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 11 U. S. dollar buys 4949 Indian rupees, 11 Indian rupee buys 22 Japanese yen, and 11 Japanese yen buys 0.01070.0107 U. S. dollars. Then, by converting currencies, a trader can start with 11 U. S. dollar and buy 49 \times 2 \times 0.0107 = 1.048649×2×0.0107=1.0486 U. S. dollars, thus turning a profit of 4.864.86 percent.

Suppose that we are given nn currencies c_1, c_2, \ldots, c_nc
1

,c
2

,…,c
n

and an n \times nn×n table RR of exchange rates, such that one unit of currency c_ic
i

buys R[i, j]R[i, j] units of currency c_jc
j

.

a. Give an efficient algorithm to determine whether or not there exists a sequence of currencies \langle c_{i_1}, c_{i_2}, \ldots, c_{i_k} \rangle⟨c
i
1



,c
i
2



,…,c
i
k



⟩ such that

R[i_1, i_2] \cdot R[i_2, i_3] \cdots R[i_{k - 1}, i_k] \cdot R[i_k, i_1] > 1.
R[i
1

,i
2

]⋅R[i
2

,i
3

]⋯R[i
k−1

,i
k

]⋅R[i
k

,i
1

]>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.

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Write a program that displays the following menu: geometry calculator 1. calculate the area of a circle 2. calculate the area of a rectangle 3. calculate the area of a triangle 4. quit enter your choice (1-4): if the user enters 1, the program should ask for the radius of the circle and then display its area. use the following formula: area = ď€(the square of r) use 3.14159 for ď€ and the radius of the circle for r. if the user enters 2, the program should ask for the length and width of the rectangle and then display the rectangle’s area. use the following formula: area = length * width if the user enters 3, the program should ask for the length of the triangle’s base and its height, and then display its area. use the following formula: area = base * height * .5 if the user enters 4, the program should end. input validation: display an error message if the user enters a number outside the range of 1 through 4 when selecting an item from the menu. do not accept negative values for the circle’s radius, the rectangle’s length or width, or the triangle’s base or height. note: if the user enters an improper menu choice (1-4), the program prints "the valid choices are 1 through 4. run the program again and select one of those." if the user enters a negative radius, the program prints "the radius can not be less than zero." if the user enters a negative value for height or base, the program prints "only enter positive values for base and height."