Given an integer array nums of 2n integers, group these integers into n pairs (a1, b1), (a2, b2), ..., (an, bn) such that the sum of min(ai, bi) for all i is maximized. Return the maximized sum.

Examples

Example 1:

Input: nums = [1,4,3,2]
Output: 4
Explanation: All possible pairings (ignoring the ordering of elements) are:
1. (1, 4), (2, 3) -> min(1,4) + min(2,3) = 1 + 2 = 3
2. (1, 3), (2, 4) -> min(1,3) + min(2,4) = 1 + 2 = 3
3. (1, 2), (3, 4) -> min(1,2) + min(3,4) = 1 + 3 = 4
So the maximum possible sum is 4.

Example 2:

Input: nums = [6,2,6,5,1,2]
Output: 9
Explanation: The optimal pairing is (2, 1), (2, 5), (6, 6). min(2, 1) + min(2, 5) + min(6, 6) = 1 + 2 + 6 = 9.

Constraints

  • 1 <= n <= 10^4
  • nums.length == 2 * n
  • -10^4 <= nums[i] <= 10^4

Thinking Process

Given an integer array nums of 2n integers, group these integers into n pairs (a1, b1), (a2, b2), ..., (an, bn) such that the sum of min(ai, bi) for all i is maximized. Return the maximized sum.

  • Greedy works when local optimal choices lead to global optimum.
  • Often sort first to make the greedy choice obvious.
  • Prove or sanity-check: would swapping two choices ever help?
Greedy choice pick locally best after sorting

Common Approaches

Typical techniques for this pattern:

Approach Time Space Notes
Sort + greedy (this problem) O(n log n) O(1) Interval scheduling, assignment
Local greedy choice O(n) O(1) Jump game, gas station
Greedy + heap O(n log n) O(n) Merge streams, room allocation
Exchange argument O(n) O(1) Prove greedy choice is safe

Solution

Solution: Greedy with Sorting

class Solution:
    def arrayPairSum(self, nums):
        nums.sort()
        total = 0

        for i in range(0, len(nums), 2):
            total += nums[i]

        return total

Solution Explanation

Approach: Sort + greedy (this problem)

Key idea: Given an integer array nums of 2n integers, group these integers into n pairs (a1, b1), (a2, b2), ..., (an, bn) such that the sum of min(ai, bi) for all i is maximized. Return the maximized sum.

How the code works:

  • Greedy works when local optimal choices lead to global optimum.
  • Often sort first to make the greedy choice obvious.
  • Prove or sanity-check: would swapping two choices ever help?

Walkthrough — input nums = [1,4,3,2], expected output 4:

All possible pairings (ignoring the ordering of elements) are:

  1. (1, 4), (2, 3) -> min(1,4) + min(2,3) = 1 + 2 = 3
  2. (1, 3), (2, 4) -> min(1,3) + min(2,4) = 1 + 2 = 3
  3. (1, 2), (3, 4) -> min(1,2) + min(3,4) = 1 + 3 = 4 So the maximum possible sum is 4.

Time: O(n log n) where n is the length of the array · Space: O(1)

Algorithm Explanation:

  1. Sort Array (Line 4):
    • Sort nums in ascending order
    • This ensures smaller numbers come first
  2. Sum Minimums (Lines 5-8):
    • Initialize: sum = 0
    • For each pair: Iterate with step 2 (i += 2)
      • After sorting, pairs are (nums[i], nums[i+1])
      • The minimum in each pair is nums[i] (smaller element)
      • Add nums[i] to sum
  3. Return (Line 9):
    • Return the accumulated sum

Example Walkthrough:

Example 1: nums = [1,4,3,2]

After sorting:

Original: [1, 4, 3, 2]
Sorted:   [1, 2, 3, 4]

Pairing and summing:

Pair 1: (1, 2) → min = 1
Pair 2: (3, 4) → min = 3
Sum: 1 + 3 = 4

Execution:

Initial: sum = 0

i=0: nums[0]=1
  sum = 0 + 1 = 1

i=2: nums[2]=3
  sum = 1 + 3 = 4

Result: 4

Example 2: nums = [6,2,6,5,1,2]

After sorting:

Original: [6, 2, 6, 5, 1, 2]
Sorted:   [1, 2, 2, 5, 6, 6]

Pairing and summing:

Pair 1: (1, 2) → min = 1
Pair 2: (2, 5) → min = 2
Pair 3: (6, 6) → min = 6
Sum: 1 + 2 + 6 = 9

Execution:

Initial: sum = 0

i=0: nums[0]=1
  sum = 0 + 1 = 1

i=2: nums[2]=2
  sum = 1 + 2 = 3

i=4: nums[4]=6
  sum = 3 + 6 = 9

Result: 9

Algorithm Breakdown

Why Greedy Works

The greedy strategy is optimal because:

  1. Minimize Loss: By pairing smallest with second smallest, we minimize the “waste” from the larger number
  2. Maximize Minimums: This strategy maximizes the sum of minimums
  3. No Better Pairing: Any other pairing would result in a smaller sum

Mathematical Proof:

  • If we pair (a, b) where a < b, we get a in the sum
  • If we pair (a, c) where c > b, we still get a, but b might be paired with a smaller number, reducing the sum
  • Therefore, pairing consecutive elements after sorting is optimal

Key Insight: Pair Consecutive Elements

After sorting, the optimal pairing is:

  • (nums[0], nums[1])
  • (nums[2], nums[3])
  • (nums[4], nums[5])
  • …

This ensures:

  • Each pair has the smallest possible difference
  • The sum of minimums is maximized

Why Not Pair Differently?

Counter-example:

nums = [1, 2, 3, 4]

Pairing 1 (optimal): (1,2), (3,4)
  Sum: min(1,2) + min(3,4) = 1 + 3 = 4

Pairing 2: (1,3), (2,4)
  Sum: min(1,3) + min(2,4) = 1 + 2 = 3

Pairing 3: (1,4), (2,3)
  Sum: min(1,4) + min(2,3) = 1 + 2 = 3

Pairing consecutive elements gives the maximum sum.

Time & Space Complexity

  • Time Complexity: O(n log n) where n is the length of the array
    • Dominated by sorting: O(n log n)
    • Iteration: O(n) - single pass with step 2
    • Total: O(n log n)
  • Space Complexity: O(1)
    • Only using a few variables
    • Sorting is in-place (or O(n) if not in-place, but typically O(1) extra space)

Key Points

  1. Sort First: Essential for greedy strategy
  2. Greedy Pairing: Pair consecutive elements after sorting
  3. Sum Minimums: Add the first element of each pair
  4. Optimal: Greedy approach finds maximum sum
  5. Simple: Straightforward implementation after sorting

Common Mistakes

  1. All same values: [1,1,1,1] → return 2 (1+1)
  2. Negative numbers: [-1,-2,-3,-4] → return -6 (-1 + -3)
  3. Mixed signs: [-1,2,-3,4] → return -1 (-3 + 2)
  4. Two elements: [1,2] → return 1

  5. Large range: [1,100,2,99] → return 3 (1+2)

  6. Not sorting: Trying to pair without sorting
  7. Wrong pairing: Not pairing consecutive elements
  8. Wrong sum: Summing both elements instead of minimum
  9. Index error: Off-by-one errors in loop
  10. Negative numbers: Forgetting to handle negative numbers correctly

Tags

Array, Greedy, Sorting, Easy

Key Takeaways

  • Greedy works when local optimal choices lead to global optimum.
  • Often sort first to make the greedy choice obvious.
  • Prove or sanity-check: would swapping two choices ever help?

References

Template Reference