Solution Approach

Let's walk through the implementation step by step:

Step 1: Extract decimal parts and calculate bounds

mi = 0 arr = [] for p in prices:  p = float(p)  mi += int(p) # Sum of all floors  if d := p - int(p): # Extract decimal part  arr.append(d)

• Convert each price string to float
• mi accumulates the sum of all floored values (minimum possible sum)
• arr collects all non-zero decimal parts (these are the prices we can choose to round up)

Step 2: Check feasibility

if not mi <= target <= mi + len(arr):  return "-1"

• The minimum sum is mi (all floored)
• The maximum sum is mi + len(arr) (all non-integers rounded up)
• If target is outside this range, it's impossible to achieve

Step 3: Determine how many to round up

d = target - mi

• Since we start with all floored (sum = mi), we need to round up exactly d numbers to reach the target
• Each rounding up increases the sum by 1

Step 4: Greedy selection

arr.sort(reverse=True) ans = d - sum(arr[:d]) + sum(arr[d:])

• Sort decimal parts in descending order
• Round up the d largest decimal parts (first d elements)
• Calculate total error:
  • For rounded up numbers (arr[:d]): error = 1 - decimal_part for each
  • Sum of errors = d - sum(arr[:d])
  • For rounded down numbers (arr[d:]): error = decimal_part for each
  • Sum of errors = sum(arr[d:])
  • Total error = d - sum(arr[:d]) + sum(arr[d:])

Step 5: Format result

return f'{ans:.3f}'

• Return the minimum error formatted to 3 decimal places

The algorithm runs in O(n log n) time due to sorting, with O(n) space for storing decimal parts.

Example Walkthrough

Let's walk through the solution with prices = ["2.700", "3.400", "5.900"] and target = 11.

Step 1: Extract decimal parts and calculate bounds

• Convert to floats: [2.7, 3.4, 5.9]
• Extract floors and decimal parts:
  • 2.7 → floor = 2, decimal = 0.7
  • 3.4 → floor = 3, decimal = 0.4
  • 5.9 → floor = 5, decimal = 0.9
• mi (sum of floors) = 2 + 3 + 5 = 10
• arr (decimal parts) = [0.7, 0.4, 0.9]

Step 2: Check feasibility

• Minimum possible sum = 10 (all floored)
• Maximum possible sum = 10 + 3 = 13 (all ceiled)
• Target = 11 is within [10, 13] ✓

Step 3: Determine how many to round up

• We need: target - mi = 11 - 10 = 1
• So we must round up exactly 1 number

Step 4: Greedy selection

• Sort decimal parts descending: [0.9, 0.7, 0.4]
• Round up the 1 largest decimal (0.9):
  • 5.9 → 6 (round up), error = |6 - 5.9| = 0.1
• Keep others floored:
  • 2.7 → 2 (round down), error = |2 - 2.7| = 0.7
  • 3.4 → 3 (round down), error = |3 - 3.4| = 0.4

Calculate total error:

• Using the formula: d - sum(arr[:d]) + sum(arr[d:])
• = 1 - sum([0.9]) + sum([0.7, 0.4])
• = 1 - 0.9 + 1.1
• = 1.2

Step 5: Format result

• Return "1.200"

Verification:

• Rounded values: [2, 3, 6]
• Sum: 2 + 3 + 6 = 11 ✓ (matches target)
• Total error: 0.7 + 0.4 + 0.1 = 1.2 ✓

Time and Space Complexity

Time Complexity: O(n log n) where n is the length of the prices array.

• Converting each price string to float and calculating the floor value takes O(n) time
• Building the array of decimal parts takes O(n) time
• Sorting the decimal parts array takes O(k log k) time where k ≤ n is the number of non-integer prices
• Computing the sum of subarrays takes O(k) time
• Since k ≤ n, the dominant operation is sorting, giving us O(n log n) overall

Space Complexity: O(n)

• The arr list stores at most n decimal parts, requiring O(n) space
• Other variables (mi, d, ans) use constant space O(1)
• The sorting operation may use O(log n) space for the recursive call stack (depending on the implementation)
• Overall space complexity is O(n)

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

Common Pitfalls

1. Floating Point Precision Issues

One of the most common pitfalls in this problem is dealing with floating-point arithmetic inaccuracies. When converting strings to floats and performing calculations, small precision errors can accumulate.

Problem Example:

price = "0.1" p = float(price) # 0.1 floor_val = int(p) # 0 decimal = p - floor_val # Might be 0.09999999999999998 instead of 0.1

Solution: Use the decimal module for precise arithmetic or round intermediate calculations appropriately:

from decimal import Decimal, ROUND_HALF_UP  price = Decimal("0.1") floor_val = int(price) decimal_part = price - floor_val # Exactly 0.1

2. Integer Prices Handling

The code might incorrectly handle prices that are already integers (like "5.000" or "3").

Problem Example:

price = "5.000" p = float(price) # 5.0 decimal = p - int(p) # 0.0 if decimal: # This condition fails, but "5.000" could still be in prices array  arr.append(decimal)

Solution: Be explicit about checking for near-zero decimals:

decimal_part = price_float - floor_value if decimal_part > 1e-9: # Use epsilon comparison  decimal_parts.append(decimal_part)

3. Incorrect Target Feasibility Check

A subtle bug can occur when all prices are integers. The maximum sum calculation assumes we can only increase the sum by the count of non-integer prices.

Problem Example:

prices = ["1", "2", "3"] # All integers target = 7 # decimal_parts = [] (empty) # min_sum = 6, max_sum = 6 + 0 = 6 # Returns "-1" even though target 7 might seem unreachable

This is actually correct behavior, but developers might misunderstand that integer prices cannot be "rounded up" since ceil(integer) = integer.

4. String Formatting Edge Cases

The final formatting might produce unexpected results for very small errors.

Problem Example:

total_error = 0.0005 return f'{total_error:.3f}' # Returns "0.001" due to rounding

Solution: Consider whether you want banker's rounding or always round down for consistency:

import math # To always round down to 3 decimals: truncated = math.floor(total_error * 1000) / 1000 return f'{truncated:.3f}'

5. Empty Decimal Parts Array

When all prices are integers, sorting an empty array and slicing it could cause issues in some implementations.

Problem Example:

decimal_parts = [] decimal_parts.sort(reverse=True) # OK, but empty sum(decimal_parts[:0]) # Returns 0, which is correct sum(decimal_parts[0:]) # Returns 0, which is correct

While Python handles this gracefully, it's worth adding an early check:

if not decimal_parts and target != min_sum:  return "-1" if not decimal_parts:  return "0.000"