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A Practical Minimum Spanning Tree Algorithm Using the Cycle Property
 IN 11TH EUROPEAN SYMPOSIUM ON ALGORITHMS (ESA), NUMBER 2832 IN LNCS
, 2003
"... We present a simple new (randomized) algorithm for computing minimum spanning trees that is more than two times faster than the best previously known algorithms (for dense, "difficult" inputs). It is of conceptual interest that the algorithm uses the property that the heaviest edge in a cycle can be ..."
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Cited by 10 (2 self)
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We present a simple new (randomized) algorithm for computing minimum spanning trees that is more than two times faster than the best previously known algorithms (for dense, "difficult" inputs). It is of conceptual interest that the algorithm uses the property that the heaviest edge in a cycle can be discarded. Previously this has only been exploited in asymptotically optimal algorithms that are considered impractical. An additional advantage is...
AllPairs Bottleneck Paths For General Graphs in Truly SubCubic Time
 STOC'07
, 2007
"... In the allpairs bottleneck paths (APBP) problem (a.k.a. allpairs maximum capacity paths), one is given a directed graph with real nonnegative capacities on its edges and is asked to determine, for all pairs of vertices s and t, the capacity of a single path for which a maximum amount of flow can b ..."
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Cited by 10 (6 self)
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In the allpairs bottleneck paths (APBP) problem (a.k.a. allpairs maximum capacity paths), one is given a directed graph with real nonnegative capacities on its edges and is asked to determine, for all pairs of vertices s and t, the capacity of a single path for which a maximum amount of flow can be routed from s to t. The APBP problem was first studied in operations research, shortly after the introduction of maximum flows and allpairs shortest paths. We present the first truly subcubic algorithm for APBP in general dense graphs. In particular, we give a procedure for computing the (max,min)product of two arbitrary matrices over R ∪ {∞, −∞} in O(n 2+ω/3) ≤ O(n 2.792) time, where n is the number of vertices and ω is the exponent for matrix multiplication over rings. Using this procedure, an explicit maximum bottleneck path for any pair of nodes can be extracted in time linear in the length of the path.
l∞approximation via subdominants
 Journal of Mathematical Psychology
"... Given a vector u and a certain subset K of a real vector space E, the problem of lapproximation involves determining an element u ^ in K nearest to u in the sense of the lerror norm. The subdominant u * of u is the upper bound (if it exists) of the set [x # K: xOu] (we let xOy if all coordinates o ..."
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Cited by 7 (5 self)
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Given a vector u and a certain subset K of a real vector space E, the problem of lapproximation involves determining an element u ^ in K nearest to u in the sense of the lerror norm. The subdominant u * of u is the upper bound (if it exists) of the set [x # K: xOu] (we let xOy if all coordinates of x are smaller than or equal to the corresponding coordinates of y). We present general conditions on K under which a simple relationship between the subdominant of u and a best lapproximation holds. We specify this result by taking as K the cone of isotonic functions defined on a poset (X, O), the cone of convex functions defined on a subset of R N, the cone of ultrametrics on a set X, and the cone of tree metrics on a set X with fixed distances to a given vertex. This leads to simple optimal algorithms for the problem of best lfitting of distances by ultrametrics and by tree metrics preserving the distances to a fixed vertex (the latter provides a 3approximation algorithm for the problem of fitting a distance by a tree metric). This simplifies the recent results of Farach, Kannan, and Warnow (1995) and of Agarwala et al. (1996). 2000 Academic Press 1.
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 3 (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
All Pairs Bottleneck Paths and MaxMin Matrix Products in Truly Subcubic Time
, 2009
"... In the all pairs bottleneck paths (APBP) problem, one is given a directed graph with real weights on its edges. Viewing the weights as capacities, one is asked to determine, for all pairs (s,t) of vertices, the maximum amount of flow that can be routed along a single path from s to t. The APBP pro ..."
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In the all pairs bottleneck paths (APBP) problem, one is given a directed graph with real weights on its edges. Viewing the weights as capacities, one is asked to determine, for all pairs (s,t) of vertices, the maximum amount of flow that can be routed along a single path from s to t. The APBP problem was first studied in operations research, shortly after the introduction of maximum flows and all pairs shortest paths. We present the first truly subcubic algorithm for APBP in general dense graphs. In particular, we give a procedure for computing the (max,min)product of two arbitrary matrices over R ∪ {∞,−∞} in O(n 2+ω/3) ≤ O(n 2.792) time, where n is the number of vertices and ω is the exponent for matrix multiplication over rings. Maxmin products can be used to compute the maximum bottleneck values for all pairs of vertices together with a “successor matrix ” from which one can extract an explicit maximum bottleneck path for any pair of vertices in time linear in the length of the path.
On Cartesian Trees and Range Minimum Queries
"... We present new results on Cartesian trees with applications in range minimum queries and bottleneck edge queries. We introduce a cacheoblivious Cartesian tree for solving the range minimum query problem, a Cartesian tree of a tree for the bottleneck edge query problem on trees and undirected graphs, ..."
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We present new results on Cartesian trees with applications in range minimum queries and bottleneck edge queries. We introduce a cacheoblivious Cartesian tree for solving the range minimum query problem, a Cartesian tree of a tree for the bottleneck edge query problem on trees and undirected graphs, and a proof that no Cartesian tree exists for the twodimensional version of the range minimum query problem. 1