Problem Description

This problem asks you to implement a simple banking system that can handle three types of transactions: transfers, deposits, and withdrawals.

You need to create a Bank class with the following features:

Initial Setup:

• The bank has n accounts numbered from 1 to n (note: 1-indexed)
• You're given an initial balance array where balance[i] represents the initial balance of account (i + 1)

Operations to Implement:

1. Constructor Bank(balance): Initialize the bank with the given balance array

2. transfer(account1, account2, money): Transfer money dollars from account1 to account2

   • Returns true if successful, false otherwise
   • The transaction is valid only if:
     • Both account numbers are between 1 and n
     • account1 has sufficient balance (≥ money)

3. deposit(account, money): Add money dollars to the specified account

   • Returns true if successful, false otherwise
   • The transaction is valid only if the account number is between 1 and n

4. withdraw(account, money): Remove money dollars from the specified account

   • Returns true if successful, false otherwise
   • The transaction is valid only if:
     • The account number is between 1 and n
     • The account has sufficient balance (≥ money)

Key Points:

• Account numbers are 1-indexed (accounts are numbered 1 to n)
• The balance array is 0-indexed (standard array indexing)
• All invalid transactions should return false without modifying any balances
• Valid transactions should update the balances and return true

Intuition

The key insight is that this is a straightforward simulation problem - we need to maintain the state of all bank accounts and update them based on valid transactions.

Since we need to track the balance of multiple accounts and access them by account number, an array is the natural choice for storing balances. The main challenge is handling the index conversion correctly: accounts are numbered from 1 to n, but arrays are 0-indexed. So account k corresponds to balance[k-1].

For each operation, we follow a simple pattern:

1. First validate - check if the transaction can proceed
2. Then execute - only modify balances if validation passes

The validation logic follows directly from the requirements:

• For any operation, the account number must be valid (between 1 and n)
• For operations that remove money (withdraw and transfer), we must ensure sufficient balance exists

Why check account validity as account > n instead of account < 1 || account > n? Since account numbers are positive integers in this context, we only need to check the upper bound. Invalid negative or zero account numbers would naturally fail this check.

The beauty of this solution is its simplicity - we don't need complex data structures or algorithms. Just an array to store balances and careful validation before each operation. Each method performs constant time operations: array access, comparison, and basic arithmetic, making the solution efficient.

The separation of validation and execution also makes the code clean and maintainable - if a transaction fails validation, we return false immediately without touching any balances, ensuring data consistency.

Solution Approach

The implementation uses a simple simulation approach with an array to store account balances.

Data Structure:

• Use an array balance to store the balance of each account
• Store the total number of accounts n as a member variable for quick validation

Implementation Details:

1. Constructor (__init__):

   • Store the input balance array directly as a member variable
   • Calculate and store n = len(balance) for boundary checking
   • This gives us O(1) access to any account's balance

2. Transfer Function:

   • Validation: Check if account1 > n or account2 > n (invalid account numbers)
   • Check if balance[account1 - 1] < money (insufficient funds)
   • If any validation fails, return false
   • Otherwise, execute the transfer:
     • balance[account1 - 1] -= money (deduct from source)
     • balance[account2 - 1] += money (add to destination)
   • Return true after successful transfer

3. Deposit Function:

   • Validation: Check if account > n (invalid account number)
   • If validation fails, return false
   • Otherwise, add money: balance[account - 1] += money
   • Return true after successful deposit

4. Withdraw Function:

   • Validation: Check if account > n (invalid account number)
   • Check if balance[account - 1] < money (insufficient funds)
   • If any validation fails, return false
   • Otherwise, deduct money: balance[account - 1] -= money
   • Return true after successful withdrawal

Index Conversion: The critical detail is converting between 1-indexed account numbers and 0-indexed array positions. For account number k, we access balance[k - 1] in the array.

Time Complexity:

• All operations (transfer, deposit, withdraw) run in O(1) time
• Each operation performs a fixed number of array accesses and comparisons

Space Complexity:

• O(n) space to store the balance array where n is the number of accounts

Example Walkthrough

Let's walk through a small example to illustrate how the banking system works.

Initial Setup:

balance = [100, 50, 200]

This creates a bank with 3 accounts:

• Account 1: $100 (stored at index 0)
• Account 2: $50 (stored at index 1)
• Account 3: $200 (stored at index 2)

Operation 1: Transfer $30 from Account 1 to Account 3

transfer(1, 3, 30)

• Validation:
  • Is account 1 valid? 1 ≤ 3 ✓
  • Is account 3 valid? 3 ≤ 3 ✓
  • Does account 1 have ≥ $30? balance[0] = 100 ≥ 30 ✓
• Execution:
  • balance[0] = 100 - 30 = 70
  • balance[2] = 200 + 30 = 230
• Result: Returns true
• New balances: [70, 50, 230]

Operation 2: Withdraw $80 from Account 2

withdraw(2, 80)

• Validation:
  • Is account 2 valid? 2 ≤ 3 ✓
  • Does account 2 have ≥ $80? balance[1] = 50 < 80 ✗
• Result: Returns false (insufficient funds)
• Balances unchanged: [70, 50, 230]

Operation 3: Deposit $25 to Account 2

deposit(2, 25)

• Validation:
  • Is account 2 valid? 2 ≤ 3 ✓
• Execution:
  • balance[1] = 50 + 25 = 75
• Result: Returns true
• New balances: [70, 75, 230]

Operation 4: Transfer $100 from Account 4 to Account 1

transfer(4, 1, 100)

• Validation:
  • Is account 4 valid? 4 > 3 ✗ (invalid account)
• Result: Returns false (invalid account)
• Balances unchanged: [70, 75, 230]

This example demonstrates the key aspects:

1. Index conversion: Account 1 maps to index 0, Account 2 to index 1, etc.
2. Validation before execution: Each operation checks validity first
3. Balance preservation: Failed operations don't modify any balances
4. Successful updates: Valid operations update the appropriate balances and return true

Solution Implementation

Time and Space Complexity

Time Complexity: Each operation (transfer, deposit, withdraw) has a time complexity of O(1). All operations perform a constant number of operations regardless of the input size - they only involve simple comparisons, arithmetic operations, and direct array access using indices, which all take constant time.

Space Complexity: The overall space complexity is O(n), where n is the length of the balance list. The class stores the balance list which contains n elements representing account balances. The additional variables self.n and temporary variables within methods use O(1) space, so the dominant factor is the storage of the balance list itself.

Learn more about how to find time and space complexity quickly.

Common Pitfalls

1. Forgetting to Check Lower Bound for Account Numbers

The current implementation only checks if account numbers exceed n but doesn't verify they're at least 1. Since accounts are 1-indexed, account number 0 or negative numbers would cause incorrect array access.

Problem Example:

bank = Bank([100, 200])
bank.withdraw(0, 50)  # Would access balance[-1], which is valid but wrong!
bank.transfer(-1, 1, 50)  # Would access balance[-2], also valid but wrong!

Solution: Add lower bound validation to all methods:

def transfer(self, account1: int, account2: int, money: int) -> bool:
    # Check both upper AND lower bounds
    if (account1 < 1 or account1 > self.n or 
        account2 < 1 or account2 > self.n or 
        self.balance[account1 - 1] < money):
        return False
    # ... rest of the code

def deposit(self, account: int, money: int) -> bool:
    if account < 1 or account > self.n:
        return False
    # ... rest of the code

def withdraw(self, account: int, money: int) -> bool:
    if account < 1 or account > self.n or self.balance[account - 1] < money:
        return False
    # ... rest of the code

2. Not Validating Negative Money Amounts

The implementation doesn't check if the money amount is negative. While some systems might want to allow this (negative deposit = withdrawal), it's typically a bug and can lead to unexpected behavior.

Problem Example:

bank = Bank([100, 200])
bank.deposit(1, -50)  # Would actually subtract money!
bank.withdraw(1, -50)  # Would actually add money!
bank.transfer(1, 2, -50)  # Would transfer money backwards!

Solution: Add validation for non-negative amounts:

def transfer(self, account1: int, account2: int, money: int) -> bool:
    if money < 0:
        return False
    # ... rest of validation
  
def deposit(self, account: int, money: int) -> bool:
    if money < 0:
        return False
    # ... rest of validation

def withdraw(self, account: int, money: int) -> bool:
    if money < 0:
        return False
    # ... rest of validation

3. Integer Overflow Concerns

In languages with fixed-size integers or when dealing with very large amounts, adding money to an account could cause integer overflow.

Problem Example:

# If using a language with fixed integer size
bank = Bank([2**31 - 100])  # Near max int
bank.deposit(1, 200)  # Could overflow in some languages

Solution: Check for potential overflow before performing operations:

def deposit(self, account: int, money: int) -> bool:
    if account < 1 or account > self.n:
        return False
  
    # Check for potential overflow (Python handles this automatically, 
    # but other languages might need explicit checks)
    if self.balance[account - 1] > sys.maxsize - money:
        return False  # Would overflow
  
    self.balance[account - 1] += money
    return True

4. Modifying the Original Balance Array

The constructor stores a reference to the input array rather than creating a copy. If the caller modifies the original array after creating the Bank object, it could corrupt the bank's state.

Problem Example:

initial_balance = [100, 200]
bank = Bank(initial_balance)
initial_balance[0] = 1000  # This modifies the bank's internal state!

Solution: Create a defensive copy of the input array:

def __init__(self, balance: List[int]):
    self.balance = balance.copy()  # Create a copy instead of using reference
    self.n = len(self.balance)