A. V. Aho
ABSTRACT
In [1], Kennedy conjectures that for every n node reducible flow graph, there is a sequence of nodes (with repetitions) of length O(nlogn) such that all acyclic paths are subsequences thereof. Such a sequence would, if it could be found easily, enable one to do various kinds of global data flow analyses quickly. We show that for all reducible flow graphs such a sequence does exist, even if the number of edges is much larger than n. If the number of edges is O(n), the node listing can be found in O(nlogn) time.
- 1.K. Kennedy, "Node listing techniques applied to data flow analysis," Proc. 2nd ACM Conference on Principles of Programming Languages, January 1975, 10-21. Google Scholar
Digital Library
- 2.F.E. Allen, "Control flow analysis," SIGPLAN Notices5:7, (July, 1970), 1-19. Google ScholarDigital Library
- 3.D. E. Knuth, "An empirical study of FORTRAN programs," Software: Practice and Experience1:2, (April, 1971), 105-134.Google ScholarCross Ref
- 4.M.S. Hecht and J. D. Ullman, "Analysis of a simple algorithm for global data flow problems," Proc. ACM Symposium on Principles of Programming Languages, October, 1973. To appear, SIAM J. Computing. Google ScholarDigital Library
- 5.J. Cocke, "Global common subexpression elimination," SIGPLAN Notices5:7 (July, 1970), 20-24. Google ScholarDigital Library
- 6.A.V. Aho and J. D. Ullman, The Theory of Parsing, Translation and Compiling, Volume 2, Compiling, Prentice-Hall, Englewood Cliffs, N. J., 1973. Google ScholarDigital Library
- 7.M. Schaefer, A Mathematical Theory of Global Flow Analysis, Prentice-Hall, Englewood Cliffs, N. J., 1973.Google Scholar
- 8.K. Kennedy, "A global flow analysis algorithm," Int. J. Comp. Math.3 (December, 1971), 5-15.Google ScholarCross Ref
- 9.G.A. Kildall, "A unified approach to global program optimization," Proc. ACM Symposium on Principles of Programming Languages, October, 1973. Google ScholarDigital Library
- 10.J. Kam and J. D. Ullman, "Global optimization problems and iterative algorithms," TR 146, Computer Science Laboratory, Princeton University, January, 1974.Google Scholar
- 11.S. L. Graham and M. Wegman, "A fast and usually linear algorithm for global flow analysis," Proc. 2nd ACM Symposium on Principles of Programming Languages, January, 1975. Google ScholarDigital Library
- 12.C. M. Geschke, Global Program Optimizations, Ph. D. Thesis, Carnegie-Mellon University, 1972. Google ScholarDigital Library
- 13.M. Newey, "Notes on a problem involving permutations as subsequences," STAN-CS-73-340, Computer Science Department, Stanford University, Stanford, California. Google ScholarDigital Library
- 14.M. S. Hecht and J. D. Ullman, "Flow graph reducibility," SIAM J. Computing1:2 (June, 1972), 188-202.Google ScholarDigital Library
- 15.M. S. Hecht and J. D. Ullman, "Characterizations of reducible flow graphs," J. ACM21:3 (July, 1974), 367-375. Google ScholarDigital Library
- 16.J. D. Ullman, "Fast algorithms for the elimination of common subexpressions," Acta Informatica2 (December, 1973), 191-213.Google ScholarDigital Library
- 17.P. M. Lewis II, R. E. Stearns, and J. Hartmanis, "Memory bounds for recognition of context-free and context-sensitive languages," IEEE Conference Record of 6th Annual ACM Symposium on Switching Circuit Theory and Logical Design, October, 1966, pp. 191-202.Google Scholar
- 18.R. E. Tarjan, "Testing flow graph reducibility," J. Computer and System Sciences9:3 (December, 1974), 355-365.Google ScholarDigital Library
- 19.G. Markowsky, Unpublished memorandum, December, 1974.Google Scholar
- 20.R. E. Tarjan, private communication, December, 1974.Google Scholar
Index Terms
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Node listings for reducible flow graphs
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Recommendations
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Characterizations of Reducible Flow Graphs
It is established that if G is a reducible flow graph, then edge (n, m) is backward (a back latch) if and only if either n = m or m dominates n in G. Thus, the backward edges of a reducible flow graph are unique.
Further characterizations of reducibility ...
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Node listings for reducible flow graphs
K. Kennedy recently conjectured that for every n node reducible flow graph, there is a sequence of nodes (with repetitions) of length O(n log n) such that all acyclic paths are subsequences thereof. Such a sequence would, if it could be found easily, ...
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Feedback vertex sets and cyclically reducible graphs
The problem of finding a minimum cardinality feedback vertex set of a directed graph is considered. Of the classic NP-complete problems, this is one of the least understood. Although Karp showed the general problem to be NP-complete, a linear algorithm ...
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