415 research outputs found
Representation of graphs by OBDDs
AbstractRecently, 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 P4s, 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
Ackermann Encoding, Bisimulations, and OBDDs
We propose an alternative way to represent graphs via OBDDs based on the
observation that a partition of the graph nodes allows sharing among the
employed OBDDs. In the second part of the paper we present a method to compute
at the same time the quotient w.r.t. the maximum bisimulation and the OBDD
representation of a given graph. The proposed computation is based on an
OBDD-rewriting of the notion of Ackermann encoding of hereditarily finite sets
into natural numbers.Comment: To appear on 'Theory and Practice of Logic Programming
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
Processing Succinct Matrices and Vectors
We study the complexity of algorithmic problems for matrices that are
represented by multi-terminal decision diagrams (MTDD). These are a variant of
ordered decision diagrams, where the terminal nodes are labeled with arbitrary
elements of a semiring (instead of 0 and 1). A simple example shows that the
product of two MTDD-represented matrices cannot be represented by an MTDD of
polynomial size. To overcome this deficiency, we extended MTDDs to MTDD_+ by
allowing componentwise symbolic addition of variables (of the same dimension)
in rules. It is shown that accessing an entry, equality checking, matrix
multiplication, and other basic matrix operations can be solved in polynomial
time for MTDD_+-represented matrices. On the other hand, testing whether the
determinant of a MTDD-represented matrix vanishes PSPACE$-complete, and the
same problem is NP-complete for MTDD_+-represented diagonal matrices. Computing
a specific entry in a product of MTDD-represented matrices is #P-complete.Comment: An extended abstract of this paper will appear in the Proceedings of
CSR 201
Quantum Algorithm for Dynamic Programming Approach for DAGs. Applications for Zhegalkin Polynomial Evaluation and Some Problems on DAGs
In this paper, we present a quantum algorithm for dynamic programming
approach for problems on directed acyclic graphs (DAGs). The running time of
the algorithm is , and the running time of the
best known deterministic algorithm is , where is the number of
vertices, is the number of vertices with at least one outgoing edge;
is the number of edges. We show that we can solve problems that use OR,
AND, NAND, MAX and MIN functions as the main transition steps. The approach is
useful for a couple of problems. One of them is computing a Boolean formula
that is represented by Zhegalkin polynomial, a Boolean circuit with shared
input and non-constant depth evaluating. Another two are the single source
longest paths search for weighted DAGs and the diameter search problem for
unweighted DAGs.Comment: UCNC2019 Conference pape
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