From 8bfab221c1354c4266cbddc1784c873651f57618 Mon Sep 17 00:00:00 2001 From: Lusina <12752833+BrianLusina@users.noreply.github.com> Date: Sun, 25 Jan 2026 13:22:04 +0300 Subject: [PATCH 1/2] feat(algorithms, backtracking): optimal account balancing --- .../optimal_account_balancing/README.md | 76 +++++++++++++++ .../optimal_account_balancing/__init__.py | 91 ++++++++++++++++++ .../optimal_account_balancing_example_1.png | Bin 0 -> 120841 bytes .../optimal_account_balancing_example_2.png | Bin 0 -> 113522 bytes .../optimal_account_balancing_example_3.png | Bin 0 -> 145616 bytes .../optimal_account_balancing_solution_1.png | Bin 0 -> 55788 bytes .../optimal_account_balancing_solution_10.png | Bin 0 -> 93079 bytes .../optimal_account_balancing_solution_11.png | Bin 0 -> 100671 bytes .../optimal_account_balancing_solution_12.png | Bin 0 -> 104211 bytes .../optimal_account_balancing_solution_13.png | Bin 0 -> 86973 bytes .../optimal_account_balancing_solution_2.png | Bin 0 -> 91794 bytes .../optimal_account_balancing_solution_3.png | Bin 0 -> 98547 bytes .../optimal_account_balancing_solution_4.png | Bin 0 -> 102483 bytes .../optimal_account_balancing_solution_5.png | Bin 0 -> 71455 bytes .../optimal_account_balancing_solution_6.png | Bin 0 -> 84147 bytes .../optimal_account_balancing_solution_7.png | Bin 0 -> 89526 bytes .../optimal_account_balancing_solution_8.png | Bin 0 -> 71056 bytes .../optimal_account_balancing_solution_9.png | Bin 0 -> 85341 bytes .../test_optimal_account_balancing.py | 36 +++++++ 19 files changed, 203 insertions(+) create mode 100644 algorithms/backtracking/optimal_account_balancing/README.md create mode 100644 algorithms/backtracking/optimal_account_balancing/__init__.py create mode 100644 algorithms/backtracking/optimal_account_balancing/images/examples/optimal_account_balancing_example_1.png create mode 100644 algorithms/backtracking/optimal_account_balancing/images/examples/optimal_account_balancing_example_2.png create mode 100644 algorithms/backtracking/optimal_account_balancing/images/examples/optimal_account_balancing_example_3.png create mode 100644 algorithms/backtracking/optimal_account_balancing/images/solutions/optimal_account_balancing_solution_1.png create mode 100644 algorithms/backtracking/optimal_account_balancing/images/solutions/optimal_account_balancing_solution_10.png create mode 100644 algorithms/backtracking/optimal_account_balancing/images/solutions/optimal_account_balancing_solution_11.png create mode 100644 algorithms/backtracking/optimal_account_balancing/images/solutions/optimal_account_balancing_solution_12.png create mode 100644 algorithms/backtracking/optimal_account_balancing/images/solutions/optimal_account_balancing_solution_13.png create mode 100644 algorithms/backtracking/optimal_account_balancing/images/solutions/optimal_account_balancing_solution_2.png create mode 100644 algorithms/backtracking/optimal_account_balancing/images/solutions/optimal_account_balancing_solution_3.png create mode 100644 algorithms/backtracking/optimal_account_balancing/images/solutions/optimal_account_balancing_solution_4.png create mode 100644 algorithms/backtracking/optimal_account_balancing/images/solutions/optimal_account_balancing_solution_5.png create mode 100644 algorithms/backtracking/optimal_account_balancing/images/solutions/optimal_account_balancing_solution_6.png create mode 100644 algorithms/backtracking/optimal_account_balancing/images/solutions/optimal_account_balancing_solution_7.png create mode 100644 algorithms/backtracking/optimal_account_balancing/images/solutions/optimal_account_balancing_solution_8.png create mode 100644 algorithms/backtracking/optimal_account_balancing/images/solutions/optimal_account_balancing_solution_9.png create mode 100644 algorithms/backtracking/optimal_account_balancing/test_optimal_account_balancing.py diff --git a/algorithms/backtracking/optimal_account_balancing/README.md b/algorithms/backtracking/optimal_account_balancing/README.md new file mode 100644 index 00000000..33c190da --- /dev/null +++ b/algorithms/backtracking/optimal_account_balancing/README.md @@ -0,0 +1,76 @@ +# Optimal Account Balancing + +Given a list of transactions, where each transaction is represented as transactions[i]=[fromi, toi, amounti], indicating +that the person fromi gave amounti to the person toi. + +Return the minimum number of transactions needed to settle all debts. + +## Constraints + +- 1 ≤ transactions.length ≤ 10 +- transactions[i].length == 3 +- 0 ≤ fromi, toi ≤ 10 +- 1 ≤ amounti ≤ 100 +- fromi ≠ toi + +## Examples + +![Example 1](./images/examples/optimal_account_balancing_example_1.png) +![Example 2](./images/examples/optimal_account_balancing_example_2.png) +![Example 3](./images/examples/optimal_account_balancing_example_3.png) + +## Solution + +This solution calculates the minimum number of transactions required to settle debts among a group of people based on an +initial set of transactions. It begins by computing each person’s net balance, indicating how much they owe or are owed. +Any individuals with a zero net balance are ignored, as they have no outstanding debts or credits. + +Once the net balances are calculated, the algorithm uses a recursive approach to settle the remaining balances. It pairs +individuals with opposite balances (i.e., those who owe with those who are owed) to reduce them to zero with the minimum +number of transactions. The solution explores various pairings and tracks each combination’s cost (number of transactions), +ensuring optimal results. If a particular path doesn’t minimize the transactions, it backtracks to explore alternative +pairings. + +Here’s the step-by-step implementation of the solution: + +- For each transaction, decrease the balance of the person who gave the money and increase the balance of the person who + received it. + +- Ignore people with a zero balance after all transactions, as they neither owe nor are owed. +- Use depth-first search (DFS) to recursively calculate the minimum number of transactions required to settle all balances. + - Base case: If the current person reaches n (number of people with non-zero balances), meaning all balances are zero + or settled, return 0. + - For each next person: + - If the current and next person have opposite sign balance: + - Temporarily add the current person’s balance to the next person’s balance. + - Recursively call DFS with the next person’s index. + - Track the minimum of the existing cost and the new cost calculated by DFS. + - **Backtrack**: Restore the balance to its original state by reversing the temporary addition. + - Return the minimum cost for all possible transaction paths. + +Let’s look at the following illustration to get a better understanding of the solution: + +![Solution 1](./images/solutions/optimal_account_balancing_solution_1.png) +![Solution 2](./images/solutions/optimal_account_balancing_solution_2.png) +![Solution 3](./images/solutions/optimal_account_balancing_solution_3.png) +![Solution 4](./images/solutions/optimal_account_balancing_solution_4.png) +![Solution 5](./images/solutions/optimal_account_balancing_solution_5.png) +![Solution 6](./images/solutions/optimal_account_balancing_solution_6.png) +![Solution 7](./images/solutions/optimal_account_balancing_solution_7.png) +![Solution 8](./images/solutions/optimal_account_balancing_solution_8.png) +![Solution 9](./images/solutions/optimal_account_balancing_solution_9.png) +![Solution 10](./images/solutions/optimal_account_balancing_solution_10.png) +![Solution 11](./images/solutions/optimal_account_balancing_solution_11.png) +![Solution 12](./images/solutions/optimal_account_balancing_solution_12.png) +![Solution 13](./images/solutions/optimal_account_balancing_solution_13.png) + +### Time Complexity + +The time complexity is O((n−1)!), where n is the number of persons. This complexity arises because, in the initial call +to dfs(0), there are n−1 possible choices for the next person, each leading to a recursive call to dfs(1), and so on. +This results in a chain of recursive calls multiplied by the choices at each level, giving a factorial pattern (n−1)!. + +### Space Complexity + +The space complexity is O(n), where n is the number of unique amounts. This is because both balance_map and balance can +hold at most n amounts. diff --git a/algorithms/backtracking/optimal_account_balancing/__init__.py b/algorithms/backtracking/optimal_account_balancing/__init__.py new file mode 100644 index 00000000..afdb1b58 --- /dev/null +++ b/algorithms/backtracking/optimal_account_balancing/__init__.py @@ -0,0 +1,91 @@ +from typing import List, DefaultDict +from collections import defaultdict + + +def min_transfers_dfs(transactions: List[List[int]]) -> int: + if not transactions: + return 0 + + # Net balances where the key is the person and the value is their balance + net_balances: DefaultDict[int, int] = defaultdict(int) + + # Populate net balances + for transaction in transactions: + sender, receiver, amount = transaction + net_balances[sender] -= amount + net_balances[receiver] += amount + + # Remove zero balances + non_zero_balances = [amt for amt in net_balances.values() if amt != 0] + number_of_non_zero_balances = len(non_zero_balances) + + if non_zero_balances == 0: + return 0 + + def dfs(current: int, balances: List[int]) -> int: + # Skip settled accounts, move to the next person with non-zero balaance + while current < number_of_non_zero_balances and balances[current] == 0: + current += 1 + # All accounts settled + if current >= number_of_non_zero_balances: + return 0 + + min_transactions = float("inf") + # Try to settle non_zero_balances[start] by paring with another opposite sign balance + for j in range(current + 1, number_of_non_zero_balances): + # One owes, the other is owed + if balances[current] * balances[j] < 0: + balances[j] += balances[current] + # recurse for remaining transactions after settling this one + min_transactions = min(min_transactions, 1 + dfs(current + 1, balances)) + # backtrack, undo the settlement + non_zero_balances[j] -= non_zero_balances[current] + + return min_transactions + + return dfs(0, non_zero_balances) + + +def min_transfers_backtrack(transactions: List[List[int]]) -> int: + if not transactions: + return 0 + + # Net balances where the key is the person and the value is their balance + net_balances: DefaultDict[int, int] = defaultdict(int) + + # Populate net balances + for transaction in transactions: + sender, receiver, amount = transaction + net_balances[sender] -= amount + net_balances[receiver] += amount + + # Remove zero balances + non_zero_balances = [amt for amt in net_balances.values() if amt != 0] + number_of_non_zero_balances = len(non_zero_balances) + + if non_zero_balances == 0: + return 0 + + def backtrack(start: int) -> int: + # Skip settled accounts + while start < number_of_non_zero_balances and non_zero_balances[start] == 0: + start += 1 + # All accounts settled + if start >= number_of_non_zero_balances: + return 0 + + min_transactions = float("inf") + # Try to settle non_zero_balances[start] by paring with another opposite sign balance + for j in range(start + 1, number_of_non_zero_balances): + # One owes, the other is owed + if non_zero_balances[start] * non_zero_balances[j] < 0: + non_zero_balances[j] += non_zero_balances[start] + # recurse for remaining transactions after settling this one + min_transactions = min(min_transactions, 1 + backtrack(start + 1)) + # backtrack, undo the settlement + non_zero_balances[j] -= non_zero_balances[start] + if non_zero_balances[j] + non_zero_balances[start] == 0: + break + return min_transactions + + return backtrack(0) diff --git a/algorithms/backtracking/optimal_account_balancing/images/examples/optimal_account_balancing_example_1.png b/algorithms/backtracking/optimal_account_balancing/images/examples/optimal_account_balancing_example_1.png new file mode 100644 index 0000000000000000000000000000000000000000..e9483a9f7ccae1cac782c757f8b50f35b929fc2e GIT binary patch literal 120841 zcmeFZbySsI*Ds6;Qi3#6(kb2DA>C2}0-Nq`5D*cRme{n?-Cas|cPiaTOP|GE-p}Lv z{r{aaUdGt$#dWQmYt8!2xd>KLkU~MiM}mQYL6Lqgt^xxC3xR=o5Q6X!Jef~^TMGk& zRAw$FrX(#UMxx|kYhrF?3Nh0PNFCh9>^q=Z*YV~c&8Ag 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z^outok(d{rL~@pZl~Gyg)V*XDovJo&d$*&}s37E&$ zgRIzJj?*t2E15tF=0rjG@!enF{P)>bFiT!c+{&Nz3x)T363I90>KONxzoP__Jgula$|D1&XoP?iI?EjyW c@crZz0@-N(d9r8b6!;@Aqb!{(_2l*c0V{Qg?f?J) literal 0 HcmV?d00001 diff --git a/algorithms/backtracking/optimal_account_balancing/test_optimal_account_balancing.py b/algorithms/backtracking/optimal_account_balancing/test_optimal_account_balancing.py new file mode 100644 index 00000000..c501f8fc --- /dev/null +++ b/algorithms/backtracking/optimal_account_balancing/test_optimal_account_balancing.py @@ -0,0 +1,36 @@ +import unittest +from typing import List +from parameterized import parameterized +from algorithms.backtracking.optimal_account_balancing import ( + min_transfers_backtrack, + min_transfers_dfs, +) + +OPTIMAL_ACCOUNT_BALANCING_TEST_CASES = [ + ([[0, 1, 10], [2, 0, 5]], 2), + ([[0, 1, 10], [1, 0, 1], [1, 2, 5], [2, 0, 5]], 1), + ([[0, 1, 40], [1, 0, 30], [1, 4, 10], [4, 5, 15], [0, 3, 10]], 3), + ([[0, 1, 10], [1, 2, 20], [2, 3, 30], [3, 4, 40], [4, 5, 50]], 5), + ([[0, 1, 10], [1, 2, 20], [2, 3, 30]], 3), + ([[0, 1, 20], [0, 1, 20], [2, 0, 30], [3, 4, 30], [4, 5, 5]], 4), + ([[0, 1, 25], [1, 2, 5], [2, 0, 15]], 2), + ([[0, 2, 5], [1, 2, 10], [1, 0, 30], [0, 1, 25]], 1), +] + + +class OptimalAccountBalancingTestCase(unittest.TestCase): + @parameterized.expand(OPTIMAL_ACCOUNT_BALANCING_TEST_CASES) + def test_min_transfers_backtrack( + self, transactions: List[List[int]], expected: int + ): + actual = min_transfers_backtrack(transactions) + self.assertEqual(expected, actual) + + @parameterized.expand(OPTIMAL_ACCOUNT_BALANCING_TEST_CASES) + def test_min_transfers_dfs(self, transactions: List[List[int]], expected: int): + actual = min_transfers_dfs(transactions) + self.assertEqual(expected, actual) + + +if __name__ == "__main__": + unittest.main() From 93ced3cfd93dcce03a32f72be031d34bba896c04 Mon Sep 17 00:00:00 2001 From: github-actions <${GITHUB_ACTOR}@users.noreply.github.com> Date: Sun, 25 Jan 2026 10:22:26 +0000 Subject: [PATCH 2/2] updating DIRECTORY.md --- DIRECTORY.md | 2 ++ 1 file changed, 2 insertions(+) diff --git a/DIRECTORY.md b/DIRECTORY.md index 77c530a6..0576d0df 100644 --- a/DIRECTORY.md +++ b/DIRECTORY.md @@ -27,6 +27,8 @@ * [Test Decode Message](https://github.com/BrianLusina/PythonSnips/blob/master/algorithms/backtracking/decode_message/test_decode_message.py) * Letter Combination * [Test Letter Combination](https://github.com/BrianLusina/PythonSnips/blob/master/algorithms/backtracking/letter_combination/test_letter_combination.py) + * Optimal Account Balancing + * [Test Optimal Account Balancing](https://github.com/BrianLusina/PythonSnips/blob/master/algorithms/backtracking/optimal_account_balancing/test_optimal_account_balancing.py) * Partition String * [Test Partition String](https://github.com/BrianLusina/PythonSnips/blob/master/algorithms/backtracking/partition_string/test_partition_string.py) * Permutations

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