Bellman-Ford deals with graphs in which the distance along a path is the sum of the distance of its component edges, but in this case the most natural path dist

Bellman-Ford deals with graphs in which the distance along a path is the sum of the distance of its component edges, but in this case the most natural path distance the product of the edge weights. An instance of the problem corresponds to a directed graph that has one node for each currency and a directed edge between every pair of distinct currencies. The weight of the edge from currency A to currency B is the value in row A, column B of the exchange data table. For instance, the weight of the EUR→JPY edge is 131.7110. We define the value of a directed simple cycle in the graph to be the product of its edge weights. For instance, the value of the USD→EUR→BTC→USD cycle is 0.7779 * 0.01125* 115.65 = 1.01209651875. An arbitrage opportunity is then a directed simple cycle with value > 1. I was stuck here until I came across the clever trick of using the logarithm, as pointed out on http://stackoverflow.com/questions/2282427/interesting-problem-currency-arbitrage http://stackoverflow.com/questions/2282427/interesting-probl... To apply Bellman-Ford to this problem we use the negative of the base-10 logarithm of each edge instead and use the sum of these new edge weights rather than the product in calculating the value of a cycle. (Any base will do, provided we use the same base for every edge.) For instance, the weight of the EUR→JPY edge is now considered to be -log(131.7110) =~ -2.1196, and thus the weight of the USD→EUR→BTC→USD cycle is now -log(0.7779) + -log(0.01125) + -log(115.65) =~ -0.00522. An arbitrage opportunity is then a directed simple cycle with negative weight, which can easily be detected by Bellman-Ford.