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40
Error Correcting Codes, Perfect Hashing Circuits, and Deterministic Dynamic Dictionaries
, 1997
"... We consider dictionaries of size n over the finite universe U = and introduce a new technique for their implementation: error correcting codes. The use of such codes makes it possible to replace the use of strong forms of hashing, such as universal hashing, with much weaker forms, such as clus ..."
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We consider dictionaries of size n over the finite universe U = and introduce a new technique for their implementation: error correcting codes. The use of such codes makes it possible to replace the use of strong forms of hashing, such as universal hashing, with much weaker forms, such as clustering. We use
Transdichotomous Results in Computational Geometry, I: Point Location in Sublogarithmic Time
, 2008
"... Given a planar subdivision whose coordinates are integers bounded by U ≤ 2 w, we present a linearspace data structure that can answer point location queries in O(min{lg n / lg lg n, √ lg U/lg lg U}) time on the unitcost RAM with word size w. Thisisthe first result to beat the standard Θ(lg n) bou ..."
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Cited by 10 (3 self)
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Given a planar subdivision whose coordinates are integers bounded by U ≤ 2 w, we present a linearspace data structure that can answer point location queries in O(min{lg n / lg lg n, √ lg U/lg lg U}) time on the unitcost RAM with word size w. Thisisthe first result to beat the standard Θ(lg n) bound for infinite precision models. As a consequence, we obtain the first o(n lg n) (randomized) algorithms for many fundamental problems in computational geometry for arbitrary integer input on the word RAM, including: constructing the convex hull of a threedimensional point set, computing the Voronoi diagram or the Euclidean minimum spanning tree of a planar point set, triangulating a polygon with holes, and finding intersections among a set of line segments. Higherdimensional extensions and applications are also discussed. Though computational geometry with bounded precision input has been investigated for a long time, improvements have been limited largely to problems of an orthogonal flavor. Our results surpass this longstanding limitation, answering, for example, a question of Willard (SODA’92).
A TradeOff For WorstCase Efficient Dictionaries
"... We consider dynamic dictionaries over the universe U = {0, 1}^w on a unitcost RAM with word size w and a standard instruction set, and present a linear space deterministic dictionary accommodating membership queries in time (log log n)^O(1) and updates in time (log n)^O(1), where n is the size of t ..."
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Cited by 7 (2 self)
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We consider dynamic dictionaries over the universe U = {0, 1}^w on a unitcost RAM with word size w and a standard instruction set, and present a linear space deterministic dictionary accommodating membership queries in time (log log n)^O(1) and updates in time (log n)^O(1), where n is the size of the set stored. Previous solutions either had query time (log n) 18 or update time 2 !( p log n) in the worst case.
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 6 (3 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.
Dynamic 3sided Planar Range Queries with Expected Doubly Logarithmic Time
 Proceedings of ISAAC, 2009
"... Abstract. We consider the problem of maintaining dynamically a set of points in the plane and supporting range queries of the type [a, b] × (−∞, c]. We assume that the inserted points have their xcoordinates drawn from a class of smooth distributions, whereas the ycoordinates are arbitrarily distr ..."
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Abstract. We consider the problem of maintaining dynamically a set of points in the plane and supporting range queries of the type [a, b] × (−∞, c]. We assume that the inserted points have their xcoordinates drawn from a class of smooth distributions, whereas the ycoordinates are arbitrarily distributed. The points to be deleted are selected uniformly at random among the inserted points. For the RAM model, we present a linear space data structure that supports queries in O(log log n + t) expected time with high probability and updates in O(log log n) expected amortized time, where n is the number of points stored and t is the size of the output of the query. For the I/O model we support queries in O(log log B n + t/B) expected I/Os with high probability and updates in O(log B log n) expected amortized I/Os using linear space, where B is the disk block size. The data structures are deterministic and the expectation is with respect to the input distribution. 1
Efficient IP table lookup via adaptive stratified trees with selective reconstructions
 12TH EUROPEAN SYMP ON ALGORITHMS
, 2004
"... IP address lookup is a critical operation for high bandwidth routers in packet switching networks such as Internet. The lookup is a nontrivial operation since it requires searching for the longest prefix, among those stored in a (large) given table, matching the IP address. Ever increasing routing ..."
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IP address lookup is a critical operation for high bandwidth routers in packet switching networks such as Internet. The lookup is a nontrivial operation since it requires searching for the longest prefix, among those stored in a (large) given table, matching the IP address. Ever increasing routing tables size, traffic volume and links speed demand new and more efficient algorithms. Moreover, the imminent move to IPv6 128bit addresses will soon require a rethinking of previous technical choices. This article describes a the new data structure for solving the IP table look up problem christened the Adaptive Stratified Tree (AST). The proposed solution is based on casting the problem in geometric terms and on repeated application of efficient local geometric optimization routines. Experiments with this approach have shown that in terms of storage, query time and update time the AST is at a par with state of the art algorithms based on data compression or string manipulations (and often it is better on some of the measured quantities).
Adjacency queries in dynamic sparse graphs
 Inf. Process. Lett
, 2007
"... We deal with the problem of maintaining a dynamic graph so that queries of the form “is there an edge between u and v? ” are processed fast. We consider graphs of bounded arboricity, i.e., graphs with no dense subgraphs, like for example planar graphs. Brodal and Fagerberg [WADS’99] described a very ..."
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We deal with the problem of maintaining a dynamic graph so that queries of the form “is there an edge between u and v? ” are processed fast. We consider graphs of bounded arboricity, i.e., graphs with no dense subgraphs, like for example planar graphs. Brodal and Fagerberg [WADS’99] described a very simple linearsize data structure which processes queries in constant worstcase time and performs insertions and deletions in O(1) and O(log n) amortized time, respectively. We show a complementary result that their data structure can be used to get O(log n) worstcase time for query, O(1) amortized time for insertions and O(1) worstcase time for deletions. Moreover, our analysis shows that by combining the data structure of Brodal and Fagerberg with efficient dictionaries one gets O(log log logn) worstcase time bound for queries and deletions and O(log log logn) amortized time for insertions, with size of the data structure still linear. This last result holds even for graphs of arboricity bounded by O(logk n), for some constant k.
A Provably Efficient Computational Model For Approximate Spatiotemporal Retrieval
"... The paper is concerned with the effective and efficient processing of spatiotemporal selection queries under varying degrees of approximation. Such queries may employ operators like overlaps, north, during, etc., and their result is a set of entities standing approximately in some spatiotemporal rel ..."
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The paper is concerned with the effective and efficient processing of spatiotemporal selection queries under varying degrees of approximation. Such queries may employ operators like overlaps, north, during, etc., and their result is a set of entities standing approximately in some spatiotemporal relation # with respect to a query object X. The contribution of our work is twofold: i) First we describe a mathematical framework for representing multidimensional relations at varying granularity levels, modelling relation approximation through the concept of relation convexity. ii) We subsequently exploit the framework for developing approximate spatiotemporal retrieval mechanisms, employing a set of existing as well as new main memory and secondary memory data structures that achieve either optimal or the best known performance in terms of time and space complexity, for both static and dynamic problems. ##2VERVIEW The handling of spatiotemporal information is an increasingly evident dem...
Geometric Retrieval for Grid Points in the RAM Model
 JOURNAL OF UNIVERSAL COMPUTER SCIENCE
"... We consider the problem of ddimensional searching (d 3) for four query types: range, partial range, exact match and partial match searching. Let N be the number of points, s be the number of keys specified in a partial match and partial range query and t be the number of points retrieved. We pre ..."
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We consider the problem of ddimensional searching (d 3) for four query types: range, partial range, exact match and partial match searching. Let N be the number of points, s be the number of keys specified in a partial match and partial range query and t be the number of points retrieved. We present a data structure with worst case time complexities O(t + log N), O(t + (d s) + log N), O(d + # log N) and O(t + (d s) + s # log N) for each of the aforementioned query types respectively. We also present a second, more concrete solution for exact and partial match queries, which achieves the same query time but has di#erent space requirements. The proposed data structures are considered in the RAM model of computation.
Lower bounds for predecessor searching in the cell probe model ∗
, 2003
"... We consider a fundamental problem in data structures, static predecessor searching: Given a subset S of size n from the universe [m], store S so that queries of the form “What is the predecessor of x in S? ” can be answered efficiently. We study this problem in the cell probe model introduced by Yao ..."
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We consider a fundamental problem in data structures, static predecessor searching: Given a subset S of size n from the universe [m], store S so that queries of the form “What is the predecessor of x in S? ” can be answered efficiently. We study this problem in the cell probe model introduced by Yao [Yao81]. Recently, Beame and Fich [BF99] obtained optimal bounds on the number of probes needed by any deterministic query scheme if the associated storage scheme uses only n O(1) cells of word size (log m) O(1) bits. We give a new lower bound proof for this problem that matches the bounds of Beame and Fich. Our lower bound proof has the following advantages: it works for randomised query schemes too, while Beame and Fich’s proof works for deterministic query schemes only. In addition, it is simpler than Beame and Fich’s proof. In fact, our lower bound for predecessor searching extends to the ‘quantum addressonly ’ query schemes that we define in this paper. In these query schemes, quantum parallelism is allowed only over the ‘address lines ’ of the queries. These query schemes subsume classical randomised query schemes, and include many quantum query algorithms like Grover’s algorithm [Gro96]. We prove our lower bound using the round elimination approach of Miltersen, Nisan, Safra and Wigderson [MNSW98]. Using tools from information theory, we prove a strong round elimination lemma for communication complexity that enables us to obtain a tight lower bound for the predecessor problem. Our strong round elimination lemma also extends to quantum communication complexity. We also use our round elimination lemma to obtain a rounds versus communication tradeoff for the ‘greaterthan’ problem, improving on the tradeoff in [MNSW98]. We believe that our round elimination lemma is of independent interest and should have other applications. 1