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86
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
Fullyfunctional succinct trees
 In Proc. 21st SODA
, 2010
"... We propose new succinct representations of ordinal trees, which have been studied extensively. It is known that any nnode static tree can be represented in 2n + o(n) bits and a large number of operations on the tree can be supported in constant time under the wordRAM model. However existing data s ..."
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Cited by 33 (13 self)
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We propose new succinct representations of ordinal trees, which have been studied extensively. It is known that any nnode static tree can be represented in 2n + o(n) bits and a large number of operations on the tree can be supported in constant time under the wordRAM model. However existing data structures are not satisfactory in both theory and practice because (1) the lowerorder term is Ω(nlog log n / log n), which cannot be neglected in practice, (2) the hidden constant is also large, (3) the data structures are complicated and difficult to implement, and (4) the techniques do not extend to dynamic trees supporting insertions and deletions of nodes. We propose a simple and flexible data structure, called the range minmax tree, that reduces the large number of relevant tree operations considered in the literature to a few primitives, which are carried out in constant time on sufficiently small trees. The result is then extended to trees of arbitrary size, achieving 2n + O(n/polylog(n)) bits of space. The redundancy is significantly lower than in any previous proposal, and the data structure is easily implemented. Furthermore, using the same framework, we derive the first fullyfunctional dynamic succinct trees. 1
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.
Data Structures for Traveling Salesmen
, 1995
"... The choice of data structure for tour representation plays a critical role in the efficiency of local improvement heuristics for the Traveling Salesman Problem. The tour data structure must permit queries about the relative order of cities in the current tour and must allow sections of the tour to b ..."
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Cited by 29 (2 self)
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The choice of data structure for tour representation plays a critical role in the efficiency of local improvement heuristics for the Traveling Salesman Problem. The tour data structure must permit queries about the relative order of cities in the current tour and must allow sections of the tour to be reversed. The traditional arraybased representation of a tour permits the relative order of cities to be determined in small constant time, but requires worstcase W(N) time (where N is the number of cities) to implement a reversal, which renders it impractical for large instances. This paper considers alternative tour data structures, examining them from both a theoretical and experimental point of view. The first alternative we consider is a data structure based on splay trees, where all queries and updates take amortized time O(logN). We show that this is close to the best possible, because in the cell probe model of computation any data structure must take worstcase amortized time W(...
Cell probe complexity  a survey
 In 19th Conference on the Foundations of Software Technology and Theoretical Computer Science (FSTTCS), 1999. Advances in Data Structures Workshop
"... The cell probe model is a general, combinatorial model of data structures. We give a survey of known results about the cell probe complexity of static and dynamic data structure problems, with an emphasis on techniques for proving lower bounds. 1 ..."
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Cited by 28 (0 self)
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The cell probe model is a general, combinatorial model of data structures. We give a survey of known results about the cell probe complexity of static and dynamic data structure problems, with an emphasis on techniques for proving lower bounds. 1
Dynamic Ordered Sets with Exponential Search Trees
 Combination of results presented in FOCS 1996, STOC 2000 and SODA
, 2001
"... We introduce exponential search trees as a novel technique for converting static polynomial space search structures for ordered sets into fullydynamic linear space data structures. This leads to an optimal bound of O ( √ log n/log log n) for searching and updating a dynamic set of n integer keys i ..."
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Cited by 26 (1 self)
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We introduce exponential search trees as a novel technique for converting static polynomial space search structures for ordered sets into fullydynamic linear space data structures. This leads to an optimal bound of O ( √ log n/log log n) for searching and updating a dynamic set of n integer keys in linear space. Here searching an integer y means finding the maximum key in the set which is smaller than or equal to y. This problem is equivalent to the standard text book problem of maintaining an ordered set (see, e.g., Cormen, Leiserson, Rivest, and Stein: Introduction to Algorithms, 2nd ed., MIT Press, 2001). The best previous deterministic linear space bound was O(log n/log log n) due Fredman and Willard from STOC 1990. No better deterministic search bound was known using polynomial space.
Succinct Dynamic Data Structures
"... We develop succinct data structures to represent (i) a sequence of values to support partial sum and select queries and update (changing values) and (ii) a dynamic array consisting of a sequence of elements which supports insertion, deletion and access of an element at any given index. For the parti ..."
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Cited by 25 (2 self)
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We develop succinct data structures to represent (i) a sequence of values to support partial sum and select queries and update (changing values) and (ii) a dynamic array consisting of a sequence of elements which supports insertion, deletion and access of an element at any given index. For the partial sums problem...
A Fully Dynamic Algorithm for Recognizing and Representing Proper Interval Graphs
 SIAM J. COMPUT
, 1999
"... In this paper we study the problem of recognizing and representing dynamically changing proper interval graphs. The input to the problem consists of a series of modifications to be performed on a graph, where a modification can be a deletion or an addition of a vertex or an edge. The objective is to ..."
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Cited by 22 (1 self)
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In this paper we study the problem of recognizing and representing dynamically changing proper interval graphs. The input to the problem consists of a series of modifications to be performed on a graph, where a modification can be a deletion or an addition of a vertex or an edge. The objective is to maintain a representation of the graph as long as it remains a proper interval graph, and to detect when it ceases to be so. The representation should enable one to efficiently construct a realization of the graph by an inclusionfree family of intervals. This problem has important applications in physical mapping of DNA. We give a nearoptimal fully dynamic algorithm for this problem. It operates in time O(log n) per edge insertion or deletion. We prove a close lower bound of\Omega\Gamma/24 n=(log log n + log b)) amortized time per operation, in the cell probe model with wordsize b. We also construct optimal incremental and decremental algorithms for the problem, which handle each edge operation in O(1) time.
Low Redundancy in Static Dictionaries with O(1) Worst Case Lookup Time
 IN PROCEEDINGS OF THE 26TH INTERNATIONAL COLLOQUIUM ON AUTOMATA, LANGUAGES AND PROGRAMMING (ICALP '99
, 1999
"... A static dictionary is a data structure for storing subsets of a nite universe U , so that membership queries can be answered efficiently. We study this problem in a unit cost RAM model with word size (log jU j), and show that for nelement subsets, constant worst case query time can be obtained us ..."
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Cited by 21 (5 self)
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A static dictionary is a data structure for storing subsets of a nite universe U , so that membership queries can be answered efficiently. We study this problem in a unit cost RAM model with word size (log jU j), and show that for nelement subsets, constant worst case query time can be obtained using B +O(log log jU j) + o(n) bits of storage, where B = dlog 2 jUj n e is the minimum number of bits needed to represent all such subsets. For jU j = n log O(1) n the dictionary supports constant time rank queries.
Certification Trails for Data Structures
 Digest of the 1991 Fault Tolerant Computing Symposium
, 1991
"... Certification trails are a recently introduced and promising approach to faultdetection and faulttolerance [19]. In this paper, we significantly generalize the applicability of the certification trail technique. Previously, certification trails had to be customized to each algorithm application, bu ..."
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Cited by 21 (8 self)
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Certification trails are a recently introduced and promising approach to faultdetection and faulttolerance [19]. In this paper, we significantly generalize the applicability of the certification trail technique. Previously, certification trails had to be customized to each algorithm application, but here we develop trails appropriate to wide classes of algorithms. These certification trails are based on common datastructure operations such as those carried out using balanced binary trees and heaps. Any algorithm using these sets of operations can therefore employ the certification trail method to achieve software fault tolerance. To exemplify the scope of the generalization of the certification trail technique provided in this paper, constructions of trails for abstract data types such as priority queues and unionfind structures will be given. These trails are applicable to any datastructure implementation of the abstract data type. It will also be shown that these ideas lead natur...