Given an undirected weighted graph with n vertices and m edges, we give the first deterministic m1+o(1)-time algorithm for constructing the cactus representation of all global minimum cuts. This improves the current n2+o(1)-time state-of-the-art deterministic algorithm, which can be obtained by combining ideas implicitly from three papers [22, 27, 12]. The known explicitly stated deterministic algorithm has a runtime of Õ(mn) [9, 34]. Using our technique, we can even speed up the fastest randomized algorithm of [23] whose running time is at least Ω(m log4 n) to O(m log3 n).
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Near-Optimal Derandomization of Medium-Width Branching Programs
We give a deterministic white-box algorithm to estimate the expectation of a read-once branching program of length n and width w in space Õ(logn+√logn·logw). In particular, we obtain an almost optimal space Õ(logn) derandomization of programs up to width w=2√logn. Previously, the best known space complexity for this problem was O(min{logn· logw,log3/2n+√logn· logw}) via the classic algorithms of Savitch (JCSS 1970) and Saks and Zhou (JCSS 1999), which only achieve space Õ(logn) for w=polylog(n). We prove this result by showing that a variant of the Saks-Zhou algorithm developed by Cohen, Doron, and Sberlo (ECCC 2022) still works without executing one of the steps in the algorithm, the so-called random shift step. This allows us to extend their algorithm from computing the nth power of a w× w stochastic matrix to multiplying n distinct w× w stochastic matrices with no degradation in space consumption. In the regime where w≥ n, we also show that our approach can achieve parameters matching those of the original Saks-Zhou algorithm (with no loglog factors). Finally, we show that for w≤ 2√logn, an algorithm even simpler than our algorithm and that of Saks and Zhou achieves space O(log3/2 n).
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- Award ID(s):
- 2310818
- PAR ID:
- 10494250
- Publisher / Repository:
- 55th Annual ACM Symposium on Theory of Computing, STOC 2023
- Date Published:
- Journal Name:
- 55th Annual ACM Symposium on Theory of Computing, STOC 2023
- Page Range / eLocation ID:
- 23-34
- Format(s):
- Medium: X
- Location:
- Orlando, Florida
- Sponsoring Org:
- National Science Foundation
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