Question & Answer: For all algorithms give the analysis for the worst-case time bound. Read the problem statements carefully. You cannot use hashin…..

For all algorithms give the analysis for the worst-case time bound. Read the problem statements carefully. You cannot use hashing

Given is an array A of size n containing integers in arbitrary order and an integer T. Determine whether there exist two indices i and j, 1 ≤ i < j ≤ n, such that 0 ≤ A[i] + A[i + 1] + . . . + A[j] ≤ T. For example, for A = [3, 5, 10, -13, 1, 4, 16] and T=2, a solution exists for i = 3 and j = 6 with 10 − 13 + 1 + 4 = 2 ≤ T.

Describe and analyze a Divide-and-Conquer algorithm solving the problem in O(n log n) time. Additional O(n) space can be used. Clearly state what a recursive call returns and how the combine step works. Note: You will get partial credit for an O(n log^2 n) time solution. No credit will be given for a quadratic time solution.

Expert Answer

Two loops in a row:

for (i = 0; i < N; i++) {
    sequence of statements
}
for (j = 0; j < M; j++) {
    sequence of statements
}

How would the complexity change if the second loop went to N instead of M?

A nested loop followed by a non-nested loop:

for (i = 0; i < N; i++) {
    for (j = 0; j < N; j++) {
        sequence of statements
    }
}
for (k = 0; k < N; k++) {
    sequence of statements
}

A nested loop in which the number of times the inner loop executes depends on the value of the outer loop index:
```
for (i = 0; i < N; i++) {
    for (j = N; j > i; j--) {
        sequence of statements
    }
}
```

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