I have a resource scheduling issue in Java where things need to be sequenced, but there are restrictions on what resources can be next to each other. A good analogy is a string of "digits", where only certain digits can be next to each other. My solution was recursive, and works fine for small strings, but run time is O(X^N), where X is the number of possible digits (the base), and N is the length of the string. It quickly becomes unmanageable.

Using the compatibility matrix below, here are a few examples of allowed strings

Length of 1: 0, 1, 2, 3, 4

Length of 2: 02, 03, 14, 20, 30, 41

Length of 3: 020, 030, 141, 202, 203, 302, 303, 414



0 1 2 3 4 --------------------- 0| 0 0 1 1 0 1| 0 0 0 0 1 2| 1 0 0 0 0 3| 1 0 0 0 0 4| 0 1 0 0 0

My solution for counting all strings of length N was start with an empty string, permute the first digit, and make a recursive call for all strings of length N-1. The recursive calls check the last digit that was added and try all permutations that can be next to that digit. There are some optimizations made so that I don't try and permute 00, 01, 04 every time, for example - only 02, 03, but performance is still poor as it scales from base 5 (the example) to base 4000.

Any thoughts on a better way to count the permutations other than trying to enumerate all of them?