CS270 1.[20 points] Solve Kleinberg and Tardos, Chapter 6, Exercise 5. Solution

(a) Define (in plain English) subproblems to be solved. (4 pts)
(b) Write a recurrence relation for the subproblems (6 pts)
(c) Using the recurrence formula in part b, write pseudocode to find the maximum total quality among all segmentation possibilities. (5 pts)
(d) Make sure you specify
i. base cases and their values (2 pts)
ii. where the final answer can be found (1 pt)
(e) What is the complexity of your solution? (2 pts)
2. [20 points] You are given an integer array a[1],...,a[n], find the contiguous subarray (containing at least one number) which has the largest sum and only returns its sum. The optimal subarray is not required to return or compute. Taking a = [5,4,−1,7,8] as an example: the subarray [5] is considered as a valid subarray with sum 5, though it only has one single element; the subarray [5,4,−1,7,8] achieves the largest sum 23; on the other hand, [5,4,7,8] is not a valid subarray as the numbers 4 and 7 are not contiguous.
(a) Define (in plain English) subproblems to be solved. (4 pts)
(b) Write a recurrence relation for the subproblems. (6 pts)
(c) Using the recurrence formula in part b, write pseudocode to find the subarray
(containing at least one number) which has the largest sum. (5 pts)
(d) Make sure you specify
i. base cases and their values (2 pts)
ii. where the final answer can be found (1 pt)
(e) What is the complexity of your solution? (2 pts)
3. [20 points] You are given an array of positive numbers a[1],...,a[n]. For a subsequence a[i1],a[i2],...,a[it] of array a (that is, i1 < i2 < ··· < it): if it is an increasing sequence of numbers, that is, a[i1],a[i2],...,a[it], its happiness score is given by
t
X
k × a[ik] k=1
Otherwise, the happiness score of this array is zero.
For example, for the input a = [22,44,33,66,55], the increasing subsequence [22,44,55] has happiness score (1) × (22) + (2) × (44) + (3) × (55) = 275; the increasing subsequence [22,33,55] has happiness score (1) × (22) + (2) × (33) + (3) × (55) = 253; the subsequence [33,66,55] has happiness score 0 as this sequence is not increasing. Please design an efficient algorithm to only return the highest happiness score over all the subsequences.
(a) Define (in plain English) subproblems to be solved. (4 pts)
(b) Write a recurrence relation for the subproblems (6 pts)
(c) Using the recurrence formula in part b, write pseudocode to find the highest happiness score over all the subsequences. (5 pts)
(d) Make sure you specify
i. base cases and their values (2 pts)
ii. where the final answer can be found (1 pt)
(e) What is the complexity of your solution? (2 pts)
(a) Define (in plain English) subproblems to be solved. (4 pts)
(b) Write a recurrence relation for the subproblems (6 pts)
(c) Using the recurrence formula in part b, write pseudocode to solve the problem. (5 pts)
(d) Make sure you specify
i. base cases and their values (2 pts)
ii. where the final answer can be found (1 pt)
(e) What is the complexity of your solution? (2 pts)