Lecture 14 : Minimum Spanning Tree I

Suppose u, v are different currencies, and the exchange rate is C(u, v) (i.e., 1 unit of v is worth C(u, v) units of u). We set the length of the edge as w(u, v) = logC(u, v). We take the log because, in currency exchange, the rates are multiplicative factors, but in shortest path, path length is defined as sum of edge lengths. By this conversion, the length of a path is log of the total exchange rate. Because of the logarithm, the edge length w(u, v) can be negative or positive. It is negative if C(u, v) is less than 1, and as we mentioned, a negative cycle in the graph means currency arbitrage. In general, one can walk along a negative cycle infinitely many times, and a path that contains the cycle has length of negative infinity. Therefore, our algorithms will try to find shortest paths, assuming there is no negative cycle.