Randomized OBDD-based graph algorithms

Implicit graph algorithms deal with the characteristic function chi(E) of the edge set E of a graph G = (V, E). Encoding the nodes by binary vectors, chi(E) can be represented by an Ordered Binary Decision Diagram (OBDD) which is a well known data structure for Boolean functions. OBDD-based graph algorithms solve graph optimization problems by mainly using functional operations and are a heuristic approach to cope with massive graphs. These algorithms heavily rely on a compact representation of the underlying Boolean functions which is why all previously known OBDD-based algorithms are deterministic since random functions are not compressible in general. Here, the first randomized OBDDbased algorithms are presented where random functions with limited independence are used to overcome the large representation size. On the theoretical part, we investigate the size of OBDD5 representing k-wise independent random functions. In addition, we give a construction of almost k-wise independent random functions by means of a random OBDD generation and show an almost matching lower bound of the OBDD size representing such random functions. On the algorithmic part, randomization often facilitates the design of simple algorithms which in the context of OBDD-based algorithms means a small number of functional operations and as few input variables of the used Boolean functions as possible. We present two randomized implicit algorithms: A randomized minimum spanning tree algorithm and a maximal matching algorithm. The matching algorithm needs O(log(3) vertical bar V vertical bar) functional operations in expectation and uses functions with at most 3 log vertical bar V vertical bar variables which is both better than the best known deterministic algorithms w.r.t. functional operations and variables. The algorithm may be of independent interest. The experimental evaluation of the algorithms shows that they outperform known OBDD-based algorithms for the maximal matching problem and the minimum spanning tree problem. (C) 2016 Elsevier B.V. All rights reserved.