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 efficiently 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-efficient 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...

We investigate a problem from computational biology: Given a constant size alphabet M with a weight function / : M--> +, find an efficient data structure and query algorithm solving the following problem: For a weight M C + and a string cr over A, decide whether cr contains a substring with weight M (ONE STRING MASS FINDING PROBLEM). If the answer is yes, then we may in addition require a witness, i.e. indices i _ i and ending at position j has weight M. We allow preprocessing of the string, and measure efficiency in two parameters: storage space required for the preprocessed data, and running time of the query algorithm for given M. We are interested in data structures and algorithms requiring subquadratic storage space and sublinear query time, where we measure the input size as the length of the input string. We present two efficient algorithms: LOOKUP solves the problem with O(,) space and (Wg ' loglog,) time; INTERVAL solves the problem for binary alphabets with O0, ) space in O(log,) time. We sketch a third al-gorithm, CLUSTER, which can be adjusted for a space time tradeoff but for which we do not yet have a resource analysis. We introduce a function on weighted strings which is closely related to the analysis of algorithms for the ONE STRING MASS FINDING PROBLEM: The number of different submasses of a weighted string. We present several properties of this function, including upper and lower bounds. Finally, we introduce two more general variants of the problem and sketch how algorithms may be extended for these variants.