Elsevier

Journal of Discrete Algorithms

Volume 16, October 2012, Pages 187-205
Journal of Discrete Algorithms

The weak-heap data structure: Variants and applications

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Abstract

The weak heap is a priority queue that was introduced as a competitive structure for sorting. Its array-based form supports the operations find-min in O(1) worst-case time, and insert and delete-min in O(lgn) worst-case time using at most lgn element comparisons. Additionally, its pointer-based form supports delete and decrease in O(lgn) worst-case time using at most lgn element comparisons. In this paper we enhance this data structure as follows:

  • 1.

    We improve the array-based form to support insert in O(1) amortized time. The main idea is to temporarily store the inserted elements in a buffer, and, once the buffer is full, to move its elements to the heap using an efficient bulk-insertion procedure. As an application, we use this variant in the implementation of adaptive heapsort. Accordingly, we guarantee, for several measures of disorder, that the formula expressing the number of element comparisons performed by the algorithm is optimal up to the constant factor of the high-order term. Unlike other previous constant-factor-optimal adaptive sorting algorithms, adaptive heapsort relying on the developed priority queue is practically workable.

  • 2.

    We improve the pointer-based form to support insert and decrease in O(1) worst-case time per operation. The expense is that delete then requires at most 2lgn element comparisons, but this is still better than the 3lgn bound known for run-relaxed heaps. The main idea is to allow some nodes to violate the weak-heap ordering; we call the resulting priority queue a relaxed weak heap. We also develop a more efficient amortized variant that provides delete guaranteeing an amortized bound of 1.5lgn element comparisons, which is better than the 2logϕn bound known for Fibonacci heaps, where ϕ is the golden ratio. As an application, we use this variant in the implementation of Dijkstraʼs shortest-paths algorithm. Experimental results indicate that weak heaps are practically efficient; they are competitive with other priority-queue structures when considering the number of element comparisons performed, and lose by a small margin when considering the actual running time.

Keywords

Data structures
Priority queues
Weak heaps
Adaptive sorting
Shortest paths
Comparison complexity

Cited by (0)

The material on adaptive sorting was presented at the 22nd International Workshop on Combinatorial Algorithms held in Victoria, Canada, in June 2011; and the material on shortest paths was presented at the 18th Computing: The Australasian Theory Symposium in Melbourne, Australia, in February 2012.