Values of the Merging Function and Algorithm Design as a Game

| Finding algorithms that are optimal for worst case input is equivalent to solving a 2-player game of perfect information. The merging game arises when nding the minimum number M(m; n) of comparisons needed to merge two sorted lists of cardi-nalities m and n. We devised a computer program capable of determining any value of M(m; n) 20. It was based on standard game solving techniques (alpha-beta search, transposition table) plus bounding , \fuzzy trans table," and \partitioning" ideas speciic to the merging game. Thanks to a new idea called \iterative widening," our game solver can also determine strong upper bounds (but not lower bounds) on M(m; n) in a much larger region. The widening/shrinking idea may have other applications in computer game playing. We present a large table of bounds on M(m; n) which were obtained by combining our new results with previous bounds on M(m; n) (which we survey).