I jumped straight to greedy, pairing the biggest creditor with the biggest debtor each round.
Start by clarifying the problem constraints and confirming that the goal is to minimize the number of transactions, not the amount transferred. Then, explain that this is a classic debt settlement problem that can be modeled as a graph where each transaction reduces the net balance of two users, and the optimal solution involves matching debtors and creditors greedily or using backtracking for exact minimization.
Pro tip: Mention that while the greedy approach (always settling the largest debt with the largest credit) often yields the minimum number of transactions, it is not always optimal; for exact minimization, backtracking with pruning is needed, but in practice, the greedy approach is efficient and widely accepted.
Ask questions to confirm input format, constraints (e.g., number of users, balance ranges), and whether the goal is to minimize the number of transactions or the total amount transferred. Also, confirm if partial settlements are allowed.
Compute the net balance for each user by summing their credits and debits. Separate users into debtors (negative balance) and creditors (positive balance). This reduces the problem to matching debtors with creditors.
Discuss greedy approach: repeatedly match the largest debtor with the largest creditor, settling the minimum of the two amounts. For exact minimization, mention backtracking: try all possible matches and prune when the current transaction count exceeds the best found.
Explain that the greedy approach runs in O(n log n) due to sorting, while backtracking can be exponential in the worst case. Discuss that greedy is not always optimal but is efficient and often used in practice; exact minimization may be NP-hard.
Walk through a small example to illustrate the algorithm. Mention edge cases: all balances zero, one debtor and one creditor, multiple users with same balances, and floating-point precision issues.
AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.
This follow-up is where things got uncomfortable.
First, clarify the problem: given a set of items with weights, determine if a balance (subset sum) can be achieved. Then, explain the O(2^N) DP over subsets by enumerating all possible subsets and checking if any subset sums to the target. Finally, construct a counterexample where the greedy approach (e.g., picking largest weights first) fails to find a valid subset even though one exists.
Pro tip: Mention that while O(2^N) is exponential, it's optimal for exact subset sum unless P=NP, and that greedy can fail due to lack of matroid structure. This shows awareness of complexity theory and algorithmic trade-offs.
Restate the problem: given a set of weights and a target balance, decide if a subset sums exactly to the target. Confirm that 'balance subsets' refers to subset sum.
Explain that you can iterate over all 2^N subsets, compute each subset's sum, and check if any equals the target. This is a brute-force DP over subsets, often implemented via bitmask or recursion.
Greedy (e.g., always pick the largest weight that fits) can fail because it makes locally optimal choices that may preclude a global solution. Provide a concrete counterexample, such as weights [4,3,3] and target 6: greedy picks 4, then can't reach 6, but 3+3 works.
Acknowledge that O(2^N) is exponential but exact; mention that pseudo-polynomial DP (O(N*sum)) exists but may be infeasible for large sums. Emphasize that greedy is not correct for subset sum in general.
AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.