Mapping binary precedence trees to hypercubes and meshes

The paper considers the mapping of communicating modules of a parallelized task to the processing elements of a parallel computer when precedence relationships among the modules is available. The goal of the mapping is to minimize the total execution time of the task, including both processing and communications time, within a processor network of limited size. This paper presents a method for contracting complete binary precedence trees with n nodes to trees with (n+1)/2 nodes with no increase in execution time. The authors then provide methods for embedding these trees into hypercubes and m-dimensional meshes. When embedded into the hypercube of dimension log(ⁿ⁺¹/₂), or into meshes with dimension m ⩾(log(ⁿ⁺¹/₂))/2, the contracted tree is embedded with unit dilation and with no increase in execution time. For meshes with dimension m<(log(ⁿ⁺¹/₂))/2, the method embeds the contracted tree with dilation O(n¹m/), and with no additional cost in processing time and limited link contention. Further contractions are carried out by contracting the hypercube or mesh. The mapping algorithms take O(n) time for the contraction and embedding steps