A parallel algorithm for bound-smoothing

Determining molecular structure from interatomic distances is an important and challenging problem. Given a molecule with n atoms, lower and upper bounds on inter-atomic distances can usually be obtained only for a small subset of the n(n-1)/2 atom pairs, using NMR. Given the bounds so obtained on the distances between some of the atom pairs, it is often useful to compute tight bounds on all the n(n-1)/2 pairwise distances. This process is referred to as bound-smoothing. The initial lower and upper bounds for the pairwise distances not measured are usually assumed to be 0 and infinity. One method for bound-smoothing is to use the limits imposed by the triangle inequality. The distance bounds so obtained can often be tightened further by applying the tetrangle inequality - the limits imposed on the six pairwise distances among a set of four atoms (instead of three for the triangle inequalities). The tetrangle inequality is expressed by the Cayley-Menger determinants. For every quadruple of atoms, each pass of the tetrangle-inequality bound-smoothing procedure finds upper and lower limits on each of the six distances in the quadruple. Applying the tetrangle inequalities lo each of the ((4)(n)) quadruples requires Theta (n(4)) time. Here, we propose a parallel algorithm for bound-smoothing employing the tetrangle inequality. Each pass of our algorithm requires Theta (n(3) log n) time on a CREW PRAM with Theta (n/log n) processors. An implementation of this parallel algorithm on the intel Paragon XP/S and its performance are also discussed.