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Data Structural Bootstrapping, Linear Path Compression, and Catenable Heap Ordered Double Ended Queues
 SIAM Journal on Computing
, 1992
"... A deque with heap order is a linear list of elements with realvalued keys which allows insertions and deletions of elements at both ends of the list. It also allows the findmin (equivalently findmax) operation, which returns the element of least (greatest) key, but it does not allow a general delet ..."
Abstract

Cited by 15 (7 self)
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A deque with heap order is a linear list of elements with realvalued keys which allows insertions and deletions of elements at both ends of the list. It also allows the findmin (equivalently findmax) operation, which returns the element of least (greatest) key, but it does not allow a general deletemin (deletemax) operation. Such a data structure is also called a mindeque (maxdeque) . Whereas implementing mindeques in constant time per operation is a solved problem, catenating mindeques in sublogarithmic time has until now remained open. This paper provides an efficient implementation of catenable mindeques, yielding constant amortized time per operation. The important algorithmic technique employed is an idea which is best described as data structural bootstrapping: We abstract mindeques so that their elements represent other mindeques, effecting catenation while preserving heap order. The efficiency of the resulting data structure depends upon the complexity of a special case of pa...
Dynamic Planar Range Maxima Queries
 In Proc. 38th International Colloquium on Automata, Languages, and Programming, vol 6755 of LNCS
"... Abstract. We consider the dynamic twodimensional maxima query problem. Let P be a set of n points in the plane. A point is maximal if it is not dominated by any other point in P. We describe two data structures that support the reporting of the t maximal points that dominate a given query point, an ..."
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Cited by 5 (1 self)
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Abstract. We consider the dynamic twodimensional maxima query problem. Let P be a set of n points in the plane. A point is maximal if it is not dominated by any other point in P. We describe two data structures that support the reporting of the t maximal points that dominate a given query point, and allow for insertions and deletions of points in P. In the pointer machine model we present a linear space data structure with O(log n + t) worst case query time and O(log n) worst case update time. This is the first dynamic data structure for the planar maxima dominance query problem that achieves these bounds in the worst case. The data structure also supports the more general query of reporting the maximal points among the points that lie in a given 3sided orthogonal range unbounded from above in the same complexity. We can support 4sided queries in O(log 2 n+t) worst case time, and O(log 2 n) worst case update time, using O(n log n) space, where t is the size of the output. This improves the worst case deletion time of the dynamic rectangular visibility query problem from O(log 3 n) to O(log 2 n). We adapt the data structure to the RAM model with word size w, where the coordinates of the points are integers in the range U={0,..., 2 w −1}. We present a linear space data structure that supports 3sided range maxima queries in log n log n O ( +t) worst case time and updates in O ( ) worst case time. log log n log log n These are the first sublogarithmic worst case bounds for all operations in the RAM model. 1
, Aarhus University
"... We study the static and dynamic planar range skyline reporting problem in the external memory model with block size B, under a linear space budget. The problem asks for an O(n/B) space data structure that stores n points in the plane, and supports reporting the k maximal input points (a.k.a. skyline ..."
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We study the static and dynamic planar range skyline reporting problem in the external memory model with block size B, under a linear space budget. The problem asks for an O(n/B) space data structure that stores n points in the plane, and supports reporting the k maximal input points (a.k.a. skyline) among the points that lie within a given query rectangle Q = [α1, α2] × [β1, β2]. When Q is 3sided, i.e. one of its edges is grounded, two variants arise: topopen for β2 = ∞ and leftopen for α1 = − ∞ (symmetrically bottomopen and rightopen) queries. We present optimal static data structures for topopen queries, for the cases where the universe is R 2, a U × U grid, and rank space [O(n)] 2. We also show that leftopen queries are harder, as they require Ω((n/B) ɛ + k/B) I/Os for ɛ> 0, when only linear space is allowed. We show that the lower bound is tight, by a structure that supports 4sided queries in matching complexities. Interestingly, these lower and upper bounds coincide with those of the planar orthogonal range reporting problem, i.e., the skyline requirement does not alter the problem difficulty at all! Finally, we present the first dynamic linear space data structure that supports topopen queries in O(log 2B ɛ n + k/B 1−ɛ) and updates in O(log 2B ɛ n) worst case I/Os, for ɛ ∈ [0, 1]. This also yields a linear space data structure for 4sided queries with optimal query I/Os and O(log(n/B)) amortized update I/Os. We consider of independent interest the main component of our dynamic structures, a new realtime I/Oefficient and catenable variant of the fundamental structure priority queue with attrition by Sundar. The full version is found on
Orthogonal Range Queries and Update Efficiency
"... We study dynamic data structures for different variants of orthogonal range reporting query problems. In particular, we consider (1) the planar orthogonal 3sided range reporting problem: given a set of points in the plane, report the points that lie within a given 3sided rectangle with one unbound ..."
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We study dynamic data structures for different variants of orthogonal range reporting query problems. In particular, we consider (1) the planar orthogonal 3sided range reporting problem: given a set of points in the plane, report the points that lie within a given 3sided rectangle with one unbounded side, (2) the planar orthogonal range maxima reporting problem: given a set of points in the plane, report the points that lie within a given orthogonal range and are not dominated by any other point in the range, and (3) the problem of designing fully persistent Btrees for external memory. Dynamic problems like the above arise in various applications of network optimization, VLSI layout design, computer graphics and distributed computing. For the first problem, we present dynamic data structures for internal and external memory that support planar orthogonal 3sided range reporting queries, and insertions and deletions of points efficiently over an average case sequence of update operations. The external memory data structures find applications in constraint and temporal databases. In particular, we assume that the coordinates