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32
Polylogarithmic deterministic fullydynamic graph algorithms I: connectivity and minimum spanning tree
 JOURNAL OF THE ACM
, 1997
"... Deterministic fully dynamic graph algorithms are presented for connectivity and minimum spanning forest. For connectivity, starting with no edges, the amortized cost for maintaining a spanning forest is O(log² n) per update, i.e. per edge insertion or deletion. Deciding connectivity between any two ..."
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Cited by 123 (6 self)
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Deterministic fully dynamic graph algorithms are presented for connectivity and minimum spanning forest. For connectivity, starting with no edges, the amortized cost for maintaining a spanning forest is O(log² n) per update, i.e. per edge insertion or deletion. Deciding connectivity between any two given vertices is done in O(log n= log log n) time. This matches the previous best randomized bounds. The previous best deterministic bound was O( 3 p n log n) amortized time per update but constant time for connectivity queries. For minimum spanning trees, first a deletionsonly algorithm is presented supporting deletes in amortized time O(log² n). Applying a general reduction from Henzinger and King, we then get a fully dynamic algorithm such that starting with no edges, the amortized cost for maintaining a minimum spanning forest is O(log^4 n) per update. The previous best deterministic bound was O( 3 p n log n) amortized time per update, and no better randomized bounds were ...
Randomized Fully Dynamic Graph Algorithms with Polylogarithmic Time per Operation
 JOURNAL OF THE ACM
, 1999
"... This paper solves a longstanding open problem in fully dynamic algorithms: We present the first fully dynamic algorithms that maintain connectivity, bipartiteness, and approximate minimum spanning trees in polylogarithmic time per edge insertion or deletion. The algorithms are designed using a new d ..."
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Cited by 53 (0 self)
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This paper solves a longstanding open problem in fully dynamic algorithms: We present the first fully dynamic algorithms that maintain connectivity, bipartiteness, and approximate minimum spanning trees in polylogarithmic time per edge insertion or deletion. The algorithms are designed using a new dynamic technique which combines a novel graph decomposition with randomization. They are LasVegas type randomized algorithms which use simple data structures and have a small constant factor. Let n denote the number of nodes in the graph. For a sequence of \Omega\Gamma m 0 ) operations, where m 0 is the number of edges in the initial graph, the expected time for p updates is O(p log 3 n) 1 for connectivity and bipartiteness. The worstcase time for one query is O(log n= log log n). For the kedge witness problem ("Does the removal of k given edges disconnect the graph?") the expected time for p updates is O(p log 3 n) and expected time for q queries is O(qk log 3 n). Given a grap...
An Approximation Algorithm for MinimumCost VertexConnectivity Problems
, 1997
"... We present an approximation algorithm for solving graph problems in which a lowcost set of edges must be selected that has certain vertexconnectivity properties. In the survivable network design problem, one is given a value r ij for each pair of vertices i and j, and must find a minimumcost set ..."
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Cited by 50 (6 self)
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We present an approximation algorithm for solving graph problems in which a lowcost set of edges must be selected that has certain vertexconnectivity properties. In the survivable network design problem, one is given a value r ij for each pair of vertices i and j, and must find a minimumcost set of edges such that there are r ij vertexdisjoint paths between vertices i and j. In the case for which r ij 2 f0; 1; 2g for all i; j, we can find a solution of cost no more than 3 times the optimal cost in polynomial time. In the case in which r ij = k for all i; j, we can find a solution of cost no more than 2H(k) times optimal, where H(n) = 1 + 1 2 + \Delta \Delta \Delta + 1 n . No approximation algorithms were previously known for these problems. Our algorithms rely on a primaldual approach which has recently led to approximation algorithms for many edgeconnectivity problems. 1 Introduction Let G = (V; E) be an undirected graph with nonnegative costs c e 0 on all edges e 2 E. In...
DynFO: A Parallel, Dynamic Complexity Class
 Journal of Computer and System Sciences
, 1994
"... Traditionally, computational complexity has considered only static problems. Classical Complexity Classes such as NC, P, and NP are defined in terms of the complexity of checking  upon presentation of an entire input  whether the input satisfies a certain property. For many applications of compu ..."
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Cited by 49 (4 self)
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Traditionally, computational complexity has considered only static problems. Classical Complexity Classes such as NC, P, and NP are defined in terms of the complexity of checking  upon presentation of an entire input  whether the input satisfies a certain property. For many applications of computers it is more appropriate to model the process as a dynamic one. There is a fairly large object being worked on over a period of time. The object is repeatedly modified by users and computations are performed. We develop a theory of Dynamic Complexity. We study the new complexity class, Dynamic FirstOrder Logic (DynFO). This is the set of properties that can be maintained and queried in firstorder logic, i.e. relational calculus, on a relational database. We show that many interesting properties are in DynFO including multiplication, graph connectivity, bipartiteness, and the computation of minimum spanning trees. Note that none of these problems is in static FO, and this f...
Lower Bounds for Fully Dynamic Connectivity Problems in Graphs
, 1998
"... We prove lower bounds on the complexity of maintaining fully dynamic kedge or kvertex connectivity in plane graphs and in (k − 1)vertex connected graphs. We show an amortized lower bound of �(log n/k(log log n + log b)) per edge insertion, deletion, or query operation in the cell probe model, whe ..."
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Cited by 32 (5 self)
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We prove lower bounds on the complexity of maintaining fully dynamic kedge or kvertex connectivity in plane graphs and in (k − 1)vertex connected graphs. We show an amortized lower bound of �(log n/k(log log n + log b)) per edge insertion, deletion, or query operation in the cell probe model, where b is the word size of the machine and n is the number of vertices in G. We also show an amortized lower bound of �(log n/(log log n + log b)) per operation for fully dynamic planarity testing in embedded graphs. These are the first lower bounds for fully dynamic connectivity problems.
An Empirical Study of Dynamic Graph Algorithms
 ACM JOURNAL ON EXPERIMENTAL ALGORITHMICS
, 1996
"... The contributions of this paper are both of theoretical and of experimental nature. From the experimental point of view, we conduct an empirical study on some dynamic connectivity algorithms which where developed recently. In particular, the following implementations were tested and compared with ..."
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Cited by 24 (4 self)
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The contributions of this paper are both of theoretical and of experimental nature. From the experimental point of view, we conduct an empirical study on some dynamic connectivity algorithms which where developed recently. In particular, the following implementations were tested and compared with simple algorithms: simple sparsification by Eppstein et al. and the recent randomized algorithm by Henzinger and King. In our experiments, we considered both random and nonrandom inputs. Moreover, we present a simplified variant of the algorithm by Henzinger and King, which for random inputs was always faster than the original implementation. For nonrandom inputs, simple sparsification was the fastest algorithm for small sequences of updates; for medium and large sequences of updates, the original algorithm by Henzinger and King was faster. From the theoretical point of view, we analyze the average case running time of simple sparsification and prove that for dynamic random graph...
Experimental Analysis of Dynamic Minimum Spanning Tree Algorithms (Extended Abstract)
, 1997
"... ) Giuseppe Amato Giuseppe Cattaneo y Giuseppe F. Italiano z Abstract We conduct an extensive empirical study on the performance of several algorithms for maintaining the minimum spanning tree of a dynamic graph. In particular, we implemented and tested Frederickson's algorithms, and spa ..."
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Cited by 15 (2 self)
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) Giuseppe Amato Giuseppe Cattaneo y Giuseppe F. Italiano z Abstract We conduct an extensive empirical study on the performance of several algorithms for maintaining the minimum spanning tree of a dynamic graph. In particular, we implemented and tested Frederickson's algorithms, and sparsification on top of Frederickson's algorithms, and compared them to other dynamic algorithms. Moreover, we propose a variant of a dynamic algorithm by Frederickson, which was in our experience always faster than the other implementations derived from the papers. In our experiments, we considered both random and nonrandom inputs, with nonrandom inputs trying to enforce bad update patterns on the algorithms. For random inputs, a simple adaptation of a partially dynamic data structure on Kruskal's algorithm was the fastest implementation. For nonrandom inputs, sparsification yielded the fastest algorithm. In both cases, the performance of our variant of the algorithm of Frederickson was clos...
Average Case Analysis of Dynamic Graph Algorithms
 PROC. 6TH SYMP. ON DISCRETE ALGORITHMS
, 1995
"... We present a model for edge updates with restricted randomness in dynamic graph algorithms and a general technique for analyzing the expected running time of an update operation. This model is able to capture the average case in many applications, since (1) it allows restrictions on the set of edges ..."
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Cited by 13 (3 self)
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We present a model for edge updates with restricted randomness in dynamic graph algorithms and a general technique for analyzing the expected running time of an update operation. This model is able to capture the average case in many applications, since (1) it allows restrictions on the set of edges which can be used for insertions and (2) the type (insertion or deletion) of each update operation is arbitrary, i.e., tot random. We use our technique to analyze existing and new dynamic algorithms for the following problems: maximum cardinality matching, minimum spanning forest, connectivity, 2edge connectivity, kedge connectivity, kvertex connectivity, and bipartiteness. Given a random graph G with m0 edges and n vertices and a sequence of 1 update operations such that the graph contains rtt edges after operation i, the expected tinhe for performing the updates for any 1 is O(/log  I= 1/x/) in the case of minimum spanning forests, connectivity, 2edge connectivity, and bipartiteness. The expected time per update operation is O(n) in the case of maximum matching. We also give improved bounds for kedge and kvertex connectivity. Additionally we give an insertionsonly algorithm for maximum cardinality matching with worstcase O(n) amortized time per insertion.
Dynamic Graph Algorithms
, 2000
"... INTRODUCTION Dynamic graph algorithms are algorithms that maintain properties of a (possibly edgeweighted) graph while the graph is changing. These algorithms are potentially useful in a number of application areas, including communication networks, VLSI design, distributed computing, and graphics, ..."
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Cited by 11 (0 self)
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INTRODUCTION Dynamic graph algorithms are algorithms that maintain properties of a (possibly edgeweighted) graph while the graph is changing. These algorithms are potentially useful in a number of application areas, including communication networks, VLSI design, distributed computing, and graphics, where the underlying graphs are subject to dynamic changes. Efficient dynamic graph algorithms are also used as subroutines in algorithms that build and modify graphs as part of larger tasks, e.g., the algorithm for constructing Voronoi diagrams by building planar subdivisions. GLOSSARY Update: an operation that changes the graph. The primitive updates considered in the literature are edge insertions and deletions and, in the case of edgeweighted graphs, changes in edge weights. Query: a request for information about the property being maintained. For example, if the property is planarity, a query simply asks whether the graph is currently
Computing Tutte Polynomials
 THE FIRST CENSUS ON THE TERTIARY INDUSTRY IN CHINA: SUMMARY STATISTICS, CHINA STATISTICAL
, 1996
"... The Tutte polynomial of a graph, also known as the partition function of the qstate Potts model, is a 2variable polynomial graph invariant of considerable importance in both combinatorics and statistical physics. It contains several other polynomial invariants, such as the chromatic polynomial an ..."
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Cited by 10 (1 self)
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The Tutte polynomial of a graph, also known as the partition function of the qstate Potts model, is a 2variable polynomial graph invariant of considerable importance in both combinatorics and statistical physics. It contains several other polynomial invariants, such as the chromatic polynomial and flow polynomial as partial evaluations, and various numerical invariants such as the number of spanning trees as complete evaluations. However despite its ubiquity, there are no widelyavailable effective computational tools able to compute the Tutte polynomial of a general graph of reasonable size. In this paper we describe the implementation of a program that exploits isomorphisms in the computation tree to extend the range of graphs for which it is feasible to compute their Tutte polynomials. We also consider edgeselection heuristics which give good performance in practice. We empirically demonstrate the utility of our program on random graphs. More evidence of its usefulness arises from our success in finding counterexamples to a conjecture of Welsh on the location of the real flow roots of a graph.