Bimonotone enumeration

Solutions of a diophantine equation f(a, b) = g(c, d), with a, b, c, d in some finite range, can be efficiently enumerated by sorting the values of f and g in ascending order and searching for collisions. This article considers functions f : N×N → Z that are bimonotone in the sense that f(a, b) ≤ f(a′, b′) whenever a ≤ a′ and b ≤ b′. A two-variable polynomial with non-negative coefficients is a typical example. The problem is to efficiently enumerate all pairs (a, b) such that the values f(a, b) appear in increasing order. We present an algorithm that is memory-efficient and highly parallelizable. In order to enumerate the first n values of f , the algorithm only builds up a priority queue of length at most √ 2n+1. In terms of bit-complexity this ensures that the algorithm takes time O(n log n) and requires memory O( √ n logn), which considerably improves on the memory bound Θ(n logn) provided by a näıve approach, and extends the semimonotone enumeration algorithm previously considered by R.L. Ekl and D.J. Bernstein.