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12
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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Cited by 43 (4 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 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...
ONLINE PLANARITY TESTING
, 1996
"... The online planaritytesting problem consists of performing the following operations on a planar graph G: (i) testing if a new edge can be added to G so that the resulting graph is itself planar; (ii) adding vertices and edges such that planarity is preserved. An efficient technique for online plan ..."
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Cited by 26 (2 self)
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The online planaritytesting problem consists of performing the following operations on a planar graph G: (i) testing if a new edge can be added to G so that the resulting graph is itself planar; (ii) adding vertices and edges such that planarity is preserved. An efficient technique for online planarity testing of a graph is presented that uses O(n) space and supports tests and insertions of vertices and edges in O(log n) time, where n is the current number of vertices of G. The bounds for tests and vertex insertions are worstcase and the bound for edge insertions is amortized. We also present other applications of this technique to dynamic algorithms for planar graphs.
Efficient Incremental Evaluation of Queries with Aggregation
 In SIGMOD
, 1994
"... We present a technique for efficiently evaluating queries on programs with monotonic aggregation, a class of programs defined by Ross and Sagiv. Our technique consists of the following components: incremental computation of aggregate functions, incremental fixpoint evaluation of monotonic programs a ..."
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Cited by 15 (3 self)
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We present a technique for efficiently evaluating queries on programs with monotonic aggregation, a class of programs defined by Ross and Sagiv. Our technique consists of the following components: incremental computation of aggregate functions, incremental fixpoint evaluation of monotonic programs and Magic Sets transformation of monotonic programs. We also present a formalization of the notion of incremental computation of aggregate functions on a multiset, and upper and lower bounds for incremental computation of a variety of aggregate functions. We describe a prooftheoretic reformulation of the monotonic semantics in terms of computations, following the approach of Beeri et al.; this reformulation greatly simplifies the task of proving the correctness of our optimizations. 1 Introduction There has been a lot of recent work in the literature on defining the semantics of complex database queries involving aggregate functions. Early work assumed some form of stratification of predic...
Dynamic Expression Trees
, 1991
"... We present a technique for dynamically maintaining a collection of arithmetic expressions represented by binary trees (whose leaves are variables and whose internal nodes are operators). A query operation asks for the value of an expression (associated with the root of a tree). Update operations inc ..."
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Cited by 11 (3 self)
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We present a technique for dynamically maintaining a collection of arithmetic expressions represented by binary trees (whose leaves are variables and whose internal nodes are operators). A query operation asks for the value of an expression (associated with the root of a tree). Update operations include changing the value of a variable and combining or decomposing expressions by linking or cutting the corresponding trees. Our dynamic data structure uses linear space and supports queries and updates in logarithmic time. An important application is the dynamic maintenance of maximum flow and shortest path in seriesparallel digraphs under a sequence of vertex and edge insertions, series and parallel compositions, and their respective inverses. Queries include reporting the maximum flow or shortest stpath in a seriesparallel subgraph.
Incremental Computation: A SemanticsBased Systematic Transformational Approach
, 1996
"... ion of a function f adds an extra cache parameter to f . Simplification simplifies the definition of f given the added cache parameter. However, as to how the cache parameter should be used in the simplification to provide incrementality, KIDS provides only the observation that distributive laws can ..."
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Cited by 9 (3 self)
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ion of a function f adds an extra cache parameter to f . Simplification simplifies the definition of f given the added cache parameter. However, as to how the cache parameter should be used in the simplification to provide incrementality, KIDS provides only the observation that distributive laws can often be applied. The Munich CIP project [BMPP89,Par90] has a strategy for finite differencing that captures similar ideas. It first "defines by a suitable embedding a function f 0 ", and then "derives a recursive version of f 0 using generalized unfold/fold strategy", but it provides no special techniques for discovering incrementality. We believe that both works provide only general strategies with no precise procedure to follow and therefore are less automatable than ours. Chapter 4 Caching intermediate results The value of f 0 (x \Phi y) may often be computed faster by using not only the return value of f 0 (x), as discussed in Chapter 3, but also the values of some subcomputation...
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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Cited by 3 (0 self)
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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...
Combine and Conquer
 Department of Computer Science, Brown University, Providence, RI
, 1992
"... We present a general technique for dynamizing a class of problems whose underlying structure is a computation graph embedded in a tree. We associate values, called attributes, with the nodes, paths, and subtrees of our trees. Path attributes form a path attribute system, if they are maintained in ..."
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Cited by 2 (0 self)
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We present a general technique for dynamizing a class of problems whose underlying structure is a computation graph embedded in a tree. We associate values, called attributes, with the nodes, paths, and subtrees of our trees. Path attributes form a path attribute system, if they are maintained in constant time under path concatenation. Additionally, attributes form a tree attribute system if the tree attributes of the tail of a path \Pi are determined in constant time from the path attributes of \Pi. We also introduce a new data structure called a linear attribute grammar. An attribute grammar is a treebased expression where the values a node are calculated from the values at the parent, siblings, and/or the children of . A linear attribute grammar, is an attribute grammar where all dependencies are linear. Our contributions can be summarized as follows. We provide a framework for maintaining attribute systems on trees in a fully dynamic environment. We show that given a ...
Incremental Updates in Structured Documents
, 1994
"... Contents 1 Introduction 1 1.1 Related work : : : : : : : : : : : : : : : : : : : : : : : : : : : 5 1.2 Outline of the thesis : : : : : : : : : : : : : : : : : : : : : : : 7 2 Lazy and incremental methods 9 2.1 Incrementality : : : : : : : : : : : : : : : : : : : : : : : : : : : 9 2.2 Laziness : : ..."
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Cited by 1 (0 self)
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Contents 1 Introduction 1 1.1 Related work : : : : : : : : : : : : : : : : : : : : : : : : : : : 5 1.2 Outline of the thesis : : : : : : : : : : : : : : : : : : : : : : : 7 2 Lazy and incremental methods 9 2.1 Incrementality : : : : : : : : : : : : : : : : : : : : : : : : : : : 9 2.2 Laziness : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 12 2.3 Discussion : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 13 3 Structured text and the HST system 15 3.1 Grammars for structure description : : : : : : : : : : : : : : : 15 3.2 Views of documents : : : : : : : : : : : : : : : : : : : : : : : : 17 3.3 The document preparation model of the HST system : : : : : 19 3.4 Lazy transformation : : : : : : : : : : : : : : : : : : : : : : : 21 3.5 Incremental updates : : : : : : :
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 1 (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 ...