Optimal Account Balancing

Hard

Google

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Dynamic ProgrammingGreedy Algorithms

You are given an array of transactions where transactions[i] = [from_i, to_i, amount_i] indicates that the person with ID from_i gave amount_i to the person with ID to_i. You need to minimize the total number of transactions to settle the debts. Return the minimum number of transactions required to settle all the debts.

For example:

Input: transactions = [[0,1,10], [2,0,5]]
- Explanation:
  - Person 0 gave person 1, $10
  - Person 2 gave person 0, $5 One way to settle the debt is:
  - Person 1 owes Person 0 $10.
  - Person 0 owes Person 2 $5. A minimum of two transactions are needed to settle the debts. Thus, the output is 2.
Input: transactions = [[0,1,10], [1,0,1], [1,2,5], [2,0,5]]
- Explanation:
  - Person 0 gave person 1, $10
  - Person 1 gave person 0, $1
  - Person 1 gave person 2, $5
  - Person 2 gave person 0, $5 Therefore,
  - Person 0 is owed a net amount of $4 (10 received - 1 given - 5 received ).
  - Person 1 is owed a net amount of -$4 (1 given - 10 received - 5 given).
  - Person 2 is owed a net amount of $0 (5 received - 5 given). One possible way to settle this requires only one transaction, where Person 1 gives $4 to Person 0. Thus, the output is 1.

Explain the naive (brute force) approach to solving this problem. What are its limitations? Then, describe an optimal solution to solve this problem. What is the Big O runtime and space complexity of this solution? List any edge cases and how to handle them. Provide code for the optimal solution.

Optimal Account Balancing

Hard

Google

11 views

Topics:

Dynamic ProgrammingGreedy Algorithms

You are given an array of transactions where transactions[i] = [from_i, to_i, amount_i] indicates that the person with ID from_i gave amount_i to the person with ID to_i. You need to minimize the total number of transactions to settle the debts. Return the minimum number of transactions required to settle all the debts.

For example:

Input: transactions = [[0,1,10], [2,0,5]]
- Explanation:
  - Person 0 gave person 1, $10
  - Person 2 gave person 0, $5 One way to settle the debt is:
  - Person 1 owes Person 0 $10.
  - Person 0 owes Person 2 $5. A minimum of two transactions are needed to settle the debts. Thus, the output is 2.
Input: transactions = [[0,1,10], [1,0,1], [1,2,5], [2,0,5]]
- Explanation:
  - Person 0 gave person 1, $10
  - Person 1 gave person 0, $1
  - Person 1 gave person 2, $5
  - Person 2 gave person 0, $5 Therefore,
  - Person 0 is owed a net amount of $4 (10 received - 1 given - 5 received ).
  - Person 1 is owed a net amount of -$4 (1 given - 10 received - 5 given).
  - Person 2 is owed a net amount of $0 (5 received - 5 given). One possible way to settle this requires only one transaction, where Person 1 gives $4 to Person 0. Thus, the output is 1.

Explain the naive (brute force) approach to solving this problem. What are its limitations? Then, describe an optimal solution to solve this problem. What is the Big O runtime and space complexity of this solution? List any edge cases and how to handle them. Provide code for the optimal solution.

Solution

Clarifying Questions

When you get asked this question in a real-life environment, it will often be ambiguous (especially at FAANG). Make sure to ask these questions in that case:

What are the possible ranges for the amounts of money involved in each transaction? Can a transaction amount be zero or negative?
What is the expected output if it's impossible to balance all accounts? Should I return an empty list, throw an exception, or indicate failure in some other way?
Can the input array of transactions be empty or null? What should I return in those cases?
If multiple optimal solutions exist (i.e., different sets of transactions that minimize the total number of transactions to balance accounts), is any valid solution acceptable, or is there a preferred solution based on some criteria (e.g., lexicographical order of involved accounts)?
Are there any constraints on the number of accounts involved? For example, is there a maximum number of unique accounts that might appear across all the transactions?

Brute Force Solution

Approach

The brute force approach tackles the account balancing problem by exhaustively exploring all possible ways to settle debts. It involves trying every combination of transactions between people to see which one results in the fewest number of transactions overall.

Here's how the algorithm would work step-by-step:

First, determine the net amount each person owes or is owed.
Then, try settling the debt between two people at a time. Explore all possible pairs of people to see if transferring money directly between them balances their accounts.
If that doesn't solve everything, try settling the debts using three people. Explore all possible groups of three people and all the different ways they could transfer money amongst themselves until their accounts are balanced.
Continue increasing the number of people involved in each transaction, exploring all possible combinations and payment arrangements.
For each possible arrangement of transactions, keep track of the total number of transactions required to settle all debts.
Finally, compare all the different arrangements you've tried, and select the one that uses the smallest number of transactions. That's the optimal solution found by brute force.

Code Implementation

def optimal_account_balancing_brute_force(transactions):
    # Calculate net balances for each person.
    person_balances = {}
    for transaction in transactions:
        debtor = transaction[0]
        creditor = transaction[1]
        amount = transaction[2]

        person_balances[debtor] = person_balances.get(debtor, 0) - amount
        person_balances[creditor] = person_balances.get(creditor, 0) + amount

    net_balances = list(person_balances.values())
    
    def settle_debts(balances):
        if not balances:
            return 0

        # Find the first non-zero balance.
        for i in range(len(balances)):
            if balances[i] != 0:
                break
        
        transactions_count = float('inf')

        # Iterate through the remaining balances.
        for j in range(i + 1, len(balances)):
            if balances[j] != 0:

                # Attempt to settle the debt between person i and j.
                amount_to_transfer = min(abs(balances[i]), abs(balances[j]))
                new_balances = balances[:]
                new_balances[i] += amount_to_transfer if balances[i] < 0 else -amount_to_transfer
                new_balances[j] -= amount_to_transfer if balances[j] > 0 else -amount_to_transfer

                # Recursively settle remaining debts and update min transactions
                transactions_count = min(transactions_count, 1 + settle_debts(new_balances))

        return transactions_count

    return settle_debts(net_balances)

Big(O) Analysis

Time Complexity

O(n!) – The brute force approach explores all possible combinations of transactions between people to settle debts. In the worst case, it considers all possible subsets of people, starting with pairs, then triplets, and so on, up to considering all n people together. The number of such combinations grows factorially with the number of people (n), as it involves exploring permutations of people and amounts in multi-person transactions. Therefore, the time complexity is approximately O(n!), reflecting the factorial growth in the number of transaction arrangements explored.

Space Complexity

O(N!) – The brute force approach described explores all possible combinations of transactions involving 2, 3, up to N people, where N is the number of people. The space complexity is primarily dictated by the implicit recursion stack used to explore all possible combinations. The maximum depth of the recursion can potentially explore all permutations of people involved in transactions, thus the auxiliary space grows factorially with N, corresponding to the number of possible permutations considered during the exhaustive search. Hence, the space complexity is O(N!).

Optimal Solution

Approach

The best way to balance accounts is to track who owes whom money and then systematically settle these debts. We focus on finding a series of direct payments that cancel out all debts, aiming to minimize the number of transactions needed.

Here's how the algorithm would work step-by-step:

First, figure out the net amount each person owes or is owed. Some people will have a positive balance (they are owed money), and others will have a negative balance (they owe money).
Focus on the people who owe money (negative balances). Pick someone who owes money and someone who is owed money (positive balance).
Have the person who owes money pay as much as they can to the person who is owed money. This could either fully settle the debt of the person who owes, fully satisfy the claim of the person who is owed, or result in both having smaller balances.
If the person who owed money now has a balance of zero, they are done. If the person who was owed money now has a balance of zero, they are also done. If both balances are not zero, then repeat the step again until one balance is zero.
Continue the process until everyone's balance is zero. Each payment is a transaction, and by always directly settling debts this way, you minimize the number of necessary transactions.

Code Implementation

def min_transactions(transactions):
    balances = {}
    for transaction in transactions:
        payer = transaction[0]
        receiver = transaction[1]
        amount = transaction[2]
        balances[payer] = balances.get(payer, 0) - amount
        balances[receiver] = balances.get(receiver, 0) + amount

    net_balances = list(balances.values())
    positive_balances = [balance for balance in net_balances if balance > 0]
    negative_balances = [balance for balance in net_balances if balance < 0]

    positive_index = 0
    negative_index = 0
    transaction_count = 0

    # Iterate until all balances are settled.
    while positive_index < len(positive_balances) and negative_index < len(negative_balances):
        payer_owes = -negative_balances[negative_index]
        receiver_owed = positive_balances[positive_index]

        # Determine the amount to transfer.
        transfer_amount = min(payer_owes, receiver_owed)

        negative_balances[negative_index] += transfer_amount
        positive_balances[positive_index] -= transfer_amount
        transaction_count += 1

        # Move to the next person if balance is zero
        if negative_balances[negative_index] == 0:
            negative_index += 1

        # Move to the next person if balance is zero.
        if positive_balances[positive_index] == 0:
            positive_index += 1

    return transaction_count

Big(O) Analysis

Time Complexity

O(n²) – The algorithm first calculates the net balance for each of the n people, which takes O(n) time. The core of the algorithm involves iterating through the balances and settling debts. In the worst-case scenario, each person who owes money might interact with every person who is owed money. This involves, for each negative balance, finding a corresponding positive balance to settle the debt. In the worst case we may need to examine all other balances. Consequently, it can involve approximately n * n/2 comparisons/operations. Therefore, the overall time complexity is O(n²).

Space Complexity

O(N) – The primary space complexity arises from storing the net balances of each person. In the worst case, each person involved in the transactions will have a unique balance, resulting in an array or list to store N balances, where N is the number of people involved in the transactions. Therefore, the auxiliary space required scales linearly with the number of people. No significant additional data structures beyond this are utilized.

Edge Cases

Case	How to Handle
Empty transactions list	Return 0 as no transactions mean no transfers are needed to balance.
Single person involved in all transactions.	The algorithm should correctly aggregate debts/credits for that single person and determine optimal transfers amongst themselves if necessary.
Transactions summing to zero across all individuals (perfectly balanced initial state).	The algorithm should return 0, indicating no transfers are needed as debts/credits are already balanced.
Large number of transactions causing potential integer overflow during aggregation.	Use long integers (or equivalent) to prevent overflow when calculating the net balance for each person.
Transactions with very large monetary values.	Ensure that the chosen data type (e.g., long or double) can accommodate these large values without loss of precision or overflow.
A cycle of debts, where A owes B, B owes C, and C owes A.	The algorithm should handle cycles by consolidating and simplifying the debt network until an optimal transfer amount is found to resolve the cycle.
Transactions involving a large number of people, potentially leading to memory issues.	Consider using a more memory-efficient data structure (e.g., adjacency list or sparse matrix) for representing the debt network if memory becomes a bottleneck.
No solution possible because of constraints (e.g., minimum transfer amount is greater than the total debt).	The algorithm should return -1 or throw an exception if the constraints make balancing impossible based on problem constraints.