The risk-free currency exchange problem offers a risk-free way to make money. suppose we have currencies cn. (for example, c1 might be dollars, c2 rubles, c3 yen, etc.) for every two currencies ci and cj there is an exchange rate ri, j such that you can exchange one unit of ci for ri, j units of cj . note that if ri, j ·rj, i > 1, then you can make money simply by trading units of currency i into units of currency j and back again. this almost never happens, but occasionally (because the updates for exchange rates do not happen quickly enough) for very short periods of time exchange traders can find a sequence of trades that can make risk-free money. that is, if there is a sequence of currencies ci1 , ci2 cik such that ri1,i2 ·ri2,i3 ·rik−1,ik ·rik, i1 > 1, then trading one unit of ci1 into ci2 and trading that into ci3 and so on will yield a profit. design an efficient algorithm to detect if a risk-free currency exchange exists. (you need not actually find it.)

When making changes to optimize part of a processor, it is often the case that speeding up one type of instruction comes at the cost of slowing down something else. for example, if we put in a complicated fast floating-point unit, that takes space, and something might have to be moved farther away from the middle to accommodate it, adding an extra cycle in delay to reach that unit. the basic amdahl's law equation does not take into account this trade-off. a. if the new fast floating-point unit speeds up floating-point operations by, on average, 2ă—, and floating-point operations take 20% of the original program's execution time, what is the overall speedup (ignoring the penalty to any other instructions)? b. now assume that speeding up the floating-point unit slowed down data cache accesses, resulting in a 1.5ă— slowdown (or 2/3 speedup). data cache accesses consume 10% of the execution time. what is the overall speedup now? c. after implementing the new floating-point operations, what percentage of execution time is spent on floating-point operations? what percentage is spent on data cache accesses?