Representation of graphs by OBDDs

Abstract

Recently, it has been shown in a series of works that the representation of graphs by Ordered Binary Decision Diagrams (OBDDs) often leads to good algorithmic behavior. However, the question for which graph classes an OBDD representation is advantageous, has not been investigated, yet. In this paper, the space requirements for the OBDD representation of certain graph classes, specifically cographs, several types of graphs with few P 4 s, unit interval graphs, interval graphs and bipartite graphs are investigated. Upper and lower bounds are proven for all these graph classes and it is shown that in most (but not all) cases a representation of the graphs by OBDDs is advantageous with respect to space requirements.

OBDD Representation of Intersection Graphs

“…Indeed, Meer and Rautenbach [19] investigate compact OBDD representations of graphs with bounded tree-and clique-width by using a label of length more than log N for each vertex. They show that the OBDD size of cographs can be improved from O(N log N) [20] to O(N), if we use vertex labels of length c · log N for some constant c > 1. However, the running time of an OBDD-based algorithm may be larger for longer vertex labels, since the worst-case OBDD size is exponentially larger in the number of bits.…”

“…As discussed in [4], [20], it is known to be important to keep the number of bits as low as possible by the following reason. Since the running time of one OBDD operation depends on the size of OBDD on which the operation is performed, the size of intermediate OBDDs during the computation of an OBDD-based algorithm should be small.…”

“…In the previous works, Nunkesser and Woelfel [20], and Gillé [10], [11] have used a log N -bit label for each vertex, which is the minimal number of bits for encoding every vertices. As discussed in [4], [20], it is known to be important to keep the number of bits as low as possible by the following reason.…”

“…For that reason, compact OBDD representations of graphs have been considered. Nunkesser and Woelfel [20] investigate the worst-case space complexities of the OBDD representations where N and M are the number of vertices and edges of a graph, respectively. Recently, Gillé [10], [11] has been improved the results as follows:…”

“…Since the adjacency matrix of a graph can be considered as a Boolean function, a graph can be implicitly represented by an OBDD that represents the adjacency function. The OBDD representation of graphs has been considered as a promising implicit representation of graphs, since it realizes compact representations as well as efficient algorithms, based on existing OBDD operations (See [20] for survey). To realize OBDD-based graph algorithms that have good running times, the size of the OBDD representation of the input graph should be small, since the running time of one OBDD operation depends on the size of OBDD on which the operation is performed.…”