Selection from heaps, row-sorted matrices, and X + Y using soft heaps

Haim Kaplan; László Kozma; Or Zamir; Uri Zwick

doi:10.4230/OASIcs.SOSA.2019.5

Selection from heaps, row-sorted matrices, and X + Y using soft heaps

Haim Kaplan, László Kozma, Or Zamir, Uri Zwick

Eindhoven University of Technology

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution › peer-review

9 Scopus citations

Abstract

We use soft heaps to obtain simpler optimal algorithms for selecting the k-th smallest item, and the set of k smallest items, from a heap-ordered tree, from a collection of sorted lists, and from X + Y , where X and Y are two unsorted sets. Our results match, and in some ways extend and improve, classical results of Frederickson (1993) and Frederickson and Johnson (1982). In particular, for selecting the k-th smallest item, or the set of k smallest items, from a collection of m sorted lists we obtain a new optimal “output-sensitive” algorithm that performs only O(m + Ʃ^m_i=1 log(k_i + 1)) comparisons, where k_i is the number of items of the i-th list that belong to the overall set of k smallest items.

Original language	English
Title of host publication	2nd Symposium on Simplicity in Algorithms, SOSA 2019 - Co-located with the 30th ACM-SIAM Symposium on Discrete Algorithms, SODA 2019
Editors	Jeremy T. Fineman, Michael Mitzenmacher
Publisher	Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
ISBN (Electronic)	9783959770996
DOIs	https://doi.org/10.4230/OASIcs.SOSA.2019.5
State	Published - Jan 2019
Event	2nd Symposium on Simplicity in Algorithms, SOSA 2019 - San Diego, United States Duration: 8 Jan 2019 → 9 Jan 2019

Publication series

Name	OpenAccess Series in Informatics
Volume	69
ISSN (Print)	2190-6807

Conference

Conference	2nd Symposium on Simplicity in Algorithms, SOSA 2019
Country/Territory	United States
City	San Diego
Period	8/01/19 → 9/01/19

Funding

Funders	Funder number
Basic Algorithms Research Copenhagen
Villum Fonden
Blavatnik Family Foundation
European Commission	16582, 617951
Israel Science Foundation	1841-14
Israeli Centers for Research Excellence	4/11

Keywords

Selection
soft heap

Access to Document

10.4230/OASIcs.SOSA.2019.5

Cite this

Kaplan, H., Kozma, L., Zamir, O., & Zwick, U. (2019). Selection from heaps, row-sorted matrices, and X + Y using soft heaps. In J. T. Fineman, & M. Mitzenmacher (Eds.), 2nd Symposium on Simplicity in Algorithms, SOSA 2019 - Co-located with the 30th ACM-SIAM Symposium on Discrete Algorithms, SODA 2019 Article 5 (OpenAccess Series in Informatics; Vol. 69). Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing. https://doi.org/10.4230/OASIcs.SOSA.2019.5

Kaplan, Haim ; Kozma, László ; Zamir, Or et al. / Selection from heaps, row-sorted matrices, and X + Y using soft heaps. 2nd Symposium on Simplicity in Algorithms, SOSA 2019 - Co-located with the 30th ACM-SIAM Symposium on Discrete Algorithms, SODA 2019. editor / Jeremy T. Fineman ; Michael Mitzenmacher. Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing, 2019. (OpenAccess Series in Informatics).

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abstract = "We use soft heaps to obtain simpler optimal algorithms for selecting the k-th smallest item, and the set of k smallest items, from a heap-ordered tree, from a collection of sorted lists, and from X + Y , where X and Y are two unsorted sets. Our results match, and in some ways extend and improve, classical results of Frederickson (1993) and Frederickson and Johnson (1982). In particular, for selecting the k-th smallest item, or the set of k smallest items, from a collection of m sorted lists we obtain a new optimal “output-sensitive” algorithm that performs only O(m + Ʃmi=1 log(ki + 1)) comparisons, where ki is the number of items of the i-th list that belong to the overall set of k smallest items.",

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note = "Publisher Copyright: {\textcopyright} Haim Kaplan, L{\'a}szl{\'o} Kozma, Or Zamir, and Uri Zwick.; 2nd Symposium on Simplicity in Algorithms, SOSA 2019 ; Conference date: 08-01-2019 Through 09-01-2019",

year = "2019",

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Kaplan, H, Kozma, L, Zamir, O & Zwick, U 2019, Selection from heaps, row-sorted matrices, and X + Y using soft heaps. in JT Fineman & M Mitzenmacher (eds), 2nd Symposium on Simplicity in Algorithms, SOSA 2019 - Co-located with the 30th ACM-SIAM Symposium on Discrete Algorithms, SODA 2019., 5, OpenAccess Series in Informatics, vol. 69, Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing, 2nd Symposium on Simplicity in Algorithms, SOSA 2019, San Diego, United States, 8/01/19. https://doi.org/10.4230/OASIcs.SOSA.2019.5

Selection from heaps, row-sorted matrices, and X + Y using soft heaps. / Kaplan, Haim; Kozma, László; Zamir, Or et al.
2nd Symposium on Simplicity in Algorithms, SOSA 2019 - Co-located with the 30th ACM-SIAM Symposium on Discrete Algorithms, SODA 2019. ed. / Jeremy T. Fineman; Michael Mitzenmacher. Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing, 2019. 5 (OpenAccess Series in Informatics; Vol. 69).

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution › peer-review

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AB - We use soft heaps to obtain simpler optimal algorithms for selecting the k-th smallest item, and the set of k smallest items, from a heap-ordered tree, from a collection of sorted lists, and from X + Y , where X and Y are two unsorted sets. Our results match, and in some ways extend and improve, classical results of Frederickson (1993) and Frederickson and Johnson (1982). In particular, for selecting the k-th smallest item, or the set of k smallest items, from a collection of m sorted lists we obtain a new optimal “output-sensitive” algorithm that performs only O(m + Ʃmi=1 log(ki + 1)) comparisons, where ki is the number of items of the i-th list that belong to the overall set of k smallest items.

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Kaplan H, Kozma L, Zamir O , Zwick U. Selection from heaps, row-sorted matrices, and X + Y using soft heaps. In Fineman JT, Mitzenmacher M, editors, 2nd Symposium on Simplicity in Algorithms, SOSA 2019 - Co-located with the 30th ACM-SIAM Symposium on Discrete Algorithms, SODA 2019. Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing. 2019. 5. (OpenAccess Series in Informatics). doi: 10.4230/OASIcs.SOSA.2019.5

Selection from heaps, row-sorted matrices, and X + Y using soft heaps

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