Optimal Account Balancing

10 Prerequisite LeetCode Problems

Here are 10 problems to prepare for this problem:

121. Best Time to Buy and Sell Stock: Basic problem to understand how to maximize profit based on given conditions.
122. Best Time to Buy and Sell Stock II: A more generalized version of the problem above that requires understanding of handling multiple transactions.
139. Word Break: Good problem to understand dynamic programming and decision making at each step.
198. House Robber: This problem will help you understand how to calculate maximum value under certain constraints.
322. Coin Change: This problem helps you understand how to minimize the number of items (coins in this case) needed to achieve a certain goal.
518. Coin Change 2: This problem extends the concept of coin change problem by adding another dimension to it.
406. Queue Reconstruction by Height: This problem deals with reconstructing an optimal solution based on certain conditions.
376. Wiggle Subsequence: This problem deals with finding an optimal subsequence which would help understand how to approach the given problem.
152. Maximum Product Subarray: This problem helps in understanding the calculation of maximum or minimum product under given constraints.
53. Maximum Subarray: This is a basic problem that helps in understanding subarray problems, which forms the base for understanding the given problem.

By solving these problems, you learn how to approach problems where you need to find an optimal solution under certain constraints.

“Optimal Account Balancing” involves optimizing the reconciliation of accounts in a transactional system, in this case represented by a series of transactions between different people. This problem can be approached as a graph problem or a variant of the Partition problem, and is usually solved with a combination of depth-first search (DFS) and memoization for efficiency.

Here are 10 problems to prepare for this problem:

207. Course Schedule: This problem involves finding cycles in a graph using DFS, a skill that can come in handy when working with transactional systems.
399. Evaluate Division: This problem involves performing calculations based on a graph of relationships, similar to calculating account balances.
133. Clone Graph: This problem helps reinforce understanding of graph traversal, which is necessary for optimal account balancing.
690. Employee Importance: This problem deals with hierarchical data, which is similar to the interconnectedness of accounts.
332. Reconstruct Itinerary: This problem involves DFS and Eulerian paths, which can be adapted for the main problem.
416. Partition Equal Subset Sum: This problem is a variant of the Partition problem, and introduces the concept of dividing a set of numbers into two equal subsets.
494. Target Sum: This problem requires finding ways to assign signs to make the sum of numbers equal to target, which is similar to balancing accounts.
130. Surrounded Regions: This problem can help you to understand more complex scenarios in graph traversal.
684. Redundant Connection: This problem requires understanding of union-find, a key concept in graph theory.
787. Cheapest Flights Within K Stops: This problem involves a combination of graph theory and dynamic programming, similar to the main problem.

These cover traversing graphs and manipulating sets of numbers, which are integral to the “Optimal Account Balancing” problem.

Problem Analysis and Key Insights

Analyzing the problem statement, we gain several key insights:

Debt Simplification: The crux of this problem lies in the simplification of debt. Multiple transactions between two people can be reduced to one. For example, if Person A gave Person B $10, and later Person B gave Person A $5, we can simplify these transactions to state that Person A owes Person B $5. Therefore, identifying such possible reductions in the transactions would be critical in minimizing the total number of transactions.
Net Balances: We should focus on the net balance of each person. For example, if a person has given out more than they received, they have a negative balance, and vice versa. A person with zero net balance doesn’t need to participate in any transactions.
Minimization Problem: It’s important to recognize this as a minimization problem where we need to find the smallest number of transactions to balance the debt.
Potential for Graph Theory: This problem could be modeled as a graph where nodes represent individuals, and edges represent transactions between them. The problem then becomes one of finding an optimized way to eliminate cycles or reduce paths in the graph, reflecting the consolidation of transactions.
Backtracking: Given the constraint that there can be at most 8 transactions, backtracking or depth-first search is a viable option to find the optimal solution. In other words, we can afford to try out different sequences of settling debts and choose the one with the minimum transactions.
Problem Complexity: The problem can potentially become complex as the number of transactions or people increases. Careful consideration of the algorithm’s time and space complexity would be necessary.
Distinct Transactions: Each transaction in the input list is distinct, and a person does not owe money to themselves. This can help simplify the problem, as we don’t need to handle such edge cases.

By synthesizing these insights, we can get a clearer idea of how to approach solving the problem.

Problem Boundary

The scope of this problem is well defined. The aim is to find the minimum number of transactions required to settle the debts between a group of people. Here are the specifics of the problem’s scope:

The problem deals with a list of transactions among a group of individuals. Each transaction includes a giver, a receiver, and an amount.
The group size is not explicitly mentioned, but it can be inferred from the constraints that it can involve up to 12 people (IDs from 0 to 11).
The total number of transactions is at most 8, each transaction involves a distinct pair of individuals, and the amount involved in a single transaction is between 1 and 100.
The transactions are not necessarily balanced, i.e., some people might have given more money than they received, and vice versa. The problem doesn’t require you to return a balanced list of transactions, but to find the minimum number of transactions required to settle the debts.
There’s no need to indicate who should pay whom in the returned result; the goal is to simply return the minimum number of transactions required.
The problem doesn’t require the solution to be the most optimized in terms of computational complexity. Given the constraints, even algorithms with higher time complexity would likely be acceptable.

In summary, the problem is primarily focused on the calculation and optimization of transactions needed to balance debts among a group of individuals, given a list of initial transactions.

Establishing the boundary of a problem involves understanding the limits and constraints that the problem operates under. This can include things like the size of the inputs, the time complexity required, and the expected format of the outputs. Here’s how to establish the boundaries for this problem, “465. Optimal Account Balancing”:

Input:
- The input is an array of transactions. Each transaction is a list containing three elements: fromi, toi, and amounti.
- The number of transactions can be anywhere from 1 to 8.
Output:
- The output is a single integer, which represents the minimum number of transactions required to settle the debt.
Constraints:
- The fromi and toi values will be different and can range from 0 to 11, inclusive.
- The amounti will be a positive integer between 1 and 100, inclusive.
Functionality:
- The function should be able to process the list of transactions and calculate the minimum number of transactions required to balance the debts.
Time complexity:
- The problem doesn’t specify a required time complexity, but given the maximum of 8 transactions, an algorithm with a time complexity of O(n^2) or even O(2^n) should still be acceptable.
Edge cases:
- All edge cases should be considered, such as a single transaction, transactions that have already been balanced, and transactions that involve the same pair of individuals but in different directions.

By understanding these boundaries, we can ensure that our solution to the problem is both correct and efficient.

Problem Classification

Problem Analysis:

This problem is a typical financial reconciliation or account balancing problem. We are given a list of transactions, where each transaction is a tuple containing information about who is the payer, who is the payee, and how much money was paid. The objective is to minimize the number of transactions needed to balance the accounts, i.e., to make sure everyone has given and received the correct amounts of money.

Domain:

The problem lies within the domain of computational finance and combinatorial optimization, as it deals with financial transactions and seeks to minimize the number of transactions.

What Components:

The key components of the problem are:

Transactions: A list of tuples, each containing information about the payer (fromi), the payee (toi), and the amount of money transferred (amounti).
Objective: To find the minimum number of transactions needed to balance all accounts.

Problem Classification:

The problem can be classified as a minimization problem, where we aim to minimize a certain quantity, in this case, the number of transactions. The problem can also be seen as a combinatorial optimization problem, as we have to find the optimal way to organize transactions to achieve our objective.

Additionally, this problem has elements of graph theory. The transactions can be seen as directed edges in a graph, with the nodes representing individuals, and the aim is to minimize the number of edges while keeping the net flow into each node the same. Therefore, it can also be classified as a minimum path cover problem in directed graphs.

Furthermore, given the constraints, this problem also lends itself to a depth-first search or backtracking based solution. We need to explore different sequences of transactions (or edges) and find the sequence with the smallest size that covers all debts (or nodes).

In summary, this problem is a computational finance problem with elements of combinatorial optimization, graph theory, and backtracking.

Distilling the Problem to Its Core Elements

Fundamental Concept:

This problem is essentially a network flow problem, where each transaction forms a directed edge in a graph. It also involves elements of graph traversal and optimization to find the minimum number of transactions needed.

Simplest Description:

Imagine a group of friends who owe each other money. Each person might owe money to some people and might be owed money by others. We want to find out the minimum number of payments required so that everyone gets paid back and no one owes anyone else.

Core Problem:

The core problem here is to minimize the number of transactions required to balance out all debts. Simplified, we need to determine the fewest number of payments such that each person either receives or pays the correct total amount of money.

Key Components: - Interpret the transactions as a directed graph where edges represent debts. - Calculate the net balance of each person (money they are owed - money they owe). - Minimize the number of transactions among people who have a net positive balance (creditors) and people who have a net negative balance (debtors).

Minimal Operations: - Calculate the net balance for each person. - Match the creditors and debtors to balance out the debts with the fewest transactions possible. This may involve traversing the graph of transactions and using a form of a greedy algorithm or depth-first search.

Visual Model of the Problem

This problem can be visualized as a graph, where each node represents a person, and the directed edges represent transactions between people. The direction of an edge indicates the direction of money flow - from the payer to the payee, and the weight of the edge corresponds to the amount of money exchanged.

The initial graph can have multiple edges between two nodes, representing multiple transactions, and loops, representing a person owing money to themselves.

After combining all transactions, the graph becomes simpler. If a person ‘A’ owes person ‘B’ $10 and person ‘B’ owes person ‘A’ $7, we can represent this as a single edge from ‘A’ to ‘B’ with a weight of $3 ($10-$7).

Now, in the final graph, a positive balance at a node means that the person has money to receive, and a negative balance means the person has money to pay. The problem now becomes to find the minimum number of transactions to make all balances zero.

For example:

For transactions = [[0,1,10],[2,0,5]], the visual representation will be:

0 ---(10)---> 1
^
|
|(5)
2

This means person 0 owes person 1 $10 and person 2 owes person 0 $5.

The goal now is to find the minimum number of transactions that balances this graph.

Problem Restatement

Let’s break down the problem statement into simpler terms:

You have a list of financial transactions between a group of people. Each transaction is represented as a triplet of numbers [from, to, amount] indicating that the person with ID ‘from’ has given ‘amount’ money to the person with ID ’to’.

The goal is to balance the books i.e., make sure that every person has given as much money as they received, with the least number of additional transactions. In other words, we need to minimize the number of extra transactions required to settle all the debts.

The constraints are:

The list of transactions contains at least one and at most eight transactions.
Each transaction involves two different people, who are identified by a unique number between 0 and 11 (inclusive).
The ‘amount’ in each transaction is a positive integer between 1 and 100 (inclusive).

Note: A person can be involved in multiple transactions, either as the one who gives money or as the one who receives money.

Abstract Representation of the Problem

Consider a directed graph where each node represents a person and each directed edge from node ‘a’ to node ‘b’ represents a transaction of some amount from person ‘a’ to person ‘b’.

In this graph, the goal is to eliminate all the edges (transactions) by adjusting the weights of the edges (transaction amounts) in such a way that the total weight (amount) of outgoing edges from any node (person) is equal to the total weight (amount) of incoming edges to that node (person).

The key constraint is to make these adjustments in the least number of steps, which represents the minimum number of additional transactions required. Each step can be seen as adding or subtracting an edge (transaction) between any two nodes (persons).

The abstract problem, therefore, can be stated as: given a directed graph with weighted edges, find the minimum number of edge weight adjustments needed to equalize the total weight of incoming and outgoing edges for each node.

Terminology

There are several terms and concepts that are key to understanding the problem and its solution:

Transaction: This term is borrowed from finance and refers to an exchange of money from one person to another. In the context of this problem, a transaction is represented as an edge in the graph from one node to another, with the weight of the edge representing the amount of money exchanged.
Debt Settlement: This is the process of clearing outstanding debts. In the context of the problem, this refers to adjusting the weights of the edges such that the total outgoing weight (amount) for each node is equal to the total incoming weight (amount).
Directed Graph: A set of nodes connected by edges, where each edge has a direction associated with it. In this problem, each node represents a person, and a directed edge from one node to another represents a transaction from the first person to the second.
Weighted Edge: In a graph, an edge that has a value (weight) associated with it. In this problem, the weight of an edge represents the amount of money transacted.
Graph Balancing: Adjusting the weights of edges in a directed graph such that for each node, the sum of the weights of the outgoing edges is equal to the sum of the weights of the incoming edges. In this problem, balancing the graph represents settling all debts.

Understanding these concepts and terms is crucial for understanding the problem and formulating a solution.

Problem Simplification and Explanation

Let’s imagine a group of friends went on a vacation together. During the vacation, some of them paid for the group expenses. At the end of the vacation, they decided to settle the accounts, i.e., those who spent less should pay back to those who spent more so that each person has paid the same amount in total. This is what we mean by settling the debt.

In the given problem, each transaction represents a situation where one person is paying an amount to another. For example, in the transaction [0,1,10], person 0 is giving $10 to person 1.

The problem requires us to settle these transactions, that is, everyone should end up with the same amount of money they started with. We need to find a way to accomplish this with the least number of transactions.

To simplify the problem, think of it as a balancing act. It’s like trying to balance weights on a scale. Each person is a point on the scale and the transactions are the weights. You want to move these weights around until both sides of the scale are balanced, i.e., no person is owing money to someone else.

To solve this problem, we would need to understand and use concepts from graph theory as the problem can be modelled as a directed graph where each node represents a person and each edge represents a transaction. The direction of the edge tells you who owes whom and the weight of the edge tells you how much. The goal is to balance this graph, which essentially means settling the debts.

Constraints

Given the problem statement and the constraints, there are several things that can be exploited to find an efficient solution:

Number of transactions is small: The number of transactions is limited to 8. This makes the problem small enough to be handled by an exhaustive search algorithm, possibly with some optimization.
Limited number of people: The IDs of people involved range from 0 to 11, which means there are at most 12 people involved in the transactions. This can be used to create a compact representation of the debts between each pair of people.
Unique transactions: Each transaction is unique, as the “from” and “to” people in a transaction are different. This means there are no repeated transactions between the same pair of people.
Arbitrary order of settlement: The order in which the debts are settled does not matter. This allows for a flexible approach in choosing the order of settlements.

Looking for patterns, we see that if a person ‘A’ owes person ‘B’, and person ‘B’ owes person ‘C’, it might be beneficial for ‘A’ to pay ‘C’ directly, if possible. This could reduce the number of transactions. Such patterns can be exploited to derive a solution.

To manipulate or interpret the data, one could construct a net debt for each person, which is the sum of money they owe minus the sum of money they are owed. The goal is then to balance these debts in the minimum number of transactions.

Analyzing the constraints of the problem can indeed provide us with useful insights:

Limited number of transactions: The number of transactions in the input is no more than 8. This is a relatively small amount and indicates that we could potentially use approaches with higher time complexity, such as exhaustive search or backtracking, and still achieve acceptable performance.
Limited number of people involved: The IDs of people range from 0 to 11, which means that we have a maximum of 12 people involved. This relatively small set of people also implies that we can efficiently process transactions, perhaps by using data structures like arrays of size 12 to represent the debts of each person.
Small amounts involved in each transaction: The amount of each transaction is between 1 and 100. This means that we don’t have to deal with large numbers in our calculations, which simplifies the problem.
Unique Transactions: The ‘from’ person and the ’to’ person in a transaction are always different. This helps us to avoid handling scenarios where a person owes money to themselves, which would unnecessarily complicate the problem.

Overall, the constraints of this problem make it suitable for solutions involving exhaustive search, backtracking, or greedy algorithms, given the small scale of transactions, people, and amounts involved.

Case Analysis

Here are some additional examples which cover different aspects of the problem:

Minimal Transaction (Edge Case)
Consider the input transactions = [[0,1,1]]. Here, person 0 owes person 1 an amount of 1. Since there’s only one transaction, this is also the minimum number of transactions needed to settle the debt. So the expected output is 1.
No Debt (Edge Case)
Consider the input transactions = [[0,1,10],[1,0,10]]. Here, person 0 owes person 1 an amount of 10 and person 1 owes person 0 the same amount. They can simply cancel each other’s debt out, and no transactions are needed. So the expected output is 0.
Circular Debt (Interesting Case)
Consider the input transactions = [[0,1,10],[1,2,10],[2,0,10]]. Here, person 0 owes person 1 an amount of 10, person 1 owes person 2 the same amount, and person 2 owes person 0 the same amount. They can simply each pay their debt to the next person in the circle, and the debts will be settled in 3 transactions. So the expected output is 3.
Multiple Persons Owe to One (Boundary Case)
Consider the input transactions = [[0,1,10],[2,1,10],[3,1,10]]. Here, persons 0, 2, and 3 each owe person 1 an amount of 10. Each of them will have to make a transaction to person 1 to settle their debt, making a total of 3 transactions. So the expected output is 3.
One Person Owes to Multiple (Boundary Case)
Consider the input transactions = [[0,1,10],[0,2,10],[0,3,10]]. Here, person 0 owes persons 1, 2, and 3 an amount of 10 each. Person 0 will have to make a transaction to each of them to settle the debt, making a total of 3 transactions. So the expected output is 3.

Remember, the edge cases usually involve minimum/maximum input size or special input values, while boundary cases involve values on or beyond the limits. Interesting cases, as the name implies, are those that are not immediately obvious but could reveal deep insights about the problem.

Analyzing different cases reveals some key insights into the problem, which are:

Cancellation of Debts: There can be cases where two persons owe each other equal amounts. In such cases, the debts can be canceled out without any transaction.
Circular Debts: There can be circular debts where person A owes person B, who owes person C, who in turn owes person A. Such cases need to be handled carefully as they can be resolved in minimum transactions if tackled properly.
Multiple Debtors or Creditors: There can be cases where one person owes multiple people or multiple people owe one person. In such cases, each debt needs to be settled individually, increasing the number of transactions.
Debt Redistribution: In some cases, the minimum number of transactions can be achieved by redistributing the debts, i.e., if A owes B and B owes C, then A can directly owe C, reducing the transactions.
Handling Edge Cases: Edge cases where there is a minimum number of transactions (one) or no transactions required (debts cancel out) need to be handled appropriately.

By analyzing these cases, we understand that the problem has to deal with the distribution and redistribution of debts in a way that the total number of transactions is minimized.

Identification of Applicable Theoretical Concepts

The “Optimal Account Balancing” problem can be seen as a problem of network flow or as a variant of the partition problem. Here are some mathematical and algorithmic concepts that can be applied:

Graph Theory: The problem can be modelled as a graph where nodes represent people and edges represent transactions. In this context, the problem becomes finding the minimum number of edges (transactions) to balance the graph (settle all debts).
Network Flow: This problem can also be seen as a problem of network flow, where the goal is to find the minimum number of transactions to ensure the flow (money) from the source (debtor) to the sink (creditor) is balanced.
Depth-First Search (DFS) or Breadth-First Search (BFS): These graph traversal algorithms can be used to navigate the graph and find the optimal path for settling the debts.
Partition Problem: This problem can be seen as a variant of the partition problem, where the goal is to divide the set of debts into two subsets such that the sum of the amounts in the two subsets is equal. This is related to the fact that the sum of debts and credits must be equal.
Greedy Algorithms: These can be used to settle the larger debts first, reducing the number of transactions needed. This would work under the assumption that settling larger debts first would lead to an optimal solution, which may not always be the case.
Backtracking: This can be used to try different possibilities for settling the debts and backtrack when a particular path does not lead to a solution.

Remember, though, that just identifying these concepts doesn’t mean they will all be useful or that they are the only concepts that could be used to solve the problem. The actual approach will depend on a detailed analysis of the problem and the constraints given.

Simple Explanation

Imagine you and your friends have been buying snacks and lending each other money for a while, and now it’s gotten a little complicated. Some people owe money to others, and it’s just a big web of who owes what to whom.

For example, let’s say your friend Alice gave $10 to your friend Bob because he forgot his wallet. And on another day, your friend Charlie gave $5 to Alice because she wanted to buy a book but didn’t have enough cash on her. Now, Alice, Bob, and Charlie owe each other money.

But instead of Alice paying Charlie back directly, she could just tell Bob to give $5 of the $10 he owes her to Charlie. Now everyone’s debts are settled, and it only took two transactions: the original ones where Alice gave money to Bob and Charlie gave money to Alice.

Your task is to figure out the smallest number of transactions it would take to make sure everyone has paid back everything they owe. This means you need to figure out who can pay whom so that the least amount of money changes hands.

This is like untangling a big knot of debts. It might seem a bit tricky, but with some careful thinking, it can be figured out!

Problem Breakdown and Solution Methodology

Let’s break it down:

Identify and Aggregate Balances: The first step to solve this problem is to understand the overall balance of each person, i.e., how much each person owes or is owed. This is done by going through each transaction, and for each person involved in a transaction, subtract the amount they gave and add the amount they received. This way, we create a balance sheet: if the final amount is positive, that person is owed that much money, if it’s negative, they owe that much money.
Create an Optimization Strategy: The next step is to minimize the number of transactions to settle the debts. The main idea here is to match the person who owes the most with the person who is owed the most. The person who owes pays back as much as they can, i.e., either the entire amount they owe, or an amount that fully reimburses the person who is owed, whichever is smaller.
To understand this, imagine you are trying to match people for carpooling. You want to find the person who lives farthest north with the person who lives farthest south, then the next farthest, and so on. By doing this, you’re making sure that each pair travels the longest possible distance together, reducing the overall number of trips.
Execute Transactions and Recalculate: Once a transaction is done, recalculate the amounts owed by each person, and repeat the process until all debts are settled.

Let’s illustrate this approach with an example:

Suppose we have the following transactions:

[[0,1,10], [2,0,5], [3,2,7], [4,3,5]]

Step 1: We create a balance sheet: person 0 owes 5, person 1 is owed 10, person 2 is owed 2, person 3 owes 2, person 4 is owed 5.

Step 2: We start settling the debts. The person who owes the most is person 0 (owes 5), and the person who is owed the most is person 1 (owed 10). Person 0 pays person 1, and person 1 is now owed 5.

Step 3: We repeat step 2. Now, the person who owes the most is person 3 (owes 2), and the person who is owed the most is person 1 (owed 5). Person 3 pays person 1, and person 1 is now owed 3.

Step 4: We repeat again. The person who owes the most is person 1 (owes 3), and the person who is owed the most is person 4 (owed 5). Person 1 pays person 4, and person 4 is now owed 2.

Step 5: Finally, the person who owes the most is person 4 (owes 2), and the person who is owed the most is person 2 (owed 2). Person 4 pays person 2, and all debts are settled.

As you can see, the total number of transactions is 4, which is the minimum number of transactions required to settle the debt.

If the number of transactions or the amounts involved in the transactions were to increase, the process would still remain the same, but it would take longer due to the increased complexity. However, the principle of matching the highest debtor with the highest creditor remains an effective strategy to minimize the number of transactions.

Inference of Problem-Solving Approach from the Problem Statement

Here are some key terms and concepts related to this problem:

Transactions: Each transaction is an exchange of money from one person to another. It is the fundamental unit of operation in this problem. A transaction record is represented as [from, to, amount], meaning that from gave amount to to.
Understanding transactions is crucial as it forms the basis for creating a balance sheet for each person. Also, the goal of this problem is to minimize the total number of transactions, so understanding what constitutes a transaction helps shape the solution.
Balances: The balance of a person represents how much they owe or are owed after all transactions. This is a derived concept, not explicitly mentioned in the problem but crucial for its solution.
The idea of balancing is a key to solving the problem because, ultimately, we aim to bring everyone’s balance to zero using the minimum number of transactions. The process of updating balances also guides the solution’s iterative nature.
Debt Settlement: Debt settlement is the process of paying off debt in an agreed-upon manner between debtor and creditor. In this problem, we aim to settle the debts of all persons using the minimum number of transactions.
The concept of debt settlement directs us towards the strategy of matching the person who owes the most with the person who is owed the most, which leads to an optimal solution.
Optimization: The problem requires finding a minimum, specifically the minimum number of transactions needed to settle the debt. This is a classic optimization problem.
Recognizing this as an optimization problem hints at certain solution strategies, like greedy algorithms or dynamic programming, that are common for such problems.
Graph Theory: Although not explicitly stated, this problem can be viewed as a graph problem, where each person is a node, and a transaction is a directed edge from one node to another. The ‘amount’ can be thought of as the weight of the edge.
Viewing the problem this way might open up new avenues for solving it using graph algorithms. For instance, the problem can be thought of as a circulation problem in network flow, where the goal is to find a circulation that minimizes the total weight (i.e., the number of transactions).

Understanding these key terms and concepts helps to identify the appropriate data structures (like hash maps for storing balances) and algorithms (like greedy algorithms for choosing transactions) to use when formulating a solution.

How did you infer from the problem statement that this problem can be solved using ?

Simple Explanation of the Proof

I’m having trouble understanding the proof of this algorithm. Could you explain it in a way that’s easy to understand?

Let’s breakdown the problem.

Create a Balances Array: The first step in our approach would be to process the transactions and calculate the balance of each person. The balance of a person is the total amount they owe subtracted from the total amount they are owed. We can create a balance array where the index represents the person’s ID and the value at that index represents their balance.
Identify Debtors and Creditors: Once we have the balance array, we can identify who are the debtors (people who owe money) and creditors (people who are owed money). We can create two separate lists for this purpose.
Settle Debts: Now, we start settling the debts. For this, we repeatedly pick the person with the maximum debt and the person with the maximum credit. The debtor pays the creditor as much as they can (i.e., the minimum of the debt and credit). Then, we update the balances and continue the process until all debts are settled.
Count Transactions: Every time a debt is settled, it counts as a transaction. We keep track of the number of transactions.

The parts of the problem that can be solved independently are creating the balance array and identifying the debtors and creditors. These can be done in parallel.

The repeatable pattern within our solution is the process of settling debts. We repeatedly perform the same operation of settling debts until all debts are settled.

Finally, it’s worth noting that this is a high-level approach. In practice, we need to be careful about edge cases. For instance, if a person’s debt or credit becomes zero during the process, we need to remove them from the respective list. Also, when counting transactions, we should make sure to only count those transactions where an actual transfer of money takes place (i.e., ignore transactions with an amount of zero).

Solution Approach and Analysis

Step 1: Calculate the net balance for each person. This is the amount they gave away minus the amount they received. For this, we initialize an array balance of size n where n is the maximum person id plus one. We traverse through the list of transactions and for each transaction fromi, toi, amounti, we decrement amounti from balance[fromi] and increment amounti to balance[toi].

Step 2: After calculating the net balance for each person, we split them into two groups - creditors and debtors. Creditors are the ones who have a positive balance (they need to receive money) and debtors are the ones with a negative balance (they need to pay money).

Step 3: Now, we need to settle the debts. We take one creditor and one debtor. The debtor pays as much as they can (i.e., the minimum of the debt and credit). If the debtor’s debt is fully paid, we move on to the next debtor. If the creditor’s credit is fully covered, we move on to the next creditor. We continue this until either all debts are paid or all credits are covered.

Step 4: Every time a debt is settled, it counts as a transaction. We keep track of the number of transactions and return this as our final result.

Let’s consider an example to demonstrate the above steps:

transactions = [[0,1,10],[2,0,5]]

Here, person 0 gave $10 to person 1, and person 2 gave $5 to person 0.

Step 1: Calculate the net balance. The balance array will be [5, -10, 5]. Here, person 0 has a positive balance of $5, person 1 has a negative balance of $10 (they owe $10), and person 2 has a positive balance of $5.

Step 2: Separate into debtors and creditors. The debtors are [1] and the creditors are [0, 2].

Step 3: Settle the debts. We take the debtor 1 and the creditor 0. The debtor pays $5 to the creditor. Now the balance array becomes [0, -5, 5]. The creditors remain the same since person 0’s credit is fully covered. But debtor 1 still owes $5. We take the next creditor 2. The debtor 1 pays $5 to the creditor. Now the balance array becomes [0, 0, 0]. All debts are paid and all credits are covered.

Step 4: Count the transactions. There were two transactions, so the final result is 2.

Changes in the problem’s parameters would affect the solution in the following ways:

Increasing the number of transactions will increase the size of the balance array and the time it takes to compute the final result.
Having large amounts in transactions will not affect the number of transactions needed to settle the debt, but it will affect the values in the balance array.
Increasing the number of people (represented by fromi and toi) will increase the size of the balance array. If there are many people but few transactions, a lot of the balance array will be zeros.

This problem is a good illustration of the principle of conservation in mathematics. In this case, the total amount of money in the system is conserved. It’s just a matter of redistributing it in a way that minimizes the number of transactions.

Identify Invariant

An invariant, in computer science and mathematics, is a condition that remains true before and after an operation or a series of operations.

In this problem, the invariant is the “net balance of all people involved in transactions”. No matter how the transactions are re-arranged or optimised to minimise the number of transactions, the net balance of all people stays constant.

In other words, the sum of all debts should always equal the sum of all credits. This is because every dollar owed by a debtor corresponds to a dollar that a creditor expects to receive.

Before we begin processing transactions, we calculate the net balance for every person. At the end of all transactions, everyone’s net balance should be zero (because all debts have been paid off, and all creditors have received the money owed to them). The sum of all balances in the system, therefore, remains zero before and after settling the transactions.

So, the “sum of the balance of all people involved in transactions” is the invariant in this problem. This holds true at all stages of the problem and hence can be considered as the invariant condition.

Identify Loop Invariant

A loop invariant is a condition that is initially true and remains true after each iteration of a loop. It is used to prove that a loop is correct, and the computation carried out in the loop is as expected.

For the problem 465. Optimal Account Balancing, a potential loop invariant would be involved in the main loop where we iterate over all the transactions. The exact loop invariant can vary based on how we implement the solution, but a common loop invariant would be:

“The total sum of all accounts is preserved after each iteration.”

This is because every transaction involves transferring money from one account to another, which doesn’t change the total amount of money across all accounts.

Also, if we are using a loop to iterate over all people who owe or are owed money, another potential loop invariant could be:

“The sum of money owed by people who are in debt is equal to the sum of money that creditors expect to receive after each iteration.”

This statement means that every dollar that a debtor owes corresponds to a dollar a creditor is owed. This total never changes, regardless of how the debts are paid off.

Again, it’s important to note that the loop invariant can vary depending on your specific implementation and solution approach.

Thought Process

Given the problem, the approach can be formed by following these steps:

Identify the problem’s nature: This problem is about finding the minimum number of transactions to settle the debt among a group of people.
Understand the transactions: Each transaction is represented by [fromi, toi, amounti] where the person with ID = fromi gave amounti $ to the person with ID = toi.
Define the aim: Our task is to minimize the number of transactions needed to settle the debt. This implies we should try to eliminate as many debts as possible in each transaction.
Formulate a strategy: Since the problem revolves around settling debts, we could calculate the net balance for each person. This net balance is the total amount a person owes subtracted from the total amount a person is owed. We can use a hashmap to store these net balances. Positive balances are amounts the person is owed, and negative balances are amounts the person owes.
Debt settlement: People with positive balances are creditors, while people with negative balances are debtors. The goal is to match these creditors and debtors in a way that minimizes transactions.
Consider the edge case: If the net balance is zero, the person is neither a creditor nor a debtor. We can ignore these people as they neither owe nor are owed any money.
Optimize transactions: One way to minimize transactions is to settle multiple debts with a single transaction whenever possible. A creditor can distribute their positive balance among multiple debtors, or a debtor can pay off their debt to multiple creditors.
Depth-First Search (DFS): This is a recursive process and it’s better to use Depth-First Search to explore all possible transaction sequences. The base case for the recursion is when there are no more debtors, which means all debts have been paid and we can return zero transactions.

Here is a Python code implementing the steps above:

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class Solution:
    def minTransfers(self, transactions: List[List[int]]) -> int:
        from collections import defaultdict
        
        bal = defaultdict(int)
        for trans in transactions:
            bal[trans[0]] -= trans[2]
            bal[trans[1]] += trans[2]

        debt = list(bal.values())

        return self.settle(0, debt)

    def settle(self, start, debt):
        while start < len(debt) and debt[start] == 0:
            start += 1
        if start == len(debt):
            return 0
        res = float('inf')
        for i in range(start + 1, len(debt)):
            if debt[start] * debt[i] < 0:  # one is debtor, another one is creditor
                debt[i] += debt[start]  # settle the debt between start and i
                res = min(res, 1 + self.settle(start + 1, debt))
                debt[i] -= debt[start]  # backtrack
        return res

The minTransfers function calculates the balance for each person. Then it finds the minimum number of transactions needed to settle all the debts by calling the settle function. The settle function tries to settle the debt of the person at index start with every person at index i where i > start. It uses DFS to explore all possible ways to settle the debts and backtracks to undo the settlement for the next iteration. It returns the minimum number of transactions found.

The key insight here is understanding that we can transform the problem into one about balancing debts. This simplifies the problem and allows us to use DFS to find the minimum number of transactions. The cues in the problem statement that lead to this approach include the facts that each person has a unique ID and that the transactions involve transferring money between two people.

Establishing Preconditions and Postconditions

Parameters:
- The method minTransfers takes one parameter, transactions, which is a list of lists. Each sublist represents a transaction with three elements: [fromi, toi, amounti].
- The method settle takes two parameters, start which is an integer, and debt which is a list of integers.
- The parameter transactions represents the transactions between people. Each sublist is a transaction where the first element is the ID of the person giving money, the second element is the ID of the person receiving money, and the third element is the amount of money being given.
- The parameter start represents the starting index for the settling process.
- The parameter debt represents the net balance of each person after all transactions.
Preconditions:
- Before minTransfers is called, it is assumed that the transactions list is properly formatted as per the problem’s constraints.
- The parameter transactions must be a list of lists, where each sublist contains three elements: two integers representing the people involved in the transaction and one integer representing the amount of money transacted.
- The method doesn’t require a specific state of the program, it can be called independently as long as the input is correctly provided.
Method Functionality:
- The minTransfers method is expected to calculate the minimum number of transactions needed to settle all debts.
- It first calculates the net balance of each person after all transactions. Then it uses the settle method to find the minimum number of transactions.
Postconditions:
- After the method has been called and has returned, the state of the program doesn’t change. All changes are local to the method.
- The return value represents the minimum number of transactions required to settle all debts.
- There are no side effects, the method doesn’t alter any of the input parameters.
Error Handling:
- The method doesn’t explicitly handle errors. If the preconditions are not met, it will likely result in an error (like an IndexError or TypeError).
- It doesn’t throw a custom exception or return a special value in case of error. The Python interpreter would raise a standard exception if there’s an error during execution.

Problem Decomposition

Problem Understanding:
- The problem is about settling debts among a group of people. Each transaction between two people is represented by [fromi, toi, amounti], where ‘fromi’ is the person giving money, ’toi’ is the person receiving money, and ‘amounti’ is the amount of money being transferred. The objective is to find the minimum number of transactions needed to settle all debts.
Initial Breakdown:
- The problem can be divided into two broad parts. First, we need to calculate the net balance for each person after all the transactions. Second, we need to find the minimum number of transactions to settle the debts.
Subproblem Refinement:
- The first part, calculating the net balance, involves going through each transaction and updating the balances of the ‘from’ and ’to’ individuals.
- The second part, finding the minimum number of transactions, involves recursively settling debts starting from each person until no more debts can be settled.
Task Identification:
- We repeatedly update the balances of individuals, and this process can be generalized into a single task.
- We also repeatedly try to settle debts, which is another task that can be generalized.
Task Abstraction:
- Updating balances and settling debts are both abstracted tasks that make sense in the context of the problem.
Method Naming:
- We could name the task of updating balances as ‘update_balances’ and the task of settling debts as ‘settle_debts’.
Subproblem Interactions:
- The ‘update_balances’ task needs to be performed first to set the stage for the ‘settle_debts’ task. There is a dependency in this order because ‘settle_debts’ requires the result of ‘update_balances’ (the net balances of each person) to work properly.

From Brute Force to Optimal Solution

Brute Force Approach:

The brute force solution to this problem would involve trying out all possible ways of settling the debts and then choosing the one that involves the minimum number of transactions. This involves generating all permutations of transactions that could settle the debt and then checking each one to see if it balances out all the debts. This approach, however, is extremely inefficient because the number of permutations grows factorially with the number of individuals, making this approach infeasible even for a small number of people.

Inefficiencies of Brute Force:

The main inefficiency of the brute force approach is that it checks all possible transaction combinations without using any insights from the problem to reduce the search space. Since the search space grows factorially, this approach becomes infeasible even for a relatively small number of people. The brute force approach also does not take into account that the order of transactions does not matter, leading to many redundant checks.

Optimization Steps:

Net balance calculation: The first optimization step is to calculate the net balance for each individual, i.e., the total amount they gave minus the total amount they received. This reduces the problem to finding the minimum number of transactions among the individuals with a non-zero net balance.
Recursive debt settling: The second optimization step is to recursively try to settle the debts, starting from each individual with a non-zero net balance. This involves trying to pair each debtor (an individual with a negative balance) with a creditor (an individual with a positive balance), and recursively trying to settle the rest of the debts. This approach utilizes a depth-first search (DFS) strategy, which efficiently explores the search space by going as deep as possible before backtracking.
Early stopping: The third optimization step is to stop the DFS as soon as we find a valid debt settling, since any other debt settling would involve the same or a larger number of transactions.

Complexity Analysis:

Time complexity: After the optimizations, the time complexity of the problem is still exponential in the number of individuals, because in the worst-case scenario we still need to explore all possible pairings of debtors and creditors. However, the optimizations significantly reduce the constant factor, making the solution feasible for a larger number of individuals.
Space complexity: The space complexity of the solution is linear in the number of individuals, as we need to store the net balance for each individual. The recursive debt settling also involves a recursion stack, but its size is also linear in the number of individuals.

Code Explanation and Design Decisions

Initial Parameters: The initial parameter to our method is the list of transactions, where each transaction is represented by a list of three elements: [fromi, toi, amounti]. Each element in the transaction represents a transfer of money from one person (fromi) to another person (toi), and the amount of money transferred (amounti). This information is essential as it forms the base data from which we calculate the net balance of each person.
Primary Loop or Iteration: The primary iteration in the solution is over each person with a non-zero balance. We are trying to settle the debts in a way that minimizes the number of transactions. The iteration over each person signifies an attempt to settle that person’s debt or credit.
Conditions or Branches within the Loop: Within the loop, we have a condition that checks whether a person’s balance can be settled with another person’s balance. If a debtor can pay back a creditor fully or partially, we proceed to adjust their balances and recursively try to settle the remaining balances. This is in line with the problem’s requirement of settling the debts.
Updates or Modifications to Parameters: Within the loop, we update the balances of the debtor and creditor when a transaction takes place. This is necessary because as we progress in settling the debts, the balances of people will change. Updating the balances ensures that we accurately track the state of our solution.
Invariant: One invariant that’s maintained throughout the code is that the sum of all the balances is always zero. This is because every transaction involves one person losing some money and another person gaining the same amount of money. This invariant helps us ensure that our transactions are correctly adjusting the balances.
Final Output: The final output of the function is the minimum number of transactions required to settle all the debts. It represents the least number of money transfers that need to happen so that all the people end up with a zero balance. This satisfies the problem’s requirements as we are asked to minimize the number of transactions.

Coding Constructs

High-Level Problem-Solving Strategies: This code uses a combination of techniques. First, it employs balance calculation, which is essentially an application of arithmetic operations to calculate the net balance for each individual. Then, it uses recursive search along with backtracking to try different possibilities of debt settling, looking for the solution that requires the minimum number of transactions.
Explaining to a Non-Programmer: This code is like a virtual debt settler. Suppose you have a group of friends who lend money to each other. At some point, they want to settle up so that everyone has lent and borrowed the same amount. This program helps them do that in the least number of payments possible.
Logical Elements or Constructs: This code uses several logical constructs, including arithmetic operations for balance calculation, a list (or array) to store balances, a loop to iterate over balances, conditional statements to check if a transaction can occur, and recursive function calls to explore different transaction possibilities.
Algorithmic Approach in Plain English: The code first calculates how much each person owes or is owed. Then it tries different combinations of payments among the group to minimize the total number of transactions. If it finds a possible payment, it adjusts the amounts owed and then tries to settle the remaining amounts.
Key Steps or Operations: The key steps this code performs include:
- Calculation of the net balance for each individual.
- Initiation of a recursive function to explore different transaction possibilities.
- A check in the recursive function to see if a person can pay another.
- Adjustment of the balances if a transaction occurs.
- Return of the number of transactions once all debts are settled.
Algorithmic Patterns or Strategies: This code employs a few common algorithmic patterns. These include accumulation (summing up balances), recursion (to explore all possible transactions), and backtracking (undoing a step if it doesn’t lead to a solution). The combination of these strategies allows for a comprehensive search of the solution space.

Language Agnostic Coding Drills

1. Identification of distinct coding concepts:

Let’s deconstruct the code into separate, language-agnostic coding concepts. Some of these concepts, in no particular order, include:

Variable Assignment: Assigning values to variables to keep track of data.
Basic Arithmetic Operations: Adding, subtracting, multiplying, and dividing numbers.
Lists/Arrays: Using a data structure to store multiple values.
Loops: Repeatedly executing a block of code while a certain condition is true.
Conditional Statements: Executing different code blocks based on whether a condition is true or false.
Function Calls: Using functions to break down the code into reusable components.
Recursion: A function calling itself to solve a problem in smaller parts.
Backtracking: Exploring all possible options and undoing a step if it doesn’t lead to a solution.

2. Classification by difficulty:

Variable Assignment: This is a basic concept that forms the foundation of most programming. It’s usually one of the first concepts taught to beginners.
Basic Arithmetic Operations: Another fundamental concept used in all kinds of programs.
Conditional Statements: Conditional statements are a bit more complex because they involve decision-making, but they’re still one of the basic building blocks of programming.
Lists/Arrays: Understanding how to work with collections of data is a little more difficult, especially when it involves indexing and manipulating the data.
Loops: Loops are more advanced because they involve repetition and often depend on understanding conditionals and variables as well.
Function Calls: Functions involve understanding how to modularize code, which requires a higher level of abstract thinking.
Recursion: This is more advanced because it involves the function calling itself, which can be a difficult concept to grasp and requires understanding of the function call stack.
Backtracking: This is one of the most complex strategies, as it involves trying different options and undoing them if they don’t lead to the solution.

3. Problem-solving approach:

The approach to solving this problem involves applying these concepts in a sequential manner:

Begin by calculating the net balance for each person using variable assignments, arithmetic operations, and lists/arrays.
Initialize a recursive function that will be used to find the minimal number of transactions.
Within this recursive function, implement a loop to iterate over the balances.
Include a conditional statement in the loop to check if a person can pay another.
If the condition is true, adjust the balances using arithmetic operations and variable assignment. This represents a decision and is an instance of backtracking if the decision doesn’t lead to a minimum number of transactions.
Continue the process by making a recursive function call, which will try to settle the remaining amounts.
Once all debts are settled, return the number of transactions. The recursion ends here, which is also part of the backtracking process.

Each of these steps contributes to the overall solution, enabling a comprehensive search for the minimal number of transactions needed to settle all debts.

Targeted Drills in Python

Concept Drills in Python

Here are some isolated examples in Python that illustrate the concepts we discussed:

Variable Assignment

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x = 5
y = 10
print(x, y)

Basic Arithmetic Operations

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sum = x + y
diff = x - y
prod = x * y
quot = x / y
print(sum, diff, prod, quot)

Conditional Statements

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if x < y:
    print("x is less than y")
elif x > y:
    print("x is greater than y")
else:
    print("x is equal to y")

Lists/Arrays

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nums = [1, 2, 3, 4, 5]
for num in nums:
    print(num)

Loops

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i = 1
while i <= 5:
    print(i)
    i += 1

Function Calls

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def greet(name):
    print(f"Hello, {name}!")

greet("Alice")

Recursion

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def factorial(n):
    if n == 0:
        return 1
    else:
        return n * factorial(n-1)

print(factorial(5))

Backtracking

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def permute(data, i, length): 
    if i==length: 
        print(''.join(data) )
    else: 
        for j in range(i,length): 
            data[i], data[j] = data[j], data[i] 
            permute(data, i+1, length) 
            data[i], data[j] = data[j], data[i]  

string = "ABC"
permute(list(string), 0, len(string))

Problem-Specific Drills

The key problem-specific concept in this case is the handling of the transactions and the way debts are settled among individuals, which is not covered by any of the basic concepts above. Here’s a simple example of a drill that reflects this concept:

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def settle_debts(transactions):
    balances = [0 for _ in range(100)]  # Assume we have 100 people

    # Calculate balances
    for transaction in transactions:
        payer, receiver, amount = transaction
        balances[payer] -= amount
        balances[receiver] += amount

    # Settle debts
    for i in range(100):
        for j in range(i+1, 100):
            if balances[i] < 0 and balances[j] > 0:
                transfer = min(-balances[i], balances[j])
                balances[i] += transfer
                balances[j] -= transfer
                print(f"Person {i} pays Person {j} {transfer} units")

transactions = [[0, 1, 10], [2, 0, 5]]  # Person 0 pays Person 1 10 units, Person 2 pays Person 0 5 units
settle_debts(transactions)

Integrating the Concepts

The final solution to the problem would involve integrating all these concepts. We would start by calculating the net balance for each person (variable assignment, arithmetic operations, lists/arrays). Then, we would implement a recursive function with a loop to iterate through the balances and check if a person can pay another (conditional statements, loops, function calls, recursion). If so, we adjust the balances, make a decision, and call the function recursively (backtracking). Finally, once all the debts are settled, we return the total number of transactions.

Q&A

Given the input [[0,1,1], [2,3,2], [4,5,3], [6,7,4], [8,9,5], [10,11,6], [12,13,7], [14,15,2], [14,16,2], [14,17,2], [14,18,2]]

The provided expected result is 14. However, there are only 11 transactions, so the upper bound of the result should be 11.

You’re correct. In the problem “Optimal Account Balancing”, the upper bound on the number of transactions would indeed be the number of input transactions. The scenario given: [[0,1,1], [2,3,2], [4,5,3], [6,7,4], [8,9,5], [10,11,6], [12,13,7], [14,15,2], [14,16,2], [14,17,2], [14,18,2]] involves 11 transactions, so the number of transactions in the optimal solution should be less than or equal to 11.

The provided expected result being 14 seems to be a mistake. It’s likely there has been a misunderstanding or error somewhere in the process. The actual expected result should indeed be less than or equal to 11, as you pointed out.

I’d suggest re-examining the problem and the approach in the context of this corrected understanding of the constraints.

Think about this case: [[0,3,2],[1,4,3],[2,3,2],[2,4,2]]

The correct answer should be 3, but the so-called standard answer is 4. Actually this question seems to be a NPC problem. And I don’t know why there is no data scale.

The problem “Optimal Account Balancing” is indeed a complex one, and it can be related to the NP-Complete problems.

Here, for the example [[0,3,2],[1,4,3],[2,3,2],[2,4,2]], you’re correct that the minimal number of transactions should be 3. One possible sequence of transactions could be:

Person 2 gives $2 to person 3.
Person 0 gives $2 to person 4.
Person 1 gives $3 to person 4.

In total, that’s 3 transactions.

As for the data scale, it’s not specified in the problem statement. However, due to the nature of the problem and the exponential increase in complexity as the number of transactions increases, it is reasonable to assume that the number of transactions is not extraordinarily large in most test cases.

The incorrect expected output could be a mistake or misunderstanding.

So, the primary challenge of this problem lies in determining an optimal way to balance the accounts, i.e., minimize the number of transactions needed to equalize the debts, which essentially makes it a combinatorial optimization problem. This problem indeed has similarities to some classic NP-Complete problems like the Travelling Salesman Problem (minimizing total distance) and the Knapsack Problem (optimizing the total value).

I headed down a Graph + Union Find + trying to union all sets with sum 0, before giving up and seeing other brute force backtrack solutions get accepted.

What are the clues we have here (apart from the limits) that brute force would be enough?

Indeed, this problem might initially seem like it requires a sophisticated algorithm involving graphs and union-find data structures, especially given the transactional nature of the data which naturally lends itself to a graph representation. However, when you see problems like this with a small data scale, it often implies that brute force or backtracking solutions could be acceptable.

The primary clue here is in the constraints: the problem statement doesn’t explicitly state a maximum limit on the number of transactions, but it does say the number of transactions will not be in the scale of thousands or millions. This means the number of transactions is likely to be reasonably small, which makes brute force and backtracking strategies more feasible.

In brute force and backtracking solutions, we typically generate all possible arrangements and then choose the best one according to the problem’s requirements. This approach can be highly inefficient for large inputs due to its high time complexity (often exponential). However, when the input size is small, these methods can be feasible and sometimes even efficient.

So, the lack of a large-scale data constraint is a key clue that a brute force or backtracking solution might be acceptable. Of course, it’s not a surefire indicator – sometimes, even with small input sizes, a more sophisticated algorithm is required. However, in the context of this problem, it does suggest that a brute force approach might be sufficient.

The limits are probably the biggest clue. Anytime I see a problem with such tight input constraints (n = 8 for transaction length) it’s a quick give away that a brute-force solution is fine.

Yes, you’re absolutely right. The problem constraints often provide a hint about the expected complexity of the solution.

The fact that the transaction length is limited to a maximum of 8 suggests that a brute-force or backtracking solution (which typically have higher time complexities) would be suitable for this problem. These methods, despite their inefficiency for larger inputs, work perfectly fine for smaller inputs. They enumerate all possible configurations and then choose the best one according to the problem’s requirements.

When you see such small constraints, it’s a good hint that the problem is designed for an exhaustive search solution like brute force or backtracking. This strategy won’t be applicable to every problem with small constraints, but it’s a good rule of thumb to keep in mind when approaching problems like this one.

When I first looked at the problem, I thought it was a graph problem so I spent quite some time to figure out building the graph, tried to clean debts by removing edges and things like that, but it is not. Sometimes when I encounter other problems I did not think it as a graph problem, then I opened discussion section, people seemed to all agree on simulating the problem as graph problem. So I guess it simply requires more practice and the ability to recognize patterns?

Absolutely, pattern recognition is a key part of becoming proficient at solving coding problems. It’s not uncommon to initially misinterpret the problem and take an approach that doesn’t work out. Don’t be too hard on yourself, this happens to everyone and it’s part of the learning process.

Graph theory is a very broad field and many problems can be modeled as a graph problem. However, it doesn’t mean that graph theory is always the best approach. Similarly, problems that appear to be about graphs might be better solved with a different approach.

It’s also important to keep in mind that there are often multiple valid ways to solve a problem, and what works best can depend on the specifics of the problem and the constraints. In some cases, a problem that can be solved with graph theory could also be solved with other methods such as dynamic programming, backtracking, greedy algorithms, and so on.

As you continue to solve more problems, you’ll naturally get better at recognizing patterns and identifying the most suitable approach. Discussing solutions with others, reading different solutions, and reflecting on what worked and what didn’t are all great ways to accelerate this learning process.

Remember, practice is key when it comes to improving your problem-solving skills, so keep practicing and don’t get discouraged when things don’t go as expected. With time and persistence, you’ll get better at recognizing the most suitable approach for different kinds of problems.

Is this resolving similar to the coin change problem where … with the mindset of “either we use that coin (in any possible number) or we don’t” ?

While both the problem we’ve discussed and the coin change problem are optimization problems, and both can be solved using similar techniques like dynamic programming or recursive backtracking, the specific nature of the problems are quite different.

Coin Change Problem: The main problem is to find the fewest number of coins that you need to make up a certain amount of change. For this problem, we typically use a dynamic programming approach, where we build up an array of the minimum number of coins needed for each possible amount up to the target amount. This approach works because of the principle of optimality - an optimal solution to a problem contains within it optimal solutions to subproblems.
Minimum Number of Transactions to Settle Debt Problem: Here, the problem involves finding the minimum number of transactions needed to settle all debts. This is a significantly more complex problem. While it may seem similar to the coin change problem on the surface, the approach to solving it will be different due to its complexity. The process involves trying all permutations of paying off debts and then backtracking to find the minimum number of transactions.

The main similarity between the two problems is that they both involve finding a minimum (minimum number of coins, minimum number of transactions), but the specific strategies to get that minimum are quite different because of the nature of the problems. So it wouldn’t be accurate to say this is a variant of the coin change problem, as the problems involve different approaches to achieve their solutions.

This is a NP complete (Subset Sum Problem) problem.

Yes, you are correct. This problem is indeed NP-Complete and is a variant of the Subset Sum Problem.

The Subset Sum Problem is defined as: given a set of integers, is there a non-empty subset whose sum is zero? This is known to be a problem in the NP-Complete class, which means it’s a decision problem whose solution can be verified in polynomial time, but no known algorithm can solve all instances of the problem in polynomial time.

In the context of this problem, the Subset Sum Problem is applied to find a subset of the debts that sum up to zero. This means finding a subset of transactions that balances the debts (sums to zero) and requires the fewest number of transactions. Solving this problem involves enumerating all possible subsets, which is an exponential-time operation, hence why the problem is NP-Complete.

However, it’s worth noting that for this problem, the input size is small (the length of the transactions list is maximum 10), so we can solve the problem using exponential time algorithms such as backtracking or depth-first search. For larger input sizes, this wouldn’t be feasible.

https://www.win.tue.nl/~wstomv/publications/settling-debts-problems.pdf

This is a paper presenting several problems and their solutions related to settling debts between a group of friends. These friends lend each other money and at the end of the year, they want to settle their debts. The aim is to figure out the most efficient way to do this, meaning reducing the number of transactions and the total amount transferred.

The paper provides a series of problems that show different scenarios of money lending and asks how these debts can be optimally settled. For example, if Alice lends Bob 10 euro and Carol lends Dick 10 euro, one possible settlement could be Bob gives Carol 10 euro and Dick gives Alice 10 euro. The paper also introduces more complex scenarios where multiple debts have to be resolved between multiple people.

After each problem, the paper provides the solution, explaining why it’s the most efficient way to settle debts. The solutions are based on the balances of each person, which is the total amount they lent others minus the total amount they borrowed.

Finally, the paper presents a general scheme for settling all debts among N people with at most N - 1 transfers and the minimum total amount transferred. This scheme uses an algorithm that keeps selecting a person with a negative balance (debtor) and a person with a positive balance (creditor) and makes a transfer from the debtor to the creditor.

The paper concludes by noting that minimising the number of transfers is equivalent to maximising the number of zero-sum groups (groups that can settle among themselves). It also points out that it is not necessarily best to combine smaller groups first and that there is no shortcut to this problem; all partitions must be tried.