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I started with the naive approach, just tracking who owes who after each expense, and it worked fine for the basic case.
Start by clarifying requirements and defining the data model for users, groups, and expenses. Then, compute net balances per user and design an algorithm to minimize the number of transactions to settle all debts, discussing trade-offs between optimality and simplicity.
Pro tip: Acknowledge that finding the absolute minimum number of transactions is NP-hard (related to the partition problem), so in practice, a greedy approach that matches largest creditors with largest debtors is often used, and mention that this is a common trade-off in real systems.
Ask about functional and non-functional requirements: number of users, frequency of expenses, need for real-time updates, persistence, and whether the settlement algorithm must be optimal or just efficient.
Define entities: User, Group, Expense (with payer, amount, participants, split method), and Balance. Consider how to store and update balances efficiently.
For each user, calculate the net amount they owe or are owed by summing all expenses they paid and their shares. This reduces the problem to settling net balances.
Use a greedy algorithm: repeatedly match the largest creditor with the largest debtor, settling the smaller amount, until all balances are zero. Discuss that this yields a minimal or near-minimal number of transactions.
Address how the solution scales with many users and expenses, potential optimizations (e.g., using heaps), and trade-offs between optimality and performance. Mention alternative approaches like graph-based or linear programming if needed.
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