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33
Synchronization and linearity: an algebra for discrete event systems
, 2001
"... The first edition of this book was published in 1992 by Wiley (ISBN 0 471 93609 X). Since this book is now out of print, and to answer the request of several colleagues, the authors have decided to make it available freely on the Web, while retaining the copyright, for the benefit of the scientific ..."
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Cited by 248 (10 self)
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The first edition of this book was published in 1992 by Wiley (ISBN 0 471 93609 X). Since this book is now out of print, and to answer the request of several colleagues, the authors have decided to make it available freely on the Web, while retaining the copyright, for the benefit of the scientific community. Copyright Statement This electronic document is in PDF format. One needs Acrobat Reader (available freely for most platforms from the Adobe web site) to benefit from the full interactive machinery: using the package hyperref by Sebastian Rahtz, the table of contents and all LATEX crossreferences are automatically converted into clickable hyperlinks, bookmarks are generated automatically, etc.. So, do not hesitate to click on references to equation or section numbers, on items of thetableofcontents and of the index, etc.. One may freely use and print this document for one’s own purpose or even distribute it freely, but not commercially, provided it is distributed in its entirety and without modifications, including this preface and copyright statement. Any use of thecontents should be acknowledged according to the standard scientific practice. The
A Note on the Height of Binary Search Trees
, 1986
"... Let H. be the height of a binary search tree with n nodes constructed by standard insertions from a random permutation of I,..., n. It is shown that HJog n + c = 4.3 I 107... in probability as n + 00, where c is the unique solution of c log((2e)lc) = 1, c 2 2. Also, for all p> 0, lim,,E(H$)/ log ..."
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Cited by 79 (23 self)
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Let H. be the height of a binary search tree with n nodes constructed by standard insertions from a random permutation of I,..., n. It is shown that HJog n + c = 4.3 I 107... in probability as n + 00, where c is the unique solution of c log((2e)lc) = 1, c 2 2. Also, for all p> 0, lim,,E(H$)/ log % = cp. Finally, it is proved that &/log n, c * = 0.3733..., in probability, where c * is defined by c log((2e)lc) = 1, c 5 1, and.S, is the saturation level of the same tree, that is, the number of full levels in the tree.
Branching processes in the analysis of the heights of trees
 Acta Informatica
, 1987
"... Summary. It is shown how the theory of branching processes can be applied in the analysis of the expected height of random trees. In particular, we will study the height of random binary search trees, random kd trees, quadtrees and unionend trees under various models of randomization. For example, ..."
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Cited by 58 (19 self)
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Summary. It is shown how the theory of branching processes can be applied in the analysis of the expected height of random trees. In particular, we will study the height of random binary search trees, random kd trees, quadtrees and unionend trees under various models of randomization. For example, for the random binary search tree constructed from a random permutation of 1,..., n, it is shown that H„/(c log (n)) tends to 1 in probability and in the mean as n oo, where H „ is the height of the tree, and c =4.31107... is a solution of the equation c log (2e / = 1. In addition, we ~c ~ show that H „clog (n) = O (/log (n) loglog (n)) in probability.
Universal Limit Laws for Depths in Random Trees
 SIAM Journal on Computing
, 1998
"... Random binary search trees, bary search trees, medianof(2k+1) trees, quadtrees, simplex trees, tries, and digital search trees are special cases of random split trees. For these trees, we o#er a universal law of large numbers and a limit law for the depth of the last inserted point, as well as a ..."
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Cited by 50 (8 self)
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Random binary search trees, bary search trees, medianof(2k+1) trees, quadtrees, simplex trees, tries, and digital search trees are special cases of random split trees. For these trees, we o#er a universal law of large numbers and a limit law for the depth of the last inserted point, as well as a law of large numbers for the height.
Minimal position and critical martingale convergence in branching random walks, and directed polymers on disordered trees
, 2007
"... ..."
On the Exponentiality of Stochastic Linear Systems under the MaxPlus Algebra
 IEEE Transactions Automatic Control
, 1994
"... In this paper, we consider stochastic linear systems under the maxplus algebra. For such a system, the states are governed by the recursive equation X n = A X n01 8 U n with the initial condition X 0 = x 0 . By transforming the linear system under the maxplus algebra into a sublinear system under ..."
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Cited by 17 (4 self)
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In this paper, we consider stochastic linear systems under the maxplus algebra. For such a system, the states are governed by the recursive equation X n = A X n01 8 U n with the initial condition X 0 = x 0 . By transforming the linear system under the maxplus algebra into a sublinear system under the usual algebra, we establish various exponential upper bounds for the tail distributions of the states X n under the i.i.d. assumption on f(A n ; U n ); n 1g and a couple of regularity conditions on (A 1 ; U 1 ) and the initial condition x 0 . These upper bounds are related to the spectral radius (or the PerronFrobenius eigenvalue) of the nonnegative matrix in which each element is the moment generating function of the corresponding element in the statefeedback matrix A 1 . In particular, we have Kingman's upper bound for GI/GI/1 queue when the system is onedimensional. We also show that some of these upper bounds can be achieved if A 1 is lower triangular. These bounds are applied to some commonly used systems to derive new results or strengthen known results.
On the probabilistic worstcase time of "FIND"
 ALGORITHMICA
, 2001
"... We analyze the worstcase number of comparisons Tn of Hoare’s selection algorithm find when the input is a random permutation, and worst case is measured with respect to the rank k. We give a new short proof that Tn/n tends to a limit distribution, and provide new bounds for the limiting distributi ..."
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Cited by 14 (0 self)
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We analyze the worstcase number of comparisons Tn of Hoare’s selection algorithm find when the input is a random permutation, and worst case is measured with respect to the rank k. We give a new short proof that Tn/n tends to a limit distribution, and provide new bounds for the limiting distribution.
How Fast Does A General Branching Random Walk Spread?
, 1997
"... New results on the speed of spread of the onedimensional spatial branching process are described. Generalizations to the multitype case and to d dimensions are discussed. The relationship of the results with deterministic theory is also indicated. Finally the theory developed is used to reprove s ..."
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Cited by 14 (2 self)
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New results on the speed of spread of the onedimensional spatial branching process are described. Generalizations to the multitype case and to d dimensions are discussed. The relationship of the results with deterministic theory is also indicated. Finally the theory developed is used to reprove smoothly (and improve slightly) results on certain datastorage algorithms arising in computer science.
Minima in branching random walks
 Annals of Probability, 37(3): 1044–1079
, 2009
"... Given a branching random walk, let Mn be the minimum position of any member of the nth generation. We calculate EMn to within O(1) and prove exponential tail bounds for P{Mn −EMn > x}, under quite general conditions on the branching random walk. In particular, together with work by Bramson [Z. Wah ..."
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Cited by 13 (1 self)
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Given a branching random walk, let Mn be the minimum position of any member of the nth generation. We calculate EMn to within O(1) and prove exponential tail bounds for P{Mn −EMn > x}, under quite general conditions on the branching random walk. In particular, together with work by Bramson [Z. Wahrsch. Verw. Gebiete 45 (1978) 89–108], our results fully characterize the possible behavior of EMn when the branching random walk has bounded branching and step size. 1. Introduction. The
Probabilistic Analysis of Tree Search
 Disorder in Physical Systems
, 1990
"... Consider the family tree of an agedependent branching process, where the branches have costs corresponding to birth times. The firstbirth problem of Hammersley (1974) then concerns the cost of an optimal (cheapest) node at depth n. Suppose that we must explore the tree so as to find an optimal or ..."
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Cited by 9 (1 self)
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Consider the family tree of an agedependent branching process, where the branches have costs corresponding to birth times. The firstbirth problem of Hammersley (1974) then concerns the cost of an optimal (cheapest) node at depth n. Suppose that we must explore the tree so as to find an optimal or nearly optimal node at depth n. We now have a suitable model for analysing the behaviour of tree search algorithms, and we may extend the investigations of Karp and Pearl (1983).