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29
Faster scaling algorithms for general graphmatching problems
 JOURNAL OF THE ACM
, 1991
"... An algorithm for minimumcost matching on a general graph with integral edge costs is presented. The algorithm runs in time close to the fastest known bound for maximumcardinality matching. Specifically, let n, m, and N denote the number of vertices, number of edges, and largest magnitude of a cost ..."
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Cited by 86 (2 self)
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An algorithm for minimumcost matching on a general graph with integral edge costs is presented. The algorithm runs in time close to the fastest known bound for maximumcardinality matching. Specifically, let n, m, and N denote the number of vertices, number of edges, and largest magnitude of a cost, respectively. The best known time bound for maximumcardinal ity matching M 0 ( Am). The new algorithm for minimumcost matching has time bound 0 ( in a ( m, n)Iog n m log ( nN)). A slight modification of the new algorithm finds a maximumcardinality matching in 0 ( fire) time. Other applications of the new algorlthm are given, mchrding an efficient implementation of Christofides ’ traveling salesman approximation algorithm and efficient solutions to update problems that require the linear programming duals for matching.
Computing MinimumWeight Perfect Matchings
 INFORMS
, 1999
"... We make several observations on the implementation of Edmonds’ blossom algorithm for solving minimumweight perfectmatching problems and we present computational results for geometric problem instances ranging in size from 1,000 nodes up to 5,000,000 nodes. A key feature in our implementation is the ..."
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Cited by 83 (2 self)
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We make several observations on the implementation of Edmonds’ blossom algorithm for solving minimumweight perfectmatching problems and we present computational results for geometric problem instances ranging in size from 1,000 nodes up to 5,000,000 nodes. A key feature in our implementation is the use of multiple search trees with an individual dualchange � for each tree. As a benchmark of the algorithm’s performance, solving a 100,000node geometric instance on a 200 Mhz PentiumPro computer takes approximately 3 minutes.
Scaling Algorithms for Network Problems
, 1985
"... This paper gives algorithms for network problems that work by scaling the numeric parameters. Assume all parameters are integers. Let n, m, and N denote the number of vertices, number of edges, and largest parameter of the network, respectively. A scaling algorithm for maximum weight matching on a b ..."
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Cited by 60 (2 self)
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This paper gives algorithms for network problems that work by scaling the numeric parameters. Assume all parameters are integers. Let n, m, and N denote the number of vertices, number of edges, and largest parameter of the network, respectively. A scaling algorithm for maximum weight matching on a bipartite graph runs in O(n3 % log N) time. For appropriate N this improves the traditional Hungarian method, whose most efftcient implementation is O(n(m + n log n)). The speedup results from finding augmenting paths in batches. The matching algorithm gives similar improvements for the following problems: singlesource shortest paths for arbitrary edge lengths (Bellman’s algorithm); maximum weight degreeconstrained subgraph; minimum cost flow on a cl network. Scaling gives a simple maximum value flow algorithm that matches the best known bound (Sleator and Tarjan’s algorithm) when log N = O(log n). Scaling also gives a good algorithm for shortest paths on a directed graph with nonnegative edge lengths (Dijkstra’s algorithm).
Undirected Single Source Shortest Paths in Linear Time
 J. Assoc. Comput. Mach
, 1997
"... The single source shortest paths problem (SSSP) is one of the classic problems in algorithmic graph theory: given a weighted graph G with a source vertex s, find the shortest path from s to all other vertices in the graph. Since 1959 all theoretical developments in SSSP have been based on Dijkstra' ..."
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Cited by 49 (3 self)
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The single source shortest paths problem (SSSP) is one of the classic problems in algorithmic graph theory: given a weighted graph G with a source vertex s, find the shortest path from s to all other vertices in the graph. Since 1959 all theoretical developments in SSSP have been based on Dijkstra's algorithm, visiting the vertices in order of increasing distance from s. Thus, any implementation of Dijkstra 's algorithm sorts the vertices according to their distances from s. However, we do not know how to sort in linear time. Here, a deterministic linear time and linear space algorithm is presented for the undirected single source shortest paths problem with integer weights. The algorithm avoids the sorting bottleneck by building a hierechical bucketing structure, identifying vertex pairs that may be visited in any order. 1 Introduction Let G = (V; E), jV j = n, jEj = m, be an undirected connected graph with an integer edge weight function ` : E ! N and a distinguished source vertex...
Salient Closed Boundary Extraction with Ratio Contour
 IEEE Trans. on Pattern Analysis and Machine Intelligence
, 2005
"... We present ratio contour, a novel graphbased method for extracting salient closed boundaries from noisy images. This method operates on a set of boundary fragments that are produced by edge detection. Boundary extraction identifies a subset of these fragments and connects them sequentially to for ..."
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Cited by 38 (9 self)
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We present ratio contour, a novel graphbased method for extracting salient closed boundaries from noisy images. This method operates on a set of boundary fragments that are produced by edge detection. Boundary extraction identifies a subset of these fragments and connects them sequentially to form a closed boundary with the largest saliency. We encode the Gestalt laws of proximity and continuity in a novel boundarysaliency measure based on the relative gap length and average curvature when connecting fragments to form a closed boundary. This new measure attempts to remove a possible bias toward short boundaries. We present a polynomialtime algorithm for finding the mostsalient closed boundary. We also present supplementary preprocessing steps that facilitate the application of ratio contour to real images. We compare ratio contour to two closely related methods for extracting closed boundaries: Elder and Zucker's method based on the shortestpath algorithm and Williams and Thornber's method based on spectral analysis and a stronglyconnectedcomponents algorithm. This comparison involves both theoretic analysis and experimental evaluation on both synthesized data and real images.
Approximating MinimumSize kConnected Spanning Subgraphs via Matching
 SIAM J. Comput
, 1998
"... Abstract: An efficient heuristic is presented for the problem of finding a minimumsize k connected spanning subgraph of an (undirected or directed) simple graph G =(V#E). There are four versions of the problem, and the approximation guarantees are as follows: minimumsize knode connected spann ..."
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Cited by 34 (3 self)
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Abstract: An efficient heuristic is presented for the problem of finding a minimumsize k connected spanning subgraph of an (undirected or directed) simple graph G =(V#E). There are four versions of the problem, and the approximation guarantees are as follows: minimumsize knode connected spanning subgraph of an undirected graph 1+[1=k], minimumsize knode connected spanning subgraph of a directed graph 1+[1=k], minimumsize kedge connected spanning subgraph of an undirected graph 1+[2=(k + 1)], and minimumsize kedge connected spanning subgraph of a directed graph 1+[4= p k].
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 27 (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.
Improved shortest paths on the word RAM
 IN: 27TH COLLOQUIUM ON AUTOMATA, LANGUAGES AND PROGRAMMING (ICALP), IN: LECTURE NOTES IN COMPUT. SCI
, 2000
"... Thorup recently showed that singlesource shortestpaths problems in undirected networks with n vertices, m edges, and edge weights drawn from {0,...,2 w − 1} can be solved in O(n + m) time and space on a unitcost randomaccess machine with a word length of w bits. His algorithm works by traversin ..."
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Cited by 24 (0 self)
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Thorup recently showed that singlesource shortestpaths problems in undirected networks with n vertices, m edges, and edge weights drawn from {0,...,2 w − 1} can be solved in O(n + m) time and space on a unitcost randomaccess machine with a word length of w bits. His algorithm works by traversing a socalled component tree. Two new related results are provided here. First, and most importantly, Thorup’s approach is generalized from undirected to directed networks. The resulting time bound, O(n + m log w), is the best deterministic linearspace bound known for sparse networks unless w is superpolynomial in log n. As an application, allpairs shortestpaths problems in directed networks with n vertices, m edges, and edge weights in {−2 w,...,2 w} can be solved in O(nm + n 2 log log n) time and O(n + m) space (not counting the output space). Second, it is shown that the component tree for an undirected network can be constructed in deterministic linear time and space with a simple algorithm, to be contrasted with a complicated and impractical solution suggested by Thorup. Another contribution of the present paper is a greatly simplified view of the principles underlying algorithms based on component trees.
Improved Algorithms for Finding Level Ancestors in Dynamic Trees
 Automata, Languages and Programming, 27th International Colloquium, ICALP 2000, number 1853 in LNCS
, 2000
"... Given a node x at depth d in a rooted tree LevelAncestor(x; i) returns the ancestor to x in depth d i. We show how to maintain a tree under addition of new leaves so that updates and level ancestor queries are being performed in worst case constant time. Given a forest of trees with n nodes wher ..."
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Cited by 16 (1 self)
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Given a node x at depth d in a rooted tree LevelAncestor(x; i) returns the ancestor to x in depth d i. We show how to maintain a tree under addition of new leaves so that updates and level ancestor queries are being performed in worst case constant time. Given a forest of trees with n nodes where edges can be added, m queries and updates take O(m(m;n)) time. This solves two open problems (P.F.
Unique Maximum Matching Algorithms
, 2002
"... We consider the problem of testing the uniqueness of maximum matchings, both in the unweighted and in the weighted case. For the unweighted case, we have two results. First, given a graph with n vertices and m edges, we can test whether the graph has a unique perfect matching, and find it if it exis ..."
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Cited by 12 (0 self)
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We consider the problem of testing the uniqueness of maximum matchings, both in the unweighted and in the weighted case. For the unweighted case, we have two results. First, given a graph with n vertices and m edges, we can test whether the graph has a unique perfect matching, and find it if it exists, in O(m log^4 n) time. This algorithm uses a recent dynamic connectivity algorithm and an old result of Kotzig characterizing unique perfect matchings in terms of bridges. For the special case of...