3086 - Minimum Moves to Pick K Ones.

Hard

You are given a binary array nums of length n, a positive integer k and a non-negative integer maxChanges.

Alice plays a game, where the goal is for Alice to pick up k ones from nums using the minimum number of moves. When the game starts, Alice picks up any index aliceIndex in the range [0, n - 1] and stands there. If nums[aliceIndex] == 1 , Alice picks up the one and nums[aliceIndex] becomes 0(this does not count as a move). After this, Alice can make any number of moves ( including zero ) where in each move Alice must perform exactly one of the following actions:

• Select any index j != aliceIndex such that nums[j] == 0 and set nums[j] = 1. This action can be performed at most maxChanges times.

• Select any two adjacent indices x and y (|x - y| == 1) such that nums[x] == 1, nums[y] == 0, then swap their values (set nums[y] = 1 and nums[x] = 0). If y == aliceIndex, Alice picks up the one after this move and nums[y] becomes 0.

Return the minimum number of moves required by Alice to pick exactly k ones.

Example 1:

Input: nums = 1,1,0,0,0,1,1,0,0,1, k = 3, maxChanges = 1

Output: 3

Explanation: Alice can pick up 3 ones in 3 moves, if Alice performs the following actions in each move when standing at aliceIndex == 1:

• At the start of the game Alice picks up the one and nums[1] becomes 0. nums becomes <code>1, **<ins>1</ins>** ,1,0,0,1,1,0,0,1</code>.

• Select j == 2 and perform an action of the first type. nums becomes <code>1, **<ins>0</ins>** ,1,0,0,1,1,0,0,1</code>

• Select x == 2 and y == 1, and perform an action of the second type. nums becomes <code>1, **<ins>1</ins>** ,0,0,0,1,1,0,0,1</code>. As y == aliceIndex, Alice picks up the one and nums becomes <code>1, **<ins>0</ins>** ,0,0,0,1,1,0,0,1</code>.

• Select x == 0 and y == 1, and perform an action of the second type. nums becomes <code>0, **<ins>1</ins>** ,0,0,0,1,1,0,0,1</code>. As y == aliceIndex, Alice picks up the one and nums becomes <code>0, **<ins>0</ins>** ,0,0,0,1,1,0,0,1</code>.

Note that it may be possible for Alice to pick up 3 ones using some other sequence of 3 moves.

Example 2:

Input: nums = 0,0,0,0, k = 2, maxChanges = 3

Output: 4

Explanation: Alice can pick up 2 ones in 4 moves, if Alice performs the following actions in each move when standing at aliceIndex == 0:

• Select j == 1 and perform an action of the first type. nums becomes <code>**<ins>0</ins>** ,1,0,0</code>.

• Select x == 1 and y == 0, and perform an action of the second type. nums becomes <code>**<ins>1</ins>** ,0,0,0</code>. As y == aliceIndex, Alice picks up the one and nums becomes <code>**<ins>0</ins>** ,0,0,0</code>.

• Select j == 1 again and perform an action of the first type. nums becomes <code>**<ins>0</ins>** ,1,0,0</code>.

• Select x == 1 and y == 0 again, and perform an action of the second type. nums becomes <code>**<ins>1</ins>** ,0,0,0</code>. As y == aliceIndex, Alice picks up the one and nums becomes <code>**<ins>0</ins>** ,0,0,0</code>.

Constraints:

• <code>2 <= n <= 10⁵</code>

• 0 <= nums[i] <= 1

• <code>1 <= k <= 10⁵</code>

• <code>0 <= maxChanges <= 10⁵</code>

• maxChanges + sum(nums) >= k