## Space Efficient Algorithms for Series-Parallel Graphs (2001)

Venue: | IN 18TH INTERNATIONAL SYMPOSIUM ON THEORETICAL ASPECTS OF COMPUTER SCIENCE (STACS), NUMBER 2010 IN LECTURE NOTES IN COMPUTER SCIENCE |

Citations: | 5 - 1 self |

### BibTeX

@INPROCEEDINGS{Jakoby01spaceefficient,

author = {Andreas Jakoby and Maciej Liskiewicz and Rüdiger Reischuk},

title = {Space Efficient Algorithms for Series-Parallel Graphs},

booktitle = {IN 18TH INTERNATIONAL SYMPOSIUM ON THEORETICAL ASPECTS OF COMPUTER SCIENCE (STACS), NUMBER 2010 IN LECTURE NOTES IN COMPUTER SCIENCE},

year = {2001},

pages = {339--352},

publisher = {Springer}

}

### Years of Citing Articles

### OpenURL

### Abstract

The subclass of directed series-parallel graphs plays an important role in computer science. To determine whether a graph is series-parallel is a well studied problem in algorithmic graph theory. Fast sequential and parallel algorithms for this problem have been developed in a sequence of papers. For series-parallel graphs methods are also known to solve the reachability and the decomposition problem time efficiently. However, no dedicated results have been obtained for the space complexity of these problems -- the topic of this paper. For this special class of graphs, we develop deterministic algorithms for the recognition, reachability, decomposition and the path counting problem that use only logarithmic space. Since for arbitrary directed graphs reachability and path counting are believed not to be solvable in log-space the main contribution of this work are novel deterministic path finding routines that work correctly in series-parallel graphs, and a characterization...

### Citations

290 |
Graph Classes: A Survey
- Brandstädt, Le, et al.
- 1999
(Show Context)
Citation Context ...otes the number of vertices V of G and m the number of edges E. A well studied subclass of graphs are the series-parallel graphs, for which different definitions and characterizations have been given =-=[6]-=-. We will consider the basic class, sometimes also called two terminal series-parallel graphs, that are most important for applications in program analysis. Definition 1. G = (V; E) is a series-parall... |

286 |
Parallel Algorithms for Shared-Memory Machines
- Karp, Ramachandran
- 1990
(Show Context)
Citation Context ...emainder Representation can be converted to the standard binary one in logarithmic space [8]. Because of the relation between L and parallel time complexity classes defined by the EREWPRAM model (see =-=[14]-=-) these new algorithms can be modified to solve these problems in logarithmic time on EREW PRAMs as well. Finally, these results can also be extended to generalizations of series-parallel graphs: mult... |

114 |
The recognition of series parallel digraphs
- Valdes, Tarjan, et al.
- 1979
(Show Context)
Citation Context ...to the class of series-parallel graphs is a basic problem in algorithmic graph theory. An optimal linear time sequential algorithm for this problem has been developed by Valdes, Tarjan, and Lawler in =-=[15]-=- long time ago. Also, fast parallel algorithms have been published. He and Yesha have presented an EREW PRAM algorithm working in time O(log 2 n) while using n+m processors [12]. Eppstein has reduced ... |

63 |
Topology of series-parallel networks
- Duffin
- 1965
(Show Context)
Citation Context ...cd and P bc . If G contains a homeomorphic image of W as a subgraph then W is called a minor of G. The following characterization of series-parallel graphs by forbidden minors has been known for long =-=[10, 15]-=-. Let G be an st-connected acyclic graph. Then G is series-parallel iff W is not a minor of G. a W : b c d Fig. 3. The forbidden minor W . H : z 4 z1 z 2 z3 Fig. 4. The forbidden induced subgraph H . ... |

50 |
Problems complete for deterministic logarithmic space
- Cook, McKenzie
- 1987
(Show Context)
Citation Context ...wn to be complete with respect to AC 0 - reductions: for example, whether a graph is a forest or even a tree, or whether in a given forest G two nodes belong to the same tree (for a complete list see =-=[9, 13]-=-). In this paper we will prove three problems for series-parallel graphs to be L-complete as well: the recognition problem, the reachability problem, and counting the number of paths mod 2. While the ... |

49 |
A very hard log-space counting class
- Àlvarez, Jenner
- 1993
(Show Context)
Citation Context ...tractable. A decomposition of such graphs can be computed within the same space bound as well. For general graphs counting the number of paths is one of the generic complete problems for the class #L =-=[2]-=-. Thus, this problem is not computable in FL, the functional deterministic log-space complexity class, unless certain hierarchies collapse. We will prove that restricting to series-parallel graphs the... |

43 | An optimal parallel algorithm for formula evaluation
- Buss, Cook, et al.
- 1992
(Show Context)
Citation Context ...n logarithmic space. Let p 1 ! p 2 ! : : : be the standard enumeration of primes. The prime number theorem implies Q p i n+m p i = e (n+m)(1+o(1)) ? #PATH(a; b) : (1) Using the log-space algorithm of =-=[7]-=- one can transform T into a binary tree T 0 of depth O(log z) representing an arithmetic expression with the same value ae as T . We evaluate T 0 mod p i using the algorithm in [4]. For p isn + m this... |

42 |
Computing Algebraic Formulas Using a Constant Number of Registers
- Ben-Or, Cleve
- 1988
(Show Context)
Citation Context ...-space algorithm of [7] one can transform T into a binary tree T 0 of depth O(log z) representing an arithmetic expression with the same value ae as T . We evaluate T 0 mod p i using the algorithm in =-=[4]-=-. For p isn + m this algorithm works in space O(log z + log(n +m))sO(log n). By inequality (1), taking all p isn +m the values ae mod p i give a Chinese Remainder Representation of ae. Using the recen... |

23 | The complexity of planarity testing
- Allender, Mahajan
(Show Context)
Citation Context .... Recently, Allender and Mahajan have made a major step in classifying the computational complexity of planarity testing showing that this problem is hard for L and belongs to SL (symmetric Logspace) =-=[3]-=-. They leave as an open problem to close the gap between the lower bound and the upper bound. In this paper we determine the computational complexity of a nontrivial subproblem of planarity testing pr... |

16 | Parallel recognition of series-parallel graphs
- Eppstein
- 1992
(Show Context)
Citation Context ...ein has reduced the time bound constructing an algorithm that takes only O(log n) steps on the stronger PRAM model with concurrent instead of exclusive read and write, that requires C(m;n) processors =-=[11]-=-. Here C(m;n) denotes the number of processors necessary to compute the connected components of a graph in logarithmic time. Finally, Bodlaender and de Fluiter have presented an EREW PRAM algorithm us... |

6 | Tree isomorphism and some other complete problems for deterministic logspace
- Jenner, Lange, et al.
(Show Context)
Citation Context ...wn to be complete with respect to AC 0 - reductions: for example, whether a graph is a forest or even a tree, or whether in a given forest G two nodes belong to the same tree (for a complete list see =-=[9, 13]-=-). In this paper we will prove three problems for series-parallel graphs to be L-complete as well: the recognition problem, the reachability problem, and counting the number of paths mod 2. While the ... |

5 |
Antwerpen-de Fluiter. Parallel algorithms for series parallel graphs and graphs with treewidth two
- Bodlaender, van
(Show Context)
Citation Context ...ry to compute the connected components of a graph in logarithmic time. Finally, Bodlaender and de Fluiter have presented an EREW PRAM algorithm using O(log n \Delta log n) time and O(n +m) operations =-=[5]-=-. The space complexity of this problem, however, has not been investigated successfully so far. In this paper we give an answer to this question. The decompositon of a series-parallel graph is quite u... |

5 |
Parallel Recognition and Decomposition of Two
- He, Yesha
- 1987
(Show Context)
Citation Context ... Tarjan, and Lawler in [15] long time ago. Also, fast parallel algorithms have been published. He and Yesha have presented an EREW PRAM algorithm working in time O(log 2 n) while using n+m processors =-=[12]-=-. Eppstein has reduced the time bound constructing an algorithm that takes only O(log n) steps on the stronger PRAM model with concurrent instead of exclusive read and write, that requires C(m;n) proc... |

1 |
NC Division, unpublished manuscript
- Chiu, Davida, et al.
- 1999
(Show Context)
Citation Context ...re with a modular representation of numbers requiring only little space, and the recent result that a Chinese Remainder Representation can be converted to the standard binary one in logarithmic space =-=[8]-=-. Because of the relation between L and parallel time complexity classes defined by the EREWPRAM model (see [14]) these new algorithms can be modified to solve these problems in logarithmic time on ER... |