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Rank-Pairing Heaps
"... We introduce the rank-pairing heap, a heap (priority queue) implementation that combines the asymptotic efficiency of Fibonacci heaps with much of the simplicity of pairing heaps. Unlike all other heap implementations that match the bounds of Fibonacci heaps, our structure needs only one cut and n ..."
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Cited by 5 (3 self)
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We introduce the rank-pairing heap, a heap (priority queue) implementation that combines the asymptotic efficiency of Fibonacci heaps with much of the simplicity of pairing heaps. Unlike all other heap implementations that match the bounds of Fibonacci heaps, our structure needs only one cut and no other structural changes per key decrease; the trees representing the heap can evolve to have arbitrary structure. Our initial experiments indicate that rank-pairing heaps perform almost as well as pairing heaps on typical input sequences and better on worst-case sequences.
Violation heaps: A better substitute for Fibonacci heaps
, 2008
"... We give a priority queue that achieves the same amortized bounds as Fibonacci heaps. Namely, find-min requires O(1) worst-case time, insert, meld and decrease-key require O(1) amortized time, and delete-min requires O(log n) amortized time. Our structure is simple and promises a more efficient pract ..."
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Cited by 3 (0 self)
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We give a priority queue that achieves the same amortized bounds as Fibonacci heaps. Namely, find-min requires O(1) worst-case time, insert, meld and decrease-key require O(1) amortized time, and delete-min requires O(log n) amortized time. Our structure is simple and promises a more efficient practical behavior compared to any other known Fibonacci-like heap.
Fibonacci heaps revisited
- CoRR
"... The Fibonacci heap is a classic data structure that supports deletions in logarithmic amortized time and all other heap operations in O(1) amortized time. We explore the design space of this data structure. We propose a version with the following improvements over the original: (i) Each heap is repr ..."
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Cited by 1 (1 self)
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The Fibonacci heap is a classic data structure that supports deletions in logarithmic amortized time and all other heap operations in O(1) amortized time. We explore the design space of this data structure. We propose a version with the following improvements over the original: (i) Each heap is represented by a single heap-ordered tree, instead of a set of trees. (ii) Each decrease-key operation does only one cut and a cascade of rank changes, instead of doing a cascade of cuts. (iii) The outcomes of all comparisons done by the algorithm are explicitly represented in the data structure, so none are wasted. We also give an example to show that without cascading cuts or rank changes, both the original data structure and the new version fail to have the desired efficiency, solving an open problem of Fredman. Finally, we illustrate the richness of the design space by proposing several alternative ways to do cascading rank changes, including a randomized strategy related to one previously proposed by Karger. We leave the analysis of these alternatives as intriguing open problems.
A back–to–basics empirical study of priority queues
, 2013
"... The theory community has proposed several new heap variants in the recent past which have remained largely untested experimentally. We take the field back to the drawing board, with straightforward implementations of both classic and novel structures using only standard, well-known optimizations. We ..."
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The theory community has proposed several new heap variants in the recent past which have remained largely untested experimentally. We take the field back to the drawing board, with straightforward implementations of both classic and novel structures using only standard, well-known optimizations. We study the behavior of each structure on a variety of inputs, including artificial workloads, workloads generated by running algorithms on real map data, and workloads from a discrete event simulator used in recent systems networking research. We provide observations about which characteristics are most correlated to performance. For example, we find that the L1 cache miss rate appears to be strongly correlated with wallclock time. We also provide observations about how the input sequence affects the relative performance of the different heap variants. For example, we show (both theoretically and in practice) that certain random insertion-deletion sequences are degenerate and can lead to misleading results. Overall, our findings suggest that while the conventional wisdom holds in some cases, it is sorely mistaken in others. 1
Hollow Heaps
"... We introduce the hollow heap, a very simple data structure with the same amortized efficiency as the classical Fibonacci heap. All heap operations except delete and delete-min take O(1) time, worst case as well as amortized; delete and delete-min take O(log n) amortized time. Hollow heaps are by far ..."
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We introduce the hollow heap, a very simple data structure with the same amortized efficiency as the classical Fibonacci heap. All heap operations except delete and delete-min take O(1) time, worst case as well as amortized; delete and delete-min take O(log n) amortized time. Hollow heaps are by far the simplest structure to achieve this. Hollow heaps combine two novel ideas: the use of lazy deletion and re-insertion to do decrease-key operations, and the use of a dag (directed acyclic graph) instead of a tree or set of trees to represent a heap. Lazy deletion produces hollow nodes (nodes without items), giving the data structure its name.
Heaps Simplified
, 2009
"... The heap is a basic data structure used in a wide variety of applications, including shortest path and minimum spanning tree algorithms. In this paper we explore the design space of comparison-based, amortized-efficient heap implementations. From a consideration of dynamic single-elimination tourn ..."
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The heap is a basic data structure used in a wide variety of applications, including shortest path and minimum spanning tree algorithms. In this paper we explore the design space of comparison-based, amortized-efficient heap implementations. From a consideration of dynamic single-elimination tournaments, we obtain the binomial queue, a classical heap implementation, in a simple and natural way. We give four equivalent ways of representing heaps arising from tour-naments, and we obtain two new variants of binomial queues, a one-tree version and a one-pass version. We extend the one-pass version to support key decrease operations, obtaining the rank-pairing heap, or rp-heap. Rank-pairing heaps combine the performance guarantees of Fibonacci heaps with simplicity approaching that of pairing heaps. Like pairing heaps, rank-pairing heaps consist of trees of arbitrary structure, but these trees are combined by rank, not by list position, and rank changes, but not structural changes, cascade during key decrease operations.
A note on meldable heaps relying on data-structural bootstrapping
, 2009
"... We introduce a meldable heap which guarantees the worst-case cost of O(1) for find-min, insert, and meld with at most 0, 3, and 3 element comparisons for the respective operations; and the worst-case cost of O(lg n) with at most 3 lgn+ O(1) element comparisons for delete. Our data structure is asymp ..."
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We introduce a meldable heap which guarantees the worst-case cost of O(1) for find-min, insert, and meld with at most 0, 3, and 3 element comparisons for the respective operations; and the worst-case cost of O(lg n) with at most 3 lgn+ O(1) element comparisons for delete. Our data structure is asymptotically optimal and nearly constant-factor optimal with respect to the comparison complexity of all the meldable-heap operations. Furthermore, the data structure is also simple and elegant.
