Results 1 
7 of
7
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 ..."
Abstract

Cited by 59 (2 self)
 Add to MetaCart
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).
SublinearTime Parallel Algorithms for Matching and Related Problems
, 1988
"... This paper presents the first sublineartime deterministic parallel algorithms for bipartite matching and several related problems, including maximal nodedisjoint paths, depthfirst search, and flows in zeroone networks. Our results are based on a better understanding of the combinatorial struc ..."
Abstract

Cited by 33 (6 self)
 Add to MetaCart
This paper presents the first sublineartime deterministic parallel algorithms for bipartite matching and several related problems, including maximal nodedisjoint paths, depthfirst search, and flows in zeroone networks. Our results are based on a better understanding of the combinatorial structure of the above problems, which leads to new algorithmic techniques. In particular, we show how to use maximal matching to extend, in parallel, a current set of nodedisjoint paths and how to take advantage of the parallelism that arises when a large number of nodes are "active" during an execution of a pushrelabel network flow algorithm. We also show how to apply our techniques to design parallel algorithms for the weighted versions of the above problems. In particular, we present sublineartime deterministic parallel algorithms for finding a minimumweight bipartite matching and for finding a minimumcost flow in a network with zeroone capacities, if the weights are polynomially ...
Optimized kShortestPaths Algorithm for Facility Restoration
 SOFTWAREPRACTICE AND EXPERIENCE
, 1994
"... This paper presents experimental studies of several wellknown shortestpaths algorithms adapted to the task of finding the ksuccessivelyshortest linkdisjoint replacement paths for restoration in a telecommunications network with n nodes. The implementations range in complexity from O(kn ) when ..."
Abstract

Cited by 17 (9 self)
 Add to MetaCart
This paper presents experimental studies of several wellknown shortestpaths algorithms adapted to the task of finding the ksuccessivelyshortest linkdisjoint replacement paths for restoration in a telecommunications network with n nodes. The implementations range in complexity from O(kn ) when based on Dijkstra's original method, through several improvements to an efficient implementation of O(kn[v+logn]) complexity, and finally to an O(kn) implementation for the special case of edgesparse graphs with small integer edge weights. Here n is the maximum degree of a node in the network. Several alternatives were tested during the course of these studies, particularly with a view to minimizing the number of heap updates. These alternatives are possible because we are searching for several paths between a given pair of nodes, rather than just one path between one or more pairs of nodes. Two fairly straightforward changes yield a decrease in execution time, whereas a more complex heap management strategy consumes as much time in the added code as it releases from the main routine. Experimental results confirm the theoretical complexity of O(kn log n) and demonstrate a speedup of nearly an order of magnitude over the simpler O(kn ) implementation in the largest networks tested. The optimized implementation is recommended for planning and operational applications of kshortest paths rerouting for telecommunications network restoration and restorable network design. If hop counts or small integer link weights can be used to measure distances, then the O(kn) implementation is recommended, as typical telecommunications networks are edgesparse
Implementations of Dijkstra's Algorithm Based on MultiLevel Buckets
, 1995
"... A 2level bucket data structure has been shown to perform well in a Dijkstra's algorithm implementation [4, 5]. In this paper we study how the implementation performance depends on the number of bucket levels used. In particular we are interested in the best number of levels to use in practice. Pa ..."
Abstract

Cited by 14 (7 self)
 Add to MetaCart
A 2level bucket data structure has been shown to perform well in a Dijkstra's algorithm implementation [4, 5]. In this paper we study how the implementation performance depends on the number of bucket levels used. In particular we are interested in the best number of levels to use in practice. Part of this work was done while the author was at Computer Science Department, Stanford University, and supported in part by NSF Grant CCR9307045. 1 Introduction The shortest paths problem is a fundamental network optimization problem. Algorithms for this problem have been studied for a long time. (See e.g. [2, 7, 8, 10, 14, 15, 16].) An important special case of the problem occurs when no arc length is negative. In this case, implementations of Dijkstra's algorithm [8] achieve the best time bounds. An implementation of [11] runs in O(m+ n log n) time. (Here n and m denote the number of nodes and arcs in the network, respectively.) An improved time bound of O(m+n log n= log log n) [12] ca...
Regular Algebra Applied to Language Problems
, 2004
"... Many functions on contextfree languages can be expressed in the form of the least xed point of a function whose de nition mimics the grammar of the given language. Examples include the function returning the length of the shortest word in a language, and the function returning the smallest num ..."
Abstract

Cited by 4 (1 self)
 Add to MetaCart
Many functions on contextfree languages can be expressed in the form of the least xed point of a function whose de nition mimics the grammar of the given language. Examples include the function returning the length of the shortest word in a language, and the function returning the smallest number of edit operations required to transform a given word into a word in a language.
A Practical Shortest Path Algorithm with Linear Expected Time
 SUBMITTED TO SIAM J. ON COMPUTING
, 2001
"... We present an improvement of the multilevel bucket shortest path algorithm of Denardo and Fox [9] and justify this improvement, both theoretically and experimentally. We prove that if the input arc lengths come from a natural probability distribution, the new algorithm runs in linear average time ..."
Abstract

Cited by 3 (1 self)
 Add to MetaCart
We present an improvement of the multilevel bucket shortest path algorithm of Denardo and Fox [9] and justify this improvement, both theoretically and experimentally. We prove that if the input arc lengths come from a natural probability distribution, the new algorithm runs in linear average time while the original algorithm does not. We also describe an implementation of the new algorithm. Our experimental data suggests that the new algorithm is preferable to the original one in practice. Furthermore, for integral arc lengths that fit into a word of today's computers, the performance is close to that of breadthfirst search, suggesting limitations on further practical improvements.