26 research outputs found
OBDD-Based Representation of Interval Graphs
A graph can be described by the characteristic function of the
edge set which maps a pair of binary encoded nodes to 1 iff the nodes
are adjacent. Using \emph{Ordered Binary Decision Diagrams} (OBDDs) to store
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 and the OBDD size of interval graphs is $O(\
| V \ | \log \ | V \ |)\Omega(\ | V \ | \log
\ | V \ |)O(\log \ | V \ |)O(\log^2 \ | V \ |)$ operations and
evaluate the algorithms empirically.Comment: 29 pages, accepted for 39th International Workshop on Graph-Theoretic
Concepts 201
Markovian Analysis of Large Finite State Machines
Regarding finite state machines as Markov chains facilitates the application of probabilistic methods to very large logic synthesis and formal verification problems. In this paper we present symbolic algorithms to compute the steady-state probabilities for very large finite state machines (up to 1027 states). These algorithms, based on Algebraic Decision Diagrams (ADD's)-an extension of BDD's that allows arbitrary values to be associated with the terminal nodes of the diagrams-determine the steady-state probabilities by regarding finite state machines as homogeneous, discrete-parameter Markov chains with finite state spaces, and by solving the corresponding Chapman-Kolmogorov equations. We first consider finite state machines with state graphs composed of a single terminal strongly connected component; for this type of system we have implemented two solution techniques: One is based on the Gauss-Jacobi iteration, the other one is based on simple matrix multiplication. Then we extend our treatment to the most general case of systems which can be modelled as finite state machines with arbitrary transition structures; here our approach exploits structural information to decompose and simplify the state graph of the machine. We report experimental results obtained for problems on which traditional methods fai
Algorithms for Approximate FSM Traversal Based on State SpaceDecomposition
This paper presents algorithms for approximate finite state machine traversal based on state space decomposition. The original finite state machine is partitioned in component submachines, and each of them is traversed separately; the result of the computation is an over-estimation of the set of reachable states of the original machine. Different traversal strategies, which reduce the effects of the degrees of freedom introduced by the decomposition, are discussed. Efficient partitioning is a key point for the performance of the traversal techniques; a method to heuristically find a good decomposition of the overall finite state machine, based on the exploration of its state variable dependency graph, is proposed. Applications of the approximate traversal methods to logic optimization of sequential circuits and behavioral verification of finite state machines are described; experimental results for such applications, together with data concerning pure traversal, are reporte
Automatic State Space Decomposition for Approximate FSM Traversal Based on Circuit Structural Analysis
Exploiting circuit structure is a key issue in the implementation of algorithms for state space decomposition when the target is approximate FSM traversal. Given the gate-level description of a sequential circuit, the information about its structure can be captured by evaluating the affinity between pairs or groups of latches. Two main factors have to be considered in carrying out the structural analysis of a sequential circuit: latch connectivity and latch correlation. The first one takes into account the mutual dependency of each memory element on the others; the second one tells us how related are the functions realized by the logic feeding each latch. In this paper we estimate the affinity of two latches by combining these two factors, and we use this measure to formulate the state space decomposition problem as a graph partitioning problem. We propose an algorithm to automatically determine "good" partitions of the latch set which induce state space decomposition, and we present approximate FSM traversal and logic optimization results for the largest ISCAS'89 sequential benchmark
A Symbolic Approach to the All-Pairs Shortest-Paths Problem
Abstract. Graphs can be represented symbolically by the Ordered Binary Decision Diagram (OBDD) of their characteristic function. To solve problems in such implicitly given graphs, specialized symbolic algorithms are needed which are restricted to the use of functional operations offered by the OBDD data structure. In this paper, a symbolic algorithm for the all-pairs shortest-paths (APSP) problem in loopless directed graphs with strictly positive integral edge weights is presented. It requires Θ ( log 2 (NB) ) OBDD-operations to obtain the lengths and edges of all shortest paths in graphs with N nodes and maximum edge weight B. It is proved that runtime and space usage are polylogarithmic w. r. t. N and B on graph sequences with characteristic bounded-width functions. This convenient property is closed under certain graph composition operations. Moreover, an alternative symbolic approach for general integral edge weights is sketched which does not behave efficiently on general graph sequences with bounded-width functions. Finally, two variants of theAPSPproblemarebrieflydiscussed.