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On graph isomorphism for restricted graph classes
- IN
, 2006
"... Graph isomorphism (GI) is one of the few remaining problems in NP whose complexity status couldn’t be solved by classifying it as being either NP-complete or solvable in P. Nevertheless, efficient (polynomial-time or even NC) algorithms for restricted versions of GI have been found over the last fo ..."
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Graph isomorphism (GI) is one of the few remaining problems in NP whose complexity status couldn’t be solved by classifying it as being either NP-complete or solvable in P. Nevertheless, efficient (polynomial-time or even NC) algorithms for restricted versions of GI have been found over the last four decades. Depending on the graph class, the design and analysis of algorithms for GI use tools from various fields, such as combinatorics, algebra and logic. In this paper, we collect several complexity results on graph isomorphism testing and related algorithmic problems for restricted graph classes from the literature. Further, we provide some new complexity bounds (as well as easier proofs of some known results) and highlight some open questions.
An Optimal Parallel Matching Algorithm for Cographs
- Journal of Parallel and Distributed Computing
, 1994
"... The class of cographs, or complement-reducible graphs, arises naturally in many different areas of applied mathematics and computer science. We show that the problem of finding a maximum matching in a cograph can be solved optimally in parallel by reducing it to parenthesis matching. With an $n$-ver ..."
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The class of cographs, or complement-reducible graphs, arises naturally in many different areas of applied mathematics and computer science. We show that the problem of finding a maximum matching in a cograph can be solved optimally in parallel by reducing it to parenthesis matching. With an $n$-vertex cograph $G$ represented by its parse tree as input, our algorithm finds a maximum matching in $G$ in O($logn$) time using O($n0$) processors in the EREW-PRAM model. Key Words: list ranking, tree contraction, matching, parenthesis matching, scheduling, operating systems, cographs, parallel algorithms, EREW-PRAM. 1. Introduction A well-known class of graphs arising in a wide spectrum of practical applications [1,2,7] is the class of cographs, or complement-reducible graphs. The cographs are defined recursively as follows: . a single-vertex graph is a cograph; . if $G$ is a cograph, then its complement $G bar$ is also a cograph; . if $G$ and $H$ are cographs, then their union is also a cog...
A No-Busy-Wait Balanced Tree Parallel Algorithmic Paradigm
, 2000
"... Suppose that a parallel algorithm can include any number of parallel threads. Each thread can proceed without ever having to busy wait to another thread. A thread can proceed till its termination, but no new threads can be formed. What kind of problems can such restrictive algorithms solve and still ..."
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Cited by 5 (3 self)
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Suppose that a parallel algorithm can include any number of parallel threads. Each thread can proceed without ever having to busy wait to another thread. A thread can proceed till its termination, but no new threads can be formed. What kind of problems can such restrictive algorithms solve and still be competitive in the total number of operations they perform with the fastest serial algorithm for the same problem? Intrigued by this informal question, we considered one of the most elementary parallel algorithmic paradigms, that of balanced binary trees. The main contribution of this paper is a new balanced (not necessarily binary) tree no-busy-wait paradigm for parallel algorithms; applications of the basic paradigm to two problems are presented: building heaps, and executing parallel tree contraction (assuming a preparatory stage); the latter is known to be applicable to evaluating a family of general arithmetic expressions. For putting things in context, we also discuss our "PRAM-o...
Systematic Derivation of Tree Contraction Algorithms
- In Proceedings of INFOCOM '90
, 2005
"... While tree contraction algorithms play an important role in e#cient tree computation in parallel, it is di#cult to develop such algorithms due to the strict conditions imposed on contracting operators. In this paper, we propose a systematic method of deriving e#cient tree contraction algorithms f ..."
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While tree contraction algorithms play an important role in e#cient tree computation in parallel, it is di#cult to develop such algorithms due to the strict conditions imposed on contracting operators. In this paper, we propose a systematic method of deriving e#cient tree contraction algorithms from recursive functions on trees in any shape. We identify a general recursive form that can be parallelized to obtain e#cient tree contraction algorithms, and present a derivation strategy for transforming general recursive functions to parallelizable form. We illustrate our approach by deriving a novel parallel algorithm for the maximum connected-set sum problem on arbitrary trees, the tree-version of the famous maximum segment sum problem.
From invariants to canonization in parallel
- PROCEEDINGS OF THE THIRD INTERNATIONAL COMPUTER SCIENCE SYMPOSIUM IN RUSSIA, VOLUME 5010 OF LECTURE NOTES IN COMPUTER SCIENCE
, 2008
"... A function f of a graph is called a complete graph invariant if two given graphs G and H are isomorphic exactly when f(G) = f(H). If additionally, f(G) is a graph isomorphic to G, then f is called a canonical form for graphs. Gurevich [Gur97] proves that any polynomial-time computable complete inva ..."
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A function f of a graph is called a complete graph invariant if two given graphs G and H are isomorphic exactly when f(G) = f(H). If additionally, f(G) is a graph isomorphic to G, then f is called a canonical form for graphs. Gurevich [Gur97] proves that any polynomial-time computable complete invariant can be transformed into a polynomial-time computable canonical form. We extend this equivalence to the polylogarithmic-time model of parallel computation for classes of graphs having either bounded rigidity index or small separators. In particular, our results apply to three representative classes of graphs embeddable into a fixed surface, namely, to 3-connected graphs admitting either a polyhedral or a large-edge-width embedding as well as to all embeddable 5-connected graphs. Another application covers graphs with treewidth bounded by a constant k. Since for the latter class of graphs a complete invariant is computable in NC, it follows that graphs of bounded treewidth have a canonical form (and even a canonical labeling) computable in NC.
Subtree Isomorphism is in DLOG for Nested Trees
- Int. J. of Foundations of Computer Science
, 1995
"... This research note shows subtree isomorphism is in DLOG, and hence NC 2 , for nested trees. To our knowledge this result provides the first interesting class of trees for which the problem is in a nonrandomized version of NC. We also show that one can determine whether or not an arbitrary tree is ..."
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This research note shows subtree isomorphism is in DLOG, and hence NC 2 , for nested trees. To our knowledge this result provides the first interesting class of trees for which the problem is in a nonrandomized version of NC. We also show that one can determine whether or not an arbitrary tree is a nested tree in DLOG. Keywords: DLOG, NC, subtree isomorphism. 1 Introduction This note addresses the well-known subtree isomorphism problem. In this problem the input consists of two (directed) trees T and T 0 , and one must determine whether T is isomorphic to any subtree of T 0 . The subtree isomorphism problem is in P [7, 9]. The best sequential algorithm for the problem requires O(n 2:5 ) time [7]. Regarding the parallel complexity, in [8] an O(log n) time, n processor CRCW PRAM algorithm for the maximal subtree isomorphism problem is given. This is a restricted version of the subtree isomorphism problem in which the first tree must be isomorphic to a maximal subtree, a node an...
The Complexity of Planar Graph Isomorphism
, 1997
"... The Graph Isomorphism problem restricted to planar graphs has been known to be solvable in polynomial time many years ago. In terms of complexity classes however, the exact complexity of the problem has been established only very recently. It was proved in [6] that planar graph isomorphism can be co ..."
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The Graph Isomorphism problem restricted to planar graphs has been known to be solvable in polynomial time many years ago. In terms of complexity classes however, the exact complexity of the problem has been established only very recently. It was proved in [6] that planar graph isomorphism can be computed within logarithmic space. Since there is a matching hardness result [12], this shows that the problem is complete for L. Although this could be considered as a result in algorithmics its proof relies on several important new developments in the area of logarithmic space complexity classes and reflects the close connections between algorithms and complexity theory. In this column we give an overview of this result mentioning the developments that led to it.
Logical complexity of graphs: a survey
- CONTEMPORARY MATHEMATICS
, 2004
"... We discuss the definability of finite graphs in first-order logic with two relation symbols for adjacency and equality of vertices. The logical depth D(G) of a graph G is equal to the minimum quantifier depth of a sentence defining G up to isomorphism. The logical width W(G) is the minimum number of ..."
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We discuss the definability of finite graphs in first-order logic with two relation symbols for adjacency and equality of vertices. The logical depth D(G) of a graph G is equal to the minimum quantifier depth of a sentence defining G up to isomorphism. The logical width W(G) is the minimum number of variables occurring in such a sentence. The logical length L(G) is the length of a shortest defining sentence. We survey known estimates for these graph parameters and discuss their relations to other topics (such as the efficiency of the Weisfeiler-Lehman algorithm in isomorphism testing, the evolution of a random graph, quantitative characteristics of the zero-one law, or the contribution of Frank Ramsey to the research on Hilbert’s Entscheidungsproblem). Also, we trace the behavior of the descriptive complexity of a graph as the logic becomes more restrictive (for example, only definitions with a bounded number of variables or quantifier alternations are allowed) or more expressible
The Isomorphism Problem for k-Trees is Complete for Logspace
, 2012
"... We show that, for k constant, k-tree isomorphism can be decided in logarithmic space by giving an O(k log n) space canonical labeling algorithm. The algorithm computes a unique tree decomposition, uses colors to fully encode the structure of the original graph in the decomposition tree and invokes L ..."
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We show that, for k constant, k-tree isomorphism can be decided in logarithmic space by giving an O(k log n) space canonical labeling algorithm. The algorithm computes a unique tree decomposition, uses colors to fully encode the structure of the original graph in the decomposition tree and invokes Lindell’s tree canonization algorithm. As a consequence, the isomorphism, the automorphism, as well as the canonization problem for k-trees are all complete for deterministic logspace. Completeness for logspace holds even for simple structural properties of k-trees. We also show that a variant of our canonical labeling algorithm runs in time O((k + 1)! n), where n is the number of vertices, yielding the fastest known FPT algorithm for k-tree isomorphism.
A log-space algorithm for canonization of planar graphs
, 2008
"... Planar graph canonization is known to be hard for L this directly follows from L-hardness of tree-canonization [Lin92]. We give a log-space algorithm for planar graph canonization. This gives completeness for log-space under AC 0 many-one reductions and improves the previously known upper bound of A ..."
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Planar graph canonization is known to be hard for L this directly follows from L-hardness of tree-canonization [Lin92]. We give a log-space algorithm for planar graph canonization. This gives completeness for log-space under AC 0 many-one reductions and improves the previously known upper bound of AC 1 [MR91]. A planar graph can be decomposed into biconnected components. We give a log-space procedure for the decomposition of a biconnected planar graph into a triconnected component tree. The canonization process is based on these decomposition steps. 1
