Superlinear bounds on matrix searching

The technique of matrix searching in totally monotone matrices and their generalizations is steadily finding ever more applications in a wide variety of areas of computer science, especially computational geometry and dynamic programming problems (see [AKMSW87], [AK87], [AP88], [AS87], [AS89], [EGG88], [KK88], [WSS]). Although an asymptotically optimal linear time algorithm is known for the most basic problem of finding row minima and maxima in totally monotone matrices [AKMSW87], for most of the generalizations of totally monotone matrices, only superlinear algorithms are known, though until now no superlinear lower bounds have been proved. This paper gives the first superlinear bound for matrix searching in two types of totally monotone partial matrices. We also give a matching upper bound for a subclass of one of them, though unfortunately the proof of the lower bound does not apply to this subclass. These types of matrices, which we refer to as v-matrices and h-matrices, respectively, were introduced by Aggarwal and Suri [AS891 who used them to find the farthest visible pair in a simple polygon. In addition, these matrix classes are natural extensions of staircase matrices which have applications in computational geometry and dynamic programming problems.