Arbitrage is the use of discrepancies in currency exchange rates to transform one unit of a currency into more than one unit of the same currency. For example, suppose that 1 U. S. dollar buys 46.4 Indian rupees, 1 Indian rupee buys 2.5 Japanese yen, and 1 Japanese yen buys 0.0091 U. S. dollars. Then, by converting currencies, a trader can start with 1 U. S. dollar and buy 46.4 × 2.5 × 0.0091 = 1.0556 U. S. dollars, thus turning a profit of 5.56 percent. Suppose that we are given n currencies c1,c2,…, cn and an n × n table R of exchange rates, such that one unit of currency ci buys R[i, j] units of currency cj. Required:
Give an efficient algorithm to determine whether or not there exists a sequence of currencies.