Encodings for Range Selection and Top-k Queries

Grossi, Roberto; Iacono, John; Navarro, Gonzalo; Raman, Rajeev; Satti, Srinivasa Rao

doi:10.1007/978-3-642-40450-4_47

Cited by 16 publications

(32 citation statements)

References 19 publications

(31 reference statements)

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“…This uses almost half of the 6n + o(n) bits used for this problem in the literature [8]. Our structure can possibly be plugged into their solution, thus reducing their space.…”

Section: Resultsmentioning

confidence: 99%

“…The effective entropy of the more general range top-k problem, or finding the indices of the k smallest elements in a given range A [i, j], was recently shown to be Ω(n log k) bits by Grossi et al [8]. However, for k = 2, their approach only shows that the effective entropy of the RT2Q problem is ≥ n/2-much less than the effective entropy of the RMQ problem.…”

Section: Introductionmentioning

confidence: 99%

“…We complement this result by giving a data structure for encoding RT2Qs that takes 3.272n + o(n) bits and answers queries in O (1) time. This structure builds upon our new 2n + o(n)-bit RMQ encoding by adding further functionality to the binary tree representation of Farzan and Munro. We note that the range top-k encoding of Grossi et al [8] builds upon an encoding that answers RT2Qs in O(1) time, but their encoding for this subproblem uses 6n + o(n) bits.…”

Section: Introductionmentioning

confidence: 99%

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Encoding range minima and range top-2 queries

Davoodi

Navarro

Raman

et al. 2014

Phil. Trans. R. Soc. A.

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We consider the problem of encoding range minimum queries (RMQs): given an array A [1.. n ] of distinct totally ordered values, to pre-process A and create a data structure that can answer the query RMQ( i , j ), which returns the index containing the smallest element in A [ i .. j ], without access to the array A at query time. We give a data structure whose space usage is 2 n + o ( n ) bits, which is asymptotically optimal for worst-case data, and answers RMQs in O (1) worst-case time. This matches the previous result of Fischer and Heun, but is obtained in a more natural way. Furthermore, our result can encode the RMQs of a random array A in 1.919 n + o ( n ) bits in expectation, which is not known to hold for Fischer and Heun’s result. We then generalize our result to the encoding range top-2 query (RT2Q) problem, which is like the encoding RMQ problem except that the query RT2Q( i , j ) returns the indices of both the smallest and second smallest elements of A [ i .. j ]. We introduce a data structure using 3.272 n + o ( n ) bits that answers RT2Qs in constant time, and also give lower bounds on the effective entropy of the RT2Q problem.

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“…This uses almost half of the 6n + o(n) bits used for this problem in the literature [8]. Our structure can possibly be plugged into their solution, thus reducing their space.…”

Section: Resultsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Encoding range minima and range top-2 queries

Davoodi

Navarro

Raman

et al. 2014

Phil. Trans. R. Soc. A.

Self Cite

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“…Previous-smaller and next-smaller queries are obvious variants that can be solved similarly. In a conference version [20,Sec. 3.1] we showed how this encoding can be used to solve top(i, j, k) queries for any 1 ≤ k ≤ κ, using O(κn) bits and O(k 2 ) time, but this is subsumed in space and time by our better top(·) solutions in this article.…”

Section: Introductionmentioning

confidence: 99%

Asymptotically Optimal Encodings of Range Data Structures for Selection and Top- k Queries

Grossi

Iacono

Navarro

et al. 2017

ACM Trans. Algorithms

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Given an array A[1, n] of elements with a total order, we consider the problem of building a data structure that solves two queries: (a) selection queries receive a range [i, j] and an integer k and return the position of the kth largest element in A[i, j]; (b) top-k queries receive [i, j] and k and return the positions of the k largest elements in A [i, j]. These problems can be solved in optimal time, O(1 + lg k/ lg lg n) and O(k), respectively, using linear-space data structures.We provide the first study of the encoding data structures for the above problems, where A cannot be accessed at query time. Several applications are interested in the relative order of the entries of A, and their positions, rather their actual values, and thus we do not need to keep A at query time. In those cases, encodings save storage space: we first show that any encoding answering such queries requires n lg k − O(n + k lg k) bits of space; then, we design encodings using O(n lg k) bits, that is, asymptotically optimal up to constant factors, while preserving optimal query time.

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“…Otherwise in many cases the encodings would be able to reconstruct A, and thus could not be small. As examples of encodings, RMQs can be solved in constant time using just 2n+o(n) bits [7], and top-k queries can be solved in O(k) time using O(n log k) bits [10].…”

Section: Introductionmentioning

confidence: 99%