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127
Cache Oblivious Search Trees via Binary Trees of Small Height
 IN PROC. ACMSIAM SYMP. ON DISCRETE ALGORITHMS
, 2002
"... We propose a version of cache oblivious search trees which is simpler than the previous proposal of Bender, Demaine and FarachColton and has the same complexity bounds. In particular, our data structure avoids the use of weight balanced Btrees, and can be implemented as just a single array of ..."
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Cited by 65 (8 self)
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We propose a version of cache oblivious search trees which is simpler than the previous proposal of Bender, Demaine and FarachColton and has the same complexity bounds. In particular, our data structure avoids the use of weight balanced Btrees, and can be implemented as just a single array of data elements, without the use of pointers. The structure also improves space utilization.
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.
Cacheoblivious algorithms and data structures
 IN LECTURE NOTES FROM THE EEF SUMMER SCHOOL ON MASSIVE DATA SETS
, 2002
"... A recent direction in the design of cacheefficient and diskefficient algorithms and data structures is the notion of cache obliviousness, introduced by Frigo, Leiserson, Prokop, and Ramachandran in 1999. Cacheoblivious algorithms perform well on a multilevel memory hierarchy without knowing any pa ..."
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Cited by 42 (2 self)
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A recent direction in the design of cacheefficient and diskefficient algorithms and data structures is the notion of cache obliviousness, introduced by Frigo, Leiserson, Prokop, and Ramachandran in 1999. Cacheoblivious algorithms perform well on a multilevel memory hierarchy without knowing any parameters of the hierarchy, only knowing the existence of a hierarchy. Equivalently, a single cacheoblivious algorithm is efficient on all memory hierarchies simultaneously. While such results might seem impossible, a recent body of work has developed cacheoblivious algorithms and data structures that perform as well or nearly as well as standard externalmemory structures which require knowledge of the cache/memory size and block transfer size. Here we describe several of these results with the intent of elucidating the techniques behind their design. Perhaps the most exciting of these results are the data structures, which form general building blocks immediately
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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Orthogonal Range Searching on the RAM, Revisited
, 2011
"... We present a number of new results on one of the most extensively studied topics in computational geometry, orthogonal range searching. All our results are in the standard word RAM model: 1. We present two data structures for 2d orthogonal range emptiness. The first achieves O(n lg lg n) space and ..."
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Cited by 38 (7 self)
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We present a number of new results on one of the most extensively studied topics in computational geometry, orthogonal range searching. All our results are in the standard word RAM model: 1. We present two data structures for 2d orthogonal range emptiness. The first achieves O(n lg lg n) space and O(lg lg n) query time, assuming that the n given points are in rank space. This improves the previous results by Alstrup, Brodal, and Rauhe (FOCS’00), with O(n lg ε n) space and O(lg lg n) query time, or with O(n lg lg n) space and O(lg 2 lg n) query time. Our second data structure uses O(n) space and answers queries in O(lg ε n) time. The best previous O(n)space data structure, due to Nekrich (WADS’07), answers queries in O(lg n / lg lg n) time. 2. We give a data structure for 3d orthogonal range reporting with O(n lg 1+ε n) space and O(lg lg n+ k) query time for points in rank space, for any constant ε> 0. This improves the previous results by Afshani (ESA’08), Karpinski and Nekrich (COCOON’09), and Chan (SODA’11), with O(n lg 3 n) space and O(lg lg n + k) query time, or with O(n lg 1+ε n) space and O(lg 2 lg n + k) query time. Consequently, we obtain improved upper bounds for orthogonal range reporting in all constant dimensions above 3.
Faster Deterministic Sorting and Searching in Linear Space
, 1995
"... We present a significant improvement on linear space deterministic sorting and searching. On a unitcost RAM with word size w, an ordered set of n wbit keys (viewed as binary strings or integers) can be maintained in O ` min ` p log n; log n log w + log log n; log w log log n " time p ..."
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Cited by 38 (7 self)
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We present a significant improvement on linear space deterministic sorting and searching. On a unitcost RAM with word size w, an ordered set of n wbit keys (viewed as binary strings or integers) can be maintained in O ` min ` p log n; log n log w + log log n; log w log log n " time per operation, including insert, delete, member search, and neighbour search. The cost for searching is worstcase while the cost for updates is amortized. For range queries, there is an additional cost of reporting the found keys. As an application, n keys can be sorted in linear space at a worstcase cost of O \Gamma n p log n \Delta . The best previous method for deterministic sorting and searching in linear space has been the fusion trees which supports queries in O(logn= log log n) amortized time and sorting in O(n log n= log log n) worstcase time. We also make two minor observations on adapting our data structure to the input distribution and on the complexity of perfect hashing. 1 I...
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
Trapezoid Graphs and Generalizations, Geometry and Algorithms
 DISCRETE APPLIED MATHEMATICS
, 1993
"... Trapezoid graphs are a class of cocomparability graphs containing interval graphs and permutation graphs as subclasses. They were introduced by Dagan, Golumbic and Pinter [DGP]. They propose an O(n²) algorithm for chromatic number and a less efficient algorithm for maximum clique on trapezoid graphs ..."
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Cited by 33 (0 self)
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Trapezoid graphs are a class of cocomparability graphs containing interval graphs and permutation graphs as subclasses. They were introduced by Dagan, Golumbic and Pinter [DGP]. They propose an O(n²) algorithm for chromatic number and a less efficient algorithm for maximum clique on trapezoid graphs. Based on a geometric representation of trapezoid graphs by boxes in the plane we design optimal, i.e., O(n log n), algorithms for chromatic number, weighted independent set, clique cover and maximum weighted clique on such graphs. We also propose generalizations of trapezoid graphs called ktrapezoidal graphs. The ideas behind the clique cover and weighted independent set algorithms for trapezoid graphs carry over to higher dimensions. This leads to O(n log k\Gamma1 n) algorithms for ktrapezoidal graphs. We also propose a new class of graphs called circle trapezoid graphs. This class contains trapezoid graphs, circle graphs and circulararc graphs as subclasses. We show that cli...
Range mode and range median queries on lists and trees
 In Proceedings of the 14th Annual International Symposium on Algorithms and Computation (ISAAC
, 2003
"... ABSTRACT. We consider algorithms for preprocessing labelled lists and trees so that, for any two nodes u and v we can answer queries of the form: What is the mode or median label in the sequence of labels on the path from u to v. 1 ..."
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Cited by 28 (3 self)
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ABSTRACT. We consider algorithms for preprocessing labelled lists and trees so that, for any two nodes u and v we can answer queries of the form: What is the mode or median label in the sequence of labels on the path from u to v. 1