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13
Finding the Hidden Path: Time Bounds for AllPairs Shortest Paths
, 1993
"... We investigate the allpairs shortest paths problem in weighted graphs. We present an algorithmthe Hidden Paths Algorithmthat finds these paths in time O(m* n+n² log n), where m is the number of edges participating in shortest paths. Our algorithm is a practical substitute for Dijkstra's ..."
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Cited by 61 (0 self)
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We investigate the allpairs shortest paths problem in weighted graphs. We present an algorithmthe Hidden Paths Algorithmthat finds these paths in time O(m* n+n² log n), where m is the number of edges participating in shortest paths. Our algorithm is a practical substitute for Dijkstra's algorithm. We argue that m* is likely to be small in practice, since m* = O(n log n) with high probability for many probability distributions on edge weights. We also prove an Ω(mn) lower bound on the running time of any pathcomparison based algorithm for the allpairs shortest paths problem. Pathcomparison based algorithms form a natural class containing the Hidden Paths Algorithm, as well as the algorithms of Dijkstra and Floyd. Lastly, we consider generalized forms of the shortest paths problem, and show that many of the standard shortest paths algorithms are effective in this more general setting.
On the Complexity of Computations Under Varying Sets of Primitives
 J. of Computer and System Sciences
, 1979
"... The principal goal of research in computational complexity is the determination of tight lower bounds on the complexity, in terms of primitive operation executions, of solving problems or performing larger operations. While algorithms now exist that yield better than naive upper bounds for various o ..."
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Cited by 38 (1 self)
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The principal goal of research in computational complexity is the determination of tight lower bounds on the complexity, in terms of primitive operation executions, of solving problems or performing larger operations. While algorithms now exist that yield better than naive upper bounds for various operations (e.g. [1,9,11,12]),
A New Approach to AllPairs Shortest Paths on RealWeighted Graphs
 Theoretical Computer Science
, 2003
"... We present a new allpairs shortest path algorithm that works with realweighted graphs in the traditional comparisonaddition model. It runs in O(mn+n time, improving on the longstanding bound of O(mn + n log n) derived from an implementation of Dijkstra's algorithm with Fibonacci heaps. Her ..."
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Cited by 26 (2 self)
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We present a new allpairs shortest path algorithm that works with realweighted graphs in the traditional comparisonaddition model. It runs in O(mn+n time, improving on the longstanding bound of O(mn + n log n) derived from an implementation of Dijkstra's algorithm with Fibonacci heaps. Here m and n are the number of edges and vertices, respectively.
Information bounds are weak in the shortest distance problem
 J. ACM
, 1980
"... ASSTRACT. In the allpair shortest distance problem, one computes the matrix D = (du), where dq is the minimum weighted length of any path from vertex i to vertexj in a directed complete graph with a weight on each edge. In all the known algorithms, a shortest path p, ~ achieving di./is also implici ..."
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Cited by 22 (1 self)
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ASSTRACT. In the allpair shortest distance problem, one computes the matrix D = (du), where dq is the minimum weighted length of any path from vertex i to vertexj in a directed complete graph with a weight on each edge. In all the known algorithms, a shortest path p, ~ achieving di./is also implicitly computed. In fact, logs(f (n)) is an informationtheoretic lower bound, wheref(n) is the total number of distinct patterns (Po) for nvertex graphs. As f(n) potentially can be as large as 2":', it would appear possible that a nontrivial lower bound can be derived this way in the decision tree model. The characterization and enumeration of realizable patterns is studied, and it is shown thatf(n) < C "~. Thus no lower bound greater than Cn 2 can be derived from this approach. It is proved as a corollary that the Triangular polyhedron T ~"~, defined in E ¢~' ~ by d,j> 0 and the triangle inequalities d~j + dik> d,k, has at most C" ' faces of all dimensions, thus resolving an open question in a similar information bound approach to the shortest distance problem.
Computing Shortest Paths with Comparisons and Additions
 SODA
, 2002
"... We present an undirected allpairs shortest paths (APSP) algorithm which runs on a pointer machine in time O(mnot(m, n)) while making O(ran log a(m, n)) comparisons and additions, where m and n are the number of edges and vertices, respectively, and a(ra, n) is Tarjan's inverseAckermann function. ..."
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Cited by 19 (7 self)
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We present an undirected allpairs shortest paths (APSP) algorithm which runs on a pointer machine in time O(mnot(m, n)) while making O(ran log a(m, n)) comparisons and additions, where m and n are the number of edges and vertices, respectively, and a(ra, n) is Tarjan's inverseAckermann function. This improves upon all previous comparison & additionbased APSP algorithms when the graph is sparse, i.e., when m = o(n log n). At the heart of our APSP algorithm is a new singlesource shortest paths algorithm which runs in time O(ma(m,n) + nloglogr) on a pointer machine, where r is the ratio of the maximumtominimum edge length. So long as r < 2 '~°(a) this algorithm is faster than any implementation of Dijkstra's classical algorithm in the comparisonaddition model. For directed graphs we give an O(ra + nlogr)time comparison & additionbased SSSP algorithm on a pointer machine. Similar algorithms assuming integer weights or the RAM model were given earlier.
Finding a maximum weight triangle in n 3−δ time, with applications
 In Proc. of STOC
, 2006
"... We present the first truly subcubic algorithms for finding a maximum nodeweighted triangle in directed and undirected graphs with arbitrary real weights. The first is an O(B · n 3+ω 2) = O(B · n 2.688) deterministic algorithm, where n is the number of nodes, ω is the matrix multiplication exponen ..."
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Cited by 17 (10 self)
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We present the first truly subcubic algorithms for finding a maximum nodeweighted triangle in directed and undirected graphs with arbitrary real weights. The first is an O(B · n 3+ω 2) = O(B · n 2.688) deterministic algorithm, where n is the number of nodes, ω is the matrix multiplication exponent, and B is the number of bits of precision. The second is a strongly polynomial randomized algorithm that runs in O(n 3+ω 2 log n) expected worstcase time. To achieve this, we show how to efficiently sample a weighted triangle uniformly at random, out of just those triangles whose total weight falls in some prescribed interval (W1, W2) for arbitrary weights W1 and W2. Previous approaches to the problem resulted in time bounds with either an exponential dependence on B, or a runtime of the form Ω(n 3 /(log n) c). The algorithms are easily extended to finding a maximum nodeweighted induced subgraph on 3k nodes in Õ(n (3+ω)k 2) = Õ(n2.688k) time. We give applications to a variety of problems, including a stable matching problem between buyers and sellers in computational economics, and discuss the possibility of extending our approach to a truly subcubic algorithm for computing allpairs shortest paths on directed graphs with arbitrary weights. ∗ Both authors were supported by the NSF ALADDIN
A shortest path algorithm for realweighted undirected graphs
 in 13th ACMSIAM Symp. on Discrete Algs
, 1985
"... Abstract. We present a new scheme for computing shortest paths on realweighted undirected graphs in the fundamental comparisonaddition model. In an efficient preprocessing phase our algorithm creates a linearsize structure that facilitates singlesource shortest path computations in O(m log α) ti ..."
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Cited by 12 (3 self)
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Abstract. We present a new scheme for computing shortest paths on realweighted undirected graphs in the fundamental comparisonaddition model. In an efficient preprocessing phase our algorithm creates a linearsize structure that facilitates singlesource shortest path computations in O(m log α) time, where α = α(m, n) is the very slowly growing inverseAckermann function, m the number of edges, and n the number of vertices. As special cases our algorithm implies new bounds on both the allpairs and singlesource shortest paths problems. We solve the allpairs problem in O(mnlog α(m, n)) time and, if the ratio between the maximum and minimum edge lengths is bounded by n (log n)O(1) , we can solve the singlesource problem in O(m + nlog log n) time. Both these results are theoretical improvements over Dijkstra’s algorithm, which was the previous best for real weighted undirected graphs. Our algorithm takes the hierarchybased approach invented by Thorup. Key words. singlesource shortest paths, allpairs shortest paths, undirected graphs, Dijkstra’s
Networks Cannot Compute Their Diameter in Sublinear Time preliminary version please check for updates
, 2011
"... We study the problem of computing the diameter of a network in a distributed way. The model of distributed computation we consider is: in each synchronous round, each node can transmit a different (but short) message to each of its neighbors. We provide an ˜ Ω(n) lower bound for the number of commun ..."
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Cited by 10 (2 self)
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We study the problem of computing the diameter of a network in a distributed way. The model of distributed computation we consider is: in each synchronous round, each node can transmit a different (but short) message to each of its neighbors. We provide an ˜ Ω(n) lower bound for the number of communication rounds needed, where n denotes the number of nodes in the network. This lower bound is valid even if the diameter of the network is a small constant. We also show that a (3/2 − ε)approximation of the diameter requires ˜ Ω ( √ n) rounds. Furthermore we use our new technique to prove an ˜ Ω ( √ n) lower bound on approximating the girth of a graph by a factor 2 − ε. Contact author:
On the ComparisonAddition Complexity of AllPairs Shortest Paths
 In Proc. 13th Int'l Symp. on Algorithms and Computation (ISAAC'02
, 2002
"... We present an allpairs shortest path algorithm for arbitrary graphs that performs O(mn log (m; n)) comparison and addition operations, where m and n are the number of edges and vertices, resp., and is Tarjan's inverseAckermann function. Our algorithm eliminates the sorting bottleneck inherent in a ..."
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Cited by 6 (5 self)
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We present an allpairs shortest path algorithm for arbitrary graphs that performs O(mn log (m; n)) comparison and addition operations, where m and n are the number of edges and vertices, resp., and is Tarjan's inverseAckermann function. Our algorithm eliminates the sorting bottleneck inherent in approaches based on Dijkstra's algorithm, and for graphs with O(n) edges our algorithm is within a tiny O(log (n; n)) factor of optimal. Our algorithm can be implemented to run in polynomial time (granted, a large polynomial). We leave open the problem of providing an efficient implementation.
Efficient Algorithms for Path Problems in Weighted Graphs
, 2008
"... Problems related to computing optimal paths have been abundant in computer science since its emergence as a field. Yet for a large number of such problems we still do not know whether the stateoftheart algorithms are the best possible. A notable example of this phenomenon is the all pairs shorte ..."
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Cited by 2 (0 self)
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Problems related to computing optimal paths have been abundant in computer science since its emergence as a field. Yet for a large number of such problems we still do not know whether the stateoftheart algorithms are the best possible. A notable example of this phenomenon is the all pairs shortest paths problem in a directed graph with real edge weights. The best algorithm (modulo small polylogarithmic improvements) for this problem runs in cubic time, a running time known since the 1960s (by Floyd and Warshall). Our grasp of many such fundamental algorithmic questions is far from optimal, and the major goal of this thesis is to bring some new insights into efficiently solving path problems in graphs. We focus on several path problems optimizing different measures: shortest paths, maximum bottleneck paths, minimum nondecreasing paths, and various extensions. For the allpairs versions of these path problems we use an algebraic approach. We obtain improved algorithms using reductions