Question

QUESTION 3 (10 Marks) Suppose you are given an array A[0..n 1 of integers, each of which may be positive, negative, or zero. A contiguous subarray A|i..j] which includes all the items starting at array index i and ending at array index j is called a positive interval if the sum of its entries is at least zero. For example let A-{-1, 3,-5, 2, 0,-4, 3,-6,-2). Ten A[3, 6) is a positive interval since its sum is 2+0+(-4)+3-1, but Al4,7isnot a positive interval since its sum is 0+(-4)+3+(-6)- We want to compute the smallest number of positive intervals that are pairwise non-intersecting and which include all non-ncgative values in A (i.c. each item Ali 0 is in one of the positive intervals). In the example above, Al3,6) and Al1, 1] cover all non-negative values and there is no other covering with fewer intervals. a. (2 marks) Let A13,-5,7,-4,1,-8, 3,-7,5, -9,5,-2,4. What is the smallest number of positive, non-intersecting intervals which together include all non-negative values? b. (4 marks) Write down a formal statement of the optimisation problem. Consider the following greedy strategy (i) Maintain a record of all non-intersecting positive sums that cover all non-negative ele- ments in A (ii) Merge two sums Ali,j] and Alk,1 together (whenever jk if Ali,(] is a positive um c. 1 mark) What is the initialisation for this stratcgy? d. 3 marks) Is this strategy guaranteed to find the optimal solution? Give reasons why or why not.

Answer 1

a) The smallest number of positive non-intersecting intervals that include all non-negative values is 3. Subarrays are:

A[0,4], A[6,6], A[8,12]

b) In the given optimization problem, we have to find the minimum number of positive, non-intersecting intervals which together include all non-negative values

Here is the formal statement:

Let A₁, A₂, . .A_k be the non-intersecting subarray of A, then minimize k such that sum(A_i)>=0 for all i from 1 to k and No element in A - (A₁, A₂, . .A_k) is non-negative.

c) The initialisation for the above strategy will be all the subarrays of A containing one positive element i.e. there will be an array corresponding to each non-negative element in A

d) No, the algorithm may fail in some cases. Consider the following example:

A = [ 2, -4, 3, -7, 5, -9, 7]

the optimal solution for the given array is 2, A[0,2], A[4,6]

Let us work as per the greedy strategy:

Initially, we have

A[0,0], A[2,2], A[4,4] and A[6,6]

Now, we can combine any of the following as per strategy:

1) A[0,0] and A[2,2] and resulting subarray will be A[0,2]

2) A[2,2] and A[4,4] and resulting subarray will be A[2,4]

3) A[4,4] and A[6,6] and resulting subarray will be A[4,6]

No, if we choose from 1 and 3, the strategy will lead to a correct answer but if we choose 2, there is no way back and strategy will not give an optimal solution.