Topics in combinatorics and algorithms

Abstract

This thesis studies several topics in theoretical computer science. First, the author shows that $5n-4$ is a tight lower bound on the number of edges in the visibility graph of n non-intersecting line segments in the plane.

Second, the author studies a new class of combinatorial structures called Generalized Latin squares which is a generalization of the classical definition of Latin squares. A perfect $\langle k,l\rangle$-Latin square is an N $\times$ N array in which any row or column contains every distinct symbol and the symbol $a\sb{ij}$ appears exactly $\kappa$ times in the $i\sp{\rm th}$ row and l times in the $j\sp{\rm th\/}$ column, or vice versa. Let A = ($a\sb{ij}$) and B = ($b\sb{ij}$) be two perfect $\langle k,l\rangle$-Latin squares of order N with the symbol set $\{1, 2, \..., D\}.$ They are said to be orthogonal, if D divides N and each of the $D\sp2$ ordered pairs of symbols $(s,t)$ $(1 \leq s,t \leq D)$ appears exactly $N\sp2/D\sp2$ times in the array C = ($(a\sb{ij} , b\sb{ij})$). The author shows some general existence and orthogonality results by presenting constructive algorithms.

Third, the author shows some new results for the problem of unbounded searching. Given a function F:N$\sp{+}\ \to\ \{X,Y\}$ with the property that if $F(n\sb{0})$ = Y then $F(n)$ = Y for all $n\ >\ n\sb{0}$, the unbounded search problem is to use tests of the form "is $F(i)$ = X?" to determine the smallest n such that F(n) = Y. The "cost" of a search algorithm is a function c(n), the number of such tests used when the location of the first Y is n. He shows that the "ultimate algorithm" of Bentley and Yao (Info. Proc. Let. 5 (1976), 82-87) is "far" from optimal in the sense that it is only the second one in an infinite sequence of search algorithms, each of which is much closer to optimality than its predecessor.

Finally, consider this problem: how should n records with $\kappa$ keys be ordered so that a search can be performed as quickly as possible under any key? Fiat et al present an $O(lgn)$ time algorithm. This is asymptotically optimal. However, it uses quite complicated encoding and decoding procedures and requires tables of enormous sizes for the encoding method to work. The author presents a simpler O(lgn) algorithm for this problem, which works for any table of reasonable size. He also introduces an $O(lg\sp{2}n)$ algorithm which has better performance than any known $O(lgn)$ algorithm when n is not extremely large.