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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
On the bitcomplexity of LempelZiv compression
"... One of the most famous and investigated lossless datacompression schemes is the one introduced by Lempel and Ziv about 30 years ago [37]. This compression scheme is known as “dictionarybased compressor ” and consists of squeezing an input string by replacing some of its substrings with (shorter) c ..."
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One of the most famous and investigated lossless datacompression schemes is the one introduced by Lempel and Ziv about 30 years ago [37]. This compression scheme is known as “dictionarybased compressor ” and consists of squeezing an input string by replacing some of its substrings with (shorter) codewords which are actually pointers to a dictionary of phrases built as the string is processed. Surprisingly enough, although many fundamental results are nowadays known about the speed and effectiveness of this compression process (see e.g. [23, 29] and references therein), “we are not aware of any parsing scheme that achieves optimality when the LZ77dictionary is in use under any constraint on the codewords other than being of equal length” [29, pag. 159]. Here optimality means to achieve the minimum number of bits in compressing each individual input string, without any assumption on its generating source. In this paper we investigate three issues pertaining to the bitcomplexity of LZbased compressors, and we design algorithms which achieve bitoptimality in the compressed output size by taking efficient/optimal time and optimal space. These theoretical results will be sustained by some experiments that will compare our novel LZbased compressors against the most popular compression tools (like gzip, bzip2) and stateoftheart compressors (like the booster of [13, 12]).
Efficient IP table lookup via adaptive stratified trees with selective reconstructions. 12th European Symp
 on Algorithms
"... 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).
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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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.
NEFOS: Rapid CacheAware Range Query Processing with Probabilistic Guarantees
"... Abstract. We present NEFOS (NEsted FOrest of balanced treeS), a new cacheaware indexing scheme that supports insertions and deletions in O(1) worstcase block transfers for rebalancing operations (given and update position) and searching in O(log B log n) expected block transfers, (B = disk block s ..."
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Abstract. We present NEFOS (NEsted FOrest of balanced treeS), a new cacheaware indexing scheme that supports insertions and deletions in O(1) worstcase block transfers for rebalancing operations (given and update position) and searching in O(log B log n) expected block transfers, (B = disk block size and n = number of stored elements). The expected search bound holds with high probability for any (unknown) realistic input distribution. Our expected search bound constitutes an improvement over the O(log B log n) expected bound for search achieved by the ISBtree (Interpolation Search Btree), since the latter holds with high probability for the class of smooth only input distributions. We define any unknown distribution as realistic if the smoothness doesn’t appear in the whole data set, still it may appear locally in small spatial neighborhoods. This holds for a variety of reallife nonsmooth distributions like skew, zipfian, powlaw, beta e.t.c.. The latter is also verified by an accompanying experimental study. Moreover, NEFOS is a Bparametrized concrete structure, which works for both I/O and RAM model, without any kind of transformation or adaptation. Also, it is the first time an expected sublogarithmic bound for search operation was achieved for a broad family of nonsmooth input distributions. Keywords: Data Structures, Data Management Algorithms. 1
LowEntropy Computational Geometry
, 2010
"... The worstcase model for algorithm design does not always reflect the real world: inputs may have additional structure to be exploited, and sometimes data can be imprecise or become available only gradually. To better understand these situations, we examine several scenarios where additional informa ..."
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The worstcase model for algorithm design does not always reflect the real world: inputs may have additional structure to be exploited, and sometimes data can be imprecise or become available only gradually. To better understand these situations, we examine several scenarios where additional information can affect the design and analysis of geometric algorithms. First, we consider hereditary convex hulls: given a threedimensional convex polytope and a twocoloring of its vertices, we can find the individual monochromatic polytopes in linear expected time. This can be generalized in many ways, eg, to more than two colors, and to the offlineproblem where we wish to preprocess a polytope so that any large enough subpolytope can be found quickly. Our techniques can also be used to give a simple analysis of the selfimproving algorithm for planar Delaunay triangulations by Clarkson and Seshadhri [58]. Next, we assume that the point coordinates have a bounded number of bits, and that we can do standard bit manipulations in constant time. Then Delaunay triangulations can be found in expected time O(n √ log log n). Our result is based on a new connection between quadtrees and Delaunay triangulations, which also lets us generalize a recent result by Löffler and Snoeyink about Delaunay triangulations for imprecise points [110]. Finally, we consider randomized incremental constructions when the input permutation is generated by a boundeddegree Markov chain, and show that the resulting running time is almost optimal for chains with a constant eigenvalue gap.
CacheOblivious Dictionaries and Multimaps with Negligible Failure Probability
"... Abstract. A dictionary (or map) is a keyvalue store that requires all keys be unique, and a multimap is a keyvalue store that allows for multiple values to be associated with the same key. We design hashingbased indexing schemes for dictionaries and multimaps that achieve worstcase optimal perfo ..."
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Abstract. A dictionary (or map) is a keyvalue store that requires all keys be unique, and a multimap is a keyvalue store that allows for multiple values to be associated with the same key. We design hashingbased indexing schemes for dictionaries and multimaps that achieve worstcase optimal performance for lookups and updates, with minimal space overhead and subpolynomial probability that the data structure will require a rehash operation. Our dictionary structure is designed for the Random Access Machine (RAM) model, while our multimap implementation is designed for the cacheoblivious external memory (I/O) model. The failure probabilities for our structures are subpolynomial, which can be useful in cryptographic or dataintensive applications. 1
Sequential Dependency Computation via Geometric Data Structures
"... We are given integers 0 ≤ G1 ≤ G2 = 0 and a sequence SN = 〈a1, a2,..., aN 〉 of N integers. The goal is to compute the minimum number of insertions and deletions necessary to transform SN into a valid sequence, where a sequence is valid if it is nonempty, all elements are integers, and all the diffe ..."
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We are given integers 0 ≤ G1 ≤ G2 = 0 and a sequence SN = 〈a1, a2,..., aN 〉 of N integers. The goal is to compute the minimum number of insertions and deletions necessary to transform SN into a valid sequence, where a sequence is valid if it is nonempty, all elements are integers, and all the differences between consecutive elements are between G1 and G2. For this problem from the database theory literature, previous dynamic programming algorithms have running times O(N 2) and O(A·N log N), for a parameter A unrelated to N. We use a geometric data structure to obtain a O(N log N log log N) running time.
Author manuscript, published in "24th International Symposium on Algorithms and Computation, HongKong: Hong Kong (2013)" Single and multiple consecutive permutation motif search
, 2013
"... Abstract: Let t be a permutation (that shall play the role of the text) on[n] anda pattern p be a sequence of m distinct integer(s) of [n], m apple n. The pattern p occurs in t in position i if and only if p1...pm is orderisomorphic to ti...ti+m 1, that is, for all 1 apple k<`apple m, pk>p ` if and ..."
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Abstract: Let t be a permutation (that shall play the role of the text) on[n] anda pattern p be a sequence of m distinct integer(s) of [n], m apple n. The pattern p occurs in t in position i if and only if p1...pm is orderisomorphic to ti...ti+m 1, that is, for all 1 apple k<`apple m, pk>p ` if and only if ti+k 1>ti+ ` 1. Searching for a pattern p in a text t consists in identifying all occurrences of p in t. We first present a forward automaton which allows us to search for p in t in O(m 2 log log m + n) time.Wethen introduce a MorrisPratt automaton representation of the forward automaton which allows us to reduce this complexity to O(m log log m + n) at the price of an additional amortized constant term by integer of the text. Both automata occupy O(m) space. We then extend the problem to search for a set of patterns and exhibit a specific AhoCorasick like ⇣algorithm. Next we present a sublinear average case search algorithm m log m running in O log log m + n log m time, that we eventually prove to be optimal on m log log m average. 1