D. E. Knuth. The Art of Computer Programming, Volume III: Sorting and Searching. Addison-Wesley, Reading, MA, 1973.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....binary search tree. 1 Introduction Quicksort, invented by Hoare [10] in the early 1960s, is one of the most widely used sorting algorithms. It has the average performance 2n log n O(n) as long as the number of comparisons is concerned, where n is the number of elements to sort (cf. Knuth [15], Sedgewick [25, 26] The large number of literature and variants for quicksort partly explain its importance and usefulness (cf. 5] Most of these variants focused on improving the average performance of quicksort or on making the quadratic worst case behavior less likely. Of these, ....

....[5] Most of these variants focused on improving the average performance of quicksort or on making the quadratic worst case behavior less likely. Of these, median of three proves the simplest and the most successful. The average number of comparisons of this variant drops to n log n O(n) cf. [15, 25, 27]) A natural question then arises: how does the standard quicksort improve from 2n log n to n log n as far as the number of comparisons is concerned Our purpose of this paper is to give a possible explanation of the underlying improvement process. For completeness, we briefly describe ....

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D. E. Knuth, The Art of Computer Programming, Volume III: Sorting and Searching , Addison Wesley, Reading, Massachusetts, second edition, 1998.

....binary search tree. 1 Introduction Quicksort, invented by Hoare [10] in the early 1960s, is one of the most widely used sorting algorithms. It has the average performance 2n log n O(n) as long as the number of comparisons is concerned, where n is the number of elements to sort (cf. Knuth [15], Sedgewick [25, 26] The large number of literature and variants for quicksort partly explain its importance and usefulness (cf. 5] Most of these variants focused on improving the average performance of quicksort or on making the quadratic worst case behavior less likely. Of these, ....

....[5] Most of these variants focused on improving the average performance of quicksort or on making the quadratic worst case behavior less likely. Of these, median of three proves the simplest and the most successful. The average number of comparisons of this variant drops to n log n O(n) cf. [15, 25, 27]) A natural question then arises: how does the standard quicksort improve from 2n log n to 7 n log n as far as the number of comparisons is concerned Our purpose of this paper is to give a possible explanation of the underlying improvement process. For completeness, we brie y describe ....

[Article contains additional citation context not shown here]

D. E. Knuth, The Art of Computer Programming, Volume III: Sorting and Searching , Addison Wesley, Reading, Massachusetts, second edition, 1998.

.... linear; Doberkat [5] derives the probability generating functions for the number of exchanges and comparisons used by Floyd s algorithm, he also finds the asymptotics of the first two moments; Sprugnoli [27] examines in more detail the average case behavior of Floyd s heap construction algorithm [17] for general values of the heap size. The method of Doberkat is via probability generating functions and that of Sprugnoli relies largely on refining the results on special sequences. We propose a new approach to the asymptotic study of the number of heaps and the cost of heap construction. It is ....

....counting argument for Young tableaux of a given shape (see [17, 5.1.4, 5.2.3] and give an explicit solution to the general recurrence (2) Lemma 1. For n 1, the solution f n of the heap recurrence (2) is given by t n j , 6) for any given sequence n#1 . Proof. As in [17], we call the nodes lying on the path from 1 to n special nodes. Let # 1 and # 2 denote the first sum and the second sum, respectively. Then # 1 counts the weights of nonspecial nodes under the cost function t n , and # 2 counts similarly the weights of special nodes. Nonspecial nodes are always ....

D. E. Knuth, The Art of Computer Programming, Volume III---Sorting and Searching, AddisonWesley, Second Edition, 1998.

....page is not yet in memory, the sorting program must wait, thus relinquishing not only the CPU but also the CPU cache. Second, the very nature of merging implies that many inputs are read, and it is necessary to determine which of the inputs must be read next, commonly known as forecasting [K 73] Note that double buffering [S 89a] for all input runs does not truly solve the problem. On one hand, it reduces the merge fan in to half, whereas good forecasting reduces the fan in only by a relatively small fixed number. Useful values are the number of disk drives if known or simply ten, ....

....different amounts of read ahead at different times during a merge step. In both cases, deep forecasting is required, i.e. forecasting that reaches beyond one asynchronous read operation and beyond finding the lowest one among the highest keys on each page currently consumed by the merge logic [K 73] Other researchers have considered technique for planning the page consumption sequence ahead of a merge step [ZL 98] or as the merge progresses [S 94] In both efforts, a separate data structure was designed to retain the highest keys in each data page. In commercial reality, however, every ....

Donald E. Knuth: The Art of Computer Programming, Volume III: Sorting and Searching. AddisonWesley 1973.

.... study of the communication and synchronization requirements of a fundamental computational problem namely, selection, along with an application to Parallel Priority Queues (PPQs) A PPQ is an abstract data structure that generalizes the well known sequential priority queue data structure [11]. A p processor PPQ implementation allows the efficient simultaneous insertion of n new items, n=p per processor, and the simultaneous deletion of the n items associated with the smallest values. As noted in [14] PPQs can be employed to efficiently implement several parallel techniques, such as ....

....; 5. do let E u j and x e 1 g ; j) fx j x 2 u j and e j Gamma1 x e j g, 8 j 2 f2; d Gamma 2g ; j and x e d Gamma1 g ; end Adjust Figure 3: Procedure Adjust. time T brd (p) and subsequently, by sorting locally these items in time max fL; O(d lg d)g [11]. The selection process (line 3) takes at most (d Gamma 1)T sel (dpr; n; p) We note that for sufficiently large d it might be more efficient to sort the dpr items instead of performing d Gamma 1 selection processes. The re labeling operation can be realized as a by product of the selection ....

D. E. Knuth. The Art of Computer Programming. Volume III: Sorting and Searching. Addison-Wesley, Reading, 1969.

.... independent study of the computation and communication requirements of a fundamental combinatorial problem namely, selection, along with an application to Parallel Priority Queues (PPQs) A PPQ is an abstract data structure that generalizes the well known sequential priority queue data structure [21]. A p processor PPQ implementation allows the efficient simultaneous insertion of n new items, n=p per processor, and the simultaneous deletion of the n items associated with the smallest values. As noted in [27] PPQs can be employed to efficiently implement several parallel techniques, such as ....

....(j) and x e 1 g ; j) fx j x 2 (j) and e j Gamma1 x e j g, 8 j 2 f2; d Gamma 2g ; u ( d) fx j x 2 u (j) and x e d Gamma1 g ; end Adjust Figure 5: Procedure Adjust, p number of processors. by sorting locally these items in time max fL; O(d lg d)g [21]. The selection process (line 2) takes time at most i=0 T sel ( d Gamma i)pr; n; p) We note that for sufficiently large d it might be more efficient to sort the dn items instead of performing d Gamma 1 selection processes. The relabeling operation can be realized as a by product of the ....

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D. E. Knuth. The Art of Computer Programming. Volume III: Sorting and Searching. Addison-Wesley, Reading, 1973.

....The most studied comparator networks are sorting and merging networks. In the early 1960 s, Batcher proposed the odd even merge algorithm to merge two sequences of n and m elements, n m, which can be implemented by a merging network of size O( m n) log m) In the early 1970 s Floyd [12] and Yao [18] proved the asymptotic optimality of Batcher s networks. The lower bound has recently been improved by Miltersen, Paterson and Tarui [13] closing the long standing factor of two gap between upper and lower bounds. It is noteworthy to recall, that merge can be solved in the comparison ....

....between upper and lower bounds. It is noteworthy to recall, that merge can be solved in the comparison tree model with a tree of depth m n 1. Batcher also showed how his merge algorithm could be used to implement sorting networks with size O(n log n) and depth O(log n) to sort n inputs [12]. For a long time, the question remained open as to whether sorting networks with size O(n log n) and depth O(log n) existed. In 1983, Ajtai, Koml os and Szemer edi [1] presented sorting networks with size O(n log n) and depth O(log n) to sort n items. This result, although partially unsatisfying ....

Donald E. Knuth. The Art of Computer Programming, Volume III: Sorting and Searching. Addison-Wesley, Reading, MA, 1973.

....Sampling access methods support some type of randomized probing operation. Augmented trees can be used, as can some variation of acceptance rejection (A R) sampling, or some combination. This subsection shows how to emulate both types of sampling. Sampling is easy using trees augmented with ranks [KNUT73] or other weighting [WONG80] information. To sample from an n record index, we choose a random number k [1, n] and return the kth record by following the pointers whose corresponding ranges contain k (see Figure 4(a) This is discussed in undergraduate textbooks [CORM90] A R sampling is more ....

D. E. Knuth, The Art of Computer Programming, Volume III: Sorting and Searching, Addison Wesley, Reading, MA, 1973.

....and jX = j small 1. determine the median m of the set fhx; ti j x 2 X g 2. X : fx 2 X j hx; ti mg, X : fx 2 X j hx; ti mg, X= fx 2 X j hx; ti = mg return Remark 3. 8 Since the median of a set can be determined in optimal (i.e. linear in the number of elements) complexity (see, e.g. [1, 9]) Algorithm 3.7 can be realized in optimal complexity. The next algorithm formalizes our procedure of the example in Section 3.2. Algorithm 3.9 (subdomain numbering) numbering(V; t; N 0 ) input: nodes V, vector t, starting number N 0 output: numbering of nodes V if jVj = 1 then label the ....

D.E. Knuth. The Art of Computer Programming, volume III: Sorting and Searching. Addison-Wesley, 1973.

....e respectively. A dominant term in an integer sorting BSP algorithm to be claimed later is contributed by sequential count sort performed independently at each processor. We charge O(n) time units for stable sorting n integers in the range [0, 1, n 1] by sequential count sort algorithm [35]. We call such an algorithm Count Sort. We also apply radix sort techniques to sort n integers in the range [1, n # ] for some positive constant #, within #lg #n # # lg n# iterations of Count Sort and call such an algorithm Radix Sort. For other operations, we charge O(1) time ....

....# lg lg n n p lg (n p) # lg n # . 25 [Case (l #= r and Nm # lg lg lg n n] The single bucket assigned to processors indexed l through r is sorted by an adaptation on the BSP model of Batcher s odd even merge sort algorithm [7] where comparators are replaced by #Nm q m # mergers [35]. Each processor is assigned an Nm q m size segment of the single bucket. Each such segment is first sorted locally in time (Nm q m ) lg (Nm q m ) For Nm # lg lg lg n n we have that lg Nm # lg lg 2 n. As l #= r we have that km q m , and therefore, by equations (9) and (16) and ....

D. E. Knuth. The Art of Computer Programming. Volume III: Sorting and Searching. AddisonWesley, Reading, 1973.

....terminal. Contains fchild i (u[i] u[i] w[i]g. high queue A priority queue for finding the maximum terminal. Contains fchild i (w[i] u[i] w[i]g. For an example refer back to Figure 2.3. Tree T 6 has N = 10; C = 59; mDeg = 2; D[1] 0; D[2] 2; D[3] 3; D[4] 4; D[5] 4; D[6] = 4; u[1] 4; u[2] 3; u[3] 1; w[1] 6; w[2] 6; w[3] 3 low queue = fchild 1 (4) child 2 (3) child 3 (1)g; high queue = fchild 1 (6) child 2 (6) child 3 (3)g: The priority queues are maintained as follows. In general, a terminal in Tm can have rank (label) arbitrarily larger than ....

D. E. Knuth, The Art of Computer Programming, Volume III: Sorting and Searching, AddisonWesley.

....them so by appending to each one their index memory location storing it. We consider, in this section, the problem of merging these two sequences into a non decreasing sequence say, Z = hz 1 ; z m k i. Linear time sequential algorithms to solve the merging 5 problem are discussed in [11]. In the following sections we present simple sub optimal and rather more complicated optimal BSP merging algorithms. We first introduce a few definitions. The rank of an element x in sequence X, denoted by rank(x : X) is the number of elements of X that are less than or equal to x. Sequence X ....

.... : hy 1 ; y rank(z i 1 :Y ) i if i = 0; hy rank(z i :Y ) 1 ; y rank(z i 1 :Y ) i if 1 i j Zj; hy rank(z i :Y ) 1 ; y k i if i = j Zj; and subsequently merges them into subsequence Z i by employing some optimal deterministic sequential merging algorithm [11]. The desired sequence merge(X; Y ) is the concatenation of sequences Z i , for all i, in increasing order of i. Analysis of Stage 5: By way of stages 1 3, the cardinality of X 0 i [ Y 0 i , for all i stored in some processor j is bounded above by d p n edn= p p n )e = n=p O(n= p ....

D. E. Knuth. The Art of Computer Programming. Volume III: Sorting and Searching. Addison-Wesley, Reading, 1973.

....The most studied comparator networks are sorting and merging networks. In the early 1960 s, Batcher proposed the odd even merge algorithm to merge two sequences of n and m elements, n m, which can be implemented by a merging network of size O( m n) log m) In the early 1970 s Floyd [12] and Yao [18] proved the asymptotic optimality of Batcher s networks. The lower bound has recently been improved by Miltersen, Paterson and Tarui [13] closing the long standing factor of two gap between upper and lower bounds. It is noteworthy to recall, that merge can be solved in the comparison ....

....between upper and lower bounds. It is noteworthy to recall, that merge can be solved in the comparison tree model with a tree of depth m n 1. Batcher also showed how his merge algorithm could be used to implement sorting networks with size O(n log 2 n) and depth O(log 2 n) to sort n inputs [12]. For a long time, the question remained open as to whether sorting networks with size O(n log n) and depth O(log n) existed. In 1983, Ajtai, Koml os and Szemer edi [1] presented sorting networks with size O(n log n) and depth O(log n) to sort n items. This result, although partially unsatisfying ....

Donald E. Knuth. The Art of Computer Programming, Volume III: Sorting and Searching. Addison-Wesley, Reading, MA, 1973. 12

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D. E. Knuth. The Art of Computer Programming, Volume III: Sorting and Searching. Addison-Wesley, Reading, MA, 1973.

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D. Knuth. The art of computer programming, Volume III: Sorting and searching. Addison-Wesley, 1973.

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D. Knuth. The art of computer programming, Volume III: Sorting and searching. Addison-Wesley, 1973.

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D. E. Knuth, The Art of Computer Programming, Volume III---Sorting and Searching, Second edition, Addison-Wesley, 1973.

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D. E. Knuth. The Art of Computer Programming. Volume III: Sorting and Searching. AddisonWesley, Reading, 1973.

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D. E. Knuth. The Art of Computer Programming, Volume III: Sorting and Searching. Addison Wesley Longman, USA, 2 edition, 1998.

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D. E. Knuth. The Art of Computer Programming. Volume III: Sorting and Searching. AddisonWesley, Reading, 1973.

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D. E. Knuth. The Art of Computer Programming. Volume III: Sorting and Searching. Addison-Wesley, Reading, 1973.

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D. E. Knuth, The art of computer programming, Volume III---sorting and searching, AddisonWesley, 1973.

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D. E. Knuth, The art of computer programming, Volume III---sorting and searching, AddisonWesley, 1973.

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Donald E. Knuth. The Art of Computer Programming, Volume III: Sorting and Searching. Addison-Wesley, 1973.

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Donald E. Knuth. The Art of Computer Programming, Volume III: Sorting and Searching. AddisonWesley, 1973.

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