TY - GEN

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 - 1991/12/1

Y1 - 1991/12/1

N2 - The upper bound on the exponent, ω, of matrix multiplication over a ring, which 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. An algorithm is given of time O(nν log 3 n), ν = (3 + ω)/2, for the case of edge lengths in {-1,0,1}. Thus, for the current known bound on ω, a bound on the exponent, ν < 2.688, is obtained. In case of integer edge lengths with absolute value bounded above by M, the time bound is O((Mn)ν log 3 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, which 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. An algorithm is given of time O(nν log 3 n), ν = (3 + ω)/2, for the case of edge lengths in {-1,0,1}. Thus, for the current known bound on ω, a bound on the exponent, ν < 2.688, is obtained. In case of integer edge lengths with absolute value bounded above by M, the time bound is O((Mn)ν log 3 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=0026375767&partnerID=8YFLogxK

M3 - Conference contribution

AN - SCOPUS:0026375767

SN - 0818624450

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

SP - 569

EP - 575

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

PB - Publ by IEEE

T2 - Proceedings of the 32nd Annual Symposium on Foundations of Computer Science

Y2 - 1 October 1991 through 4 October 1991

ER -