Solution Explanation for Optimal Account Balancing

This problem asks for the minimum number of transactions needed to settle debts between people. The core idea is to model the problem as a subset sum problem. We first calculate the net balance for each person. If a person has a positive balance, they are owed money; a negative balance means they owe money. Then, we use dynamic programming (DP) and bit manipulation to find the minimum number of transactions.

1. Net Balance Calculation:

The solution starts by calculating the net balance for each person. It iterates through the transactions: For each transaction [from, to, amount], it subtracts amount from the balance of from and adds amount to the balance of to. This accurately reflects the change in each person's balance after all transactions.

2. Subset Sum Problem Formulation:

After calculating the net balances, we are left with a list of numbers representing the net balance of each person with a non-zero balance. The goal is to find the minimum number of groups of numbers that sum to zero. Each group corresponds to a transaction, where the people in the group settle their debts among themselves.

This is essentially a subset sum problem: find the minimum number of subsets whose elements sum to zero. The problem's constraint of at most 8 transactions (implied by transactions.length <= 8) allows for a relatively efficient solution using bit manipulation and dynamic programming.

3. Dynamic Programming and Bit Manipulation:

The DP solution uses a bitmask to represent the subset of people included in a transaction. A bitmask of length m (where m is the number of people with non-zero balances) is used. If the i-th bit is 1, the i-th person is included; if it's 0, they are not.

The DP state f[mask] stores the minimum number of transactions required to settle the debts for the subset of people represented by mask. The base case f[0] = 0 because no transactions are needed if no one has any debt.

The algorithm iterates through all possible bitmasks (subsets of people). For each bitmask i:

• It calculates the sum (s) of the net balances of the people in the subset represented by i.
• If s == 0, it means the subset's debts can be settled with Integer.bitCount(i) - 1 transactions (one less than the number of people, as they can settle among themselves).
• If s != 0, it means this subset cannot be a valid transaction, so it's skipped.
• It then uses bit manipulation to efficiently find all the submasks of i and use the DP results of those submasks to compute the minimum transactions for i. (This optimizes the calculation, avoiding unnecessary iterations).

4. Time Complexity Analysis:

The time complexity is dominated by the DP loop, which iterates through all possible bitmasks (2m where m is the number of people with non-zero balances). The inner loop iterates through all submasks, which is also bounded by 2m in the worst case. Thus, the overall time complexity is O(3m), which is acceptable because m is small (at most 8 due to problem constraints).

5. Space Complexity Analysis:

The space complexity is O(2m) due to the DP table f.

In summary: The solution leverages bit manipulation to efficiently represent subsets of people and dynamic programming to find the minimum number of transactions needed to settle all debts. The algorithm's efficiency is heavily dependent on the relatively small number of people involved (at most 8 with non-zero balances), which makes the exponential time complexity manageable.