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41
Subcubic Equivalences Between Path, Matrix, and Triangle Problems
"... We say an algorithm on n × n matrices with entries in [−M,M] (or n-node 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 n-node 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 all-pairs 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 edge-weighted graph. • Finding a minimum weight cycle in a graph of non-negative 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
Finding time-dependent shortestpaths over large graphs,” in
- Proc. 11th EDBT,
, 2008
"... ABSTRACT The spatial and temporal databases have been studied widely and intensively over years. In this paper, we study how to answer queries of finding the best departure time that minimizes the total travel time from a place to another, over a road network, where the traffic conditions dynamical ..."
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Cited by 36 (1 self)
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ABSTRACT The spatial and temporal databases have been studied widely and intensively over years. In this paper, we study how to answer queries of finding the best departure time that minimizes the total travel time from a place to another, over a road network, where the traffic conditions dynamically change from time to time. We study a generalized form of this problem, called the time-dependent shortest-path problem. A time-dependent graph GT is a graph that has an edge-delay function, wi,j(t), associated with each edge (vi, v j ), to be stored in a database. The edge-delay function w i,j (t) specifies how much time it takes to travel from node v i to node v j , if it departs from v i at time t. A user-specified query is to ask the minimum-travel-time path, from a source node, vs, to a destination node, ve, over the time-dependent graph, GT , with the best departure time to be selected from a time interval T . We denote this user query as LTT(v s , v e , T ) over G T . The challenge of this problem is the added complexity due to the time dependency in the time-dependent graph. That is, edge delays are not constants, and can vary from time to time. In this paper, we propose a novel algorithm to find the minimum-travel-time path with the best departure time for a LTT(v s , v e , T ) query over a large graph G T . Our approach outperforms existing algorithms in terms of both time complexity in theory and efficiency in practice. We will discuss the design of our algorithm, together with its correctness and complexity. We conducted extensive experimental studies over large graphs and will report our findings.
A simple and linear time randomized algorithm for computing sparse . . .
"... Let G = (V, E) be an undirected weighted graph on |V | = n vertices, and |E | = m edges. A t-spanner of the graph G, for any t ≥ 1, is a subgraph (V, ES), ES ⊆ E, such that the distance between any pair of vertices in the subgraph is at most t times the distance between them in the graph G. Comput ..."
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Cited by 34 (5 self)
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Let G = (V, E) be an undirected weighted graph on |V | = n vertices, and |E | = m edges. A t-spanner of the graph G, for any t ≥ 1, is a subgraph (V, ES), ES ⊆ E, such that the distance between any pair of vertices in the subgraph is at most t times the distance between them in the graph G. Computing a t-spanner of minimum size (number of edges) has been a widely studied and well motivated problem in computer science. In this paper we present the first linear time randomized algorithm that computes a t-spanner of a given weighted graph. Moreover, the size of the t-spanner computed essentially matches the worst case lower bound implied by a 43 years old girth conjecture made independently by Erdős [26], Bollobás [19], and Bondy & Simonovits [21]. Our algorithm uses a novel clustering approach that avoids any distance computation altogether. This feature is somewhat surprising since all the previously existing algorithms employ computation of some sort of local or global distance information which involves growing either breadth first search trees up to θ(t)-levels or full shortest path trees on a large fraction of vertices. The truly local approach of our algorithm also leads to equally simple and efficient algorithms for computing spanners in other important computational environments like distributed, parallel, and external memory.
Regularity lemmas and combinatorial algorithms
- In Proc. FOCS
"... Abstract — We present new combinatorial algorithms for Boolean matrix multiplication (BMM) and preprocessing a graph to answer independent set queries. We give the first asymptotic improvements on combinatorial algorithms for dense BMM in many years, improving on the “Four Russians ” O(n 3 /(w log n ..."
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Abstract — We present new combinatorial algorithms for Boolean matrix multiplication (BMM) and preprocessing a graph to answer independent set queries. We give the first asymptotic improvements on combinatorial algorithms for dense BMM in many years, improving on the “Four Russians ” O(n 3 /(w log n)) bound for machine models with wordsize w. (For a pointer machine, we can set w = log n.) The algorithms utilize notions from Regularity Lemmas for graphs in a novel way. • We give two randomized combinatorial algorithms for BMM. The first algorithm is essentially a reduction from BMM to the Triangle Removal Lemma. The best known bounds for the Triangle Removal Lemma only imply an O ` (n 3 log β)/(βw log n) ´ time algorithm for BMM where β = (log ∗ n) δ for some δ> 0, but improvements on the Triangle Removal Lemma would yield corresponding runtime improvements. The second algorithm applies the Weak Regularity Lemma of Frieze and Kannan along with “ several information compression ideas, running in O n 3 (log log n) 2 /(log n) 9/4 ”) time with probability exponentially “ close to 1. When w ≥ log n, it can be implemented in O n 3 (log log n) 2 /(w log n) 7/6 ”) time. Our results immediately imply improved combinatorial methods for CFG parsing, detecting triangle-freeness, and transitive closure. Using Weak Regularity, we also give an algorithm for answering queries of the form is S ⊆ V an independent set? in a graph. Improving on prior work, we show how to randomly preprocess a graph in O(n 2+ε) time (for all ε> 0) so that with high probability, all subsequent batches of log n independent “ set queries can be answered deterministically in O n 2 (log log n) 2 /((log n) 5/4 ”) time. When w ≥ log n, w queries can be answered in O n 2 (log log n) 2 /((log n) 7/6 ” time. In addition to its nice applications, this problem is interesting in that it is not known how to do better than O(n 2) using “algebraic ” methods. 1.
Fault-tolerant spanners for general graphs
- in STOC’09, 2009
"... The paper concerns graph spanners that are resistant to ver-tex or edge failures. Given a weighted undirected n-vertex graph G = (V,E) and an integer k ≥ 1, the subgraph H = (V,E′), E ′ ⊆ E, is a spanner of stretch k (or, a k-spanner) of G if δH(u, v) ≤ k · δG(u, v) for every u, v ∈ V, where δG′(u ..."
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The paper concerns graph spanners that are resistant to ver-tex or edge failures. Given a weighted undirected n-vertex graph G = (V,E) and an integer k ≥ 1, the subgraph H = (V,E′), E ′ ⊆ E, is a spanner of stretch k (or, a k-spanner) of G if δH(u, v) ≤ k · δG(u, v) for every u, v ∈ V, where δG′(u, v) denotes the distance between u and v in G Graph spanners were extensively studied since their intro-duction over two decades ago. It is known how to efficiently construct a (2k−1)-spanner of size O(n1+1/k), and this size-stretch tradeoff is conjectured to be tight. The notion of fault tolerant spanners was introduced a decade ago in the geometric setting [Levcopoulos et al., STOC’98]. A subgraph H is an f-vertex fault tolerant k-spanner of the graph G if for any set F ⊆ V of size at most f and any pair of vertices u, v ∈ V \ F, the distances in H satisfy δH\F (u, v) ≤ k · δG\F (u, v). Levcopoulos et al. presented an efficient algorithm that given a set S of n points in Rd, constructs an f-vertex fault tolerant geometric (1+)-spanner for S, that is, a sparse graph H such that for every set F ⊆ S of size f and any pair of points u, v ∈ S \F, δH\F (u, v) ≤ (1+)|uv|, where |uv | is the Euclidean distance between u and v. A fault tolerant geometric spanner with optimal maximum degree and total weight was presented in [Czumaj & Zhao, SoCG’03]. This paper also raised as an open problem the question whether it is possible to obtain a fault tolerant spanner for an arbitrary undirected weighted graph. The current paper answers this question in the affirmative, presenting an f-vertex fault tolerant (2k−1)-spanner of size
Additive Spanners and (α, β)-Spanners
"... An (α, β)-spanner of an unweighted graph G is a subgraph H that distorts distances in G up to a multiplicative factor of α and an additive term β. It is well known that any graph contains a (multiplicative) (2k − 1, 0)-spanner of size O(n 1+1/k) and an (additive) (1, 2)-spanner of size O(n 3/2). How ..."
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Cited by 14 (3 self)
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An (α, β)-spanner of an unweighted graph G is a subgraph H that distorts distances in G up to a multiplicative factor of α and an additive term β. It is well known that any graph contains a (multiplicative) (2k − 1, 0)-spanner of size O(n 1+1/k) and an (additive) (1, 2)-spanner of size O(n 3/2). However no other additive spanners are known to exist. In this paper we develop a couple of new techniques for constructing (α, β)-spanners. Our first result is an additive (1, 6)-spanner of size O(n 4/3). The construction algorithm can be understood as an economical agent that assigns costs and values to paths in the graph, purchasing affordable paths and ignoring expensive ones, which are intuitively well-approximated by paths already purchased. We show that this path buying algorithm can be parameterized in different ways to yield other sparseness-distortion tradeoffs. Our second result addresses the problem of which (α, β)-spanners can be computed efficiently, ideally in linear time. We show that for any k, a (k, k − 1)-spanner with size O(kn 1+1/k) can be found in linear time, and further, that in a distributed network the algorithm terminates in a constant number of rounds. Previous spanner constructions with similar performance had roughly twice the multiplicative distortion.
Popular conjectures imply strong lower bounds for dynamic problems
- CoRR
"... Abstract—We consider several well-studied problems in dy-namic algorithms and prove that sufficient progress on any of them would imply a breakthrough on one of five major open problems in the theory of algorithms: 1) Is the 3SUM problem on n numbers in O(n2−ε) time for some ε> 0? 2) Can one dete ..."
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Abstract—We consider several well-studied problems in dy-namic algorithms and prove that sufficient progress on any of them would imply a breakthrough on one of five major open problems in the theory of algorithms: 1) Is the 3SUM problem on n numbers in O(n2−ε) time for some ε> 0? 2) Can one determine the satisfiability of a CNF formula on n variables and poly n clauses in O((2 − ε)npoly n) time for some ε> 0? 3) Is the All Pairs Shortest Paths problem for graphs on n vertices in O(n3−ε) time for some ε> 0? 4) Is there a linear time algorithm that detects whether a given graph contains a triangle? 5) Is there an O(n3−ε) time combinatorial algorithm for n×n Boolean matrix multiplication? The problems we consider include dynamic versions of bipartite perfect matching, bipartite maximum weight matching, single source reachability, single source shortest paths, strong connec-tivity, subgraph connectivity, diameter approximation and some nongraph problems such as Pagh’s problem defined in a recent paper by Pǎtraşcu[STOC 2010]. Index Terms—dynamic algorithms; all pairs shortest paths; 3SUM; lower bounds; I.
Distributed Algorithms for Ultrasparse Spanners and Linear Size Skeletons
"... We present efficient algorithms for computing very sparse low distortion spanners in distributed networks and prove some non-trivial lower bounds on the tradeoff between time, sparseness, and distortion. All of our algorithms assume a synchronized distributed network, where relatively short messages ..."
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We present efficient algorithms for computing very sparse low distortion spanners in distributed networks and prove some non-trivial lower bounds on the tradeoff between time, sparseness, and distortion. All of our algorithms assume a synchronized distributed network, where relatively short messages may be communicated in each time step. Our first result is a fast distributed algorithm for finding an O(2 log ∗ n log n)-spanner with size O(n). Besides being nearly optimal in time and distortion, this algorithm appears to be the first that constructs an O(n)-size skeleton without requiring unbounded length messages or time proportional to the diameter of the network. Our second result is a new class of efficiently constructible (α, β)-spanners called Fibonacci spanners whose distortion improves with the distance being approximated. At their sparsest Fibonacci spanners can have nearly linear size O(n(log log n) φ) where φ = 1+ √ 5 2 is the golden ratio. As the distance increases the Fibonacci spanner’s multiplicative distortion passes through four discrete stages, moving from logarithmic to log-logarithmic, then into a period where it is constant, tending to 3, followed by another period tending to 1. On the lower bound side we prove that many recent sequential spanner constructions have no efficient counterparts in distributed networks, even if the desired distortion only needs to be achieved on the average or for a tiny fraction of the vertices. In particular, any distance preservers, purely additive spanners, or spanners with sublinear additive distortion must either be very dense, slow to construct, or have very weak guarantees on distortion.
Dynamic shortest paths and transitive closure: algorithmic techniques and data structures
- J. DISCR. ALGOR
, 2006
"... In this paper, we survey fully dynamic algorithms for path problems on general directed graphs. In particular, we consider two fundamental problems: dynamic transitive closure and dynamic shortest paths. Although research on these problems spans over more than three decades, in the last couple of ye ..."
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Cited by 9 (1 self)
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In this paper, we survey fully dynamic algorithms for path problems on general directed graphs. In particular, we consider two fundamental problems: dynamic transitive closure and dynamic shortest paths. Although research on these problems spans over more than three decades, in the last couple of years many novel algorithmic techniques have been proposed. In this survey, we will make a special effort to abstract some combinatorial and algebraic properties, and some common data-structural tools that are at the base of those techniques. This will help us try to present some of the newest results in a unifying framework so that they can be better understood and deployed also by non-specialists.
Faster Approximation Schemes for Fractional Multicommodity Flow Problems via Dynamic Graph Algorithms
"... We combine the work of Garg and Könemann, and Fleischer with ideas from dynamic graph algorithms to obtain faster (1 − ε)-approximation schemes for various versions of the multicommodity flow problem. In particular, if ε is moderately small and the size of every number used in the input instance is ..."
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Cited by 8 (1 self)
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We combine the work of Garg and Könemann, and Fleischer with ideas from dynamic graph algorithms to obtain faster (1 − ε)-approximation schemes for various versions of the multicommodity flow problem. In particular, if ε is moderately small and the size of every number used in the input instance is polynomially bounded, the running times of our algorithms match – up to poly-logarithmic factors and some provably optimal terms – the Ω(mn) flow-decomposition barrier for single-commodity flow. 1
