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34
Marked Ancestor Problems
, 1998
"... Consider a rooted tree whose nodes can be marked or unmarked. Given a node, we want to find its nearest marked ancestor. This generalises the wellknown predecessor problem, where the tree is a path. ..."
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Cited by 50 (6 self)
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Consider a rooted tree whose nodes can be marked or unmarked. Given a node, we want to find its nearest marked ancestor. This generalises the wellknown predecessor problem, where the tree is a path.
Lower bounds for UnionSplitFind related problems on random access machines
, 1994
"... We prove \Omega\Gamma p log log n) lower bounds on the random access machine complexity of several dynamic, partially dynamic and static data structure problems, including the unionsplitfind problem, dynamic prefix problems and onedimensional range query problems. The proof techniques include a ..."
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Cited by 49 (3 self)
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We prove \Omega\Gamma p log log n) lower bounds on the random access machine complexity of several dynamic, partially dynamic and static data structure problems, including the unionsplitfind problem, dynamic prefix problems and onedimensional range query problems. The proof techniques include a general technique using perfect hashing for reducing static data structure problems (with a restriction of the size of the structure) into partially dynamic data structure problems (with no such restriction), thus providing a way to transfer lower bounds. We use a generalization of a method due to Ajtai for proving the lower bounds on the static problems, but describe the proof in terms of communication complexity, revealing a striking similarity to the proof used by Karchmer and Wigderson for proving lower bounds on the monotone circuit depth of connectivity. 1 Introduction and summary of results In this paper we give lower bounds for the complexity of implementing several dynamic and sta...
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...
Logarithmic lower bounds in the cellprobe model
 SIAM Journal on Computing
"... Abstract. We develop a new technique for proving cellprobe lower bounds on dynamic data structures. This enables us to prove Ω(lg n) bounds, breaking a longstanding barrier of Ω(lg n/lg lg n). We can also prove the first Ω(lgB n) lower bound in the external memory model, without assumptions on the ..."
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Cited by 34 (4 self)
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Abstract. We develop a new technique for proving cellprobe lower bounds on dynamic data structures. This enables us to prove Ω(lg n) bounds, breaking a longstanding barrier of Ω(lg n/lg lg n). We can also prove the first Ω(lgB n) lower bound in the external memory model, without assumptions on the data structure. We use our technique to prove better bounds for the partialsums problem, dynamic connectivity and (by reductions) other dynamic graph problems. Our proofs are surprisingly simple and clean. The bounds we obtain are often optimal, and lead to a nearly complete understanding of the problems. We also present new matching upper bounds for the partialsums problem. Key words. cellprobe complexity, lower bounds, data structures, dynamic graph problems, partialsums problem AMS subject classification. 68Q17
Incremental Validation of XML Documents
"... We investigate the incremental validation of XML documents with respect to DTDs and XML Schemas, under updates consisting of element tag renamings, insertions and deletions. DTDs are modeled as extended contextfree grammars and XML Schemas are abstracted as "specialized DTDs", allowing to decouple ..."
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Cited by 31 (2 self)
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We investigate the incremental validation of XML documents with respect to DTDs and XML Schemas, under updates consisting of element tag renamings, insertions and deletions. DTDs are modeled as extended contextfree grammars and XML Schemas are abstracted as "specialized DTDs", allowing to decouple element types from element tags. For DTDs, we exhibit an O(m log n) incremental validation algorithm using an auxiliary structure of size O(n), where n is the size of the document and m the number of updates. For specialized DTDs, we provide an O(m log² n) incremental algorithm, again using an auxiliary structure of size O(n). This is a significant improvement over bruteforce revalidation from scratch.
Complexity of Consistent Query Answering in Databases under CardinalityBased and Incremental Repair Semantics
 In ICDT
, 2007
"... Abstract. Consistent Query Answering (CQA) is the problem of computing from a database the answers to a query that are consistent with respect to certain integrity constraints that the database, as a whole, may fail to satisfy. Consistent answers have been characterized as those that are invariant u ..."
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Cited by 30 (9 self)
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Abstract. Consistent Query Answering (CQA) is the problem of computing from a database the answers to a query that are consistent with respect to certain integrity constraints that the database, as a whole, may fail to satisfy. Consistent answers have been characterized as those that are invariant under certain minimal forms of restoration of the database consistency. In this paper we investigate algorithmic and complexity theoretic issues of CQA under database repairs that minimally departwrt the cardinality of the symmetric difference from the original database. Research on this kind of repairs has been suggested in the literature, but no systematic study had been done. Here we obtain first tight complexity bounds. We also address, considering for the first time a dynamic scenario for CQA, the problem of incremental complexity of CQA, that naturally occurs when an originally consistent database becomes inconsistent after the execution of a sequence of update operations. Tight bounds on incremental complexity are provided for various semantics under denial constraints, e.g. (a) minimum tuplebased repairs wrt cardinality, (b) minimal tuplebased repairs wrt set inclusion, and (c) minimum numerical aggregation of attributebased repairs. Fixed parameter tractability is also investigated in this dynamic context, where the size of the update sequence becomes the relevant parameter. 1
On the Cell Probe Complexity of Polynomial Evaluation
, 1995
"... We consider the cell probe complexity of the polynomial evaluation problem with preprocessing of coefficients, for polynomials of degree at most n over a finite field K. We show that the trivial cell probe algorithm for the problem is optimal if K is sufficiently large compared to n. As an applicati ..."
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Cited by 18 (4 self)
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We consider the cell probe complexity of the polynomial evaluation problem with preprocessing of coefficients, for polynomials of degree at most n over a finite field K. We show that the trivial cell probe algorithm for the problem is optimal if K is sufficiently large compared to n. As an application, we give a new proof of the fact that P 6= incrTIME(o(log n= log log n)). 1 Introduction Let K be a field. We consider the polynomial evaluation problem with preprocessing of coefficients. This problem is as follows: Given a polynomial f(X) 2 K[X], preprocess it, so that later, for any field element a 2 K, f(a) can be computed efficiently. It is a classical problem in the theory of algebraic complexity and has been intensively investigated in the model of arithmetic straight line programs. In this model, a solution for the polynomials of degree at most n is given by two objects: ffl A map OE from the set of polynomials of degree at most n into K s , where s is any integer, called t...
Lower bounds for dynamic connectivity
 STOC
, 2004
"... We prove an Ω(lg n) cellprobe lower bound on maintaining connectivity in dynamic graphs, as well as a more general tradeoff between updates and queries. Our bound holds even if the graph is formed by disjoint paths, and thus also applies to trees and plane graphs. The bound is known to be tight fo ..."
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Cited by 15 (0 self)
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We prove an Ω(lg n) cellprobe lower bound on maintaining connectivity in dynamic graphs, as well as a more general tradeoff between updates and queries. Our bound holds even if the graph is formed by disjoint paths, and thus also applies to trees and plane graphs. The bound is known to be tight for these restricted cases, proving optimality of these data structures (e.g., Sleator and Tarjan’s dynamic trees). Our tradeoff is known to be tight for trees, and the best two data structures for dynamic connectivity in general graphs are points on our tradeoff curve. In this sense these two data structures are optimal, and this tightness serves as strong evidence that our lower bounds are the best possible. From a more theoretical perspective, our result is the first logarithmic cellprobe lower bound for any problem in the natural class of dynamic language membership problems, breaking the long standing record of Ω(lg n / lg lg n). In this sense, our result is the first datastructure lower bound that is “truly ” logarithmic, i.e., logarithmic in the problem size counted in bits. Obtaining such a bound is listed as one of three major challenges for future research by Miltersen [13] (the other two challenges remain unsolved). Our techniques form a general framework for proving cellprobe lower bounds on dynamic data structures. We show how our framework also applies to the partialsums problem to obtain a nearly complete understanding of the problem in cellprobe and algebraic models, solving several previously posed open problems.
Decremental Dynamic Connectivity
 In Proceedings of the 8th ACMSIAM Symposium on Discrete Algorithms (SODA
, 1997
"... We consider Las Vegas randomized dynamic algorithms for online connectivity problems with deletions only. In particular, we show that starting from a graph with m edges and n nodes, we can maintain a spanning forest during m deletions in O(minfn 2 ; m log ng+ p nm log 2:5 n) expected total ti ..."
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Cited by 13 (1 self)
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We consider Las Vegas randomized dynamic algorithms for online connectivity problems with deletions only. In particular, we show that starting from a graph with m edges and n nodes, we can maintain a spanning forest during m deletions in O(minfn 2 ; m log ng+ p nm log 2:5 n) expected total time. This is amortized constant time per operation if we start with a complete graph. The deletions may be interspersed with connectivity queries, each of which is answered in constant time. The previous best bound was O(m log 2 n) by Henzinger and Thorup (1996), which covered both insertions and deletions. Our bound is stronger for m=n = !(log n). The result is based on a general randomized reduction of many deletionsonly queries to few deletions and insertions queries. Similar results are thus derived for 2edgeconnectivity, bipartiteness, and qweights minimum spanning tree. For the decremental dynamic kedgeconnectivity problem of deleting the edges of a graph starting with m edges ...
Dynamic Tree Isomorphism via Firstorder Updates to a Relational Database
 In PODS'98
"... In databases, as in other computational settings, one would like to efficiently update answers to queries while minor changes are being made to the data. Dynamic complexity asks what resources are required to perform such updates. In this paper our main focus will be a particular dynamic graph prob ..."
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Cited by 13 (0 self)
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In databases, as in other computational settings, one would like to efficiently update answers to queries while minor changes are being made to the data. Dynamic complexity asks what resources are required to perform such updates. In this paper our main focus will be a particular dynamic graph problem, tree isomorphism, and the efficiency of our update scheme will be measured in terms of how expressive a query language is required to express our update. Working in the framework developed by [DS93, PI97] for dynamic query evaluation, we show that dynamic tree isomorphism can be performed via firstorder updates to a relational database (in [DS93] this framework is called a firstorder incremental evaluation system, and in [PI97] it is called DynFO). In [EI95] it was shown that treeisomorphism can not be expressed in firstorder logic augmented with a transitive closure operator and counting, (FO + TC + COUNT) (but without ordering). We thus obtain a first example of a graph problem...