Maximize memory capacity with primary-backup servers

You are helping to optimize the capacity of a cloud system.

There are n servers, where n is always even. The memory capacity of the i-th server is given by an integer array memory of length n.

When there are n = 2 * x servers:

• Exactly x servers will be chosen as primary servers.
• The remaining x servers will be backup servers.

For every primary server P, there must exist a distinct backup server B such that:

memory[B] >= memory[P]

(Each server can be used as either primary or backup in exactly one such pair, so the n servers are partitioned into x (primary, backup) pairs.)

The system memory capacity is defined as the sum of the memory capacities of all primary servers:

capacity = sum(memory[P] over all primary servers P)

Your task is to determine the maximum possible system memory capacity that can be obtained by:

• Partitioning the n servers into n/2 disjoint (primary, backup) pairs, and
• Respecting the constraint memory[backup] >= memory[primary] within each pair.

Input

• An even integer n (number of servers).
• An integer array memory of length n, where memory[i] is the memory capacity of the i-th server.

You may assume:

• 2 <= n <= 2 * 10^5, and n is even.
• 1 <= memory[i] <= 10^9.

Output

• Return a single integer: the maximum possible system memory capacity.

Complexity requirement

Design an algorithm efficient for n up to about 2 * 10^5 (i.e., significantly better than O(n^2) time).

Maximize Primary Server Memory Capacity

Pair servers so each backup has memory at least its primary, and maximize the sum of primary memories.

Examples

Example 1

Input: ([1,2,3,4],)
Output: 4

Explanation: Pair adjacent sorted servers.

Example 2

Input: ([1,1,100,100],)
Output: 101

Explanation: Large equal backups.

Hints

1. Choose a representation that makes the requested operation direct.
2. Handle empty inputs and boundary cases first.