The average size of ordered binary subgraphs
To analyse the demands made on the garbage collector in a graph reduction system, the change in size of an average graph is studied when an arbitrary edge is removed. In ordered binary trees the average number of deleted nodes as a result of cutting a single edge is equal to the average size of a subtree. Under the assumption that all trees with n nodes are equally likely to occur, the expected size of a subtree is found to be approximately √πn. The enumeration procedure can be applied to graphs by considering spanning trees in which the nodes that were shared in the graph are marked in the spanning tree. A correction to the calculation of the average is applied by ignoring subgraphs that have a marked root. Under the same assumption as above the average size of a subgraph is approximately √πn-2(m+1), where m represents the number of shared nodes and m≪n.
Key wordsbinary graphs Catalan statistics combinator graph reduction subgraphs
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- 1.H. P. Barendregt, The lambda calculus, its syntax and semantics, North Holland, Amsterdam (1984).Google Scholar
- 2.P. J. Landin, “The mechanical evaluation of expressions,” Computer Journal 6(4) pp. 308–320 (Jan. 1964).Google Scholar
- 3.C. P. Wadsworth, Semantics and pragmatics of the lambda calculus, Oxford University, U.K. (1971). PhD. ThesisGoogle Scholar
- 4.P. H. Hartel, “A comparative study of garbage collection algorithms,” PRM project internal report D-23, Computing Science Department, University of Amsterdam (Feb. 1988).Google Scholar
- 5.J. Cohen, “Garbage collection of linked structures”, Computing Surveys 13(3) pp. 341–367 (Sep. 1981).Google Scholar
- 6.N. G. de Bruijn, D. E. Knuth, and S. O. Rice, “The average height of planted plane trees,” pp. 15–22 in Graph Theory and Computing, ed. R.C. Read, Academic Press, London, U.K. (1972).Google Scholar
- 7.P. H. Hartel and A. H. Veen, “Statistics on graph reduction of SASL programs,” Software practice and experience 18(3) pp. 239–253 (Mar. 1988).Google Scholar
- 8.D. E. Knuth, The art of computer programming, volume 1: Fundamental algorithms, Addison Wesley, Reading, Massachusetts (1973). second editionGoogle Scholar
- 9.H. W. Gould and J. Kaucky, “Evaluation of a class of binomial coefficient summations,” Journal of Combinatorial theory 1(2) pp.233–247 (Sep. 1966).Google Scholar
- 10.D. E. Knuth, The art of computer programming, volume 2: Seminumerical algorithms, Addison Wesley, Reading, Massachusetts (1980). second editionGoogle Scholar