arXiv subjects

Stav Ben-Nun

Publications and source records attributed to Stav Ben-Nun.

2 recordsLinked to original sources

arXiv2020

An $O(\log^{3/2}n)$ Parallel Time Population Protocol for Majority with $O(\log n)$ States

In population protocols, the underlying distributed network consists of $n$ nodes (or agents), denoted by $V$, and a scheduler that continuously selects uniformly random pairs of nodes to interact. When two nodes interact, their states are updated by applying a state transition function that depends only on the states of the two nodes prior to the interaction. The efficiency of a population protocol is measured in terms of both time (which is the number of interactions until the nodes collectively have a valid output) and the number of possible states of nodes used by the protocol. By convention, we consider the parallel time cost, which is the time divided by $n$. In this paper we consider the majority problem, where each node receives as input a color that is either black or white, and the goal is to have all of the nodes output the color that is the majority of the input colors. We design a population protocol that solves the majority problem in $O(\log^{3/2}n)$ parallel time, both with high probability and in expectation, while using $O(\log n)$ states. Our protocol improves on a recent protocol of Berenbrink et al. that runs in $O(\log^{5/3}n)$ parallel time, both with high probability and in expectation, using $O(\log n)$ states.

cs.DC↗

arXiv2020

Time-Space Tradeoffs for Finding a Long Common Substring

We consider the problem of finding, given two documents of total length $n$, a longest string occurring as a substring of both documents. This problem, known as the Longest Common Substring (LCS) problem, has a classic $O(n)$-time solution dating back to the discovery of suffix trees (Weiner, 1973) and their efficient construction for integer alphabets (Farach-Colton, 1997). However, these solutions require $Θ(n)$ space, which is prohibitive in many applications. To address this issue, Starikovskaya and Vildhøj (CPM 2013) showed that for $n^{2/3} \le s \le n^{1-o(1)}$, the LCS problem can be solved in $O(s)$ space and $O(\frac{n^2}{s})$ time. Kociumaka et al. (ESA 2014) generalized this tradeoff to $1 \leq s \leq n$, thus providing a smooth time-space tradeoff from constant to linear space. In this paper, we obtain a significant speed-up for instances where the length $L$ of the sought LCS is large. For $1 \leq s \leq n$, we show that the LCS problem can be solved in $O(s)$ space and $\tilde{O}(\frac{n^2}{L\cdot s}+n)$ time. The result is based on techniques originating from the LCS with Mismatches problem (Flouri et al., 2015; Charalampopoulos et al., CPM 2018), on space-efficient locally consistent parsing (Birenzwige et al., SODA 2020), and on the structure of maximal repetitions (runs) in the input documents.

cs.DS↗