TY - GEN

T1 - Efficient parallel algorithms for testing connectivity and finding disjoint s-t paths in graphs

AU - Khuller, Samir

AU - Schieber, Baruch

PY - 1989/11/1

Y1 - 1989/11/1

N2 - An efficient parallel algorithm for testing whether a graph G is k-vertex connected, for any fixed k, is presented. The algorithm runs in O(log n) time and uses nC(n, m) processors on a concurrent-read, concurrent-write parallel random-access machine (CRCW PRAM), where n and m are the number of vertices and edges of G and C(n, m) is the number of processors required to compute the connected components of G in logarithmic time. An optimal speedup algorithm for computing connected components would induce an optimal speedup algorithm for testing k-vertex connectivity, for any k > 4. To develop the algorithm, an efficient parallel algorithm is designed for the following disjoint s-t paths problem: Given a graph G and two specified vertices s and t, find k-vertex disjoint paths between s and t, if they exist. If no such paths exist, find a set of at most k - 1 vertices whose removal disconnects s and t. The parallel algorithm for this problem runs in O(log n) time using C(n, m) processors. It is shown how to modify the algorithm to find k-edge disjoint paths, if they exist. This yields an efficient parallel algorithm for testing whether a graph G is k-edge connected, for any fixed k. The algorithm runs in O(log n) time and uses nC (n, n) processors on a CRCW PRAM. Again, an optimal speedup algorithm for computing connected components would induce an optimal speedup algorithm for testing k-edge connectivity. Further applications of the disjoint s-t paths algorithm are described.

AB - An efficient parallel algorithm for testing whether a graph G is k-vertex connected, for any fixed k, is presented. The algorithm runs in O(log n) time and uses nC(n, m) processors on a concurrent-read, concurrent-write parallel random-access machine (CRCW PRAM), where n and m are the number of vertices and edges of G and C(n, m) is the number of processors required to compute the connected components of G in logarithmic time. An optimal speedup algorithm for computing connected components would induce an optimal speedup algorithm for testing k-vertex connectivity, for any k > 4. To develop the algorithm, an efficient parallel algorithm is designed for the following disjoint s-t paths problem: Given a graph G and two specified vertices s and t, find k-vertex disjoint paths between s and t, if they exist. If no such paths exist, find a set of at most k - 1 vertices whose removal disconnects s and t. The parallel algorithm for this problem runs in O(log n) time using C(n, m) processors. It is shown how to modify the algorithm to find k-edge disjoint paths, if they exist. This yields an efficient parallel algorithm for testing whether a graph G is k-edge connected, for any fixed k. The algorithm runs in O(log n) time and uses nC (n, n) processors on a CRCW PRAM. Again, an optimal speedup algorithm for computing connected components would induce an optimal speedup algorithm for testing k-edge connectivity. Further applications of the disjoint s-t paths algorithm are described.

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M3 - Conference contribution

SN - 0818619821

T3 - Annual Symposium on Foundations of Computer Science (Proceedings)

SP - 288

EP - 293

BT - Annual Symposium on Foundations of Computer Science (Proceedings)

PB - Publ by IEEE

T2 - 30th Annual Symposium on Foundations of Computer Science

Y2 - 30 October 1989 through 1 November 1989

ER -