By C. J. Colbourn
The scope of the amount comprises all algorithmic and computational facets of analysis on combinatorial designs. Algorithmic elements contain iteration, isomorphism and research suggestions - either heuristic equipment utilized in perform, and the computational complexity of those operations. The scope inside layout conception contains all features of block designs, Latin squares and their variations, pairwise balanced designs and projective planes and comparable geometries.
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To describe these properties we need to consider the data structures used to represent subsets. ,n . A k-subset S of an n-set can be represented as a 6itwctor (61, 62, ... ,6,), where 6, is 1 if z is in S and 0 if z is not in S. Alternatively, if S={aI, 82, ... ,ak} where 81 < 82 < ... >8 k ) - (Aside : AU the algorithms above can be implemented using either data structure. For testing each algorithm was implemented using the data structure which made it faster: bitvectors were used for BER and EE, all the others used ordered arrays.
If the processing time dominates the generation time, then the processing time also determines the size of the largest problem that can be tackled. However, if the processing time is about the same or less than the generation time, then the generation time imposes a limit on the largest problem which can be tackled: for instance, in an hour of CPU time on the Cyber172/2, LS can process every 15subset of a 30-set. Hand optimized assembler, or a supercomputer, could improve this limit, but not significantly.
Shimon Even, Algorithmic Combinnforb, hlacmillan (1973). Clement W. H. Lam and Leonard H. Soicher, “Three New Combination Algorithms with the Minimal Change Property”, Communications of the Association ,’or Computing Machinery, 25 (1982)555-559. Edward M. Reingold, Jurg Nicvergelt and Narsingh Deo, Combinatorial Algorithme, T h e o y and Practice, Prentice Hall (1977). [lo] Albert Nijenhuis and Herbert S. Wilf, Combinaton’al Algorithms, Monographs in Computer Science and Applied Mathematics, Academic Press, (1975).