a.

Applying backtracking to solve the subset sum problem with A = {1, 3, 4, 5} and d = 11:

Start with an empty subset and check if it can sum up to the desired target (d = 11).

Explore all possible combinations by either including or excluding each element from the set A.

Here is the step-by-step process:

• Start with an empty subset and the target sum (11). Current subset: [] Remaining set: {1, 3, 4, 5}

• Include the first element, 1: Current subset: [1] Remaining set: {3, 4, 5} Calculate the sum: 1 Since the sum (1) is less than the target (11), we continue.

• Include the second element, 3: Current subset: [1, 3] Remaining set: {4, 5} Calculate the sum: 4 Since the sum (4) is less than the target (11), we continue.

• Include the third element, 4: Current subset: [1, 3, 4] Remaining set: {5} Calculate the sum: 8 Since the sum (8) is less than the target (11), we continue.

• Include the fourth element, 5: Current subset: [1, 3, 4, 5] Remaining set: {} Calculate the sum: 13 Since the sum (13) is greater than the target (11), we backtrack.

• Exclude the fourth element, 5: Current subset: [1, 3, 4] Remaining set: {} Calculate the sum: 8 Since the sum (8) is less than the target (11), we continue.

• Exclude the third element, 4: Current subset: [1, 3] Remaining set: {} Calculate the sum: 4 Since the sum (4) is less than the target (11), we continue.

• Exclude the second element, 3: Current subset: [1] Remaining set: {} Calculate the sum: 1 Since the sum (1) is less than the target (11), we continue.

• Include the second element, 5: Current subset: [1, 5] Remaining set: {} Calculate the sum: 6 Since the sum (6) is less than the target (11), we continue.

• Exclude the second element, 5: Current subset: [1] Remaining set: {} Calculate the sum: 1 Since the sum (1) is less than the target (11), we continue.

• Include the first element, 4: Current subset: [4] Remaining set: {} Calculate the sum: 4 Since the sum (4) is less than the target (11), we continue.

• Exclude the first element, 4: Current subset: [] Remaining set: {} Calculate the sum: 0 Since the sum (0) is less than the target (11), we continue.

• No more elements remaining. The subset [1, 3, 4, 5] sums up to the target (11). Solution found!

b.

If use just one of the two inequalities to terminate a node as nonpromising, the backtracking algorithm may not work correctly. Both inequalities in the backtracking algorithm serve different purposes to efficiently prune the search space.

• The first inequality (currentSum > target) allows the algorithm to stop exploring further if the current subset's sum exceeds the target. This is crucial because any subsequent inclusion of elements will only increase the sum further, making it impossible to reach the target.
• The second inequality (currentSum + remainingSum < target) allows the algorithm to avoid unnecessary explorations. If the sum of the current subset plus the sum of the remaining elements is already less than the target, it is impossible to reach the target even if we include all the remaining elements. Thus, it can safely terminate the exploration at that node.
• Using just one inequality could lead to either unnecessary explorations or incorrect termination conditions, resulting in incorrect or suboptimal solutions. Therefore, it is essential to use both inequalities in the backtracking algorithm for the subset sum problem to ensure correct and efficient exploration of the solution space.