An Improved Hypercube Bound for Multisearching and Its Applications

We give a result that implies an improvement by a factor or IogIog n in the hypercube bounds for the geometric problems of hatched planar point location I trapezoidal decomposition, and polygon triangulation. The improvements are achieved through a better solution to the multisearch problem on a hypercube, a parallel search problem where the elements in the data structure S to be searched arc totally ordered, but where it is not possible to compare in constant time any two given queries q and q'. Whereas the previous best solution to this problem took O(logn(loglogn)3) time on an n-processor hypercube, the solution given here takes O(logn(loglogn)2) time on an n-processor hypercube. The hypercube model for which we claim our bounds is the standard one, SIMD, with 0(1) memory registers per processor, and with one-port communication. Each register can store O(logn) bits, so that a processor knows its ID.