# On the shortest path and minimum spanning tree problems

 dc.contributor.advisor Ramachandran, Vijaya en dc.creator Pettie, Seth en dc.date.accessioned 2008-08-28T21:37:15Z en dc.date.available 2008-08-28T21:37:15Z en dc.date.issued 2003 en dc.description text en dc.description.abstract The shortest path and minimum spanning tree problems are two of the classic textbook problems in combinatorial optimization. They are simple to describe and admit simple polynomial-time algorithms. However, despite years of concerted research effort, the asymptotic complexity of these problems remains unresolved. The main contributions of this dissertation are a number of asymptotically faster algorithms for the minimum spanning tree and shortest path problems. Of equal interest, we provide some clues as to why these problems are so diffcult. In particular, we show why certain modern approaches to the problems are doomed to have super-linear complexity. A sampling of our results are listed below. We emphasize that all of our algorithms work with general graphs, and make no restrictive assumptions on the numerical representation of edge-lengths. A provably optimal deterministic minimum spanning tree algorithm. (We give a constructive proof that the algorithmic complexity of the minimum spanning tree problem is equivalent to its decision-tree complexity.) An all-pairs shortest path algorithm for general graphs running in time O(mn + n2 log log n), where m and n are the number of edges and vertices. This provides the first improvement over approaches based on Dijkstra's algorithm. An all-pairs shortest path algorithm for undirected graphs running in O(mn log α ) time, where α = α (m, n) is the inverse-Ackermann function. A single-source shortest path algorithm running in O(m α +min{n log log r, n log n}) time, where r bounds the ratio of any two edge lengths. For r polynomial in n this is O(m + n log log n), an improvement over Dijkstra's algorithm. An inverse-Ackermann style lower bound for the online minimum spanning tree veri cation problem. This is the rst inverse-Ackermann type lower bound for a comparison-based problem. An Ω (m + n log n) lower bound on any hierarchy-type single-source shortest path algorithm, implying that this type of algorithm cannot improve upon Dijkstra's algorithm. (All of our shortest path algorithms are of the hierarchy type.) The first parallel minimum spanning tree algorithm that is optimal w.r.t. to both time and work. Our algorithm is for the EREW PRAM model. A parallel, expected linear-work minimum spanning tree algorithm using only a polylogarithmic number of random bits. An O(mn log α ) bound on the comparison-addition complexity of all-pairs shortest paths. This is within a tiny log α factor of optimal when m = O(n). dc.description.department Computer Science dc.format.medium electronic en dc.identifier b57185980 en dc.identifier.oclc 56889714 en dc.identifier.proqst 3118057 en dc.identifier.uri http://hdl.handle.net/2152/859 en dc.language.iso eng en dc.rights Copyright is held by the author. Presentation of this material on the Libraries' web site by University Libraries, The University of Texas at Austin was made possible under a limited license grant from the author who has retained all copyrights in the works. en dc.subject.lcsh Spanning trees (Graph theory) en dc.subject.lcsh Paths and cycles (Graph theory) en dc.subject.lcsh Combinatorial optimization en dc.title On the shortest path and minimum spanning tree problems en dc.type.genre Thesis en thesis.degree.department Computer Sciences en thesis.degree.discipline Computer Sciences en thesis.degree.grantor The University of Texas at Austin en thesis.degree.level Doctoral en thesis.degree.name Doctor of Philosophy en

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