arXiv subjects

S. Cliff Liu

Publications and source records attributed to S. Cliff Liu.

8 recordsLinked to original sources

arXiv2025

Connected Components in Linear Work and Near-Optimal Time

Computing the connected components of a graph is a fundamental problem in algorithmic graph theory. A major question in this area is whether we can compute connected components in $o(\log n)$ parallel time. Recent works showed an affirmative answer in the Massively Parallel Computation (MPC) model for a wide class of graphs. Specifically, Behnezhad et al. (FOCS'19) showed that connected components can be computed in $O(\log d + \log \log n)$ rounds in the MPC model. More recently, Liu et al. (SPAA'20) showed that the same result can be achieved in the standard PRAM model but their result incurs $Θ((m+n) \cdot (\log d + \log \log n))$ work which is sub-optimal. In this paper, we show that for graphs that contain \emph{well-connected} components, we can compute connected components on a PRAM in sub-logarithmic parallel time with \emph{optimal}, i.e., $O(m+n)$ total work. Specifically, our algorithm achieves $O(\log(1/λ) + \log \log n)$ parallel time with high probability, where $λ$ is the minimum spectral gap of any connected component in the input graph. The algorithm requires no prior knowledge on $λ$. Additionally, based on the \textsc{2-Cycle} Conjecture we provide a time lower bound of $Ω(\log(1/λ))$ for solving connected components on a PRAM with $O(m+n)$ total memory when $λ\le (1/\log n)^c$, giving conditional optimality to the running time of our algorithm as a parameter of $λ$.

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arXiv2023

Space-Efficient Interior Point Method, with applications to Linear Programming and Maximum Weight Bipartite Matching

We study the problem of solving linear program in the streaming model. Given a constraint matrix $A\in \mathbb{R}^{m\times n}$ and vectors $b\in \mathbb{R}^m, c\in \mathbb{R}^n$, we develop a space-efficient interior point method that optimizes solely on the dual program. To this end, we obtain efficient algorithms for various different problems: * For general linear programs, we can solve them in $\widetilde O(\sqrt n\log(1/ε))$ passes and $\widetilde O(n^2)$ space for an $ε$-approximate solution. To the best of our knowledge, this is the most efficient LP solver in streaming with no polynomial dependence on $m$ for both space and passes. * For bipartite graphs, we can solve the minimum vertex cover and maximum weight matching problem in $\widetilde O(\sqrt{m})$ passes and $\widetilde O(n)$ space. In addition to our space-efficient IPM, we also give algorithms for solving SDD systems and isolation lemma in $\widetilde O(n)$ spaces, which are the cornerstones for our graph results.

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arXiv2021

Connected Components on a PRAM in Log Diameter Time

We present an $O(\log d + \log\log_{m/n} n)$-time randomized PRAM algorithm for computing the connected components of an $n$-vertex, $m$-edge undirected graph with maximum component diameter $d$. The algorithm runs on an ARBITRARY CRCW (concurrent-read, concurrent-write with arbitrary write resolution) PRAM using $O(m)$ processors. The time bound holds with good probability. Our algorithm is based on the breakthrough results of Andoni et al. [FOCS'18] and Behnezhad et al. [FOCS'19]. Their algorithms run on the more powerful MPC model and rely on sorting and computing prefix sums in $O(1)$ time, tasks that take $Ω(\log n / \log\log n)$ time on a CRCW PRAM with $\text{poly}(n)$ processors. Our simpler algorithm uses limited-collision hashing and does not sort or do prefix sums. It matches the time and space bounds of the algorithm of Behnezhad et al., who improved the time bound of Andoni et al. It is widely believed that the larger private memory per processor and unbounded local computation of the MPC model admit algorithms faster than that on a PRAM. Our result suggests that such additional power might not be necessary, at least for fundamental graph problems like connected components and spanning forest.

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arXiv2020

Chain, Generalization of Covering Code, and Deterministic Algorithm for k-SAT

We present the current fastest deterministic algorithm for $k$-SAT, improving the upper bound $(2-2/k)^{n + o(n)}$ dues to Moser and Scheder [STOC'11]. The algorithm combines a branching algorithm with the derandomized local search, whose analysis relies on a special sequence of clauses called chain, and a generalization of covering code based on linear programming. We also provide a more ingenious branching algorithm for $3$-SAT to establish the upper bound $1.32793^n$, improved from $1.3303^n$.

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arXiv2020

Simple Concurrent Labeling Algorithms for Connected Components

We study a class of simple algorithms for concurrently computing the connected components of an $n$-vertex, $m$-edge graph. Our algorithms are easy to implement in either the COMBINING CRCW PRAM or the MPC computing model. For two related algorithms in this class, we obtain $Θ(\lg n)$ step and $Θ(m \lg n)$ work bounds. For two others, we obtain $O(\lg^2 n)$ step and $O(m \lg^2 n)$ work bounds, which are tight for one of them. All our algorithms are simpler than related algorithms in the literature. We also point out some gaps and errors in the analysis of previous algorithms. Our results show that even a basic problem like connected components still has secrets to reveal.

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arXiv2020

Simpler Partial Derandomization of PPSZ for $k$-SAT

We give a simpler derandomization of the best known $k$-SAT algorithm PPSZ [FOCS'97, JACM'05] for $k$-SAT with \emph{sub-exponential} number of solutions. The existing derandomization uses a complicated construction of small sample space, while we only use \emph{hashing}. Our algorithm and theorem also have a nice byproduct: It outperforms the current fastest deterministic $k$-SAT algorithm when the formula has \emph{moderately exponential} number of solutions.

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arXiv2019

Lower Bounds for Small Ramsey Numbers on Hypergraphs

The Ramsey number $r_k(p, q)$ is the smallest integer $N$ that satisfies for every red-blue coloring on $k$-subsets of $[N]$, there exist $p$ integers such that any $k$-subset of them is red, or $q$ integers such that any $k$-subset of them is blue. In this paper, we study the lower bounds for small Ramsey numbers on hypergraphs by constructing counter-examples and recurrence relations. We present a new algorithm to prove lower bounds for $r_k(k+1, k+1)$. In particular, our algorithm is able to prove $r_5(6,6) \ge 72$, where there is only trivial lower bound on $5$-hypergraphs before this work. We also provide several recurrence relations to calculate lower bounds based on lower bound values on smaller $p$ and $q$. Combining both of them, we achieve new lower bounds for $r_k(p, q)$ on arbitrary $p$, $q$, and $k \ge 4$.

math.CO↗

arXiv2019

The Curse and Blessing of Not-All-Equal in k-Satisfiability

As a natural variant of the $k$-SAT problem, NAE-$k$-SAT additionally requires the literals in each clause to take not-all-equal (NAE) truth values. In this paper, we study the worst-case time complexities of solving NAE-$k$-SAT and MAX-NAE-$k$-SAT approximation, as functions of $k$, the number of variables $n$, and the performance ratio $δ$. The latter problem asks for a solution of at least $δ$ times the optimal. Our main results include: (1) A deterministic algorithm for NAE-$k$-SAT that is faster than the best deterministic algorithm for $k$-SAT on all $k \ge 3$. Previously, no NAE-$k$-SAT algorithm is known to be faster than $k$-SAT algorithms. For $k = 3$, we achieve an upper bound of $1.326^n$. The corresponding bound for $3$-SAT is $1.328^n$. (2) A randomized algorithm for MAX-NAE-$k$-SAT approximation, with upper bound $(2 - ε_k(δ))^n$ where $ε_k(δ) > 0$ only depends on $k$ and $δ$. Previously, no upper bound better than the trivial $2^n$ is known for MAX-NAE-$k$-SAT approximation on $k \ge 4$. For $δ= 0.9$ and $k = 4$, we achieve an upper bound of $1.947^n$. (3) A deterministic algorithm for MAX-NAE-$k$-SAT approximation. For $δ= 0.9$ and $k = 3$, we achieve an upper bound of $1.698^n$, which is better than the upper bound $1.731^n$ of the exact algorithm for MAX-NAE-$3$-SAT. Our finding sheds new light on the following question: Is NAE-$k$-SAT easier than $k$-SAT? The answer might be affirmative at least on solving the problems exactly and deterministically, while approximately solving MAX-NAE-$k$-SAT might be harder than MAX-$k$-SAT on $k \ge 4$.

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