Given two ranges of positive integers x: [1 ... n] and y: [1 ... m] and random real R from 0 to 1, I need to find the pair of elements (i,j) from x and y such that x_i / y_j is closest to R.
What is the most efficient way to find this pair?
This is a simple and mathematically beautiful algorithm to solve this: run a binary search, where on each iteration the next number is given by the mediant formula (below). By the properties of the Farey sequence that number is the one with the smallest denominator within that interval. Consequently this sequence will always converge and never 'miss' a valid solution.
In pseudocode:
input: m, n, R
a_num = 0, a_denom = 1
b_num = 1, b_denom = 1
repeat:
-- interestingly c_num/c_denom is already in reduced form
c_num = a_num + b_num
c_denom = a_denom + b_denom
-- if the numbers are too big, return the closest of a and b
if c_num > n or c_denom > m then
if R - a_num/a_denom < b_num/b_denom - R then
return a_num, a_denom
else
return b_num, b_denom
-- adjust the interval:
if c_num/c_denom < R then
a_num = c_num, a_denom = c_denom
else
b_num = c_num, b_denom = c_denom
goto repeat
Even though it's fast on average (my educated guess that it's O(log max(m,n))), it can still be slow if R is close to a fraction with a small denominator. For example finding an approximation to 1/1000000 with m = n = 1000000 will take a million iterations.