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

....frequently used in applications concerning with priorities and partial orderings. It first appeared in Williams Heapsort algorithm [19] which is the first in place O(n log n) sorting algorithm. Besides its original application to sorting, heap has wide applications to algorithmic design (cf. [2, 12, 16, 14]) for example, minimum spanning trees, Hu#man coding, job scheduling, numerical computations, tape mergings, etc. A (min ) heap is an array with elements a[j] 1 n, satisfying the path monotonic property: a[j] a[#j 2#] j = 2, 3, n, where #x# denotes the integral part of x. It ....

....greater than that of its children. We are concerned in this correspondence with one of the basic operations on heaps: insertion of a new element into a heap (the operation being completed by retaining the heap property) While its worst case time complexity is O(log n) for a heap of size n (cf. [12, 6]) the average case is O(1) under a suitable uniform probability model (cf. 17, 4] and 2.1) As in Knuth [12] let us call the path formed by the numbers a[n 1] a[# . a[1] the special path. The algorithm analyzed in [17, 4] and Algorithm A in this correspondence) is a ....

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

....# n = K K q n (n 1 = 0, where q n : m [ it) log Q n (e ) With the aid of symbolic computation systems, the methods in [1] and this note can be used to establish the following results. Theorem 2 For n 1, the cumulants of order 3 and 4 satisfy the exact formulae: [3] n = nD(log 2 n) 6 103 252 180 42 8 and [4] n = nE(log 2 n) 26 2419 4692 5400 3360 1118 336 48 where the two functions D(u) and E(u) are continuous and periodic with period 1 which admit the Fourier series expansions D(u) # # (0) ....

D. E. Knuth, The Art of Computer Programming, Vol. III---Sorting and Searching, AddisonWesley, 1973.

....paradigm naturally introduces, for a cost measure f n , the mergesort recurrence: f n = f f 2) 1) with f 1 given, for some specified sequence n#2 . For example, the number of comparisons f n used by the top down mergesort satisfies (1) with di#erent e n in di#erent cases (cf. [11, 3]) e n = # # # # # # # best case; #n 2# 1 #n 2# 1 , average case; 1, worst case. # This work was partially supported by the ESPRIT Basic Research Action No. 7141 (ALCOM II) while the author was at Ecole Polytechnique, France. As divide and conquer is widely employed in algorithm ....

D. E. Knuth, The art of computer programming, vol. III---sorting and searching, Addison-Wesley, 1973.

....( n = K K q n (n 2) 1 = 0; where q n : m [ it) log Q n (e ) With the aid of symbolic computation systems, the methods in [1] and this note can be used to establish the following results. Theorem 2 For n 1, the cumulants of order 3 and 4 satisfy the exact formulae: [3] n = nD(log 2 n) 6 Gamma 103 252 180 42 8 and [4] n = nE(log 2 n) Gamma 26 Gamma 2419 4692 5400 3360 1118 336 48 where the two functions D(u) and E(u) are continuous and periodic with period 1 which admit the Fourier series ....

D. E. Knuth, The Art of Computer Programming, Vol. III---Sorting and Searching, AddisonWesley, 1973.

....frequently used in applications concerning with priorities and partial orderings. It first appeared in Williams Heapsort algorithm [19] which is the first in place O(n log n) sorting algorithm. Besides its original application to sorting, heap has wide applications to algorithmic design (cf. [2, 12, 16, 14]) for example, minimum spanning trees, Huffman coding, job scheduling, numerical computations, tape mergings, etc. A (min ) heap is an array with elements a[j] 1 j n, satisfying the path monotonic property: a[j] a[bj=2c] j = 2; 3; n, where bxc denotes the integral part of x. It can ....

....greater than that of its children. We are concerned in this correspondence with one of the basic operations on heaps: insertion of a new element into a heap (the operation being completed by retaining the heap property) While its worst case time complexity is O(log n) for a heap of size n (cf. [12, 6]) the average case is O(1) under a suitable uniform probability model (cf. 17, 4] and x2.1) As in Knuth [12] let us call the path formed by the numbers a[n 1] a[b 2 c] a[b 4 c] a[1] the special path. The algorithm analyzed in [17, 4] and Algorithm A in this correspondence) ....

[Article contains additional citation context not shown here]

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

....with the corresponding (derived) values for the bounds c and c . 2 (Figure 1(b) will be discussed in Section 3.1. 1 What we describe here is essentially a partial sum tree [KNUT75, WONG80] The term ranks arose from the use of cardinality information in cumulative partial sum trees [KNUT73] and comes to the database literature via [OLKE89] 2 In Figure 1(a) the values for c and c are computed using the example formula from [ANTO92] using parameter values A = 1 2 and Q = 1 2. 5 node c S c 0 S c S u S c (a) 10 17 31 26 5 (c) 8 13 22 14 1 (b) 12 15 22 10 ....

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

....given by the sequence of pairs and on which to apply this operation. The number of states in the problem is given by the possible ways in which the entries can be ordered; this is for SORTN( The optimal cost of these problems is known for small values of only ( according to [7]) GPT find optimal solutions, using A , in a couple of minutes for s up to . Fig. 4 shows the codification of sorting networks; note how we use the axiom to prune from the initial states those arrays with repeated integers. Instead of pruning such bad arrays, we can just not generate them ....

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

....[n] n] For 2 P let T be the set of bijective poset morphisms ( n] these are the Young tableaux of shape . Put Pn = f 2 P j jj = n g, and let RSn : Sn P 2Pn T Theta T denote the ordinary Robinson Schensted correspondence (using row insertion) see for instance [Sche] [Knu2], vLee3] 9 3.3 Insertion and extraction using . It will be convenient to represent a total ordering on a skew diagram by the unique poset isomorphism ff: n] this is essentially a reading of [FoGr] compatibility of with . is expressed by the fact that ff is also ....

....by these symmetries. The answer will involve the Schutzenberger correspondence, an algorithmically defined shape preserving transformation of Young tableaux; we shall denote it by Sn : Tn Tn , where n 2 N and Tn = S 2Pn T . It was first defined in [Schu1] where it is called I) see also [Knu2] (the operation P 7 P S ) and [vLee3] It has a definition and some properties of a type similar to those of the Robinson Schensted algorithm, and there is a strong connection between the correspondences defined by the two algorithms, that we shall now formulate. Let n 2 Sn be the unique ....

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D. E. Knuth, The art of Computer programming, Vol. III Sorting and Searching, pp. 48--72, Addison-Wesley, 1975.

....Sorting 5 Elements h = 0 h = longest h = decomposition quicksort Figure 15: Performance of rtdp bel controller for Sorting problem with n = 5 using various heuristics. Top flat curve is for Quicksort. From [10] as 13 some of the open conjectures in lower bound theory could be settled (see [37]) 6 Discussion We have presented a unified approach for modeling and solving planning problems that is based on state models that handle various types of dynamics (deterministic, non deterministic, and probabilistic) and sensor feedback (null, partial, and complete) The approach combines ....

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

....with arrow. depending on the input numbers so that a larger number is placed in the direction of an arrow. A different way is used in Figure 14 to represent the same network than in Figure 13. RR n2077 24 T. Hr uz et D. Fortin 6 r r r r r r Figure 15: The sorting network [14] for four numbers. 3.4 Sorting networks A sorting network is a special kind of a comparison network. Input to the network is a n tuple of integers (a 1 ; a 2 ; a n ) and output are the same numbers sorted in ascending or descending order. One such network which sorts four numbers is in ....

....This network sorts four numbers so that on the output the lowest line contains the largest number. A different sorting network is in Figure 15 which cannot be effectively represented in the way used in Figure 13. There is a well established theory of sorting networks (for the reference see [14]) and they are widely applied in the development of algorithms for modern parallel machines. In the theory two main groups of questions are still open: firstly, there is a question to what extent a number of comparison modules can be minimized. Secondly, when one looks at the Figures 14 and 15 it ....

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Donald E. Knuth, The Art of Computer Programming. Vol. III: Sorting and Searching, Addison Wesley, Reading, Mass., 1973.

....seekand latencycosts even if requests were smaller than 30 blocks. 2 The SCSI standard defines the commandREAD REVERSE but its implementation by tape drive manufacturers is left optional. On a historical note, Knuth also assumes bi directional tape drives in his work on tape sorting [10]. S i half the original size. This in turn doubles the number of iterations needed and doubles the number of times R is scanned. Furthermore, the average utilization of available buffer space is only 50 assuming that the writer and reader processes operate at equal speeds. A better approach is ....

D. Knuth. The Art of Computer Programming, Vol. III: Sorting and Searching. Addison-Wesley Publishing Co., Redwood City, CA, 1973.

....both directions. It has an internal read ahead, speed matching buffer which allows the system to transfer data in requests as small as one block without performance deterioration. The tape drive tries to keep its read ahead buffer full at all times by reading as many blocks from tape as possible [3,9,14]. When the buffer fills up or a high water mark is hit, the tape drive slows down and stops (ramp down) and the tape is repositioned so that the read write head is at the first tape block that did not fit into the buffer. A subsequent read request removes blocks from the buffer and releases ....

....of the previous cycle have barrier synchronized. Each cycle has an associated time constraint: Reading R from disk should be faster than reading S i from tape. If the constraint holds, the tape drive is streaming and the join is bound by the speed of the tape drive. NBT uses relation rocking [8, 9] to save on disk transfers. Relation R is read in alternating directions, and when changing from one direction to the other, the last MR blocks of R are already in memory. 5.2.1 NBT with Memory Buffering In NBT with memory buffering (NBT MB) two memory buffers I0 and I1 are assigned to hold the ....

D. Knuth. The Art of Computer Programming, Vol. III: Sorting and Searching. Addison-Wesley Publishing Co., Redwood City, CA, 1973.

....node the difference of maximal lengths of paths in the subtrees starting in the successors must be at most one. The ordering on 2 Q required for this search tree is the one obtained from regarding the bitmaps as binary representation of numbers. For a detailed explanation of balanced trees see [Knu73]. For the representation of rest a stack is used. The components of both the tree and the stack are pointers to structures that contain all the information for one DFA state. For the complexity we note that there are actually examples in which the outermost loop is performed 2 n times, where n ....

D.E. Knuth, The Art of Computer Programming, Vol. III: Sorting and Searching, Addison-Wesley 1973, [pp. 455--457].

....respecting parity constraints, the complexity of sorting with k reversals is equivalent to sorting with (n Gamma k) reversals. Previous work on sorting with fixed length reversals has focused on the special case where k = 2. Thus each reversal simply transposes adjacent elements. Bubble sort [19] sorts any permutation using exactly one transposition for each inversion in , thus minimizing the number of reversals. Jerrum [15] presented a polynomial algorithm for the much more difficult problem of sorting circular permutations using a minimum number of transpositions. Gates and ....

....Q and Q = P . We introduce the relation = to derive useful transformations from k reversal. If P ( Q, then any permutation that can be sorted by P can be sorted by Q too. The simplest transformation capable of sorting is the 2 reversal or adjacent transposition. Bubble sort (or insertion sort) [19] demonstrates that each permutation can be sorted in O(n 2 ) steps of 2 reversals. We also can use it to establish whether a set of generators is sufficient for sorting. Lemma 1 A set of generators P is sufficient for sorting iff P = 2 Gammareversal Proof: 2 reversal is sufficient to sort ....

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

....key values than the latest tuple written out in the current run. When none of the tuples in the heap satisfy this condition, the current run ends and a new run is started. On the average, the length of the runs produced by replacement selection is twice the memory allocated for the split phase [Knut73], i.e. twice as long as the runs generated with Quicksort. Hence, replacement selection creates only half as many runs as Quicksort. This could significantly shorten the merge phase that follows. A nice discussion of the details involved in implementing replacement selection can be found in ....

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

....with respect to an indexing g, if the v plane is ascendingly arranged and every block of V P (i) is sorted. 3 1 1 Sorting In the whole paper we discuss oblivious sorting algorithms, i.e. the algorithm performs always the same compare and exchange steps. Therefore we can use the 0 1 principle [3]: To prove the correctness of an oblivious sorting algorithm it is sufficient to show that the algorithm sorts an arbitrary input of zeros and ones. In this context we use the concept of dirty and clean areas. Clean areas are areas which consist either only of zeros or only of ones. Dirty areas ....

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

....respecting parity constraints, the complexity of sorting with k reversals is equivalent to sorting with (n Gamma k) reversals. Previous work on sorting with fixed length reversals has focused on the special case where k = 2. Thus each reversal simply transposes adjacent elements. Bubble sort [20] sorts any permutation using exactly one transposition for each inversion in , thus minimizing the number of reversals. 20 1 5 3 6 7 8 9 10 11 12 13 14 15 16 17 18 19 After Flip Top Spin puzzle After Shift Left FLIP FLIP 18 3 17 9 7 5 6 13 11 20 19 18 17 16 14 15 12 10 9 8 7 5 3 1 6 8 10 ....

....Q and Q = P . We introduce the relation = to derive useful transformations from k reversal. If P ( Q, then any permutation that can be sorted by P can be sorted by Q too. The simplest transformation capable of sorting is the 2 reversal or adjacent transposition. Bubble sort (or insertion sort) [20] demonstrates that each permutation can be sorted in O(n 2 ) steps of 2 reversals. We also can use it to establish whether a set of generators is sufficient for sorting. Lemma 1 A set of generators P is sufficient for sorting iff P = 2 Gamma reversal Proof: 2 reversal is sufficient to sort ....

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

....key values than the latest tuple written out in the current run. When none of the tuples in the heap satisfy this condition, the current run ends and a new run is started. On the average, the length of the runs produced by replacement selection is twice the memory allocated for the split phase [Knut73], i.e. twice 51 as long as the runs generated with Quicksort. Hence, replacement selection creates only half as many runs as Quicksort. This could significantly shorten the merge phase that follows. A nice discussion of the details involved in implementing replacement selection can be found in ....

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

.... and notation follows [ANTO92] An easy and intuitive way to construct a hierarchical histogram from a tree index is to augment every non leaf node entry with a cardinality count (i.e. the total number of leaf records in the specified subtree) Such counts are commonly called ranks (cf. [KNUT73, OLKE86]) Inserting or deleting a record results in node modifications from leaf to root because any such update changes the cardinality of every subtree containing that record. This is generally considered to be impractical in a production DBMS (though bulk update, common in data warehouses, can reduce ....

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

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

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

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

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D. E. Knuth, The Art of Computer Programming. Vol. III: Sorting and Searching, Addison-Wesley, Reading, Massachusetts, 1973. 11

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

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

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