On the OBDD Size for Graphs of Bounded Tree- and Clique-Width

Meer, Klaus; Rautenbach, Dieter

doi:10.1007/11847250_7

Cited by 5 publications

(6 citation statements)

References 14 publications

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“…OBDD's are a generalization of union trees in the setting of boolean functions. In [21] authors gave a O(n k 2 log k )-sized, O(log n)-depth OBDD with an encoding size of O(log k log n)-bits. This scheme is based on a bottom up tree decomposition approach originally introduced in [22].…”

Section: Previous and Related Workmentioning

confidence: 99%

An adjacency labeling scheme based on a tree-decomposition

Banerjee¹

2022

Preprint

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In this paper we look at the problem of adjacency labeling of graphs. Given a family of undirected graphs the problem is to determine an encoding-decoding scheme for each member of the family such that we can decode the adjacency information of any pair of vertices only from their encoded labels. Further, we want the length of each label to be short (logarithmic in n, the number of vertices) and the encoding-decoding scheme to be computationally efficient. We proposed a simple tree-decomposition based encoding scheme and used it give an adjacency labeling of size O(k log k log n)-bits. Here k is the clique-width of the graph family. We also extend the result to a certain family of k-probe graphs.

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Section: Previous and Related Workmentioning

confidence: 99%

An adjacency labeling scheme based on a tree-decomposition

Banerjee¹

2022

Preprint

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“…It should be noticed that increasing the length of labels may be valid for compact OBDD representations. 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.…”

Section: Length Of Labelsmentioning

confidence: 99%

OBDD Representation of Intersection Graphs

Takaoka

Tayu

Ueno

2015

IEICE Trans. Inf. & Syst.

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Asahi TAKAOKA†a) , Student Member, Satoshi TAYU †b) , Member, and Shuichi UENO †c) , Fellow SUMMARY Ordered Binary Decision Diagrams (OBDDs for short) are popular dynamic data structures for Boolean functions. In some modern applications, we have to handle such huge graphs that the usual explicit representations by adjacency lists or adjacency matrices are infeasible. To deal with such huge graphs, OBDD-based graph representations and algorithms have been investigated. Although the size of OBDD representations may be large in general, it is known to be small for some special classes of graphs. In this paper, we show upper bounds and lower bounds of the size of OBDDs representing some intersection graphs such as bipartite permu-

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“…Using a larger domain for the labels also possibly increases the size of the data structure storing the valid labels which is often needed in implicit algorithms. Aiming for a good compression of χ E , Meer and Rautenbach [24] investigated graphs with bounded clique-width or tree-width and increases the number of bits used for the node labeling to c • log N with constant c and were able to improve for instance the OBDD size of cographs from O(N log N ) [27] to O(N ).…”

Section: Introductionmentioning

confidence: 99%

OBDD-Based Representation of Interval Graphs

Gillé

2013

Graph-Theoretic Concepts in Computer Science

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A graph G = (V, E) can be described by the characteristic function of the edge set χ E which maps a pair of binary encoded nodes to 1 iff the nodes are adjacent. Using Ordered Binary Decision Diagrams (OBDDs) to store χ E can lead to a (hopefully) compact representation. Given the OBDD as an input, symbolic/implicit OBDD-based graph algorithms can solve optimization problems by mainly using functional operations, e. g., quantification or binary synthesis. While the OBDD representation size can not be small in general, it can be provable small for special graph classes and then also lead to fast algorithms. In this paper, we show that the OBDD size of unit interval graphs is O(|V |/ log |V |) and the OBDD size of interval graphs is O(|V | log |V |) which both improve a known result from Nunkesser and Woelfel (2009). Furthermore, we can show that using our variable order and node labeling for interval graphs the worst-case OBDD size is Ω(|V | log |V |). We use the structure of the adjacency matrices to prove these bounds. This method may be of independent interest and can be applied to other graph classes. We also develop a maximum matching algorithm on unit interval graphs using O(log |V |) operations and a coloring algorithm for unit and general intervals graphs using O(log 2 |V |) operations and evaluate the algorithms empirically. * Supported by Deutsche Forschungsgemeinschaft, grant BO 2755/1-1.

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On the OBDD Size for Graphs of Bounded Tree- and Clique-Width

Cited by 5 publications

References 14 publications

An adjacency labeling scheme based on a tree-decomposition

An adjacency labeling scheme based on a tree-decomposition

OBDD Representation of Intersection Graphs

OBDD-Based Representation of Interval Graphs

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