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 49 Indian rupees, 1 Indian rupee buys 2 Japanese yen, and 1 Japanese yen buys 0.0107 U.S. dollars. Then, by converting currencies, a trader can start with 1 U.S. dollar and buy 49 x 2 x 0.0107 = 1.0486 U.S. dollars, thus turning a profit of 4.86 percent. Suppose that we are given n currencies C1,C2,..., Cn and an n x n table R of exchange rates, such that one unit of currency ci buys R [i,j] units of currency Cj. (a) Give an efficient algorithm to determine whether or not there exists a sequence of currencies Cil, Ciz, . *. , Cin such that R [11, 12] · R[i2, 13] ..... R[ix-1, ik] · R [ik, 21] > 1 Analyze the running time of your algorithm. (b) Give an efficient algorithm to print out such a sequence if one exists. Analyze the running time of your algorithm.
