Dissertation•
Efficient graph algorithms for sequential and parallel computers
01 Feb 1987-
TL;DR: This thesis proves lower bounds on the parallel complexity of the maximal independent set problem and the problem of 2-coloring a rooted tree, and introduces a frame work that allows the generalization of the maximum flow techniques to the minimum-cost flow problem.
Abstract: In this thesis we study graph algorithms, both in sequential and parallel contexts. In the following outline of the thesis, algorithms complexities are stated in terms of the number of vertices n, the number of edges m, the largest absolute value of capacities U, and the largest value of costs C. In Chapter 1 we introduce a new approach to the maximum flow problem that leads to better algorithms for the problem. These algorithms include an O(nmlog(n /m)) time sequential algorithm, an O(n logn) time parallel algorithm that uses O(n) processors and O(m) memory, and both synchronous and asynchronous distributed algorithms. Chapter 2 is devoted to the minimum cost flow problem, which is a generalization of the maximum flow problem. We introduce a frame work that allows the generalization of the maximum flow techniques to the minimum-cost flow problem. We exhibit O(nmlog(n)log(nC)), O(n m log(nC)), and O(nnnlog(nC)) time sequential algorithms as well as parallel and distributed algorithms. In Chapter 3 we address implementation of parallel algorithms through a case-study implementation of a parallel maximum flow algorithm. Parallel prefix operations play an important role in our implementation. We present experimental results achieved by the implementation. Parallel symmetry-breaking techniques are the main topic of Chapter 4. We give an O(lg*n) algorithm for 3-coloring a rooted tree. This algorithm is used to improve several parallel algorithms, including algorithms for +1-coloring and finding maximal independent set in constant-degree graphs, 5-coloring planar graphs, and finding a maximal matching in planar graphs. We also prove lower bounds on the parallel complexity of the maximal independent set problem and the problem of 2-coloring a rooted tree.
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TL;DR: An alternative method based on the preflow concept of Karzanov, which runs as fast as any other known method on dense graphs, achieving an O(n) time bound on an n-vertex graph and faster on graphs of moderate density.
Abstract: All previously known efficient maximum-flow algorithms work by finding augmenting paths, either one path at a time (as in the original Ford and Fulkerson algorithm) or all shortest-length augmenting paths at once (using the layered network approach of Dinic). An alternative method based on the preflow concept of Karzanov is introduced. A preflow is like a flow, except that the total amount flowing into a vertex is allowed to exceed the total amount flowing out. The method maintains a preflow in the original network and pushes local flow excess toward the sink along what are estimated to be shortest paths. The algorithm and its analysis are simple and intuitive, yet the algorithm runs as fast as any other known method on dense graphs, achieving an O(n3) time bound on an n-vertex graph. By incorporating the dynamic tree data structure of Sleator and Tarjan, we obtain a version of the algorithm running in O(nm log(n2/m)) time on an n-vertex, m-edge graph. This is as fast as any known method for any graph density and faster on graphs of moderate density. The algorithm also admits efficient distributed and parallel implementations. A parallel implementation running in O(n2log n) time using n processors and O(m) space is obtained. This time bound matches that of the Shiloach-Vishkin algorithm, which also uses n processors but requires O(n2) space.
1,700 citations
Cites methods from "Efficient graph algorithms for sequ..."
...A third method giving an O(n3) bound is the wave method, described in [ 15 ] and [ 171....
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01 Nov 1986
TL;DR: By incorporating the dynamic tree data structure of Sleator and Tarjan, a version of the algorithm running in O(nm log(n'/m)) time on an n-vertex, m-edge graph is obtained, as fast as any known method for any graph density and faster on graphs of moderate density.
Abstract: All previously known efftcient maximum-flow algorithms work by finding augmenting paths, either one path at a time (as in the original Ford and Fulkerson algorithm) or all shortest-length augmenting paths at once (using the layered network approach of Dinic). An alternative method based on the preflow concept of Karzanov is introduced. A preflow is like a flow, except that the total amount flowing into a vertex is allowed to exceed the total amount flowing out. The method maintains a preflow in the original network and pushes local flow excess toward the sink along what are estimated to be shortest paths. The algorithm and its analysis are simple and intuitive, yet the algorithm runs as fast as any other known method on dense. graphs, achieving an O(n)) time bound on an n-vertex graph. By incorporating the dynamic tree data structure of Sleator and Tarjan, we obtain a version of the algorithm running in O(nm log(n'/m)) time on an n-vertex, m-edge graph. This is as fast as any known method for any graph density and faster on graphs of moderate density. The algorithm also admits efticient distributed and parallel implementations. A parallel implementation running in O(n'log n) time using n processors and O(m) space is obtained. This time bound matches that of the Shiloach-Vishkin
1,374 citations
Cites methods from "Efficient graph algorithms for sequ..."
...A third method giving an O(n3) bound is the wave method, described in [15] and [ 171....
[...]
Book•
01 Jan 1985TL;DR: Kurskod av teknisk-naturvetenskapliga fakultetsnämnden Kursplan giltig från: 2012, vecka 10 Ansvarig enhet: Inst för datavetenskap SCB-ämnesrubrik: Informatik/Dataoch systemvetenskapskap Huvudområden och successiv fördjupning.
Abstract: Kurskod: 5DV050 Inrättad: 2008-03-31 Inrättad av: teknisk-naturvetenskapliga fakultetsnämnden Reviderad: 2012-02-29 Reviderad av: teknisk-naturvetenskapliga fakultetsnämnden Kursplan giltig från: 2012, vecka 10 Ansvarig enhet: Inst för datavetenskap SCB-ämnesrubrik: Informatik/Dataoch systemvetenskap Huvudområden och successiv fördjupning: Beräkningsteknik: Avancerad nivå, har endast kurs/er på grundnivå som förkunskapskrav (A1N) , Datavetenskap: Avancerad nivå, har endast kurs/er på grundnivå som förkunskapskrav (A1N) Betygsskala: För denna kurs ges betygen 5 Med beröm godkänd, 4 Icke utan beröm godkänd, 3 Godkänd, VG Väl godkänd, G Godkänd, U Underkänd Utbildningsnivå: Avancerad nivå
712 citations
TL;DR: A massively parallelizable algorithm for the classical assignment problem was proposed in this article, where unassigned persons bid simultaneously for objects thereby raising their prices. Once all bids are in, objects are awarded to the highest bidder.
Abstract: We propose a massively parallelizable algorithm for the classical assignment problem. The algorithm operates like an auction whereby unassigned persons bid simultaneously for objects thereby raising their prices. Once all bids are in, objects are awarded to the highest bidder. The algorithm can also be interpreted as a Jacobi — like relaxation method for solving a dual problem. Its (sequential) worst — case complexity, for a particular implementation that uses scaling, is O(NAlog(NC)), where N is the number of persons, A is the number of pairs of persons and objects that can be assigned to each other, and C is the maximum absolute object value. Computational results show that, for large problems, the algorithm is competitive with existing methods even without the benefit of parallelism. When executed on a parallel machine, the algorithm exhibits substantial speedup.
649 citations
Cites background from "Efficient graph algorithms for sequ..."
...It was first analyzed in the context of the more general min-cost flow problem in [36] where polynomial complexity results were given that were more fully established in [33], [34]....
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