Computational complexity of a longest path algorithm witn a recursive method

Question

I wrote a code segment to determine the longest path in a graph. Following is the code. But I don\'t know how to get the computational complexity in it because of the recurs

Answer 1

Your recurrence relation is T(n, m) = mT(n, m-1) + O(n), where n denotes number of nodes and m denotes number of unvisited nodes (because you call longestPath m times, and there is a loop which executes the visited test n times). The base case is T(n, 0) = O(n) (just the visited test).

Solve this and I believe you get T(n, n) is O(n * n!).

EDIT

Working:

T(n, n) = nT(n, n-1) + O(n) 
        = n((n-1)T(n, n-2) + O(n)) + O(n) = ...
        = n(n-1)...1T(n, 0) + O(n)(1 + n + n(n-1) + ... + n(n-1)...2)
        = O(n)(1 + n + n(n-1) + ... + n!)
        = O(n)O(n!) (see http://oeis.org/A000522)
        = O(n*n!)