Optimizing Binary Heaps

A priority queue—a data structure supporting, inter alia, the operations minimum (top), insert (push), and extract-min (pop)—is said to operate in-place if it uses O(1) extra space in addition to the n elements stored at the beginning of an array. Prior to this work, no in-place priority queue was known to provide worst-case guarantees on the number of element comparisons that are optimal up to additive constant terms for both insert and extract-min. In particular, for the standard implementation of binary heaps, insert and extract-min operate in logarithmic time while involving at most dlg ne and 2 lg n [could possibly be reduced to lg lg n + O(1) and lg n + log∗ n + O(1)] element comparisons, respectively. In this paper we propose a variant of a binary heap that operates in-place, executes minimum and insert in O(1) worst-case time, and extract-min in O(lg n) worst-case time while involving at c © 2017 Springer Science+Business Media, New York. This is the authors’ version of the work. The final publication is available at Springer via http: //dx.doi.org/10.1007/s00224-017-9760-2 A preliminary version of this paper entitled “An in-place priority queue with O(1) time for push and lgn + O(1) comparisons for pop” [19] was presented at the 10th International Computer Science Symposium in Russia held in Listvyanka in July 2015. Stefan Edelkamp Faculty 3—Mathematics and Computer Science, University of Bremen, PO Box 330 440, 28334 Bremen, Germany E-mail: edelkamp@tzi.de Amr Elmasry Department of Computer Engineering and Systems, Alexandria University, Alexandria 21544, Egypt E-mail: elmasry@alexu.edu.eg Jyrki Katajainen Department of Computer Science, University of Copenhagen, Universitetsparken 5, 2100 Copenhagen East, Denmark E-mail: jyrki@di.ku.dk most lg n+O(1) element comparisons. These efficiencies surpass lower bounds known for binary heaps, thereby resolving a long-standing theoretical debate.