Optimal Account Balancing

class Solution {
    public int optimiseTransfers(int[][] transactions) {
        Map<Integer, Integer> balanceMap = new HashMap<>();
        for (int[] t : transactions) {
            balanceMap.put(t[0], balanceMap.getOrDefault(t[0], 0) + t[2]);
            balanceMap.put(t[1], balanceMap.getOrDefault(t[1], 0) - t[2]);
        }
        
        balanceList = new ArrayList<>();
        for (int balance : balanceMap.values()) {
            if (balance != 0) {
                balanceList.add(balance);
            }
        }
        
        int n = balanceList.size();
        int[] memo = new int[1 << n];
        Arrays.fill(memo, -1);
        memo[0] = 0;
        return n - compute((1 << n) - 1, memo);
    }
    
    List<Integer> balanceList;
    private int compute(int mask, int[] memo) {
        if (memo[mask] != -1) {
            return memo[mask];
        }
        int sum = 0, res = 0;
        
        for (int i = 0; i < balanceList.size(); i++) {
            int currentMask = 1 << i;
            if ((mask & currentMask) != 0) {
                sum += balanceList.get(i);
                res = Math.max(res, compute(mask ^ currentMask, memo));
            }
        }

        memo[mask] = res + (sum == 0 ? 1 : 0);
        return memo[mask];
    }
}

This Java solution focuses on optimally balancing transactions in an account by minimizing the number of transfers necessary to balance debts and credits among multiple parties. Here's a breakdown of how the solution is structured and functions:

• Balance Map Creation: Begin with creating a Map<Integer, Integer> named balanceMap. For each transaction, update this map such that each account's net balance is adjusted accordingly—credits increase and debts decrease the balance.

• Filter Non-Zero Balances: Iterate through the values of balanceMap to filter out accounts that have a nonzero final balance and store these in balanceList. This list represents the net amount each account needs to either pay or receive.

• Memoization Setup: Introduce a memoization array memo to store the minimum number of transactions needed to achieve balance. Use bitwise representation to track subsets of accounts, where each bit in an integer represents whether a corresponding account in balanceList is included in the subset.

• Compute Function: Implement a private method compute using depth-first search to determine the minimum number of transactions required. This function:

  • Checks if the result is already computed for the given subset mask using memoization.
  • Iterates through each account in balanceList, toggling the account's inclusion in the current subset mask.
  • Recursively calculates the result excluding one account at a time, and checks if the cumulative balance of the remaining accounts is zero.
  • Memoizes and returns the maximum result from the recursive calls adjusted by one if the total sum of accounts in the subset is zero (which means one less transaction is needed).

• Return Statement: Finally, return the difference between the total number of accounts with non-zero balances and the result of the compute function for the full set mask. This gives the total number of optimal balance-restoring transactions needed.

The overall goal of the code is to reduce the complexity of the problem using bitwise operations and memoization, effectively tracking the combinations of account adjustments that lead to balances being cleared with the fewest transactions possible. This approach scales well with the number of individuals or accounts involved in the transactions.

class Solution:
    def minimumTransfers(self, transaction_data: List[List[int]]) -> int:
        account_balance = collections.defaultdict(int)
        for debtor, creditor, money in transaction_data:
            account_balance[debtor] += money
            account_balance[creditor] -= money
        
        balances = [balance for balance in account_balance.values() if balance]
        count = len(balances)
        
        dp = [-1] * (1 << count)
        dp[0] = 0
        
        def solve(mask):
            if dp[mask] != -1:
                return dp[mask]
            sum_balances, min_transfers = 0, 0

            for i in range(count):
                bit = 1 << i
                if mask & bit:
                    sum_balances += balances[i]
                    min_transfers = max(min_transfers, solve(mask ^ bit))

            dp[mask] = min_transfers + (sum_balances == 0)
            return dp[mask]
        
        return count - solve((1 << count) - 1)

The Python solution provided aims at optimizing money transfers among accounts to settle debts, based on a list of transactions. The method minimumTransfers calculates the minimal number of transactions required to balance all accounts, given a list of individual transactions formatted as [debtor, creditor, amount].

• First, the method uses a defaultdict to compute the net balance for each account. This balance is calculated from the series of debtor-creditor-money transactions provided. An account's balance increases when receiving money and decreases when owing money.
• From these balances, a new list is composed, containing only non-zero balances indicating accounts that are either owed money or owe money.
• A dynamic programming table, dp, is initialized to store the minimum number of transactions required for various subsets (combinations) of accounts, with each subset represented as a bitmask. The size of this bitmask is determined by the power set of accounts with non-zero balances.
• The function solve recursively calculates the minimum number of transactions by considering different combinations of accounts. The goal is to find subgroups of balances where the sum is zero, indicating they can be settled among themselves without interaction with other accounts.
• The solution updates the dp table for each combination of accounts, tracking the minimum transactions needed if the sum of the combination equals zero.
• The minimum number of total transactions required to balance all given accounts is returned after all potential combinations are evaluated.

The main challenges handled by this function include systematically reducing the complexity of multi-party transactions to simple pairwise balance adjustments and using bitmasking with dynamic programming to efficiently explore possible settlements.