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On the Computational Complexity of Dynamic Graph Problems
 THEORETICAL COMPUTER SCIENCE
, 1996
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Complexity Models for Incremental Computation
, 1994
"... We present a new complexity theoretic approach to incremental computation. We define complexity classes that capture the intuitive notion of incremental efficiency and study their relation to existing complexity classes. We show that problems that have small sequential space complexity also have sma ..."
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We present a new complexity theoretic approach to incremental computation. We define complexity classes that capture the intuitive notion of incremental efficiency and study their relation to existing complexity classes. We show that problems that have small sequential space complexity also have small incremental time complexity. We show that all common LOGSPACEcomplete problems for P are also incrPOLYLOGTIMEcomplete for P. We introduce a restricted notion of completeness called NRPcompleteness and show that problems which are NRPcomplete for P are also incrPOLYLOGTIMEcomplete for P. We also give incrementally complete problems for NLOGSPACE, LOGSPACE, and nonuniform NC¹. We show that under certain restrictions problems which have efficient dynamic solutions also have efficient parallel solutions. We also consider a nonuniform model of incremental computation and show that in this model most problems have almost linear complexity. In addition, we present some techniques f...
Computing the WellFounded Semantics Faster
, 1995
"... . We address methods of speeding up the calculation of the wellfounded semantics for normal propositional logic programs. We first consider two algorithms already reported in the literature and show that these, plus a variation upon them, have much improved worstcase behavior for special cases ..."
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. We address methods of speeding up the calculation of the wellfounded semantics for normal propositional logic programs. We first consider two algorithms already reported in the literature and show that these, plus a variation upon them, have much improved worstcase behavior for special cases of input. Then we propose a general algorithm to speed up the calculation for logic programs with at most two positive subgoals per clause, intended to improve the worst case performance of the computation. For a logic program P in atoms A, the speed up over the straight Van Gelder alternating fixed point algorithm (assuming worstcase behavior for both algorithms) is approximately (jPj=jAj) (1=3) . For jPj jAj 4 , the algorithm runs in time linear in jPj. 1 Introduction Logic programming researchers have, over the last several years, proposed many logicbased declarative semantics for various sorts of logic programming. The hope is that, if they can be efficiently implemented, t...
Fully Dynamic Planarity Testing with Applications
"... The fully dynamic planarity testing problem consists of performing an arbitrary sequence of the following three kinds of operations on a planar graph G: (i) insert an edge if the resultant graph remains planar; (ii) delete an edge; and (iii) test whether an edge could be added to the graph without ..."
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Cited by 7 (0 self)
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The fully dynamic planarity testing problem consists of performing an arbitrary sequence of the following three kinds of operations on a planar graph G: (i) insert an edge if the resultant graph remains planar; (ii) delete an edge; and (iii) test whether an edge could be added to the graph without violating planarity. We show how to support each of the above operations in O(n2=3) time, where n is the number of vertices in the graph. The bound for tests and deletions is worstcase, while the bound for insertions is amortized. This is the first algorithm for this problem with sublinear running time, and it affirmatively answers a question posed in [11]. The same data structure has further applications in maintaining the biconnected and triconnected components of a dynamic planar graph. The time bounds are the same: O(n2=3) worstcase time per edge deletion, O(n2=3) amortized time per edge insertion, and O(n2=3) worstcase time to check whether two vertices are either biconnected or triconnected.
Stochastic Graphs Have Short Memory: Fully Dynamic Connectivity in PolyLog Expected Time
, 1995
"... This paper presents an average case analysis of fully dynamic graph connectivity (when the operations are edge insertions and deletions). To this end we introduce the model of stochastic graph processes (i.e. dynamically changing random graphs with random equiprobable edge insertions and deletions). ..."
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Cited by 4 (2 self)
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This paper presents an average case analysis of fully dynamic graph connectivity (when the operations are edge insertions and deletions). To this end we introduce the model of stochastic graph processes (i.e. dynamically changing random graphs with random equiprobable edge insertions and deletions). As the process continues indefinitely, all potential edge locations (in V \Theta V ) may be repeatedly inspected (and learned) by the algorithm. This learning of the structure seems to imply that traditional random graph analysis methods cannot be employed (since an observed edge is not a random event anymore). However, we show that a small (logarithmic) number of dynamic random updates are enough to allow our algorithm to reexamine edges as if they were random with respect to certain events (i.e. the graph "forgets" its structure). This short memory property of the stochastic graph process enables us to present an algorithm for graph connectivity which admits an amortized expected cost of...
Lower And Upper Bounds For Incremental Algorithms
, 1992
"... An incremental algorithm (also called a dynamic update algorithm) updates the answer to some problem after an incremental change is made in the input. We examine methods for bounding the performance of such algorithms. First, quite general but relatively weak bounds are considered, along with a care ..."
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An incremental algorithm (also called a dynamic update algorithm) updates the answer to some problem after an incremental change is made in the input. We examine methods for bounding the performance of such algorithms. First, quite general but relatively weak bounds are considered, along with a careful examination of the conditions under which they hold. Next, a more powerful proof method, the Incremental Relative Lower Bound is presented, along with its application to a number of important problems. We then examine an alternative approach, deltaanalysis, which had been proposed previously, apply it to several new problems and show how it can be extended. For the specific problem of updating the transitive closure of an acyclic digraph, we present the first known incremental algorithm that is efficient in the deltaanalysis sense. Finally, we criti...
A Complexity Theoretic Approach to Incremental Computation
 IN: PROC. 10TH ANN. SYMP. THEORETICAL ASPECTS OF COMPUTER SCIENCE
, 1993
"... We present a new complexity theoretic approach to incremental computation. We define complexity classes that capture the intuitive notion of incremental efficiency and study their relation to existing complexity classes. We show that all common LOGSPACEcomplete problems for P are incrPOLYLOGTIMEc ..."
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Cited by 2 (0 self)
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We present a new complexity theoretic approach to incremental computation. We define complexity classes that capture the intuitive notion of incremental efficiency and study their relation to existing complexity classes. We show that all common LOGSPACEcomplete problems for P are incrPOLYLOGTIMEcomplete for P. This suggests that nonredundant problems that seem inherently unparallelizable also seem hard to dynamize. We show that a form of transitive closure is complete under incremental reduction for NLOGSPACE and give similar problems which are incrementally complete for the classes LOGSPACE and nonuniform NC¹. We show that under certain restrictions problems which have efficient dynamic solutions also have efficient parallel solutions. We also look at the time complexity of circuit value and network stability problems restricted to comparator gates. We show that the dynamic version of the comparatorcircuit value problem and "LexFirst Maximal Matching" problem is in LOGSPACE ...
Data Structures
 ACM Computer Surveys
, 1996
"... Introduction The study of data structures, i.e., methods for organizing data that are suitable for computer processing, is one of the classic topics of computer science. At the hardware level, a computer views storage devices such as internal memory and disk as holders of elementary data units (byt ..."
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Introduction The study of data structures, i.e., methods for organizing data that are suitable for computer processing, is one of the classic topics of computer science. At the hardware level, a computer views storage devices such as internal memory and disk as holders of elementary data units (bytes), each accessible through its address (an integer). When writing programs, instead of manipulating the data at the byte level, it is convenient to organize them into higher level entities, called data structures. Most data structures can be viewed as containers that store a collection of objects of a given type, called the elements of the container. Often a total order is defined among the elements (e.g., alphabetically ordered names, points in the plane ordered by xcoordinate). A data structure has an associated repertory of operations, classified into queries, which retrieve information on the dat
Incremental Algorithms for Some Network Flow Problems
, 2001
"... In many network flow problems an incremental algorithm yields an enormous saving in computation time. The goal of such an algorithm is to update the solution to an instance of a problem after a unit change is made in the input. In this thesis the maxflow problem and shortest path problem are consid ..."
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In many network flow problems an incremental algorithm yields an enormous saving in computation time. The goal of such an algorithm is to update the solution to an instance of a problem after a unit change is made in the input. In this thesis the maxflow problem and shortest path problem are considered. An incremental
Incremental Algorithm for Maintaining DFS Tree for Undirected Graphs
"... Abstract. Depth First Search (DFS) tree is a fundamental data structure for graphs used in solving various algorithmic problems. However, very few results are known for maintaining DFS tree in a dynamic environment insertion or deletion of edges. The only nontrivial result for this problem is by ..."
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Abstract. Depth First Search (DFS) tree is a fundamental data structure for graphs used in solving various algorithmic problems. However, very few results are known for maintaining DFS tree in a dynamic environment insertion or deletion of edges. The only nontrivial result for this problem is by Franciosa et al. [4]. They showed that, for a directed acyclic graph on n vertices, a DFS tree can be maintained in O(n) amortized time per edge insertion. They stated it as an open problem to maintain a DFS tree dynamically in an undirected graph or general directed graph. We present the first algorithm for maintaining a DFS tree for an undirected graph under insertion of edges. For processing any arbitrary online sequence of edge insertions, this algorithm takes total O(n2) time.