5. (3 points) The following routine takes as input a list of n numbers, and returns...

 

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5. (3 points) The following routine takes as input a list of n numbers, and returns the first value of i for which L[i]<L[i-1], or n if no such number exists. int firstDecrease(int * L, int n) { for (int i = 2; i <= n && L[i] >= L[i-1]; i++) { } return i; } 5a) What is the big-O runtime for the routine, measured as a function of its return value i? 5b) If the numbers are chosen independently at random, then the probability that firstDecrease(L) returns i is (i 1)/i!, except for the special case of i=n+1 for which the probability is 1/n! Use this fact to write an expression for the expected value returned by the algorithm. (Your answer can be expressed as a sum, it does not have to be solved in closed form. Do not use O-notation.) 5c) What is the big-O average case running time of the routine? Hint: Simplify the previous summation until you see a common taylor series. 5. (3 points) The following routine takes as input a list of n numbers, and returns the first value of i for which L[i]<L[i-1], or n if no such number exists. int firstDecrease(int * L, int n) { for (int i = 2; i <= n && L[i] >= L[i-1]; i++) { } return i; } 5a) What is the big-O runtime for the routine, measured as a function of its return value i? 5b) If the numbers are chosen independently at random, then the probability that firstDecrease(L) returns i is (i 1)/i!, except for the special case of i=n+1 for which the probability is 1/n! Use this fact to write an expression for the expected value returned by the algorithm. (Your answer can be expressed as a sum, it does not have to be solved in closed form. Do not use O-notation.) 5c) What is the big-O average case running time of the routine? Hint: Simplify the previous summation until you see a common taylor series.