Williams, J. W. J. Algorithm 232: Heapsort. Communications of the ACM 7 (1964), 347-348.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....the cost of heap construction using Floyd s algorithm and other variants is asymptotically normal. 1 Introduction Heap is an elementary data structure often used in applications concerned with priority queues and partial (as well as total) ordering. It rst appeared in Williams Heapsort algorithm [30], which happened to be the rst in place O(n log n) sorting algorithm. Besides its original applications to sorting, heap has wide applications in algorithm design, see Aho et al. 1] It serves as the prototype, both conceptually and in actual implementations, of many complex data structures in ....

....algorithms in [3, 19, 12, 29] the basic ideas of improvement being more or less due to Floyd. Our approach also gives in most cases more precise quantitative results in the form of (22) In connection with this, it should be pointed out that the original on line algorithm proposed by Williams [30] to construct a heap is not linear in the worst case and that the randomness is not guaranteed (see [24] in each step. The average case analysis of its behavior is more dicult; see [2, 8, 12] Likewise, a precise analysis of the expected behavior of heapsort is very involved since successive ....

J. W. J. Williams, Algorithm 232: Heapsort, Communications of the ACM, 7 (1964) 347-348. 22

....search engines, and reliability ratings for internet sites. ffl The core of job schedulers in operating systems for thread or process management, and event schedulers in production and simulation applications are priority queues. Heaps are ideally suited to implement sequential priority queues [33]. A heap is a binary tree with the property that the values stored at any node are always greater than or equal to values stored at both child nodes (if they exist) All levels, except the last, are completely filled. Several parallel priority queues, such as calendar queues, have been proposed ....

J.W. Williams. Algorithm 232: Heapsort. Comm. ACM, 7, pp. 347--348, 1964.

....cost of heap construction using Floyd s algorithm and other variants is asymptotically normal. 1 Introduction Heap is an elementary data structure often used in applications concerned with priority queues and partial (as well as total) ordering. It first appeared in Williams Heapsort algorithm [30], which happened to be the first in place O(n log n) sorting algorithm. Besides its original applications to sorting, heap has wide applications in algorithm design, see Aho et al. 1] It serves as the prototype, both conceptually and in actual implementations, of many complex data structures in ....

....algorithms in [3, 19, 12, 29] the basic ideas of improvement being more or less due to Floyd. Our approach also gives in most cases more precise quantitative results in the form of (22) In connection with this, it should be pointed out that the original on line algorithm proposed by Williams [30] to construct a heap is not linear in the worst case and that the randomness is not guaranteed (see [24] in each step. The average case analysis of its behavior is more di#cult; see [2, 8, 12] Likewise, a precise analysis of the expected behavior of heapsort is very involved since successive ....

J. W. J. Williams, Algorithm 232: Heapsort, Communications of the ACM, 7 (1964) 347--348. 22

....Mauchly (1946) for inventing this organization of the keys. Also, the keys can be suitable permuted for obtaining the heap of Williams and Floyd (1964) thus permitting to identify the current maximum key in constant time and supporting insertions and deletions of individual keys in O(log n) time [26, 10]. Indeed, sorted arrays and heaps are well known examples of dictionaries. In the dictionary problem a set of n distinct keys a 1 , a 2 , an is maintained over a total order, in which the only operations allowed on the keys are reads writes and comparisons using the standard RAM model of ....

J. W. J. Williams. Algorithm 232: Heapsort. Communications of the ACM, 7:347-348, 1964.

....model, in order to sort and array of n elements, an algorithm must perform at least n log n n log e comparisons in the worst case. For the number of moves the lower bound in the worst case is #3 2n# [MR96] An in place algorithm uses O(1) auxiliary memory cells. The well known Heapsort [Wil64] was the first in place sorting algorithm performing O(n log n) comparisons in the worst case. The number of element moves is O(n log n) and so it is not optimal. On the other hand, the classical selection sort and the algorithm of Munro and Raman [MR92] are in place and perform O(n) data moves ....

J. W. J. Williams. Algorithm 232: Heapsort. Communications of the ACM, 7:347--348, 1964.

....arrive at a signi cantly simpler structure. Another improvement in our data structure is to avoid the use of pointers. The term implicit is often used for pointer free implementations of trees and other data structures which are normally pointer based. One early example is the heap of Williams [23]. There is a large body of work dealing with implicit data structures, see e.g. 7, 11, 18] and the references therein. In that work, the term implicit is often de ned as using only space for the n elements stored, plus O(1) additional space. In the present paper, we will abuse the terminology a ....

....for a complete tree with height four. We now discuss how to calculate the position of the children of a node v at position i in the implicit layouts. For the BFS layout, the children are at position 2i and 2i 1 a fact exploited already in the 1960s in the design of the implicit binary heap [23]. For the DFS layout, the two children are at DFS 2 3 4 5 6 7 8 9 10 11 12 13 inorder 4 2 1 3 6 5 7 12 10 9 11 14 13 15 BFS 2 4 8 9 5 10 11 3 6 12 13 7 van Emde Boas 2 4 5 6 7 8 9 3 10 11 12 13 14 15 Figure 3: The DFS, inorder, BFS, and van Emde Boas ....

J. W. J. Williams. Algorithm 232: Heapsort. Communications of the ACM, 7(6):347-348, 1964. 20

....arrive at a significantly simpler structure. Another improvement in our data structure is to avoid the use of pointers. The term implicit is often used for pointer free implementations of trees and other data structures which are normally pointer based. One early example is the heap of Williams [23]. There is a large body of work dealing with implicit data structures, see e.g. 7, 11, 18] and the references therein. In that work, the term implicit is often defined as using only space for the n elements stored, plus O(1) additional space. In the present paper, we will abuse the terminology a ....

....4 5 6 7 8 9 3 10 11 12 13 14 15 Figure 3: The DFS, inorder, BFS, and van Emde Boas layouts for a complete tree with height 4. Numbers designate positions in the array of node records position 2i and 2i 1 a fact exploited already in the 1960s in the design of the implicit binary heap [23]. For the DFS layout, the two children are at positions i 1 and i 2 , and in the inorder layout the two children are at positions i and i 2 . For the implicit van Emde Boas layout the computations are more involved. Our solution is based on the fact that if we for a node in the ....

J. W. J. Williams. Algorithm 232: Heapsort. Communications of the ACM, 7(6):347--348, 1964. 20

....structures A heap ordered tree is a tree in which each node stores an element and has a key. Every node other than the root satisfies that its key is greater than or equal to the key at its parent. Heap ordered trees have been widely studied and are the basic structure of many priority queues [156, 42, 150, 44]. In this section we utilize heap ordered trees to construct two data structures, the heap tree and the colored heap tree, that are useful in our application of finding pairs with lower bounded gap but might also have applications elsewhere. A heap tree stores a collection of elements with ....

J. W. J. Williams. Algorithm 232: Heapsort. Communications of the ACM, 7(6):347--348, 1964.

....and returns the element. Delete(Q; e) Deletes element e from priority queue Q provided that it is known where e is stored in Q (priority queues do not support the searching for an element) The implementation of priority queues is a classical problem in data structures. A few references are [13, 12, 8, 7, 5, 6, 10]. In the amortised sense, 11] the best performance is achieved by binomial heaps [12] They support Delete and DeleteMin in amortised time O(log n) and all other operations in amortised constant time. If we want to perform Insert in worst case constant time two efficient data structures exist. ....

J. W. J. Williams. Algorithm 232: Heapsort. Communications of the ACM, 7(6):347--348,

....su#x tree. A heap ordered tree is a tree in which each node stores an element and has a key. Every node other than the root satisfies that its key is greater than or equal to the key at its parent. Heap ordered trees have been widely studied and are the basic structure of many priority queues [204, 56, 195, 62]. In this section we utilize heap ordered trees to construct two data structures, the heap tree and the colored heap tree, that are useful in our application of finding pairs with lower bounded gap but might also have applications elsewhere. A heap tree stores a collection of elements with ....

J. W. J. Williams. Algorithm 232: Heapsort. Communications of the ACM, 7(6):347--348, 1964.

....leaves have depth w, if the root has depth zero. The leaves are numbered from left to right [2 w ] and the internal nodes of T are numbered 1; n 1. The root is the first node and the children of node v are nodes 2v and 2v 1, i.e. like the numbering of nodes in an implicit binary heap [11, 25]. Figure 1 shows the numbering of the nodes for the case w = 4. The tree T has the following properties (see [15] Fact 1 The depth of an internal node v is msb(v) and the d th ancestor of v is v # d, for 0 d depth(v) The parent of leaf a is the internal node 2 w 1 (a # 1) for 0 a ....

J. W. J. Williams. Algorithm 232: Heapsort. Communications of the ACM, 7(6):347--348, 1964. 12

....children of the node, except the heavy child, with priorities equal to their sizes. The priority queues should support the insertion of an element with arbitrary priority and the deletion of the element with maximum priority in logarithmic time, and construction in linear time, e.g. binary heaps [20, 54]. The total time for constructing the initial priority queues at the nodes is O(n) 4.1 Separator Trees 39 T 1 R bot R top r w 3 z w 2 w 1 u T 2 T k Figure 4.2: The separator node u on the heavy path R = R top [ fug [ R bot , and the nodes w1 ; w where to update the ....

J.W.J. Williams. Algorithm 232: Heapsort. Communications of the ACM, 7(6):347-348, 1964.

....arrive at a significantly simpler structure. Another improvement in our data structure is to avoid the use of pointers. The term implicit is often used for pointer free implementations of trees and other data structures which are normally pointer based. One early example is the heap of Williams [23]. There is a large body of work dealing with implicit data structures, see e.g. 7, 11, 18] and the references therein. In that work, the term implicit is often defined as using only space for the n elements stored, plus O(1) additional space. In the present paper, we will abuse the terminology a ....

....15 Figure 2: The DFS, inorder, BFS, and van Emde Boas layouts for a complete tree with height 4. Numbers designate positions in the array of node records For the BFS layout, the children are at position 2i and 2i 1 a fact exploited already in the 1960s in the design of the implicit binary heap [23]. For the DFS layout, the two children are at positions i 1 and i 2 h(v) 1 , and in the inorder layout the two children are at positions i 2 h(v) 2 and i 2 h(v) 2 . For the implicit van Emde Boas layout the computations are more involved. Our solution is based on the fact that if ....

J. W. J. Williams. Algorithm 232: Heapsort. Communications of the ACM, 7(6):347--348, 1964.

....queue, to support the operations of the fast process while Sect. 4 concludes the paper. 2 Previous Work A number of solutions for the priority queue problem can be used to solve the time queue problem for one process with only small modifications if any. The standard heap described by Williams [18] can be modified to use fingers by adding a dictionary that stores the position in the heap for each element. The heap solution (heap in Table 1) even works if the maximum duration is unbounded and it only needs O(N) space. The model used is the pointer machine model [11] Van Emde Boas et al. ....

J. W. J. Williams. Algorithm 232: Heapsort. Communications of the ACM, 7:347-- 348, 1964.

....of the node, except the heavychild, with priorities equal to their sizes. The priority queues should support insertion of an element with arbitrary priority and deletion of the element with maximum priority in logarithmic time, and construction of a queue in linear time, as e.g. binary heaps [9, 26] do. The total time for constructing the initial priority queues at the nodes is O(n) We nd the root of the 1=2 separator tree S T using the search tree R storing the heavy path starting at the root r of T . We rst observe that jrj is the maximal key in R, which can be found in time O(log n) ....

J. W. J. Williams. Algorithm 232: Heapsort. Communications of the ACM, 7(6):347348, 1964. 25

....queue abstract data type in the heap data structure. One way to present the heap sort algorithm is to show that we can construct a heap storing n keys (or keyelement pairs) in O(n log n) time by means of n successive insertion operations, each taking O(log n) time, starting from an empty heap [14]. However, if all the keys to be stored in the heap are known in advance, there is an alternative bottom up construction method, which runs in O(n) time [6] This is a more efficient construction algorithm that can be included as one of the constructors in a Heap class. Bottom up heap ....

J. W. J. Williams. Algorithm 232: Heapsort. Communications of the ACM, 7(6):347--348, 1964.

....order of the tasks on the various priority queues remains unchanged. Therefore, a priority queue data structure that allows the actual key values to change relative to one another without requiring the data structure to be updated is most appropriate for dynamic priority scheduling, e.g. a heap [40]. While such a data structure may resolve the priority queue synchronization problem, it does not address the problem that the task priority values may be continually changing. One scheme to solve the shifting priorities problem is to assign priority values that do not need to change when new ....

....be a doubly linked list. While Max Min is constant time, insert and delete have O(n) worst case execution time, where n is the number of nodes, which is unacceptable for reasonably sized queues. Heap is a complete binary tree such that every parent node has higher priority than its children [40]. A heap is typically implemented using an array, with the root of the tree as the rst element of the array. Max Min is constant time, and insert and delete have O(lg n) worst case execution time. Ecient implementations yielding reasonably good performance exist for heaps. D tree is an extension ....

J. W. J. Williams. Algorithm 232: Heapsort. Commun. ACM, 7:347-348, 1964.

....that A 3ph Gamma ALA 1 1= L Gammadlog(n dlog ne GammaL)e Gamma1 . Now, let us consider the time and space complexities. Phase 1 can be implemented in O(n log n) time and O(n) space using Hu Tucker s algorithm [6] Phase 2 can be implemented in O(n log n) time and O(n) space using Heapsort [19] for sorting and BRCI algorithm to generarate the codewords lenghts. Finally, phase 3 can be implemented in both O(n) time and space. An immediate consequence of the previous theorem is an upper bound for the inefficiency of optimal L restricted ABPC rather than Huffman codes. Since ALA A 3ph , ....

J. W. J. Williams. Algorithm 232 : HEAPSORT. Comm. ACM, 7:347--348, 1964. 5

....should be doubly linked. Unfortunately, despite the fact that Max Min is constant time, i.e. the rst element in the list, insert and delete have O(n) worst case execution time, where n is the number of nodes. This performance is not acceptable for reasonable sized queues. Heaps A binary heap [67] based priority queue is a complete binary tree satisfying the property that every parent node has higher priority than its children. A heap is typically implemented using an array, with the root of the tree as the rst element of the array. With a heap, Max Min is constant time, i.e. the rst ....

Williams, J. W. J. Algorithm 232: Heapsort. Communications of the ACM 7 (1964), 347-348.

....that a full binary tree is of cardinality divisible by 3 iff its height is odd. This is because of quite a simple number theoretic fact, that 2 n Gamma 1 is divisible by 3 iff n is even. Hence, the above reasoning proves the following: 1 The idea of this representation actually goes back to [17]. Lemma 1 If the property of full binary trees to be of even height is not expressible in the language of first order logic over the signature h ; D;D 0 i, then the property of finite segments of natural numbers to be of cardinality divisible by 3 is not expressible over the signature Theta ....

Williams, J. W. J., Algorithm 232: Heapsort, Comm. ACM 7:6, 347--348, 1964.

....Saarbrucken, Germany. This work was partially supported by the EU ESPRIT LTR project No. 20244 (ALCOM IT) Email: shiva mpi sb.mpg.de. z Tata Institute of Fundamental Research, Mumbai, India. Email: jaikumar tcs.tifr.res.in. Implementation Insert Delete FindMin Doubly linked list 1 1 n Heap [8] log n log n 1 Search tree [5, 7] log n 1 1 Priority queue [2, 3, 4] 1 log n 1 Figure 1: Worst case asymptotic time bounds for different set implementations. the worst case times of the two update operations Insert, Delete and the query operation FindMin. We prove the following lower bound on ....

J. W. J. Williams. Algorithm 232: Heapsort. Communications of the ACM, 7(6):347--348, 1964.

....e and it is known where e is stored in Q. DeleteMin(Q) deletes and returns the minimum element from priority queue Q. Delete(Q; e) deletes element e from priority queue Q provided it is known where e is stored in Q. The construction of priority queues is a classical topic in data structures [1, 2, 3, 4, 5, 6, 7, 10, 12, 15, 16, 17]. A historical overview of implementations can be found in [11] There are many applications of priority queues. Two of the most prominent examples are sorting problems and network optimization problems [13] This work was partially supported by the ESPRIT II Basic Research Actions Program of ....

J. W. J. Williams. Algorithm 232: Heapsort. Communications of the ACM, 7(6):347--348, 1964.

....we show that for deterministic algorithms, the two problems are essentially equally hard. We THE RANDOMIZED COMPLEXITY OF MAINTAINING THE MINIMUM 3 Table I: Worst case asymptotic time bounds for different set implementations. Implementation Insert Delete FindMin Doubly linked list 1 1 n Heap [10] log n log n 1 Search tree [6, 8] log n 1 1 Priority queue [3, 4, 5] 1 log n 1 show that any deterministic algorithm with amortized update time at most t requires n=2 4t 3 Gamma 1 comparisons for some FindAny operation. This lower bound is proved using an explicit adversary argument, similar ....

J. W. J. Williams. Algorithm 232: Heapsort. Communications of the ACM, 7(6):347-- 348, 1964.

....basic operations are given in section 1.1. In serial, PQ are usually represented by heap. There exist several algorithms which manage a heap : D heap [22] leftist heap [22] Skew heap [47] Binomial queue [4] Pairing heap [15, 45] The most popular one is the D heap 2 as used in the heapsort ([49]) This is the oldest PQ implementation with O(log(n) performance. In a heap, the priority structure is represented as a binary tree that obeys the heap invariant, which holds that each item always has a higher priority than its children. In a D heap, the tree is embedded in an array, using the ....

J. Williams. Algorithm 232: Heapsort. CACM, 7:347--348, 1964.

....routing on trees under the matching model. Algorithms for routing permutations on trees under different routing models have been presented by Borodin, Rabani and Schieber [4] hot potato routing model) and Symvonis [10] simplified routing model) The notion of the heap was introduced by Williams [11]. Floyd [6] described how to create a heap efficiently. More details can be found in Knuth s book [8] Heaps are also discussed in the context of the Parallel Random Access Machine (PRAM) model [5, 9, 12] Rao and W. Zhang [9] and W. Zhang and Korf [12] described how to construct a heap ....

J.W.J Williams. Algorithm 232: Heapsort. Communications of ACM, 7:347--348, 1964.

....allowing different location access) and for reliability (allowing node failure) 3 Tree Topologies In [12] Peleg introduced a distributed heap: the m items are perfectly load balanced on a (minimum distance) spanning tree of the n processors network. The two alternatives of the heap invariant, [16], are considered: all items stored at one vertex have a larger, respectively a lower) priority than all those stored at any of its children. The heap invariant is maintained after each query (this is denoted as re heapification in the following) The merit of trees of processors lies in the fact ....

J.W.J. Williams. Algorithm 232: heapsort. CACM, 7:347--348, 1964.

....of integers greater than one defines a generalization, and different sequences result in different worst case bounds. By proper choice of sequence, one can adjust the data structure to the particular needs of an application. This situation is comparable to that of d heaps (an implicit heap [15], based on a d ary tree instead of a binary tree) which support Insert at O(log d n) cost and Delete min at O(d log d n) cost. Here, however, the tradeoff possible between the bounds is not only in terms of constant factors. Rather, asymptotically better worst case bounds for Insert and ....

J. W. J. Williams. Algorithm 232: Heapsort. Comm. of the ACM, 7:347--348, 1964.

....insert(i,h) insert a new item i with predefined priority Key i into priority queue h, deletemin(h) delete an item of minimum value (highest priority) from h and return it. Numerous subtle priority queue implementations have been investigated to improve specific applications [10] implicit heap [31], leftist heap [4] binomial queue [2] skew heap [29] etc. In parallel, efficient applications require that several processors should be able to manipulate the priority queue simultaneously. Various efforts to implement concurrent accesses for the basic operations for tightly coupled ....

....here is our point of departure for developing a solution in a more general environment. In [24] Peleg introduced a distributed heap tree: the m items are perfectly load balanced on a (minimum distance) spanning tree of the n processors network. The two alternatives of the heap invariant, [31], are considered: all items stored at a vertex have a larger, respectively a lower) priority than all those stored at any of its children. The heap invariant is maintained after each query (this is denoted as re heapification in the following) The root labeled 0 is the center of the network. The ....

J.W.J. Williams. Algorithm 232: heapsort. CACM, 7:347--348, 1964.

....weighted graph. The best known algorithms for solving this problem, due to Kruskal [Kru56] respectively Prim [Pri57] are covered in most algorithms textbooks. We have chosen to develop Kruskal s algorithm for its use of non trivial data structures such as priority queues, implemented as heaps [Wil64], and tree representations of disjoint sets [Tar75] This should be contrasted with the simple data structures (scalars and arrays) usually employed in safety critical applications. The complexity there lies rather in the large number of variables and the size of the applications themselves. The ....

....for a data structure allowing to insert elements into a set and also to nd and delete the smallest element of the set. A data structure providing for these two operations is called a priority queue. In this section we show how to implement Min Weight Edge as a priority queue and also use heaps [Wil64] as an eOEcient implementation of priority queues. 5.1 Priority Queues Actually, the min weight edge operation is required to return not only the minimum weight but also the edge for which this minimum is reached. For this reason, the speci cation of priority queues will consider a function valq ....

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J.W.J. Williams. Algorithm 232: Heapsort. Communications of the ACM, 7:347348, 1964.

....problem which has an amortized cost per operation of at most 3 comparisons. Our proofs use various averaging arguments which are used to derive general combinatorial properties of trees. These are presented in Sect. 2.2. Implementation Insert Delete FindMin Doubly linked list 1 1 n Heap [8] log n log n 1 Search tree [5, 7] log n 1 1 Priority queue [2, 3, 4] 1 log n 1 Fig. 1. Worst case asymptotic time bounds for different set implementations. 2 Preliminaries 2.1 Definitions and notation For a rooted tree T , let leaves(T ) be the set of leaves of T . For a vertex, v in T , ....

J. W. J. Williams. Algorithm 232: Heapsort. Communications of the ACM, 7(6):347--348, 1964.

....search engines, and reliability ratings for internet sites. ffl The core of job schedulers in operating systems for thread or process management, and event schedulers in production and simulation applications are priority queues. Heaps are ideally suited to implement sequential priority queues [29]. A heap is a binary tree with the property that the values stored at any node are always greater than or equal to values stored at both child nodes (if they exist) All levels, except the last, are completely filled. Several parallel priority queues, such as calendar queues, have been proposed ....

Williams, J.W. Algorithm 232: Heapsort. Comm. ACM, 7, 1964, pp. 347--348.

....but increases T (j) by one. This means that T (j) Phi(j) 2(n i Gamma j) 1. Since Phi(1) 0, we get that T (1) 2n i , which provides the desired complexity. 2. By Lemma 5.2 Steps 1, 2 and 3 can be completed in O(n) time. Now by using an appropriate priority queue, such as a heap (see [11]) we can perform Steps 4 and 5 in O(log r) time per iteration. Since there are at most n iterations of Steps 4 and 5, we obtain the following: Theorem 5.3 Algorithm 1 runs in time O(n log n) Algorithm 1 can be easily modified to an algorithm for generating all optimal forward rankings. It is ....

J.W.J. Williams, Algorithm 232: Heapsort. Comm. ACM 7 (1964) 347-348.

....generalization, and different sequences result in different worst case bounds. By choosing the right sequence, one is able to adjust the data structure to the particular needs of an application. This situation is comparable to the use of d heaps (a simple generalization of the implicit heap [16], based on a d ary tree instead of a binary tree) which E mail address: rolf imada.ou.dk. Supported by SNF grant 11 0575. support Insert at O(log d n) cost and Delete min at O(d log d n) cost, thus allowing a tradeoff between the two operations by varying d the difference is that in this ....

J. W. J. Williams. Algorithm 232: Heapsort. Comm. of the ACM, 7:347--348, 1964.

....standard heap organization and locking the root before all updates creates a congestion problem at that node and forces sequentiality in the access to the data structure. Previous work on priority queues in an environment with asynchronous parallel processes [5, 13, 23] all use some form of a heap [27]. Thus, the root has to be locked at the beginning of each deletemin operation, thereby preventing other processes from accessing the data structure during that time. In fact, 5] also locks the last node at the beginning of a deletemin. In [13, 23] the root is also locked at the beginning of an ....

J. W. J. Williams. Algorithm 232: Heapsort. Comm. ACM, 7:347--348, 1964.

....Heapsort. Heapsort is a widely used sorting algorithm. It is the first algorithm that sorts n numbers in place with running time guaranteed to be of order n log n. Here in place means it does not require extra nontrivial memory space. The method was first discovered by J.W.J. Williams [21] and subsequently improved by R.W. Floyd [7] The heapsort algorithm works in two steps. First it converts the input into a heap. Then it sorts the input by repeatedly deleting the root (smallest element) and restoring the heap. It is well known that we can build a heap from an array of n integers ....

J.W.J. Williams, Algorithm 232: HEAPSORT, Comm. ACM, 7(1964), 347348.

....and the same as the worst case bounds for the other operations. All operations but ChangeKey are implemented as in section 3 with the obvious modifications. The ChangeKey operation can be implemented in several ways. A simple implementation is to use a bubbleup and trickle down technique (see [14]) where the trickle down operation is implemented in a buttomup fashion. The bubble up operation can be replaced by a cut and rebalance technique (see section 3) The trickle down operation can be replaced by a cut andrebalance technique combined with a repeated un link technique (see the ....

J. W. J. Williams. Algorithm 232: Heapsort. Communications of the ACM, 7:347--348, 1964.

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Williams, J. W. J. Algorithm 232: Heapsort. Communications of the ACM 7 (1964), 347-348.

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J.W.J. Williams. Algorithm 232: Heapsort. Communications of the ACM, 7:347--348, 1964. 12

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J. W. J. Williams. Algorithm 232: Heapsort. Comm. of the ACM, 7:347--348, 1964. 10

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J. W. J. Williams, Algorithm 232: Heapsort, Comm. ACM 7 (1964), 347#348.

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J. W. J Williams. Algorithm 232: Heapsort. Comm. ACM, 7(6):347--348, June 1964.

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J.W.J. Williams, Algorithm 232: Heapsort. Comm. ACM 7 (1964) 347-348.

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