TY - JOUR

T1 - On the Exponent of the All Pairs Shortest Path Problem

AU - Alon, Noga

AU - Galil, Zvi

AU - Margalit, Oded

N1 - Funding Information:
* Research supported by a United States Israel BSF Grant. -Work partially supported by NSF Grants CCR-8814977 and CCR-9014605. Mainly affiliated with Columbia University, New York.

PY - 1997/1/1

Y1 - 1997/1/1

N2 - The upper bound on the exponent, ω, of matrix multiplication over a ring that was three in 1968 has decreased several times and since 1986 it has been 2.376. On the other hand, the exponent of the algorithms known for the all pairs shortest path problem has stayed at three all these years even for the very special case of directed graphs with uniform edge lengths. In this paper we give an algorithm of time O(nv log3 n), v = (3 + ω )/2, for the case of edge lengths in {-1, 0, 1}. Thus, for the current known bound on ω, we get a bound on the exponent, v < 2.688. In case of integer edge lengths with absolute value bounded above by M, the time bound is O((Mn)v log3 n) and the exponent is less than 3 for M = O(nα), for α < 0.116 and the current bound on ω.

AB - The upper bound on the exponent, ω, of matrix multiplication over a ring that was three in 1968 has decreased several times and since 1986 it has been 2.376. On the other hand, the exponent of the algorithms known for the all pairs shortest path problem has stayed at three all these years even for the very special case of directed graphs with uniform edge lengths. In this paper we give an algorithm of time O(nv log3 n), v = (3 + ω )/2, for the case of edge lengths in {-1, 0, 1}. Thus, for the current known bound on ω, we get a bound on the exponent, v < 2.688. In case of integer edge lengths with absolute value bounded above by M, the time bound is O((Mn)v log3 n) and the exponent is less than 3 for M = O(nα), for α < 0.116 and the current bound on ω.

UR - http://www.scopus.com/inward/record.url?scp=0031125393&partnerID=8YFLogxK

U2 - 10.1006/jcss.1997.1388

DO - 10.1006/jcss.1997.1388

M3 - Article

AN - SCOPUS:0031125393

VL - 54

SP - 255

EP - 262

JO - Journal of Computer and System Sciences

JF - Journal of Computer and System Sciences

SN - 0022-0000

IS - 2

ER -