H.M. Mahmoud, Evolution of Random Search Trees, Wiley, New York, 1992.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....value max R m = 1 ln ln 2 1 0:0860713320 which is quite negligible. In this paper we shall follow Fedotov and Ryabko and design simple and randomized search algorithms that achieve a low redundancy. Such algorithms can be represented by digital trees such as tries and PATRICIA tries (cf. [7, 8, 11]) In tries the words of W are stored in leaves while internal nodes are used for branching. In standard tries the branching process at level k is determined by the kth symbol of the input word. If the 5000 4000 3000 2000 1000 0.08 0.06 0.04 0.02 0 Figure 1: The average redundancy of the optimal ....

....their result. Our analysis of randomized tries is based on solving exactly and asymptotically a certain recurrence on the average search cost which is equivalent to the average typical cost in a trie. Tries, and in general digital trees, were extensively analyzed over the last twenty years (cf. [7, 8, 11]) for memoryless, Markov and mixing sources. However, the probabilistic model introduced by Fedotov and Ryabko [3] which we further call the uniform model, di ers from all the others in that m words of length n are chosen uniformly over all 2 possible words. Although this introduces additional ....

H. Mahmoud, Evolution of Random Search Trees, John Wiley & Sons, New York, 1992.

....distance in the tree between successive external nodes. # n (# ) measures a property of b(# ) that we call the nearly half measure, H n (b(# ) it is the size of the largest subtree with not more than half the external nodes. Though trees have been studied intensively (e.g. 3] 4] 8] [11], and [12] we are unaware of any previous work on these two features. Theorems 1 and 2 and Lemmas 1 and 2 thus appear to express interesting, new facts about trees, as well as about triangulations. In Section 2 we translate the functions # n and # n into the context of binary trees. We also ....

....to leaves of b(# ) Remark 2. If we regard the trees in B n as binary search trees generated by permutations of 1, n 2, each permutation being equally likely, the bijection gives the (binary search tree) probability #,onT n . Trees in B n are well studied in this model (e.g. 4] 6] [11], and [13] In contrast to the situation in the uniform distribution, the vertex degrees in this model are not identically distributed. Actually #(log n) as is familiar from [4] and [6] We can prove that in this distribution # n log n c 1 in probability. It seems difficult to analyze ....

H. Mahmoud. Evolution of Random Search Trees. Wiley, New York, 1992.

....n O(1) persists for T n = n (n) where (n) o(n) n) n 1; this re ects the robustness of the limit laws. Paged trees. Fix a page (or bucket) size b 1. Cut all nodes with subtree sizes b. The resulting tree is called the b index of the tree; see Flajolet et al. 27] and Mahmoud [47]. What is the size of a random b index And what is the total path length Obviously, both random variables satisfy (1) with di erent initial conditions) The asymptotic normality of the size was established for xed b by Flajolet et al. 27] with mean equal to 2(n 1) b 2) 1. The variance is ....

H. M. Mahmoud, Evolution of Random Search Trees, John Wiley & Sons, New York, 1992.

....S nk : I n0 I nk , the total number of elements sorted up to level k 1. Another way of looking at this quantity is via the so called 3 locally balanced (or fringe balanced) binary search trees: it is the number of internal nodes whose distances to the root is less than or equal to k (cf. [28, 23, 18, 21]) For reader s convenience, we recall the definitions of the following notations used in the last section: # = k L, # = #(2# 1) 12, # = sgn(# #(#) #(u) log u, w 1 = 3 2 # 1 48u . Theorem 2 For k #L, we have #(# 1 # k) #(#) F 1 (#) O n#(# # k) ....

H. M. Mahmoud, Evolution of Random Search Trees, John Wiley & Sons, New York, 1992.

....m 1 pivots (m 2) and to partition the input into m sub les according to the pivots chosen; the 32 pivots are in their nal position after partition and the elements falling between two pivots are sorted recursively. The search tree version of such a sorting scheme is called m ary search tree. See [53, 54, 68, 77, 109, 111] for more information. In its simplest version, the cost measures X n of quicksort using m 1 pivots (when given a random permutation) have moment generating functions of the form Q n (y) n1 nm=n m 1 P n 1 (y) P nm (y) n m) with suitable initial conditions for P n (y) n m, ....

....1 (1 o(1) uniformly for jw 1j , where K(w) is analytic for jw 1j . Then the techniques used in [19] can be further applied to derive the asymptotics of E(X n;k ) w ]M(w; z) See also [19] for an analysis of the levelwise improvements of quicksort and other related problems, and [54, 77] for more examples of the same type. 4.11 Introspective quicksort and balanced BSTs Introspective sort (see [87, 113] is a variant of quicksort in which sub les whose recursion depths (number of nodes between the root and the node itself in terms of BST) exceed a given threshold, say cblog 2 nc, ....

H. M. Mahmoud, Evolution of Random Search Trees, John Wiley & Sons, New York, 1992.

....in [17] One may conjecture that similar results as those in this correspondence hold under this model, the proof being, however, more di#cult. Another generalization of this work is to consider similar problem for other data structures like binary search trees, digital search trees, tries, etc. [13]. Other operations, like deletion, may also be considered. Acknowledgement The author is indebted to the referee for many useful suggestions and references on coding for parametric families. ....

H. M. Mahmoud, Evolution of random search trees, John Wiley & Sons, New York, 1992. 9

....in distribution. Corollary 1. For m 26, the limiting distribution of the random variables (X n not exist. Our starting point is, as in Mahmoud and Pittel [31] the bivariate generating function of X n F (z, y) E(y Xn which satisfies the nonlinear di#erential equation (see [31, 28]) # F (z, y) m 1) yF (z, y) F (z, y) 1 yz yz yz . 3) Except for m = 2 and m = 3 (see Section 7) this di#erential equation seems hard to be solved exactly. The approach by Mahmoud and Pittel for the mean of X n consists first in taking derivative with respect to ....

....the relation (8) we can express (12) as #(#)G 1 (z) where #(#) 0 is the indicial equation of the equation L[G]Y = 0: #(#) # # k ) the # k s, 1 1, being the zeros of #(#) These zeros are simple and #(2) 0. More properties of this polynomials can be found in Mahmoud [28]. Arrange these zeros by descending order of their real parts: 2 = # 1 3 ) # # #(#m 1 ) See Figure 2 for a plot of the distribution of the zeros. 10 20 30 40 10 5 5 2 1.8 1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 10 5 0 5 10 Figure 2: Distribution of the zeros of #(#) for m from 5 to 40; zeros ....

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H. M. Mahmoud, Evolution of Random Search Trees, John Wiley & Sons, New York, 1992.

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H.M. Mahmoud, Evolution of Random Search Trees, Wiley, New York, 1992.

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H. M. Mahmoud, Evolution of Random Search Trees, John Wiley & Sons, New York, 1992.

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H. M. Mahmoud, Evolution of Random Search Trees, John Wiley & Sons, New York, 1992.

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H. M. Mahmoud, Evolution of Random Search Trees, John Wiley & Sons, New York, 1992.

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H. M. Mahmoud, Evolution of Random Search Trees (John Wiley & Sons, New York, 1992).

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Hosam M. Mahmoud. Evolution of random search trees. John Wiley & Sons Inc., New York, 1992. A Wiley-Interscience Publication.

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Mahmoud, H. M. (1992) Evolution of random search trees. Wiley-Interscience Series in Discrete Mathematics and Optimization. John Wiley & Sons, Inc., New York.

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H. M. Mahmoud. Evolution of random search trees. John Wiley & Sons Inc., New York, 1992. A Wiley-Interscience Publication.

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H. Mahmoud, Evolution of Random Search Trees , John Wiley & Sons, New York, 1992.

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H. Mahmoud, Evolution of Random Search Trees , John Wiley & Sons, New York, 1992.

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H. M. Mahmoud. Evolution of random search trees. John Wiley & Sons Inc., New York, 1992. A Wiley-Interscience Publication.

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H.M. Mahmoud. Evolution of random search trees, John Wiley & Sons, 1992.

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Mahmoud, H. Evolution of Random Search Trees. John Wiley, New York, 1992.

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Mahmoud, H. M. (1992) Evolution of Random Search Trees. John Wiley, New York.

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H. M. Mahmoud, Evolution of Random Search Trees, John Wiley & Sons, New York, 1992.

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H.M. Mahmoud, Evolution of Random Search Trees. Wiley, New York, 1992.

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H. M. Mahmoud, Evolution of Random Search Trees, John Wiley & Sons, New York, 1992.

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H. M. Mahmoud, Evolution of Random Search Trees, John Wiley & Sons, New York, 1992.

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