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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.
Undirected Single Source Shortest Paths in Linear Time
 J. Assoc. Comput. Mach
, 1997
"... The single source shortest paths problem (SSSP) is one of the classic problems in algorithmic graph theory: given a weighted graph G with a source vertex s, find the shortest path from s to all other vertices in the graph. Since 1959 all theoretical developments in SSSP have been based on Dijkstra& ..."
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Cited by 49 (3 self)
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The single source shortest paths problem (SSSP) is one of the classic problems in algorithmic graph theory: given a weighted graph G with a source vertex s, find the shortest path from s to all other vertices in the graph. Since 1959 all theoretical developments in SSSP have been based on Dijkstra's algorithm, visiting the vertices in order of increasing distance from s. Thus, any implementation of Dijkstra 's algorithm sorts the vertices according to their distances from s. However, we do not know how to sort in linear time. Here, a deterministic linear time and linear space algorithm is presented for the undirected single source shortest paths problem with integer weights. The algorithm avoids the sorting bottleneck by building a hierechical bucketing structure, identifying vertex pairs that may be visited in any order. 1 Introduction Let G = (V; E), jV j = n, jEj = m, be an undirected connected graph with an integer edge weight function ` : E ! N and a distinguished source vertex...
A Pragmatic Implemention of Monotone Priority Queues
 In DIMACS’96 implementation challenge
, 1996
"... Introduction Recently there have been several theoretical improvements in the area of sorting, priority queues, and searching [1, 2, 8, 9]. All these improvements use indirect addressing to surpass the comparisonbased lower bounds. Inspired by these advances, and by the fact that algorithms based ..."
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Cited by 1 (0 self)
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Introduction Recently there have been several theoretical improvements in the area of sorting, priority queues, and searching [1, 2, 8, 9]. All these improvements use indirect addressing to surpass the comparisonbased lower bounds. Inspired by these advances, and by the fact that algorithms based on indirect addressing have proven to be efficient in many practical applications, we have implemented a triebased priority queue as part of the DIMACS implementation challenge. Having Dijkstra's single source shortest path algorithm in mind, we decided to restrict our attention to monotone priority queues, as defined in [9]. A monotone priority queue, is a priority queue where the minimum is nondecreasing  the minimum of an empty monotone priority queue is defined to be 0. The monotonicity condition is not a problem for greedy algorithms such as Dijkstra's single source shortest paths algorithm. Also, monotonicity is satisfied in eventsimulations. According to t
Point Location in Ó ÐÓ � Ò Time, Voronoi Diagrams in Ó Ò ÐÓ � Ò Time, and Other Transdichotomous Results in Computational Geometry
"... Given Ò points in the plane with integer coordinates bounded by Í � Û, we show that the Voronoi diagram can be constructed in Ç Ñ�Ò�Ò ÐÓ � Ò � ÐÓ � ÐÓ � Ò � Ò Ô ÐÓ � Í� expected time by a randomized algorithm on the unitcost RAM with word size Û. Similar results are also obtained for many other fun ..."
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Given Ò points in the plane with integer coordinates bounded by Í � Û, we show that the Voronoi diagram can be constructed in Ç Ñ�Ò�Ò ÐÓ � Ò � ÐÓ � ÐÓ � Ò � Ò Ô ÐÓ � Í� expected time by a randomized algorithm on the unitcost RAM with word size Û. Similar results are also obtained for many other fundamental problems in computational geometry, such as constructing the convex hull of adimensional point set, computing the Euclidean minimum spanning tree of a planar point set, triangulating a polygon with holes, and finding intersections among a set of line segments. These are the first results to beat the ª Ò ÐÓ � Ò algebraicdecisiontree lower bounds known for these problems. The results are all derived from a new twodimensional version of fusion trees that can answer point location queries in Ç Ñ�Ò�ÐÓ � Ò � ÐÓ � ÐÓ � Ò � Ô ÐÓ � Í � time with linear space. Higherdimensional extensions and applications are also mentioned in the paper. 1.