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Randomized Binary Search Trees
 Journal of the ACM
, 1997
"... In this paper we present randomized algorithms over binary search trees such that: a) the insertion of a set of keys, in any fixed order, into an initially empty tree always produces a random binary search tree; b) the deletion of any key from a random binary search tree results in a random binary s ..."
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Cited by 22 (2 self)
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In this paper we present randomized algorithms over binary search trees such that: a) the insertion of a set of keys, in any fixed order, into an initially empty tree always produces a random binary search tree; b) the deletion of any key from a random binary search tree results in a random binary search tree; c) the random choices made by the algorithms are based upon the sizes of the subtrees of the tree; this implies that we can support accesses by rank without additional storage requirements or modification of the data structures; and d) the cost of any elementary operation, measured as the number of visited nodes, is the same as the expected cost of its standard deterministic counterpart; hence, all search and update operations have guaranteed expected cost O(log n), but now irrespective of any assumption on the input distribution. 1. Introduction Given a binary search tree (BST, for short), common operations are the search of an item given its key and the retrieval of the inform...
On Automated Verification of Probabilistic Programs
"... Abstract. We introduce a simple procedural probabilistic programming language which is suitable for coding a wide variety of randomised algorithms and protocols. This language is interpreted over finite datatypes and has a decidable equivalence problem. We have implemented an automated equivalence c ..."
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Cited by 11 (6 self)
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Abstract. We introduce a simple procedural probabilistic programming language which is suitable for coding a wide variety of randomised algorithms and protocols. This language is interpreted over finite datatypes and has a decidable equivalence problem. We have implemented an automated equivalence checker, which we call apex, for this language, based on game semantics. We illustrate our approach with three nontrivial case studies: (i) Herman’s selfstabilisation algorithm; (ii) an analysis of the average shape of binary search trees obtained by certain sequences of random insertions and deletions; and (iii) the problem of anonymity in the Dining Cryptographers protocol. In particular, we record an exponential speedup in the latter over stateoftheart competing approaches. 1
THE EVOLUTION OF TWO STACKS IN BOUNDED SPACE AND RANDOM WALKS IN A TRIANGLE
"... We analyse a ~imple storage allocation scheme in which two stacks grow and shrink inside a shared memory area. To that purpose, we provide analytic expressions for the number of 2dimensional random walks in a triangle with two reflecting barriers and one absorbing barrier. We obtain probability dis ..."
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Cited by 5 (0 self)
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We analyse a ~imple storage allocation scheme in which two stacks grow and shrink inside a shared memory area. To that purpose, we provide analytic expressions for the number of 2dimensional random walks in a triangle with two reflecting barriers and one absorbing barrier. We obtain probability distributions and expectations of characteristic parameters of that shared memory scheme, namely the sizes of the stacks and the time until the system runs out of memory. This provides a complete solution to an open problem posed by Knuth in "The Art of Computer Programming", Vol. I, 1968 [Ex. 2.2.2.13].
Deletion Algorithms for Binary Search Trees
, 1994
"... The effect of updating (deletions/insertions) on binary search trees has been an interesting research topic for almost three decades, but in the last five years there have been a few contributions, due partially to the intrinsic difficulty of the involved analysis. Since the problem is quite difficu ..."
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Cited by 2 (2 self)
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The effect of updating (deletions/insertions) on binary search trees has been an interesting research topic for almost three decades, but in the last five years there have been a few contributions, due partially to the intrinsic difficulty of the involved analysis. Since the problem is quite difficult to be solved in a general fashion, we have restricted ourselves to solve a simpler problem, which shall be considered as an important and necessary basis for further developments. In this paper, we have faced the systematization of the study of deletion algorithms, and deduced the effect that a single random deletion produces in the probability distribution of binary trees of arbitrary size for a wide variety of deletion algorithms. Furthermore, to carry on the analysis, some new tools have been introduced, such as the concepts of strong and weak invariance of probability functions induced by an algorithm. Among others, we have been able to derive interesting results such as an extension ...
Asymmetry in Binary Search Tree Update Algorithms
, 1994
"... In this paper we explore the relationship between asymmetries in deletion algorithms used in updating binary search trees, and the resulting long term behavior of the search trees. We show that even what would appear to be negligible asymmetric effects accumulate to cause long term degeneration. ..."
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Cited by 1 (0 self)
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In this paper we explore the relationship between asymmetries in deletion algorithms used in updating binary search trees, and the resulting long term behavior of the search trees. We show that even what would appear to be negligible asymmetric effects accumulate to cause long term degeneration. This persists even in the face of other effects that would appear to counteract the long term effects. On the other hand, eliminating the asymmetry completely seems to give us trees that have a smaller IPL than is expected for trees built by a random sequence of insertions. But even then there are surprises in that the backbone becomes longer than expected. 1 Introduction Binary search trees are one of the oldest and most frequently used data structures for solving the dictionary and other problems [2, 11, 6, 9]. The average case efficiency of these structures has been well studied, when only insertions are involved. The usual insertion algorithm simply inserts new values at the leaf...
Analysis of the standard deletion algorithms in exact fit domain binary search trees
 Algorithmica
, 1990
"... Abstract. It is well known that the expected search time in an N node binary search tree generated by a random sequence of insertions is O(log N). Little has been published about the asymptotic cost when insertions and deletions are made following the usual algorithms with no attempt to retain balan ..."
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Cited by 1 (1 self)
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Abstract. It is well known that the expected search time in an N node binary search tree generated by a random sequence of insertions is O(log N). Little has been published about the asymptotic cost when insertions and deletions are made following the usual algorithms with no attempt to retain balance. We show that after a sufficient number of updates, each consisting of choosing an element at random, removing it, and reinserting the same value, that the average search cost is O(N1/2).
SOME INFORMATION ABOUT THE BINOMIAL TRANSFORM
, 1993
"... A few days ago I saw the paper [4]. I think I can make some additional remarks that might not be totally useless for the Fibonacci Community! Let (an) be a given sequence and sn = S^=0(?J^. Denoting the respective (ordinary) generating functions by A(x) and S(x), the paper in question mainly deals w ..."
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A few days ago I saw the paper [4]. I think I can make some additional remarks that might not be totally useless for the Fibonacci Community! Let (an) be a given sequence and sn = S^=0(?J^. Denoting the respective (ordinary) generating functions by A(x) and S(x), the paper in question mainly deals with the consequences of the formula Knuth [7] has introduced the binomial transform by and it is clear that this is the situation from above. But Philippe Flajolet and the present writer agreed about ten years ago that there are just exponential generating functions hidden! They have a convolution formula and upon choosing the hk's to be equal to 1, we have the old situation. So, denoting the exponential generating functions by A(x) mdS(x), we have the even simpler formula S(x) = e x A(x). This can readily be inverted as A(x) = e ~ x S(x), whence These facts about exponential generating functions are of course folklore; one particular reference is [3]. Flajolet & Richmond [2], Schmid [8], and Kirschenhofer & Prodinger [6] all made heavy use of (1). Schmid observed (among other writers) that an exponential generating function will be transformed into an ordinary generating function by the Bore I transform. Now the generalization translates into Since S(x) = e bx A(cx). A(x) = ei x S[± 412 [NOV. SOME INFORMATION ABOUT THE BINOMIAL TRANSFORM we find the inversion formula n s \ k=0 y The discussion in Theorem 2 becomes quite transparent, considering exponential generating functions. It is asked whenever we have p