Dominance made simple

Saxena, Sanjeev

doi:10.1016/j.ipl.2008.12.006

Cited by 16 publications

(5 citation statements)

References 10 publications

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“…For an array A [1, n] of n objects from a totally ordered universe and two indices i and j with 1 ≤ i ≤ j ≤ n, a Range Minimum Query 1 rmq A (i, j) returns the position of a minimum element in the sub-array A[i, j]; in symbols: rmq A (i, j) = argmin i≤k≤j A [k] . Given the ubiquity of arrays and the fundamental nature of this question, it is not surprising that RMQs have a wide range of applications in various fields of computing: text indexing [23,52], pattern matching [2,12], string mining [21,34], text compression [9,45], document retrieval [42,53,59], trees [4,6,38], graphs [28,49], bioinformatics [58], and in other types of range queries [10,56], to mention just a few.…”

Section: Introductionmentioning

confidence: 99%

Space-Efficient Preprocessing Schemes for Range Minimum Queries on Static Arrays

Fischer¹,

Heun²

2011

SIAM J. Comput.

221

233

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Given a static array of n totally ordered objects, the range minimum query problem is to build a data structure that allows to answer subsequent on-line queries of the form "what is the position of a minimum element in the sub-array ranging from i to j?" efficiently. We focus on two settings, where (1) the input array is available at query time, and (2) the input array is only available at construction time. In setting (1), we show new data structures (a) of size 2n c(n) − Θ n lg lg n c(n) lg n bits and query time O(c(n)) for any positive integer function c(n) ∈ O n ε for an arbitrary constant 0 < ε < 1, or (b) with O(nH k) + o(n) bits and O(1) query time, where H k denotes the empirical entropy of k'th order of the input array. In setting (2), we give a data structure of size 2n + o(n) bits and query time O(1). All data structures can be constructed in linear time and almost in-place.

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Section: Introductionmentioning

confidence: 99%

Space-Efficient Preprocessing Schemes for Range Minimum Queries on Static Arrays

Fischer¹,

Heun²

2011

SIAM J. Comput.

221

233

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“…2-dimensional dominance queries can be solved with a dynamic linear-size data structure with log n update and query time. Saxena [25] shows how static 3-dimensional dominance reporting can be solved using O(n log n) space and construction time with O(log n + S) query time where S is the size of the output. This approach can be easily adapted to provide the element with the maximal z-coordinate instead but there is few hope for linear space and O(log n) update time.…”

Section: Building the Height Partitionmentioning

confidence: 99%

Preprocessing Ambiguous Imprecise Points

van der Hoog,

Kostitsyna,

Löffler

et al. 2019

Preprint

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Let R = {R 1 , R 2 , . . . , R n } be a set of regions and let X = {x 1 , x 2 , . . . , x n } be an (unknown) point set with x i ∈ R i . Region R i represents the uncertainty region of x i . We consider the following question: how fast can we establish order if we are allowed to preprocess the regions in R? The preprocessing model of uncertainty uses two consecutive phases: a preprocessing phase which has access only to R followed by a reconstruction phase during which a desired structure on X is computed. Recent results in this model parametrize the reconstruction time by the ply of R, which is the maximum overlap between the regions in R. We introduce the ambiguity A(R) as a more fine-grained measure of the degree of overlap in R. We show how to preprocess a set of d-dimensional disks in O(n log n) time such that we can sort X (if d = 1) and reconstruct a quadtree on X (if d ≥ 1 but constant) in O(A(R)) time. If A(R) is sub-linear, then reporting the result dominates the running time of the reconstruction phase. However, we can still return a suitable data structure representing the result in O(A(R)) time.In one dimension, R is a set of intervals and the ambiguity is linked to interval entropy, which in turn relates to the well-studied problem of sorting under partial information. The number of comparisons necessary to find the linear order underlying a poset P is lower-bounded by the graph entropy of P . We show that if P is an interval order, then the ambiguity provides a constant-factor approximation of the graph entropy. This gives a lower bound of Ω(A(R)) in all dimensions for the reconstruction phase (sorting or any proximity structure), independent of any preprocessing; hence our result is tight. Finally, our results imply that one can approximate the entropy of interval graphs in O(n log n) time, improving the O(n 2.5

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“…In more detail, we use a method similar to that used to answer three-sided range queries [10,9,8], where we have to find all points with x-coordinate between i and j and y-coordinate greater than k.…”

Section: Reporting Cut Setsmentioning

confidence: 99%