The Arithmetic of Recursively Run-Length Compressed Natural Numbers

We study arithmetic properties of a new tree-based canonical number representation, recursively run-length compressed natural numbers, defined by applying recursively a run-length encoding of their binary digits. We design arithmetic operations with recursively run-length compressed natural numbers that work a block of digits at a time and are limited only by the representation complexity of their operands, rather than their bitsizes. As a result, operations on very large numbers exhibiting a regular structure become tractable. In addition, we ensure that the average complexity of our operations is still within constant factors of the usual arithmetic operations on binary numbers.