Alternative Approach to Achieve a Solution of Derangement Problems by Dynamic Programming

Abstract

Derangement is one well-known problem in the filed of probability theory. An instance of a derangement problem contains a finite collection C of n paired objects, C = {(x1, y1), …, (xn, yn)}. The derangement problem asks how many ways to generate a new collection C′ ≠ C such that for each (xi, yj ) ∈ C′, i ≠ j. We propose an efficient dynamic programming algorithm that divides an instance of the derangement problem into several subproblems. During a recursive process of unrolling a subproblem, there exists a repeated procedure that allows us to make a use of a subsolution that has already been computed. We present the methodology to formulate a concept of this subproblem as well as parts of designing and analyzing an efficiency of the proposed algorithm.