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Optimal Bounds for the Predecessor Problem and Related Problems
 Journal of Computer and System Sciences
, 2001
"... We obtain matching upper and lower bounds for the amount of time to find the predecessor of a given element among the elements of a fixed compactly stored set. Our algorithms are for the unitcost word RAM with multiplication and are extended to give dynamic algorithms. The lower bounds are proved ..."
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We obtain matching upper and lower bounds for the amount of time to find the predecessor of a given element among the elements of a fixed compactly stored set. Our algorithms are for the unitcost word RAM with multiplication and are extended to give dynamic algorithms. The lower bounds are proved for a large class of problems, including both static and dynamic predecessor problems, in a much stronger communication game model, but they apply to the cell probe and RAM models.
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 44 (2 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.
Deterministic Dictionaries
, 2001
"... It is shown that a static dictionary that offers constanttime access to n elements with wbit keys and occupies O(n) words of memory can be constructed deterministically in O(n log n) time on a unitcost RAM with word length w and a standard instruction set including multiplication. Whereas a rando ..."
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Cited by 41 (4 self)
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It is shown that a static dictionary that offers constanttime access to n elements with wbit keys and occupies O(n) words of memory can be constructed deterministically in O(n log n) time on a unitcost RAM with word length w and a standard instruction set including multiplication. Whereas a randomized construction working in linear expected time was known, the running time of the best previous deterministic algorithm was Ω(n²). Using a standard dynamization technique, the first deterministic dynamic dictionary with constant lookup time and sublinear update time is derived. The new algorithms are weakly nonuniform; i.e., they require access to a fixed number of precomputed constants dependent on w. The main technical tools employed are unitcost errorcorrecting codes, word parallelism, and derandomization using conditional expectations.
Tight(er) worstcase bounds on dynamic searching and priority queues
 in ‘‘Proceedings of the ThirtySecond Annual ACM Symposium on Theory of Computing
, 2000
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Integer sorting in O(n √ log log n) expected time and linear space
 In Proc. 33rd IEEE Symposium on Foundations of Computer Science (FOCS
, 2012
"... We present a randomized algorithm sorting n integers in O(n p log logn) expected time and linear space. This improves the previous O(n log logn) bound by Anderson et al. from STOC’95. As an immediate consequence, if the integers are bounded by U, we can sort them in O(n p log logU) expected time. Th ..."
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Cited by 33 (4 self)
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We present a randomized algorithm sorting n integers in O(n p log logn) expected time and linear space. This improves the previous O(n log logn) bound by Anderson et al. from STOC’95. As an immediate consequence, if the integers are bounded by U, we can sort them in O(n p log logU) expected time. This is the first improvement over the O(n log logU) bound obtained with van Emde Boas ’ data structure from FOCS’75. At the heart of our construction, is a technical deterministic lemma of independent interest; namely, that we split n integers into subsets of size at most pn in linear time and space. This also implies improved bounds for deterministic string sorting and integer sorting without multiplication. 1
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 24 (4 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).
Hashing, Randomness and Dictionaries
, 2002
"... This thesis is centered around one of the most basic information retrieval problems, namely that of storing and accessing the elements of a set. Each element in the set has some associated information that is returned along with it. The problem is referred to as the dictionary problem, due to the si ..."
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This thesis is centered around one of the most basic information retrieval problems, namely that of storing and accessing the elements of a set. Each element in the set has some associated information that is returned along with it. The problem is referred to as the dictionary problem, due to the similarity to a bookshelf dictionary, which contains a set of words and has an explanation associated with each word. In the static version of the problem the set is fixed, whereas in the dynamic version, insertions and deletions of elements are possible. The approach
Computational Geometry through the Information Lens
, 2007
"... revisits classic problems in computational geometry from the modern algorithmic ..."
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revisits classic problems in computational geometry from the modern algorithmic
Dynamic Set Intersection
"... Abstract. Consider the problem of maintaining a family F of dynamic sets subject to insertions, deletions, and setintersection reporting queries: given S, S ′ ∈ F, report every member of S ∩S ′ in any order. We show that in the word RAM model, where w is the word size, given a cap d on the maximum ..."
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Abstract. Consider the problem of maintaining a family F of dynamic sets subject to insertions, deletions, and setintersection reporting queries: given S, S ′ ∈ F, report every member of S ∩S ′ in any order. We show that in the word RAM model, where w is the word size, given a cap d on the maximum size of any set, we can support set intersection queries in O ( d w / log2 w) expected time, and updates in O(1) expected time. Using this algorithm we can list all t triangles of a graph G = (V,E) in O(m+ mα w / log2 w + t) expected time, where m = E  and α is the arboricity of G. This improves a 30year old triangle enumeration algorithm of Chiba and Nishizeki running in O(mα) time. We provide an incremental data structure on F that supports intersection witness queries, where we only need to find one e ∈ S ∩ S′. Both queries and insertions take O N w / log2 w expected time, where
Tight(er) Worstcase Bounds on Dynamic Searching andPriority Queues
"... PREPRINT. Proc. STOC 2000 ..."
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