A Faster Implementation of a Parallel Tree Contraction Scheme and Its Application on Distance-Hereditary Graphs

We consider a parallel tree contraction scheme which in each contraction phase removes leaves and nodes in the maximal chains. Let T(n) and P(n) denote the time and processor complexity required to compute the all nearest smaller values (ANSV) and the minimum of n values for input elements drawn from the integer domain 1?n]. In this paper, we give a faster implementation of the tree contraction scheme which takes O(logn·T(n)) time using P(n) processors on an arbitrary CRCW PRAM. The current best results of T(n) and P(n) are O(logloglogn) and O(n/logloglogn), respectively. To our knowledge, the previously best known implementation needs O(log2n) time using O(n/logn) processors on an EREW PRAM. The faster parallel implementation of the tree contraction scheme may be of interests by itself. We then show the above scheme can be utilized to solve problems on distance-hereditary graphs. We provide a data structure to represent a connected distance-hereditary graph in the form of a rooted tree. By applying the above tree contraction scheme on our data structure together with graph theoretical properties, we solve the problems of finding a minimum connected ?-dominating set and finding a minimum ?-dominating clique on a distance-hereditary graph in O(logn·T(n)) time using O(P(n)+(n+m)/T(n)) processors on an arbitrary CRCW PRAM, where n and m are the number of vertices and edges of the given graph, respectively. The above result implies several other problems related to the minimum ?-dominating clique problem can be solved with the same parallel complexities.

[1]  Tsan-sheng Hsu,et al.  Efficient parallel algorithms on distance-hereditary graphs , 1997, Proceedings of the 1997 International Conference on Parallel Processing (Cat. No.97TB100162).

[2]  M. Golumbic.  Algorithmic graph theory and perfect graphs , 1980 .

[3]  J. Van Leeuwen,et al.  Handbook of theoretical computer science - Part A: Algorithms and complexity; Part B: Formal models and semantics , 1990 .

[4]  Feodor F. Dragan,et al.  Dominating Cliques in Distance-Hereditary Graphs , 1994, SWAT.

[5]  Philip N. Klein,et al.  Efficient parallel algorithms for chordal graphs , 1988, [Proceedings 1988] 29th Annual Symposium on Foundations of Computer Science.

[6]  Richard Cole,et al.  Parallel merge sort , 1988, 27th Annual Symposium on Foundations of Computer Science (sfcs 1986).

[7]  Richard M. Karp,et al.  Parallel Algorithms for Shared-Memory Machines , 1991, Handbook of Theoretical Computer Science, Volume A: Algorithms and Complexity.

[8]  Wojciech Rytter,et al.  Optimal Parallel Algorithm for Dynamic Expression Evaluation and Context-Free Recognition , 1989, Inf. Comput..

[9]  Claude Berge,et al.  Graphs and Hypergraphs , 2021, Clustering.

[10]  Peter L. Hammer,et al.  Completely separable graphs , 1990, Discret. Appl. Math..

[11]  Gary L. Miller,et al.  Parallel Tree Contraction, Part 2: Further Applications , 1991, SIAM J. Comput..

[12]  Feodor F. Dragan,et al.  HT-graphs: centers, connected r-domination and Steiner trees , 1993, Comput. Sci. J. Moldova.

[13]  Elias Dahlhaus.  Optimal (Parallel) Algorithms for the All-to-All Vertices Distance Problem for Certain Graph Classes , 1992, WG.

[14]  Tsan-sheng Hsu,et al.  Efficient Parallel Algorithms on Distance Hereditary Graphs , 1999, Parallel Process. Lett..

[15]  David G. Kirkpatrick,et al.  A Simple Parallel Tree Contraction Algorithm , 1989, J. Algorithms.

[16]  A. Brandstädt,et al.  A linear-time algorithm for connected r-domination and Steiner tree on distance-hereditary graphs , 1998 .

[17]  Richard Cole,et al.  Approximate Parallel Scheduling. II. Applications to Logarithmic-Time Optimal Parallel Graph Algorithms , 1991, Inf. Comput..

[18]  Vijaya Ramachandran,et al.  Parallel Algorithms for Reducible Flow Graphs , 1997, J. Algorithms.

[19]  Hans-Jürgen Bandelt,et al.  Distance-hereditary graphs , 1986, J. Comb. Theory B.

[20]  Moni Naor,et al.  Fast parallel algorithms for chordal graphs , 1987, SIAM J. Comput..

[21]  E. Howorka.  A CHARACTERIZATION OF DISTANCE-HEREDITARY GRAPHS , 1977 .

[22]  Joseph JáJá,et al.  An Introduction to Parallel Algorithms , 1992 .

[23]  Gerard J. Chang,et al.  Labeling algorithms for domination problems in sun-free chordal graphs , 1988, Discret. Appl. Math..

[24]  Gary L. Miller,et al.  Parallel tree contraction and its application , 1985, 26th Annual Symposium on Foundations of Computer Science (sfcs 1985).

[25]  Yossi Matias,et al.  Triply-Logarithmic Parallel Upper and Lower Bounds for Minimum and Range Minima over Small Domains , 1998, J. Algorithms.

[26]  Marina Moscarini,et al.  Distance-Hereditary Graphs, Steiner Trees, and Connected Domination , 1988, SIAM J. Comput..

[27]  Uzi Vishkin,et al.  Optimal Doubly Logarithmic Parallel Algorithms Based on Finding All Nearest Smaller Values , 1993, J. Algorithms.

[28]  S. Teng,et al.  Optimal Tree Contraction in the EREW Model , 1988 .

[29]  Elias Dahlhaus,et al.  Efficient Parallel Recognition Algorithms of Cographs and Distance Hereditary Graphs , 1995, Discret. Appl. Math..

[30]  Xin He,et al.  Binary Tree Algebraic Computation and Parallel Algorithms for Simple Graphs , 1988, J. Algorithms.

[31]  Peter Damaschke,et al.  The Parallel Solution of Domination Problems on Chordal and Strongly Chordal Graphs , 1994, Discret. Appl. Math..