Philippe Flajolet, Paolo Sipala, and Jean-Marc Steyaert. Analytic variations on the common subexpression problem. In Proc. Automata, Languages, and Programming, volume 443 of Lecture Notes in Computer Science, pages 220-- 234. Springer-Verlag, 1990.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....realize finite and infinite terms with sharing. As to the complexity of collapsing, arbitrary term graphs can be made fully collapsed in time O(n log n) where n is the size of term graphs. This bound reduces to O(n) for term graphs over finite sets of function symbols and variables. See [30, 47, 38]. 15 4 Term Graph Rewriting In this section we define the transformation of term graphs by applications of term rewrite rules, introducing the notion of term graph rewriting. A fundamental property of this computational model is its soundness with respect to term rewriting. We also consider the ....

Philippe Flajolet, Paolo Sipala, and Jean-Marc Steyaert. Analytic variations on the common subexpression problem. In Proc. Automata, Languages, and Programming, volume 443 of Lecture Notes in Computer Science, pages 220-- 234. Springer-Verlag, 1990.

....realize finite and infinite terms with sharing. As to the complexity of collapsing, arbitrary term graphs can be made fully collapsed in time O(n log n) where n is the size of term graphs. This bound reduces to O(n) for term graphs over finite sets of function symbols and variables. See [30,47,38]. 1.4 Term Graph Rewriting In this section we define the transformation of term graphs by applications of term rewrite rules, introducing the notion of term graph rewriting. A fundamental property of this computational model is its soundness with respect to term rewriting. We also consider the ....

Philippe Flajolet, Paolo Sipala, and Jean-Marc Steyaert. Analytic variations on the common subexpression problem. In Proc. Automata, Languages, and Programming, volume 443 of Lecture Notes in Computer Science, pages 220--234. Springer-Verlag, 1990.

....of size n is asymptotically linear as n tends to infinity. Many other statistics have been done on trees using this techniques. The tree compaction problem was studied in [CDV85, CDSV83] where it was introduced a nice generation of compacted trees. The analytical value was finally obtained in [FSS90] where they prove that the expected size of the maximally compacted representation of a random tree of size n is asymptotically c n p log n (1 O(1= log n) In the case of the average height of trees, Equation (2.19) introduces an iteration scheme, where B(z) is the fixed point. Some ....

P. Flajolet, P. Sipala, and J-M. Steyaert. Analytic variations on the common subexpression problem. In M.S. Paterson, editor, 17th. Int. Colloquium on Automata, Languages and Programming, volume 20, pages 220--234. SpringerVerlag, Lecture Notes in Computer Science, 1990.

....of T as a directed acyclic graph. We call the link to the representative of a tree U the signature of U . Recognizing identical substructures is called common subexpression elimination. 1 Common subexpression elimination in a tree T can be performed in time O(jT j) DST80] See also [FSS90] The image of the root of P determines unambiguously the images of all the other pattern nodes in an ordered child embedding of P in T . This means that there are at most n ordered child pattern embeddings into a target of size n. Assume that common subexpression elimination has been performed ....

P. Flajolet, P. Sipala, and J.-M. Steyaert. Analytic variations on the common subexpression problem. In Automata, Languages and Programming, pages 220--234. Springer-Verlag, 1990.

....This work resorts to a combination of algebraic and analytic generating function methods which is versatile and widely applicable. The recent paper [131] deals with patterns in binary search trees. There are some earlier papers dealing with the problem of patterns in several families of trees, [15, 28, 31, 44, 52, 68, 73, 91]. Let (u) be the probability for a given unlabelled tree u to have the shape of a random binary search tree of size juj, and [t] be the number of occurrences of the (fixed) pattern u in the binary search tree t. As expected, the solution involves bivariate generating function like this one F ....

P. Flajolet, P. Sipala, and J-M. Steyaert. Analytic variations on the common subexpression problem. In Automata, Languages, and Programming, volume 443 of Lecture Notes in Computer Science, pages 220--234, 1990.

....then nothing but the directed acyclic graph (dag) representation of this tree obtained in the usual way by sharing repeated subtrees and representing them only once. It is classically known that such a dag representation of a tree of size N can always be constructed in time O(N ) see for instance [5] for a discussion. Here, one has N = 2 n for functions in n , so that the sharing algorithm approach is of exponential time complexity when f is given by its truth table or, equivalently, by its tree bdt(f ) In many cases, fortunately, one can operate with polynomial time complexity. Here is ....

Flajolet (Philippe), Sipala (Paolo), and Steyaert (Jean-Marc). { Analytic variations on the common subexpression problem. In Paterson (M. S.) (editor), Automata, languages and programming. Lecture Notes in Computer Science, vol. 443, pp. 220-234. { Springer, New York, 1990. Proceedings of the 17th ICALP Conference, Warwick, July 1990.

....path length is of order n 3=2 on average. Variations around this theme have led to the analysis of a large number of tree algorithms in the context of symbolic manipulations. We shall cite here: pattern matching [76, 2] simplification [10] unification [1] common subexpression factorization [34], and term rewriting techniques [11] See also [9] for an interesting survey. Another important case of application is to the Cayley function y = ze y . This function shows up in the enumeration of labelled trees, random mappings [28] in the analysis of hashing with linear probing [55] in ....

Flajolet, P., Sipala, P., and Steyaert, J.-M. Analytic variations on the common subexpression problem. In Automata, Languages, and Programming (1990), M. S. Paterson, Ed., vol. 443 of Lecture Notes in Computer Science, pp. 220--234. Proceedings of the 17th ICALP Conference, Warwick, July 1990.

....n 4 2:64726 Delta 0:96378 n 5 2:33576 Delta 0:98297 n 10 2:01784 Delta 0:99950 n We empirically observe that c k 2 and ae k 1 Gamma 2 Gammak Gamma1 . This can be proved: longest runs are discussed in Feller s book [3] and with more precise estimates for varying k and n by Knuth [7] in connection with carry propagation in certain adders. The end result is that the length of the longest run is on average log 2 n O(1) see the last section of this chapter for more details. 4.7 Meromorphic Functions An expansion very similar to that of Theorem 4.5 given for rational ....

....(ae k ) Gamma1 O(2 Gammak ) From there it results easily that the longest run in a random binary string of size n has expected length log 2 n O(1) Exercise 25. Establish precise estimates for ae k and P 0 (ae k ) when k gets large. Prove the log 2 n O(1) result mentioned in the text [7]. 4.12 Notes Given a generating function, valuable estimates incorporating the essential exponential growth are simply obtained from the location of dominant singularities. The method of exponential bounds applies to many function even if they are only defined implicitly through a functional ....

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Flajolet, P., Sipala, P., and Steyaert, J.-M. Analytic variations on the common subexpression problem. In Automata, Languages, and Programming (1990), M. S. Paterson, Ed., vol. 443 of Lecture Notes in Computer Science, pp. 220--234. Proceedings of the 17th ICALP Conference, Warwick, July 1990.

.... with excluded patterns (for instance no sequence of 3 identical characters in a row) have exponentially small probability, with the exponential rates being given as roots of correlation polynomials [15, 24, 40, 45] Similar properties hold true for the combinatorial tree model as established in [18, 41]. 1 As usual, a subtree of a tree t is defined by a node of t together with all its descendents. We are thus counting here occurrences of terminal subtrees. Such analyses, apart from being of combinatorial interest, are relevant to efficient storage representations of trees. A ....

.... (Similar ideas are often used to obtain compacted form of digital trie dictionaries, see for instance [37] Pushed to its limits, the pointer free representation gives rise to the directed acyclic graph (DAG) representation, also known in parsing and compiling as common subexpression factoring [18]. Naturally, this technique would not apply directly to BSTs, where internal nodes contain important informations, but, with minor adjustments, it is applicable to parse trees statistically governed by the BST model. We show here that the expected size Kn of the DAG associated to a BST of size n ....

[Article contains additional citation context not shown here]

Flajolet, P., Sipala, P., and Steyaert, J.-M. Analytic variations on the common subexpression problem. In Automata, Languages, and Programming (1990), M. S. Paterson, Ed., vol. 443 of Lecture Notes in Computer Science, pp. 220--234. Proceedings of the 17th ICALP Conference, Warwick, July 1990.

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