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...(k, j)}. Clearly computing min-max product is a special case of edge-APBP on a 3-layer graph constructed as in the proof of Theorem 2. The converse can be shown as well using a method from Aho et al. =-=[1]-=-, Section 5.9, Corollary 2. The min-max product problem resembles the min-plus product problem where one defines C(i, j) = A(i, k) + B(k, j). The fastest published algorithm for the latter, due to Cha...

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...atrix and an n s ×n t Boolean matrix can be computed in O(n ω(r,s,t) ) time. The exponent ω = ω(1, 1, 1) is usually called the exponent of fast Boolean matrix multiplication. Coppersmith and Winograd =-=[6]-=- proved that ω < 2.376. Let µ be the solution to ω(1, µ, 1) = 1 + 2µ. The results from [5] and [12] show that µ < 0.575. 1.1 The new results For a directed graph with n vertices, an APBP matrix is an ...

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...ed and that all edge weights are distinct. Let T = (V, E ′ ) be a spanning tree of G of maximum weight. The tree T can be easily computed in O(m + n log n) = O(n 2 ) time using, say, Prim’s algorithm =-=[16, 8]-=-. We claim that c(u, v) = cT (u, v), for every u, v ∈ V . Clearly, c(u, v) ≥ cT (u, v), for every u, v ∈ V , as T is a subgraph of G. Let p be a path from u to v in G. Each edge e on p closes a cycle ...

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...ed and that all edge weights are distinct. Let T = (V, E ′ ) be a spanning tree of G of maximum weight. The tree T can be easily computed in O(m + n log n) = O(n 2 ) time using, say, Prim’s algorithm =-=[16, 8]-=-. We claim that c(u, v) = cT (u, v), for every u, v ∈ V . Clearly, c(u, v) ≥ cT (u, v), for every u, v ∈ V , as T is a subgraph of G. Let p be a path from u to v in G. Each edge e on p closes a cycle ...

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... 1, 1) is usually called the exponent of fast Boolean matrix multiplication. Coppersmith and Winograd [6] proved that ω < 2.376. Let µ be the solution to ω(1, µ, 1) = 1 + 2µ. The results from [5] and =-=[12]-=- show that µ < 0.575. 1.1 The new results For a directed graph with n vertices, an APBP matrix is an n × n matrix C with rows and columns indexed by the vertices, and C(u, v) = c(u, v). If there is no...

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... capacity of a path from u to v having length d(u, v). The APBSP problem is to compute sc(u, v) for all ordered pairs of vertices u, v (again, we have two versions: open-APBSP and closedAPBSP). Zwick =-=[21]-=- gave an O(n 2+µ ) time algorithm for unweighted APSP. (Note, however, that we do not claim that unweighted APSP is computationally equivalent to the problems of Theorem 2.) Theorem 1 gives an O(n 2+µ...

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.... 3sA simple randomized algorithm for computing (not necessarily maximum) witnesses for Boolean matrix multiplication, in essentially the same time required to perform the product, is given by Seidel =-=[17]-=-. Alon and Naor [2] gave a deterministic algorithm for the problem. An alternative, slightly slower, deterministic algorithm was given by Galil and Margalit [10]. However, computing the maximum witnes...

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...ω = ω(1, 1, 1) is usually called the exponent of fast Boolean matrix multiplication. Coppersmith and Winograd [6] proved that ω < 2.376. Let µ be the solution to ω(1, µ, 1) = 1 + 2µ. The results from =-=[5]-=- and [12] show that µ < 0.575. 1.1 The new results For a directed graph with n vertices, an APBP matrix is an n × n matrix C with rows and columns indexed by the vertices, and C(u, v) = c(u, v). If th...

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...zed algorithm for computing (not necessarily maximum) witnesses for Boolean matrix multiplication, in essentially the same time required to perform the product, is given by Seidel [17]. Alon and Naor =-=[2]-=- gave a deterministic algorithm for the problem. An alternative, slightly slower, deterministic algorithm was given by Galil and Margalit [10]. However, computing the maximum witness matrix seems to b...

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...e APBP problem has several interesting applications. In Section 2 we describe three such applications, and show that they can be reduced to special cases of APBP. The first application, considered in =-=[3]-=-, is All-Pairs Lowest Common Ancestors in directed acyclic graphs. The fastest algorithm for this problem, due to Kowaluk and Lingas [13, 14], runs in O(n 2+µ ) time. We show that this problem can be ...

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...versions of many other graph optimization problems were considered before. The bottleneck spanning tree problem, for example, can be easily solved, deterministically, in O(m+n) time. Gabow and Tarjan =-=[11]-=- considered bottleneck versions of the directed spanning tree problem and the weighted matching problem. Before presenting our main result, we need a few definitions. Let ω(r, s, t) be the minimal exp...

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...Section 5.9, Corollary 2. The min-max product problem resembles the min-plus product problem where one defines C(i, j) = A(i, k) + B(k, j). The fastest published algorithm for the latter, due to Chan =-=[4]-=-, runs minn k=1 in O(n3 / log n) time (a mild improvement having running time O(n3 (log log n/ log n) 5/4 ) was recently announced by Y. Han). • Our algorithm for the APBP problem runs in time O(n 2+µ...

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...of G then for every (u, v) �∈ E we have C(u, v) = C ′ (uout, vin). Let f(n) = O(n ω ) be the complexity of Boolean matrix multiplication, and of transitive closure; the two are known to be equivalent =-=[7, 9, 15]-=-. Let g(n) = O(n 2+µ ) be the complexity of computing maximum Witnesses for Boolean matrix multiplication. By definition, f(n) ≤ g(n), and the proof of Theorem 1 gives the following corollary. Claim 3...

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...of G then for every (u, v) �∈ E we have C(u, v) = C ′ (uout, vin). Let f(n) = O(n ω ) be the complexity of Boolean matrix multiplication, and of transitive closure; the two are known to be equivalent =-=[7, 9, 15]-=-. Let g(n) = O(n 2+µ ) be the complexity of computing maximum Witnesses for Boolean matrix multiplication. By definition, f(n) ≤ g(n), and the proof of Theorem 1 gives the following corollary. Claim 3...

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... to special cases of APBP. The first application, considered in [3], is All-Pairs Lowest Common Ancestors in directed acyclic graphs. The fastest algorithm for this problem, due to Kowaluk and Lingas =-=[13, 14]-=-, runs in O(n 2+µ ) time. We show that this problem can be easily reduced to a special case of closed-APBP. The second application, first considered by Vassilevska and Williams [18], is the Largest We...

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...luk and Lingas [13, 14], runs in O(n 2+µ ) time. We show that this problem can be easily reduced to a special case of closed-APBP. The second application, first considered by Vassilevska and Williams =-=[18]-=-, is the Largest Weighted Triangle problem. Given a real vertex-weighted graph, find a triangle (if one exists) with maximum total weight. The fastest algorithm for this problem, due to Vassilevska, W...

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...ighted Triangle problem. Given a real vertex-weighted graph, find a triangle (if one exists) with maximum total weight. The fastest algorithm for this problem, due to Vassilevska, Williams and Yuster =-=[19]-=-, runs in O(n 2+µ ) time. Clearly, if we can find, for each pair of vertices, the maximum total weight of a 2-path, i.e., a path of length 2, connecting them, we can also find a largest weight triangl...

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...erform the product, is given by Seidel [17]. Alon and Naor [2] gave a deterministic algorithm for the problem. An alternative, slightly slower, deterministic algorithm was given by Galil and Margalit =-=[10]-=-. However, computing the maximum witness matrix seems to be a more difficult problem. Kowaluk and Lingas [13, 14] proved the following. Theorem 5 (Kowaluk and Lingas [13, 14]) A maximum witness matrix...

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...ortest paths between pairs of vertices, there is, in many cases, more than one solution. Which solution (namely, which shortest path) should be preferred? Such problems have been considered in, e.g., =-=[20]-=-. In Section 4, we look, among all shortest paths between a pair of vertices, for the one having maximum bottleneck weight. More formally, if G = (V, E, w) and w : V → R, let d(u, v) denote the (unwei...

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...of G then for every (u, v) �∈ E we have C(u, v) = C ′ (uout, vin). Let f(n) = O(n ω ) be the complexity of Boolean matrix multiplication, and of transitive closure; the two are known to be equivalent =-=[7, 9, 15]-=-. Let g(n) = O(n 2+µ ) be the complexity of computing maximum Witnesses for Boolean matrix multiplication. By definition, f(n) ≤ g(n), and the proof of Theorem 1 gives the following corollary. Claim 3...

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... to special cases of APBP. The first application, considered in [3], is All-Pairs Lowest Common Ancestors in directed acyclic graphs. The fastest algorithm for this problem, due to Kowaluk and Lingas =-=[13, 14]-=-, runs in O(n 2+µ ) time. We show that this problem can be easily reduced to a special case of closed-APBP. The second application, first considered by Vassilevska and Williams [18], is the Largest We...