3, 0) FTABFS structure rooted at s with at most 4n edges (improving by an O(log n) factor on the neartight result of [3] for the special case of edge failures). Assuming at most f edge failures, for constant integer f> 1, we prove that there exists a (polytime constructible) (3(f + 1), (f + 1) log n) FTABFS structure with O(fn) edges. We then consider additive (1, β) FTABFS structures. In contrast to the linear size of (α, 0) FTABFS structures, we show that for every β ∈ [1, O(log n)] there exists an nvertex graph G with a source s for which any (1, β) FTABFS structure rooted at s has Ω(n1+(β)) edges, for some function (β) ∈ (0, 1). In particular, (1, 3) FTABFS structures admit a lower bound of Ω(n5/4) edges. These lower bounds demonstrate an interesting dichotomy between multiplicative and additive
Vertex Fault Tolerant Additive Spanners
, 2014
"... A faulttolerant structure for a network is required to continue functioning following the failure of some of the network’s edges or vertices. In this paper, we address the problem of designing a faulttolerant additive spanner, namely, a subgraph H of the network G such that subsequent to the failu ..."
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A faulttolerant structure for a network is required to continue functioning following the failure of some of the network’s edges or vertices. In this paper, we address the problem of designing a faulttolerant additive spanner, namely, a subgraph H of the network G such that subsequent to the failure of a single vertex, the surviving part of H still contains an additive spanner for (the surviving part of) G, satisfying dist(s, t,H \ {v}) ≤ dist(s, t,G \ {v}) + β for every s, t, v ∈ V. Recently, the problem of constructing faulttolerant additive spanners resilient to the failure of up to f edges has been considered [8]. The problem of handling vertex failures was left open therein. In this paper we develop new techniques for constructing additive FTspanners overcoming the failure of a single vertex in the graph. Our first result is an FTspanner with additive stretch 2 and Õ(n5/3) edges. Our second result is an FTspanner with additive stretch 6 and Õ(n3/2) edges. The construction algorithm consists of two main components: (a) constructing an FTclustering graph and (b) applying a modified pathbuying procedure suitably adopted to failure prone settings. Finally, we also describe two constructions for faulttolerant multisource additive spanners, aiming to guarantee a bounded additive stretch following a vertex failure, for every pair of vertices in S×V for a given subset of sources S ⊆ V. The additive stretch bounds of our constructions are 4 and 8 (using a different number of edges).
Replacement Paths via Row Minima of Concise Matrices
, 2014
"... Matrix M is kconcise if the finite entries of each column of M consist of k or fewer intervals of identical numbers. We give an O(n + m)time algorithm to compute the row minima of any O(1)concise n×m matrix. Our algorithm yields the first O(n+m)time reductions from the replacementpaths problem ..."
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Matrix M is kconcise if the finite entries of each column of M consist of k or fewer intervals of identical numbers. We give an O(n + m)time algorithm to compute the row minima of any O(1)concise n×m matrix. Our algorithm yields the first O(n+m)time reductions from the replacementpaths problem on an nnode medge undirected graph (respectively, directed acyclic graph) to the singlesource shortestpaths problem on an O(n)node O(m)edge undirected graph (respectively, directed acyclic graph). That is, we prove that the replacementpaths problem is no harder than the singlesource shortestpaths problem on undirected graphs and directed acyclic graphs. Moreover, our lineartime reductions lead to the first O(n + m)time algorithms for the replacementpaths problem on the following classes of nnode medge graphs: (1) undirected graphs in the wordRAM model of computation, (2) undirected planar graphs, (3) undirected minorclosed graphs, and (4) directed acyclic graphs.
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"... source distance oracle for planar digraphs avoiding a failed node or link ..."
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source distance oracle for planar digraphs avoiding a failed node or link