Fast and Scalable Parallel Algorithms for Knapsack-like Problems

We present two new algorithms for searching in sorted X+Y +R+S, one based on heaps and the other on sampling. Each of the algorithms runs in time O(n 2 logn) (n being the size of the sorted arrays X, Y , R and S). Hence in each case, by constructing arrays of size n = O(2 s=4), we obtain a new algorithm for solving certain NP-Complete problems such as Knapsack on s data items in time equal (up to a constant factor) to the best algorithm currently known. Each of the algorithms is capable of being eeciently implemented in parallel and so solving large instances of these NP-Complete problems fast on coarse-grained distributed memory parallel computers. The parallel version of the heap based algorithm is communication-eecient and exhibits optimal speedup for a number of processors less than n using O(n) space in each one; the sampling based algorithm exhibits optimal speedup for any number of processors up to n using O(n) space in total provided that the architecture is capable of logarithmic time sorting.