OBDDs and (Almost) $k$-wise Independent Random Variables

TL;DR

This paper introduces a randomized OBDD-based maximal matching algorithm with improved efficiency and investigates the OBDD complexity of (almost) $k$-wise independent variables, providing new constructions and bounds.

Contribution

It presents the first randomized OBDD-based maximal matching algorithm with logarithmic complexity and analyzes the OBDD size for (almost) $k$-wise independent variables, including new constructions and lower bounds.

Findings

The maximal matching algorithm runs in expected $O( ext{log}^3 |V|)$ functional operations.

Constructed a $O(n)$ size OBDD for 3-wise independent variables.

Proved a lower bound of $2^{ ext{Omega}(n)}$ for $k extgreater= 4$ on OBDD size.

Abstract

OBDD-based graph algorithms deal with the characteristic function of the edge set E of a graph $G = (V,E)$ which is represented by an OBDD and solve optimization problems by mainly using functional operations. We present an OBDD-based algorithm which uses randomization for the first time. In particular, we give a maximal matching algorithm with $O(\log^3 \vert V \vert)$ functional operations in expectation. This algorithm may be of independent interest. The experimental evaluation shows that this algorithm outperforms known OBDD-based algorithms for the maximal matching problem. In order to use randomization, we investigate the OBDD complexity of $2^n$ (almost) $k$-wise independent binary random variables. We give a OBDD construction of size $O(n)$ for $3$-wise independent random variables and show a lower bound of $2^{\Omega(n)}$ on the OBDD size for $k \geq 4$. The best known lower bound was $\Omega(2^n/n)$ for $k \approx \log n$ due to Kabanets. We also give a very simple construction of $2^n$ $(\varepsilon, k)$-wise independent binary random variables by constructing a random OBDD of width $O(n k^2/\varepsilon)$.