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...tice [2]. Fredman and Tarjan [11] invented the Fibonacci heap specifically to support key decrease operations in O(1) time, which allows efficient implementation of Dijkstra’s shortest path algorithm =-=[3, 11]-=-, Edmonds’ minimum branching algorithm [6, 13], and certain minimum spanning tree algorithms [11, 13]. Fibonacci heaps support deletion of the minimum or of an arbitrary item in O(log n) amortized tim...

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...ed to our work. The binomial queue of Vuillemin [29] supports all the heap operations in O(log n) worst-case time per operation. This structure performs quite well in practice [2]. Fredman and Tarjan =-=[11]-=- invented the Fibonacci heap specifically to support key decrease operations in O(1) time, which allows efficient implementation of Dijkstra’s shortest path algorithm [3, 11], Edmonds’ minimum branchi...

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...tion of a heap. (See Figure 1c.) In the half-ordered representation we do not need to move items among nodes as matches take place, so the items can be the nodes: the data structure can be endogenous =-=[27]-=-. Henceforth we shall assume an endogenous representation. We obtain our fourth and final representation, the heap-ordered representation, by viewing a half tree as the binary tree representation of a...

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...lly presented in the heap-ordered representation. In their paper on pairing heaps, Fredman et al. [10] described the half-ordered representation and observed that it is the binary tree representation =-=[20]-=- of a heap-ordered tree. Later, Dutton [5] used the half-ordered representation in his weak-heap data structure, and Høyer [15] proposed various kinds of half-ordered balanced trees as heaps. The vers...

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...) amortized time, and each of the other supported heap operations takes O(1) amortized time. These bounds match the lower bound. (The logarithmic lower bound can be beaten by using multiway branching =-=[12, 14]-=-.) Many heap implementations have been proposed over the years. See e.g. [19]. We mention only those directly related to our work. The binomial queue of Vuillemin [29] supports all the heap operations...

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...g x. We shall assume that all keys are distinct; if they are not, we can break ties using any total order of the items. We allow only binary comparisons of keys, and we study the amortized efficiency =-=[28]-=- of heap operations. To obtain a bound on amortized efficiency, we assign to each configuration of the data structure a non-negative potential, initially zero. We define the amortized time of an opera...

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... Fibonacci heap specifically to support key decrease operations in O(1) time, which allows efficient implementation of Dijkstra’s shortest path algorithm [3, 11], Edmonds’ minimum branching algorithm =-=[6, 13]-=-, and certain minimum spanning tree algorithms [11, 13]. Fibonacci heaps support deletion of the minimum or of an arbitrary item in O(log n) amortized time and the other heap operations in O(1) amorti...

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... by using multiway branching [12, 14].) Many heap implementations have been proposed over the years. See e.g. [19]. We mention only those directly related to our work. The binomial queue of Vuillemin =-=[29]-=- supports all the heap operations in O(log n) worst-case time per operation. This structure performs quite well in practice [2]. Fredman and Tarjan [11] invented the Fibonacci heap specifically to sup...

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...10]. Pairing heaps support all the heap operations in O(log n) amortized time and were conjectured to support key decrease in O(1) amortized time. Despite empirical evidence supporting the conjecture =-=[16, 22, 26]-=-, Fredman [9] showed that it is not true: pairing heaps and related data structures that do not store subtree size information require Ω(log log n) amortized time per key decrease. Whether pairing hea...

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...ortized time and the other heap operations in O(1) amortized time. They do not perform well in practice, however [22, 23]. As a result, a variety of alternatives to Fibonacci heaps have been proposed =-=[4, 7, 15, 18, 19, 24]-=-, including a self-adjusting structure, the pairing heap [10]. Pairing heaps support all the heap operations in O(log n) amortized time and were conjectured to support key decrease in O(1) amortized t...

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...perform well in practice, however [22, 23]. As a result, a variety of alternatives to Fibonacci heaps have been proposed [4, 7, 15, 18, 19, 24], including a self-adjusting structure, the pairing heap =-=[10]-=-. Pairing heaps support all the heap operations in O(log n) amortized time and were conjectured to support key decrease in O(1) amortized time. Despite empirical evidence supporting the conjecture [16...

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...nly those directly related to our work. The binomial queue of Vuillemin [29] supports all the heap operations in O(log n) worst-case time per operation. This structure performs quite well in practice =-=[2]-=-. Fredman and Tarjan [11] invented the Fibonacci heap specifically to support key decrease operations in O(1) time, which allows efficient implementation of Dijkstra’s shortest path algorithm [3, 11],...

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...10]. Pairing heaps support all the heap operations in O(log n) amortized time and were conjectured to support key decrease in O(1) amortized time. Despite empirical evidence supporting the conjecture =-=[16, 22, 26]-=-, Fredman [9] showed that it is not true: pairing heaps and related data structures that do not store subtree size information require Ω(log log n) amortized time per key decrease. Whether pairing hea...

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...ing the data structure more complicated: run-relaxed heaps [4] and fat heaps [18] achieve these bounds except for melding, which takes O(log n) time worst-case; a very complicated structure of Brodal =-=[1]-=- achieves these bounds for all the heap operations. Our goal is to systematically explore the design space of amortized-efficient heaps and thereby discover the simplest possible data structures. As a...

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...) amortized time, and each of the other supported heap operations takes O(1) amortized time. These bounds match the lower bound. (The logarithmic lower bound can be beaten by using multiway branching =-=[12, 14]-=-.) Many heap implementations have been proposed over the years. See e.g. [19]. We mention only those directly related to our work. The binomial queue of Vuillemin [29] supports all the heap operations...

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... Fibonacci heaps support deletion of the minimum or of an arbitrary item in O(log n) amortized time and the other heap operations in O(1) amortized time. They do not perform well in practice, however =-=[22, 23]-=-. As a result, a variety of alternatives to Fibonacci heaps have been proposed [4, 7, 15, 18, 19, 24], including a self-adjusting structure, the pairing heap [10]. Pairing heaps support all the heap o...

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...port all the heap operations in O(log n) amortized time and were conjectured to support key decrease in O(1) amortized time. Despite empirical evidence supporting the conjecture [16, 22, 26], Fredman =-=[9]-=- showed that it is not true: pairing heaps and related data structures that do not store subtree size information require Ω(log log n) amortized time per key decrease. Whether pairing heaps meet this ...

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...and melds occur, one needs a separate data structure to maintain the partition of items into heaps. With such a data structure, the time to find the heap containing a given item is small but not O(1) =-=[17]-=-. Fibonacci heaps [11] were invented specifically to support key decrease efficiently, but they require two extra pointers per node (in the heap-ordered representation, to the parent and previous sibl...

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... data structures that do not store subtree size information require Ω(log log n) amortized time per key decrease. Whether pairing heaps meet this bound is open; the best upper bound is O(22 √ lg lgn) =-=[25]-=-5. Very recently ElMasry [8] proposed a more-complicated alternative to pairing heaps that does have an O(log log n) amortized bound per key decrease. Fredman’s result gives a time-space trade-off bet...

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...ortized time and the other heap operations in O(1) amortized time. They do not perform well in practice, however [22, 23]. As a result, a variety of alternatives to Fibonacci heaps have been proposed =-=[4, 7, 15, 18, 19, 24]-=-, including a self-adjusting structure, the pairing heap [10]. Pairing heaps support all the heap operations in O(log n) amortized time and were conjectured to support key decrease in O(1) amortized t...

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...ortized time and the other heap operations in O(1) amortized time. They do not perform well in practice, however [22, 23]. As a result, a variety of alternatives to Fibonacci heaps have been proposed =-=[4, 7, 15, 18, 19, 24]-=-, including a self-adjusting structure, the pairing heap [10]. Pairing heaps support all the heap operations in O(log n) amortized time and were conjectured to support key decrease in O(1) amortized t...

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... Fibonacci heaps support deletion of the minimum or of an arbitrary item in O(log n) amortized time and the other heap operations in O(1) amortized time. They do not perform well in practice, however =-=[22, 23]-=-. As a result, a variety of alternatives to Fibonacci heaps have been proposed [4, 7, 15, 18, 19, 24], including a self-adjusting structure, the pairing heap [10]. Pairing heaps support all the heap o...

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...ortized time and the other heap operations in O(1) amortized time. They do not perform well in practice, however [22, 23]. As a result, a variety of alternatives to Fibonacci heaps have been proposed =-=[4, 7, 15, 18, 19, 24]-=-, including a self-adjusting structure, the pairing heap [10]. Pairing heaps support all the heap operations in O(log n) amortized time and were conjectured to support key decrease in O(1) amortized t...

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...tized time. These bounds match the lower bound. (The logarithmic lower bound can be beaten by using multiway branching [12, 14].) Many heap implementations have been proposed over the years. See e.g. =-=[19]-=-. We mention only those directly related to our work. The binomial queue of Vuillemin [29] supports all the heap operations in O(log n) worst-case time per operation. This structure performs quite wel...

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...ation. In their paper on pairing heaps, Fredman et al. [10] described the half-ordered representation and observed that it is the binary tree representation [20] of a heap-ordered tree. Later, Dutton =-=[5]-=- used the half-ordered representation in his weak-heap data structure, and Høyer [15] proposed various kinds of half-ordered balanced trees as heaps. The version of Fredman et al. [10] matches Knuth’s...

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...ortized time and the other heap operations in O(1) amortized time. They do not perform well in practice, however [22, 23]. As a result, a variety of alternatives to Fibonacci heaps have been proposed =-=[4, 7, 15, 18, 19, 24]-=-, including a self-adjusting structure, the pairing heap [10]. Pairing heaps support all the heap operations in O(log n) amortized time and were conjectured to support key decrease in O(1) amortized t...

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...store subtree size information require Ω(log log n) amortized time per key decrease. Whether pairing heaps meet this bound is open; the best upper bound is O(22 √ lg lgn) [25]5. Very recently ElMasry =-=[8]-=- proposed a more-complicated alternative to pairing heaps that does have an O(log log n) amortized bound per key decrease. Fredman’s result gives a time-space trade-off between the number of bits per ...