## Strict Fibonacci Heaps

### BibTeX

@MISC{Brodal_strictfibonacci,

author = {Gerth Stølting Brodal and George Lagogiannis and Robert E. Tarjan},

title = {Strict Fibonacci Heaps},

year = {}

}

### OpenURL

### Abstract

Wepresentthefirstpointer-basedheapimplementationwith time bounds matching those of Fibonacci heaps in the worst case. We support make-heap, insert, find-min, meld and decrease-key in worst-case O(1) time, and delete and deletemin in worst-case O(lgn) time, where n is the size of the heap. The data structure uses linear space. A previous, very complicated, solution achieving the same time bounds in the RAM model made essential use of arrays and extensive use of redundant counter schemes to maintain balance. Our solution uses neither. Our key simplification is to discard the structure of the smaller heap when doing a meld. We use the pigeonhole principle in place of the redundant counter mechanism.

### Citations

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Citation Context ...ior specific permission and/or a fee. STOC.12, May 19–22, 2012, New York, New York, USA. Copyright 2012 ACM 978-1-4503-1245-5/12/05 ...$10.00. 1. INTRODUCTION Williams in 1964 introduced binary heaps =-=[25]-=-. Since then the design and analysis of heaps has been thoroughly investigated. The most common operations supported by the heaps in the literature are those listed below. We assume that each item sto... |

90 | A data structure for manipulating priority queues
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Citation Context ...ve logarithmic time for all the operations. Early examples are the implicit binary heaps of Williams [25], the leftist heaps of Crane [5] as modified by Knuth[20], and the binomial heaps of Vuillemin =-=[24]-=-. The introduction of Fibonacci heaps [15] by Fredman and Tarjan was a breakthrough since they achieved O(1) amortized time for all the operations above except for delete and delete-min, which require... |

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Citation Context ...trees give heap implementations that achieve logarithmic time for all the operations. Early examples are the implicit binary heaps of Williams [25], the leftist heaps of Crane [5] as modified by Knuth=-=[20]-=-, and the binomial heaps of Vuillemin [24]. The introduction of Fibonacci heaps [15] by Fredman and Tarjan was a breakthrough since they achieved O(1) amortized time for all the operations above excep... |

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Citation Context ...ed time for decrease-key. Heaps having amortized performance matching the amortized time complexities of Fibonacci heaps have also been presented. In particular, Driscoll, Gabow, Shrairman and Tarjan =-=[6]-=- proposed rank-relaxed heaps, Kaplan and Tarjan [19] presented thin heaps, Chan [4] introduced quake heaps, Haeupler, Sen and Tarjan introduced rank-pairing heaps [16], and Elmasry introduced violatio... |

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Citation Context ...t does notmaintain structuralinformation (like size or height) within its nodes, but still can adjust itself to perform efficiently. Within this framework, Sleator and Tarjan introduced the skew heap =-=[23]-=-, which was an amortized version of the leftist heap. They matched the complexity of Fibonacci heaps on all the operations except for decrease-key, which takes O(lgn) amortized time. The pairing heap,... |

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Citation Context .... Early examples are the implicit binary heaps of Williams [25], the leftist heaps of Crane [5] as modified by Knuth[20], and the binomial heaps of Vuillemin [24]. The introduction of Fibonacci heaps =-=[15]-=- by Fredman and Tarjan was a breakthrough since they achieved O(1) amortized time for all the operations above except for delete and delete-min, which require O(lgn) amortized time, where n is the num... |

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Citation Context ...i heaps, i.e. the time bounds of Fibonacci heaps should ideally be matched in the worst case. The next improvement after binomial heaps came with the the implicit heaps of Carlsson, Munro and Poblete =-=[3]-=- supporting worstcase O(1) time insertions and O(lgn) time deletions on a single heap stored in an array. Run-relaxedheaps [6] achieve the amortized bounds given by Fibonacci heaps in the worst case, ... |

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Citation Context ...ts parent. Heap-ordered trees give heap implementations that achieve logarithmic time for all the operations. Early examples are the implicit binary heaps of Williams [25], the leftist heaps of Crane =-=[5]-=- as modified by Knuth[20], and the binomial heaps of Vuillemin [24]. The introduction of Fibonacci heaps [15] by Fredman and Tarjan was a breakthrough since they achieved O(1) amortized time for all t... |

14 |
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Citation Context ...version of the binomial heap, andit achievedthesame time complexityas Fibonacci heaps except again for the decrease-key, the time complexity of which remained unknown for many years. In 1999, Fredman =-=[13]-=- proved that the lower bound for the decrease-key operation on pairing heaps is Ω(lglgn); thus the amortized performance of pairing heaps does not match the amortized performance of Fibonacci heaps. I... |

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Citation Context ... versions of pairing heaps, skew heaps, and skew-pairing heaps [8]. Some researchers, aiming to match the amortized bounds of Fibonacci heaps in a simpler way, followed different directions. Peterson =-=[21]-=- presented a structure based on AVL trees and Høyer [17] presented several structures, including ones based on red-black trees, AVL trees, and (a,b)-trees. Let us now review the progress on this probl... |

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Citation Context ...given by Fibonacci heaps in the worst case, with the exception of the meld operation, which is supported in O(lgn) time in the worst case. The same result was later also achieved by Kaplan and Tarjan =-=[18]-=- with fat heaps. Fat heaps without meld can be implemented on a pointer machine, but to support meld in O(lgn) time arrays are required. The meld operation was the next target for achieving constant t... |

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Citation Context ... in O(lgn) time arrays are required. The meld operation was the next target for achieving constant time in the worst case framework, in order to match the time complexities of Fibonacci heaps. Brodal =-=[1]-=- achieved O(1) worst case time for the meld operation on a pointer machine, but not for the decrease-key operation. It then became obvious that although the decrease-key and the meld operation can be ... |

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Citation Context ...aps remainedopen until now, and progress within the worst case framework has been accomplished only in other directions. In particular, Elmasry, Jensen and Katajainen presented two-tier relaxed heaps =-=[12]-=- in which the number of key comparisons is reduced to lgn + 3lglgn + O(1) per delete operation. They also presented [11] anew idea(which we adapt in this paper) for handling decrease-key operations by... |

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Citation Context ...though the decrease-key and the meld operation can be achieved in O(1) worst case time separately, it is very difficult to achieve constant time for both operations in the same data structure. Brodal =-=[2]-=- managed to solve this problem, but his solution is very complicated and requires the use of (extendable) arrays. For the pointer machine model of computation, the problem of matchingthetimeboundsofFi... |

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Citation Context ...e lower bound for the decrease-key operation on pairing heaps is Ω(lglgn); thus the amortized performance of pairing heaps does not match the amortized performance of Fibonacci heaps. In 2005, Pettie =-=[22]-=- proved that the time complexity of the decrease-key operation is 2 O(√ lglgn) . Later, Elmasry [9] gave a variant of pairing heaps that needs only O(lglgn) amortized time for decrease-key. Heaps havi... |

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Citation Context ... heaps [8]. Some researchers, aiming to match the amortized bounds of Fibonacci heaps in a simpler way, followed different directions. Peterson [21] presented a structure based on AVL trees and Høyer =-=[17]-=- presented several structures, including ones based on red-black trees, AVL trees, and (a,b)-trees. Let us now review the progress on this problem, based on the worst-case approach. The goal of worst-... |

6 |
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Citation Context ...ake heaps, Haeupler, Sen and Tarjan introduced rank-pairing heaps [16], and Elmasry introduced violation heaps [10]. Elmasry improved the number of comparisons of Fibonacci heaps by a constant factor =-=[7]-=- and also examined versions of pairing heaps, skew heaps, and skew-pairing heaps [8]. Some researchers, aiming to match the amortized bounds of Fibonacci heaps in a simpler way, followed different dir... |

5 |
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Citation Context ...introduced violation heaps [10]. Elmasry improved the number of comparisons of Fibonacci heaps by a constant factor [7] and also examined versions of pairing heaps, skew heaps, and skew-pairing heaps =-=[8]-=-. Some researchers, aiming to match the amortized bounds of Fibonacci heaps in a simpler way, followed different directions. Peterson [21] presented a structure based on AVL trees and Høyer [17] prese... |

5 | Pairing heaps with O(log log n) decrease cost
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Citation Context ...mance of pairing heaps does not match the amortized performance of Fibonacci heaps. In 2005, Pettie [22] proved that the time complexity of the decrease-key operation is 2 O(√ lglgn) . Later, Elmasry =-=[9]-=- gave a variant of pairing heaps that needs only O(lglgn) amortized time for decrease-key. Heaps having amortized performance matching the amortized time complexities of Fibonacci heaps have also been... |

3 |
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(Show Context)
Citation Context ... time complexities of Fibonacci heaps have also been presented. In particular, Driscoll, Gabow, Shrairman and Tarjan [6] proposed rank-relaxed heaps, Kaplan and Tarjan [19] presented thin heaps, Chan =-=[4]-=- introduced quake heaps, Haeupler, Sen and Tarjan introduced rank-pairing heaps [16], and Elmasry introduced violation heaps [10]. Elmasry improved the number of comparisons of Fibonacci heaps by a co... |

2 |
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Citation Context ...d rank-relaxed heaps, Kaplan and Tarjan [19] presented thin heaps, Chan [4] introduced quake heaps, Haeupler, Sen and Tarjan introduced rank-pairing heaps [16], and Elmasry introduced violation heaps =-=[10]-=-. Elmasry improved the number of comparisons of Fibonacci heaps by a constant factor [7] and also examined versions of pairing heaps, skew heaps, and skew-pairing heaps [8]. Some researchers, aiming t... |

2 |
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Citation Context ...coll, Gabow, Shrairman and Tarjan [6] proposed rank-relaxed heaps, Kaplan and Tarjan [19] presented thin heaps, Chan [4] introduced quake heaps, Haeupler, Sen and Tarjan introduced rank-pairing heaps =-=[16]-=-, and Elmasry introduced violation heaps [10]. Elmasry improved the number of comparisons of Fibonacci heaps by a constant factor [7] and also examined versions of pairing heaps, skew heaps, and skew-... |

1 | On the power of structural violations in priority queues
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(Show Context)
Citation Context ... In particular, Elmasry, Jensen and Katajainen presented two-tier relaxed heaps [12] in which the number of key comparisons is reduced to lgn + 3lglgn + O(1) per delete operation. They also presented =-=[11]-=- anew idea(which we adapt in this paper) for handling decrease-key operations by introducing structural violations instead of heap order violations. 1.1 Our contribution In this paper we present the f... |

1 |
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Citation Context ...They matched the complexity of Fibonacci heaps on all the operations except for decrease-key, which takes O(lgn) amortized time. The pairing heap, introduced by Fredman, Sedgewick, Sleator and Tarjan =-=[14]-=-, was the amortized version of the binomial heap, andit achievedthesame time complexityas Fibonacci heaps except again for the decrease-key, the time complexity of which remained unknown for many year... |

1 |
and Robert Endre Tarjan. Thin heaps, thick heaps
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(Show Context)
Citation Context ...rformance matching the amortized time complexities of Fibonacci heaps have also been presented. In particular, Driscoll, Gabow, Shrairman and Tarjan [6] proposed rank-relaxed heaps, Kaplan and Tarjan =-=[19]-=- presented thin heaps, Chan [4] introduced quake heaps, Haeupler, Sen and Tarjan introduced rank-pairing heaps [16], and Elmasry introduced violation heaps [10]. Elmasry improved the number of compari... |