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45
Integer Priority Queues with Decrease Key in . . .
 STOC'03
, 2003
"... We consider Fibonacci heap style integer priority queues supporting insert and decrease key operations in constant time. We present a deterministic linear space solution that with n integer keys support delete in O(log log n) time. If the integers are in the range [0,N), we can also support delete i ..."
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Cited by 33 (2 self)
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We consider Fibonacci heap style integer priority queues supporting insert and decrease key operations in constant time. We present a deterministic linear space solution that with n integer keys support delete in O(log log n) time. If the integers are in the range [0,N), we can also support delete in O(log log N) time. Even for the special case of monotone priority queues, where the minimum has to be nondecreasing, the best previous bounds on delete were O((log n) 1/(3−ε) ) and O((log N) 1/(4−ε)). These previous bounds used both randomization and amortization. Our new bounds a deterministic, worstcase, with no restriction to monotonicity, and exponentially faster. As a classical application, for a directed graph with n nodes and m edges with nonnegative integer weights, we get single source shortest paths in O(m + n log log n) time, or O(m + n log log C) ifC is the maximal edge weight. The later solves an open problem of Ahuja, Mehlhorn, Orlin, and
Topk document retrieval in optimal time and linear space
 In Proc. 22nd Annual ACMSIAM Symposium on Discrete Algorithms (SODA 2012
, 2012
"... We describe a data structure that uses O(n)word space and reports k most relevant documents that contain a query pattern P in optimal O(P  + k) time. Our construction supports an ample set of important relevance measures, such as the frequency of P in a document and the minimal distance between t ..."
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Cited by 29 (17 self)
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We describe a data structure that uses O(n)word space and reports k most relevant documents that contain a query pattern P in optimal O(P  + k) time. Our construction supports an ample set of important relevance measures, such as the frequency of P in a document and the minimal distance between two occurrences of P in a document. We show how to reduce the space of the data structure from O(n log n) to O(n(log σ+log D+log log n)) bits, where σ is the alphabet size and D is the total number of documents. 1
SpaceEfficient Topk Document Retrieval
"... Supporting topk document retrieval queries on general text databases, that is, finding the k documents where a given pattern occurs most frequently, has become a topic of interest with practical applications. While the problem has been solved in optimal time and linear space, the actual space usag ..."
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Cited by 14 (8 self)
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Supporting topk document retrieval queries on general text databases, that is, finding the k documents where a given pattern occurs most frequently, has become a topic of interest with practical applications. While the problem has been solved in optimal time and linear space, the actual space usage is a serious concern. In this paper we study various reducedspace structures that support topk retrieval and propose new alternatives. Our experimental results show that our novel structures and algorithms dominate almost all the space/time tradeoff.
Range Majority in Constant Time and Linear Space
, 2011
"... Given an array A of size n, we consider the problem of answering range majority queries: given a query range [i..j] where 1 ≤ i ≤ j ≤ n, return the majority element of the subarray A[i..j] if it exists. We describe a linear space data structure that answers range majority queries in constant time. W ..."
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Cited by 13 (6 self)
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Given an array A of size n, we consider the problem of answering range majority queries: given a query range [i..j] where 1 ≤ i ≤ j ≤ n, return the majority element of the subarray A[i..j] if it exists. We describe a linear space data structure that answers range majority queries in constant time. We further generalize this problem by defining range αmajority queries: given a query range [i..j], return all the elements in the subarray A[i..j] with frequency greater than α(j − i + 1). We prove an upper bound on the number of αmajorities that can exist in a subarray, assuming that query ranges are restricted to be larger than a given threshold. Using this upper bound, we generalize our range majority data structure to answer range αmajority queries in O ( 1α) time using O(n lg ( 1α + 1)) space, for any fixed α ∈ (0, 1). This result is interesting since other similar range query problems based on frequency have nearly logarithmic lower bounds on query time when restricted to linear space.
Persistent Predecessor Search and Orthogonal Point Location on the Word RAM
"... We answer a basic data structuring question (for example, raised by Dietz and Raman back in SODA 1991): can van Emde Boas trees be made persistent, without changing their asymptotic query/update time? We present a (partially) persistent data structure that supports predecessor search in a set of int ..."
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Cited by 13 (4 self)
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We answer a basic data structuring question (for example, raised by Dietz and Raman back in SODA 1991): can van Emde Boas trees be made persistent, without changing their asymptotic query/update time? We present a (partially) persistent data structure that supports predecessor search in a set of integers in {1,..., U} under an arbitrary sequence of n insertions and deletions, with O(log log U) expected query time and expected amortized update time, and O(n) space. The query bound is optimal in U for linearspace structures and improves previous nearO((log log U) 2) methods. The same method solves a fundamental problem from computational geometry: point location in orthogonal planar subdivisions (where edges are vertical or horizontal). We obtain the first static data structure achieving O(log log U) worstcase query time and linear space. This result is again optimal in U for linearspace structures and improves the previous O((log log U) 2) method by de Berg, Snoeyink, and van Kreveld (1992). The same result also holds for higherdimensional subdivisions that are orthogonal binary space partitions, and for certain nonorthogonal planar subdivisions such as triangulations without small angles. Many geometric applications follow, including improved query times for orthogonal range reporting for dimensions ≥ 3 on the RAM. Our key technique is an interesting new vanEmdeBoas–style recursion that alternates between two strategies, both quite simple.
Searching for Frequent Colors in Rectangles
"... We study a new variant of colored orthogonal range searching problem: given a query rectangle Q all colors c, such that at least a fraction o / of all points in Q are of color c, must be reported. We describe several data structures for that problem that use pseudolinear space and answer queries i ..."
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Cited by 12 (2 self)
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We study a new variant of colored orthogonal range searching problem: given a query rectangle Q all colors c, such that at least a fraction o / of all points in Q are of color c, must be reported. We describe several data structures for that problem that use pseudolinear space and answer queries in polylogarithmic time.
Improved Bounds for Finger Search on a RAM
 In Algorithms – ESA 2003, LNCS Vol. 2832 (Springer 2003
, 2003
"... We present a new finger search tree with O(1) worstcase update time and O(log log d) expected search time with high probability in the Random Access Machine (RAM) model of computation for a large class of input distributions. The parameter d represents the number of elements (distance) between ..."
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Cited by 10 (9 self)
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We present a new finger search tree with O(1) worstcase update time and O(log log d) expected search time with high probability in the Random Access Machine (RAM) model of computation for a large class of input distributions. The parameter d represents the number of elements (distance) between the search element and an element pointed to by a finger, in a finger search tree that stores n elements. For the need of the analysis we model the updates by a "balls and bins" combinatorial game that is interesting in its own right as it involves insertions and deletions of balls according to an unknown distribution.
Simple Deterministic Algorithms for Fully Dynamic Maximal Matching [Extended Abstract]
"... A maximal matching can be maintained in fully dynamic (supporting both addition and deletion of edges) nvertex graphs using a trivial deterministic algorithm with a worstcase update time of O(n). No deterministic algorithm that outperforms the naïve O(n) one was reported up to this date. The only p ..."
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Cited by 9 (1 self)
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A maximal matching can be maintained in fully dynamic (supporting both addition and deletion of edges) nvertex graphs using a trivial deterministic algorithm with a worstcase update time of O(n). No deterministic algorithm that outperforms the naïve O(n) one was reported up to this date. The only progress in this direction is due to Ivković and Lloyd [14], who in 1993 devised a deterministic algorithm with an amortized update time of O((n + m) √ 2/2), where m is the number of edges. In this paper we show the first deterministic fully dynamic algorithm that outperforms the trivial one. Specifically, we provide a deterministic worstcase update time of O ( √ m). Moreover, our algorithm maintains a matching which is in fact a 3/2approximate maximum cardinality matching (MCM). We remark that no fully dynamic algorithm for maintaining (2 − ɛ)approximate MCM improving upon the naïve O(n) was known prior to this work, even allowing amortized time bounds and randomization. For low arboricity graphs (e.g., planar graphs and graphs excluding fixed minors), we devise another simple deterministic algorithm with sublogarithmic update time. Specifically, it maintains a fully dynamic maximal matching with amortized update time of O(log n / log log n). This result addresses an open question of Onak and Rubinfeld [19]. We also show a deterministic algorithm with optimal space usage of O(n + m), that for arbitrary graphs maintains a maximal matching with amortized update time of O ( √ m).
Lower Bound Techniques for Data Structures
, 2008
"... We describe new techniques for proving lower bounds on datastructure problems, with the following broad consequences:
â¢ the first Î©(lgn) lower bound for any dynamic problem, improving on a bound that had been standing since 1989;
â¢ for static data structures, the first separation between linea ..."
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Cited by 8 (0 self)
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We describe new techniques for proving lower bounds on datastructure problems, with the following broad consequences:
â¢ the first Î©(lgn) lower bound for any dynamic problem, improving on a bound that had been standing since 1989;
â¢ for static data structures, the first separation between linear and polynomial space. Specifically, for some problems that have constant query time when polynomial space is allowed, we can show Î©(lg n/ lg lg n) bounds when the space is O(n Â· polylog n).
Using these techniques, we analyze a variety of central datastructure problems, and obtain improved lower bounds for the following:
â¢ the partialsums problem (a fundamental application of augmented binary search trees);
â¢ the predecessor problem (which is equivalent to IP lookup in Internet routers);
â¢ dynamic trees and dynamic connectivity;
â¢ orthogonal range stabbing;
â¢ orthogonal range counting, and orthogonal range reporting;
â¢ the partial match problem (searching with wildcards);
â¢ (1 + Îµ)approximate near neighbor on the hypercube;
â¢ approximate nearest neighbor in the lâ metric.
Our new techniques lead to surprisingly nontechnical proofs. For several problems, we obtain simpler proofs for bounds that were already known.
P.K.: Dynamic range majority data structures
 In: Proceedings of the 22nd International Symposium on Algorithms and Computation
, 2011
"... Abstract. Given a set P of n coloured points on the real line, we study the problem of answering range αmajority (or “heavy hitter”) queries on P. More specifically, for a query range Q, we want to return each colour that is assigned to more than an αfraction of the points contained in Q. We prese ..."
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Cited by 7 (2 self)
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Abstract. Given a set P of n coloured points on the real line, we study the problem of answering range αmajority (or “heavy hitter”) queries on P. More specifically, for a query range Q, we want to return each colour that is assigned to more than an αfraction of the points contained in Q. We present a new data structure for answering range αmajority queries on a dynamic set of points, where α ∈ (0, 1). Our data structure uses O(n) space, supports queries in O((lgn)/α) time, and updates in O((lgn)/α) amortized time. If the coordinates of the points are integers, then the query time can be improved to O(lgn/(α lg lgn)). For constant values of α, this improved query time matches an existing lower bound, for any data structure with polylogarithmic update time. We also generalize our data structure to handle sets of points in ddimensions, for d ≥ 2, as well as dynamic arrays, in which each entry is a colour. 1