Hardware Index to Set Partition Converter

Abstract

We demonstrate, for the first time, high-speed circuits that generate partitions on a set S of n objects. We offer two versions. In the first, partitions are produced in lexicographical order in response to successive clock pulses. In the second, an index input determines the set partition produced. Such circuits are needed in the hardware implementation of the optimum distribution of tasks to processors. Our circuits are combinational. For large n, they can have large delay. However, one can easily pipeline them to produce one set partition per clock period. We show 1) analytical and 2) experimental time/complexity results that quantify the efficiency of our designs. Our results show that a hardware partition generator running on a 100 MHz FPGA produces partitions at a rate that is approximately 10 times the rate of a software implementation on a processor running at 2.26 GHz.