Affiliation:
1. University of Illinois at Urbana-Champaign, Urbana, IL
2. MIT, Cambridge, MA
Abstract
We show how to solve all-pairs shortest paths on
n
nodes in
deterministic
n
3>
/2>
Ω
( √ log
n
) time, and how to count the pairs of orthogonal vectors among
n
0−1 vectors in
d
=
c
log
n
dimensions in
deterministic
n
2−1/
O
(log
c
)
time. These running times essentially match the best known randomized algorithms of Williams [46] and Abboud, Williams, and Yu [8], respectively, and the ability to count was open even for randomized algorithms. By reductions, these two results yield faster deterministic algorithms for many other problems. Our techniques can also be used to deterministically count
k
-satisfiability (
k
-SAT) assignments on
n
variable formulas in 2
n
-
n
/
O
(
k
)
time, roughly matching the best known running times for detecting satisfiability and resolving an open problem of Santhanam [24].
A key to our constructions is an efficient way to deterministically simulate certain probabilistic polynomials critical to the algorithms of prior work, carefully applying small-biased sets and modulus-amplifying polynomials.
Funder
David Morgenthaler II Faculty Fellowship
Sloan Fellowship
NSF
Publisher
Association for Computing Machinery (ACM)
Subject
Mathematics (miscellaneous)
Cited by
10 articles.
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