P. Zimmermann and W. Zimmermann. The Automatic Complexity Analysis of Divide-and-Conquer Programs. Res. Rep. 1149, INRIA, France, Dec. 1989. 144  Home/Search   Document Details and Download   Summary   Related Articles   Check

....this is rare. A key problem is how to characterize the input data and exploit this information in the analysis. In traditional complexity analysis, inputs are characterized by their size. Accommodating this requires manual or semi automatic transformation of the time or space function [17, 29]. The analysis is mainly asymptotic. A challenging problem that arises in this approach is optimizing the time bound or space bound function to a closed form in terms of the input size [17, 24, 7] But closed forms are known only for subclasses of functions. Thus, such optimization can not be ....

P. Zimmermann and W. Zimmermann. The automatic complexity analysis of divide-and-conquer algorithms. In Computer and Information Sciences VI. Elsevier, 1991.

....Performance 1. INTRODUCTION Analysis of program running time is important for reactive systems, interactive environments, compiler optimizations, performance evaluation, and many other computer applications. It has been extensively studied in many fields of computer science: algorithms [22, 13, 14, 41], programming languages [39, 23, 32, 36, 35] and systems [37, 30, 34, 33] Being able to predict accurate time bounds automatically and efficiently is particularly important for many applications, such as reactive systems. It is also particularly desirable to be able to do so for high level ....

....them in the time function. In general, due to imperfect knowledge about the input, the time function is transformed into a time bound function. In algorithm analysis, inputs are characterized by their size; accommodating this requires manual or semi automatic transformation of the time function [39, 23, 41]. The analysis is mainly asymptotic, and primitive parameters are considered independent of the input size, i.e. are constants while the computation iterates or recurses. Whatever values of the primitive parameters are assumed, a second problem arises, and it is theoretically challenging: ....

[Article contains additional citation context not shown here]

P. Zimmermann and W. Zimmermann. The automatic complexity analysis of divide-and-conquer algorithms. In Computer and Information Sciences VI. Elsevier, 1991.

....on any input in that set. A key problem is how to characterize the input data and exploit this information in the analysis. In traditional complexity analysis, inputs are characterized by their size. Accommodating this requires manual or semiautomatic transformation of the time or space function [18, 29]. The analysis is mainly asymptotic. A theoretically challenging problem that arises in this approach is optimizing the time bound or space bound function to a closed form in terms of the input size [18, 24, 6] Although much progress has been made in this area, closed forms are known only for ....

P. Zimmermann and W. Zimmermann. The automatic complexity analysis of divide-and-conquer algorithms. In Computer and Information Sciences VI. Elsevier, 1991. 10

....is how to characterize the input data and exploit this information in the analysis. In traditional analysis of the time and space complexity of algorithms, inputs are characterized by their size. Accommodating this requires manual or semi automatic transformation of the time or space function [53, 32, 55]. The analysis is mainly asymptotic. A theoretically challenging problem that arises in this approach is optimizing the time bound or space bound function to a closed form in terms of the input size [53, 5, 32, 45, 12] Although much progress has been made in this area, closed forms are known only ....

P. Zimmermann and W. Zimmermann. The automatic complexity analysis of divide-and-conquer algorithms. In Computer and Information Sciences VI. Elsevier, 1991. 31 of 31

....G omez s address: Computer Science Department, Indiana University, Bloomington, IN 47405 7104. Corresponding author: Yanhong A. Liu. Email: liu cs.sunysb.edu. Tel: 631 632 8463. Fax: 631 632 8334. URL: http: www.cs.sunysb.edu liu . 1 studied in many elds of computer science: algorithms [25, 16, 17, 53], programming languages [50, 26, 41, 44] and systems [46, 37, 43, 42] It is particularly important for many applications, such as real time systems and embedded systems, to be able to predict accurate time bounds and space bounds automatically and eciently, and it is particularly desirable to be ....

....function. In general, due to imperfect knowledge about the input, the cost function is transformed into a cost bound function. In algorithm analysis, inputs are characterized by their size; accommodating this requires manual or semi automatic transformation of the cost (time or space) function [50, 26, 53]. The analysis is mainly asymptotic, and primitive cost parameters are considered independent of input size, i.e. are constants while the computation iterates or recurses. Whatever values of the primitive cost parameters are assumed, a second problem arises, and it is theoretically 3 ....

[Article contains additional citation context not shown here]

P. Zimmermann and W. Zimmermann. The automatic complexity analysis of divide-and-conquer algorithms. In Computer and Information Sciences VI. Elsevier, 1991. 31

....is how to characterize the input data and exploit this information in the analysis. In traditional analysis of the time and space complexity of algorithms, inputs are characterized by their size. Accommodating this requires manual or semi automatic transformation of the time or space function [53, 32, 55]. The analysis is mainly asymptotic. A theoretically challenging problem that arises in this approach is optimizing the time bound or space bound function to a closed form in terms of the input size [53, 5, 32, 45, 12] Although much progress has been made in this area, closed forms are known only ....

P. Zimmermann and W. Zimmermann. The automatic complexity analysis of divide-and-conquer algorithms. In Computer and Information Sciences VI. Elsevier, 1991. 28 of 28

....A. Liu November 1999 1 Introduction Analysis of program running time is important for reactive systems, interactive environments, compiler optimizations, performance evaluation, and many other computer applications. It has been extensively studied in many fields of computer science: algorithms [21, 12, 13, 40], programming languages [38, 22, 31, 35, 34] and systems [36, 29, 33, 32] Being able to predict accurate time bounds automatically and efficiently is particularly important for many applications, such as reactive systems. It is also particularly desirable to be able to do so for high level ....

....them in the time function. In general, due to imperfect knowledge about the input, the time function is transformed into a time bound function. In algorithm analysis, inputs are characterized by their size; accommodating this requires manual or semi automatic transformation of the time function [38, 22, 40]. The analysis is mainly asymptotic, and primitive parameters are considered independent of the input size, i.e. are constants while the computation iterates or recurses. Whatever values of the primitive parameters are assumed, a second problem arises, and it is theoretically challenging: ....

[Article contains additional citation context not shown here]

P. Zimmermann and W. Zimmermann. The automatic complexity analysis of divide-and-conquer algorithms. In Computer and Information Sciences VI. Elsevier, 1991. 15

....complexity analysis. Given a program, such a system attempts to automatically derive complexity bounds for this program. This analysis is undecidable, by the halting problem. Systems have been developed primarily for first order serial call by value functional languages with recursive datatypes [17, 18, 26, 34, 35, 36], although one has used parallelism with arrays [37, 38, 39] During this analysis, the program is translated into one which computes the cost of executing the original program. Later systems defined these translations via cost models based on operational semantics, although earlier systems simply ....

Paul Zimmermann and Wolf Zimmermann. The automatic complexity analysis of divide-andconquer algorithms. Computer and Information Sciences VI, 1991.

....complexity analysis. Given a program, such a system attempts to automatically derive complexity bounds for this program. This analysis is undecidable, by the halting problem. Systems have been developed primarily for first order serial call by value functional languages with recursive datatypes [17, 18, 26, 34, 35, 36], although one has used parallelism with arrays [37, 38, 39] During this analysis, the program is translated into one which computes the cost of executing the original program. Later systems defined these translations via cost models based on operational semantics, although earlier systems simply ....

Paul Zimmermann and Wolf Zimmermann. The automatic complexity analysis of divide-andconquer algorithms. Technical Report 1149, Institut National de R'echerche en Informatique et en Automatique, Rocquencourt, December 1989.

....the calculated bounds. 1 Introduction Analysis of program running time is important for real time systems, interactive environments, compiler optimizations, performance evaluation, and many other computer applications. It has been extensively studied in many fields of computer science: algorithms [20, 12, 13, 41], programming languages [38, 21, 30, 33] and systems [35, 28, 32, 31] It is particularly important for many applications, such as real time systems, to be able to predict accurate time bounds automatically and efficiently, and it is particularly desirable to be able to do so for high level ....

....in the timing function. In general, due to imperfect knowledge about the input, the timing function is transformed into a time bound function. In algorithm analysis, inputs are characterized by their size; accommodating this requires manual or semi automatic transformation of the timing function [38, 21, 41]. The analysis is mainly asymptotic, and primitive parameters are considered independent of input size, i.e. are constants while the computation iterates or recurses. Whatever values of the primitive parameters are assumed, a second problem arises, and it is theoretically challenging: optimizing ....

[Article contains additional citation context not shown here]

P. Zimmermann and W. Zimmermann. The automatic complexity analysis of divide-and-conquer algorithms. In Computer and Information Sciences VI. Elsevier, 1991.

....3.3 A Calculus for Proving Complexity Propositions We extend the classical many sorted logic to a logic for reasoning about complexity propositions. The rules in the calculus are similar to that in [25, 8, 27] The technique of [27] is an extension of [25] to general divide and conquer algorithms [26]. The imperative rules are similar to those by Hickey and Cohen[8] However their rules are for average case analysis of programs while ours are for worst case analysis. In contrast to the above systems, our system is not designed for automatic complexity analysis but just for proving complexity ....

....S i ) m( b S i goto Gamma S i 1 ) Observe that S 0 = I H and S 1 = I H2 . Defining S 0 = I H1 and c S 0 = FH1 completes the proof. Xi For methods, we aim at worst case complexities. The rules in Table 6 are the same as in the work on automatic complexity analysis based on recurrences [25, 8, 27, 26]. In case of recursion we obtain usually a linear recurrence system. Solution methods of these systems are described in [6, 7, 27] A generalization of these rules to methods with more than one parameter is straightforward. Fig. 6 shows parts of the complexity proof of push. The termination ....

P. Zimmermann and W. Zimmermann. The automatic complexity analysis of divide-andconquer algorithms. In Proceedings of the Sixth International Symposium on Computer and Information Sciences, pages 395--404, 1991.

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P. Zimmermann and W. Zimmermann. The Automatic Complexity Analysis of Divide-and-Conquer Programs. Res. Rep. 1149, INRIA, France, Dec. 1989. 144

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Zimmermann, P., and Zimmermann, W. Automatic complexity analysis of divide-and-conquer algorithms. In Proceedings of the 6th International Symposium on Computer and Information Sciences (Antalya, Turkey, 1991), M. Baray and B. Ozguc, Eds., Elsevier Science Publishers, pp. 395--404. A Proofs Proof of Theorem 33 on page 7. Assume that the #'s are labeled in such

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Zimmermann, P., and Zimmermann, W. Automatic complexity analysis of divide-and-conquer algorithms. In Proceedings of the 6th International Symposium on Computer and Information Sciences (Antalya, Turkey, 1991), M. Baray and B. Ozguc, Eds., Elsevier Science Publishers, pp. 395--404. A Proofs Proof of Theorem 33 on page 7. Assume that the #'s are labeled in such

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Zimmermann,P., Zimmermann,W. (1989). The Automatic Complexity Analysis of Divide-and-Conquer Programs. Res. Rep. 1149, INRIA, France, December.

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