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### Symbolic topological sorting with OBDDs

Philipp Woelfel
2006 Journal of Discrete Algorithms
We present a symbolic OBDD algorithm for topological sorting which requires O(log 2 |V |) OBDD operations.  ...  This is the first true runtime analysis of a symbolic OBDD algorithm for a fundamental graph problem, and it demonstrates that one can hope that the algorithm behaves well for sufficiently structured inputs  ...  We use the symbol for the corresponding relation, i.e., v w if v precedes w. In the following we describe a symbolic OBDD-algorithm which topologically sorts the vertices according to the relation .  ...

### Symbolic Topological Sorting with OBDDs [chapter]

Philipp Woelfel
2003 Lecture Notes in Computer Science
We present a symbolic OBDD algorithm for topological sorting which requires O(log 2 |V |) OBDD operations.  ...  This is the first true runtime analysis of a symbolic OBDD algorithm for a fundamental graph problem, and it demonstrates that one can hope that the algorithm behaves well for sufficiently structured inputs  ...  We use the symbol for the corresponding relation, i.e., v w if v precedes w. In the following we describe a symbolic OBDD-algorithm which topologically sorts the vertices according to the relation .  ...

### On symbolic OBDD-based algorithms for the minimum spanning tree problem

Beate Bollig
2012 Theoretical Computer Science
Furthermore, answering an open problem posed by Sawitzki  it is shown that every symbolic OBDD-based algorithm for the minimum spanning tree problem needs exponential space (with respect to the OBDD  ...  Since in some applications graphs become larger and larger, a research branch has emerged which is concerned with the design and analysis of so-called symbolic algorithms for classical graph problems on  ...  In [36, 43] symbolic algorithms for maximum flow in 0-1 networks and topological sorting have been presented and afterward analyzed for a restricted graph class.  ...

### What Graphs can be Efficiently Represented by BDDs?

C. Dong, P. Molitor
2007 2007 International Conference on Computing: Theory and Applications (ICCTA'07)
For randomly generated dense graphs, the gain, i. e., the ratio of the number of graph edges to the OBDD size, increases with the number of vertices and the density of the graphs.  ...  We have carried out experimental research into implicit representation of large graphs using reduced ordered binary decision diagrams (OBDDs).  ...  That is the reason that researchers began to think about OBDD based symbolic graph algorithms, e. g., symbolic topological sorting  , symbolic computation of strongly connected components  and  ...

### Subject Index

2006 Journal of Discrete Algorithms
sorting Symbolic topological sorting with OBDDs, 51 Traveling Salesman problem An improved approximation algorithm for the asymmetric TSP with strengthened triangle inequality, 623 Turing reduction Polynomial  ...  few vertices of large degree, 201 Implicit graph algorithms Symbolic topological sorting with OBDDs, 51 Information visualization A framework and algorithms for circular drawings of graphs, 25  ...

### Symbolic Graphs: Linear Solutions to Connectivity Related Problems

Raffaella Gentilini, Carla Piazza, Alberto Policriti
2007 Algorithmica
The importance of symbolic data structures such as Ordered Binary Decision Diagrams (OBDD) is rapidly growing in many areas of Computer Science where the large dimensions of the input models is a challenging  ...  However, OBDD representations pose strict constraints in the algorithm design issue.  ...  Finally, the authors in [33, 34, 44] , developed a number of symbolic graph algorithms (topological sorting  , max-flow on 0-1 networks, all pair shortest paths problem  ) all relying on symbolic  ...

### A Symbolic Approach to the All-Pairs Shortest-Paths Problem [chapter]

Daniel Sawitzki
2004 Lecture Notes in Computer Science
Graphs can be represented symbolically by the Ordered Binary Decision Diagram (OBDD) of their characteristic function.  ...  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.  ...  used in the analysis of symbolic topological sorting  and maximum flow algorithms    .  ...

### On Symbolic Representations of Maximum Matchings and (Un)directed Graphs [chapter]

Beate Bollig
2010 IFIP Advances in Information and Communication Technology
Here, it is shown that symbolic OBDD-based algorithms for the maximum matching problem need exponential space (with respect to the OBDD size of the input graph).  ...  Since in some applications graphs become larger and larger, a research branch has emerged which is concerned with the theoretical design and analysis of so-called symbolic algorithms for classical graph  ...  In  and  symbolic algorithms for maximum flow in 0-1 networks and topological sorting have been presented which have polylogarithmic running time with respect to the number of vertices of a  ...

### Design and Evaluation of a Symbolic and Abstraction-Based Model Checker [chapter]

2004 Lecture Notes in Computer Science
In case of properties expressed with a linear time logic, it appears that the second step is often more time consuming than the first one.  ...  In this work, we present a mixed solution which builds an observation graph represented in a non symbolic way but where the nodes are essentially symbolic set of states.  ...  Then we check the existence of dead states with the help of the symbolic operation Enab (via DetectDead) and the existence of a loop of unobserved events by applying a kind of topologic sort of the underlying  ...

### Lower Bounds on the OBDD Size of Graphs of Some Popular Functions [chapter]

Daniel Sawitzki
2005 Lecture Notes in Computer Science
Therefore, we consider fundamental arithmetic and storage access functions with exponential OBDD size and transfer these results to the graphs of these functions.  ...  In order to investigate the limits of symbolic graph algorithms which work on OBDD-represented graph instances, it is useful to have simply-structured graphs whose OBDD representation has exponential size  ...  Until now, symbolic methods for flow maximization [11, 22] , topological sorting  , shortest paths computation [21, 24] , and component analysis [7, 8] have been presented.  ...

### OBDD-Based Representation of Interval Graphs [article]

Marc Gillé
2013 arXiv   pre-print
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.  ...  In this paper, we show that the OBDD size of unit interval graphs is O( | V | / | V |) and the OBDD size of interval graphs is O(| V | | V |) which both improve a known result from Nunkesser and Woelfel  ...  Acknowledgements I thank Beate Bollig, Melanie Schmidt and Chris Schwiegelshohn for the valuable discussions and, together with the anonymous referees, for their comments on the presentation of the paper  ...

### OBDD-Based Representation of Interval Graphs [chapter]

Marc Gillé
2013 Lecture Notes in Computer Science
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.  ...  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  ...  Acknowledgements I thank Beate Bollig, Melanie Schmidt and Chris Schwiegelshohn for the valuable discussions and, together with the anonymous referees, for their comments on the presentation of the paper  ...

### Treewidth in Verification: Local vs. Global [chapter]

Andrea Ferrara, Guoqiang Pan, Moshe Y. Vardi
2005 Lecture Notes in Computer Science
Cutwidth boundedness is used also to improve symbolic simulation and Boolean satisfiability in [4, 29] .  ...  For example, it is easy to see that the graphs underlying systems with two counters are essentially grids, which are known to have high treewidth  .  ...  OBDDs are often used to perform symbolic image operations, which requires applying existential quantification to OBDDs  .  ...

### An efficient algorithm for computing bisimulation equivalence

Agostino Dovier, Carla Piazza, Alberto Policriti
2004 Theoretical Computer Science
The proposed algorithm is then re-elaborated to produce a symbolic version.  ...  We propose an e cient algorithmic solution to the problem of determining a Bisimulation Relation on a ÿnite structure working both on the explicit and on the implicit (symbolic) representation.  ...  Acknowledgements We thank Nadia Ugel for her C implementation and many useful discussions, and Ra aella Gentilini and Elio Panegai for their contribution in the development and implementation of the Symbolic  ...

### Combinational logic synthesis for LUT based field programmable gate arrays

Jason Cong, Yuzheng Ding
1996 ACM Transactions on Design Automation of Electronic Systems
and 4-LUTs, which can also be combined to form 5-LUTs; the AT&T ORCA2C [AT&T 1995] supplies 4-LUTs that can be combined to form 5-LUTs and 6-LUTs), most studies assume a homogeneous LUT architecture with  ...  OBDD Based Extraction. If the node function f is represented by OBDD, let v be a nonterminal with associated subfunction f v . The sub-OBDD rooted at v defines f v .  ...  Similarly, if the function is represented by an OBDD, decomposition can be performed by subgraph splitting. We regard such operations as symbolic decompositions in general.  ...
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