# Combinatorial Search Problems: Lectures held at the by Gyula Katona

By Gyula Katona

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Example text

Exactly two elements 41 (0)3) In this case at each test we divide X into dis- joint subsets and the result of the test shows us which one (or which ones) includes the unknown element(s). If the number of disjoint subsets is at each test a constant (say r ), then many problems can be solved (and they are done) in the same way as for (0~). We do not want to repeat them. It may occur that the number r of subsets depends on the situation, that is, on the previous tests and previous r~ sults. Picard (1965) generalized Theorem 2 (Huffman-algorithm) toward this case.

Find conditions under which it is possible to determine the optimal average length (see Theorems 3,4,5 and 6). 3. Generalize the results for alphabetical codes (see Theorems 9, 3, 4, 5 and 6). 4. Generalize the Huffman algorithm for the case if we can use only subsets with size ~k (k<~)(see Theorems 10 and 11). 5. 26)). 4. 48 Random search 6. :: k (i. = 1, ... ,m, k fixed < ~ ) • Theorem 11 gives good estimation for this minimwn. 28) for the case (0,&) when Aw··,Am are partitions into r parts and the sizes of the first r- 1 parts are bounded.

21, 27-33 (1971): Foundations of Probability. Holden-Day, San Francisco. SANDELIUS, M. (1961): On an optimal search procedure. Amer. Math. Monthly, 68, 138-154. SCHREIER, ]. (1932): On tournament elimination system. (Polish). Mathesis Polska, 7, 154-160. S. (196o): Problem E 1399. Amer. Math. Monthly, 67, 82. SLUPECKI, ]. , ~' 286--290. of tournaments. Colloq. B. (1947): The counterfeit coin problem. Math. Gaz. 31, 31-39. SOBEL, M. (1960): Group testing to classify efficiently all defectives in a binomial sample.