Minimum Moves to Pick K Ones

Description

You are given a 0-indexed binary array nums of length n, a positive integer k and a non-negative integer maxChanges.

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

• Select any index j != dylanIndex 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 == dylanIndex, Dylan picks up the one after this move and nums[y] becomes 0.

Return the minimum number of moves required by Dylan 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: Dylan can pick up 3 ones in 3 moves, if Dylan performs the following actions in each move when standing at dylanIndex == 1:

•  At the start of the game Dylan picks up the one and nums[1] becomes 0. nums becomes [1,**0**,1,0,0,1,1,0,0,1].
• Select j == 2 and perform an action of the first type. nums becomes [1,**0**,1,0,0,1,1,0,0,1]
• Select x == 2 and y == 1, and perform an action of the second type. nums becomes [1,**1**,0,0,0,1,1,0,0,1]. As y == dylanIndex, Dylan picks up the one and nums becomes [1,**0**,0,0,0,1,1,0,0,1].
• Select x == 0 and y == 1, and perform an action of the second type. nums becomes [0,**1**,0,0,0,1,1,0,0,1]. As y == dylanIndex, Dylan picks up the one and nums becomes [0,**0**,0,0,0,1,1,0,0,1].

Note that it may be possible for Dylan 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: Dylan can pick up 2 ones in 4 moves, if Dylan performs the following actions in each move when standing at dylanIndex == 0:

• Select j == 1 and perform an action of the first type. nums becomes [**0**,1,0,0].
• Select x == 1 and y == 0, and perform an action of the second type. nums becomes [**1**,0,0,0]. As y == dylanIndex, Dylan picks up the one and nums becomes [**0**,0,0,0].
• Select j == 1 again and perform an action of the first type. nums becomes [**0**,1,0,0].
• Select x == 1 and y == 0 again, and perform an action of the second type. nums becomes [**1**,0,0,0]. As y == dylanIndex, Dylan picks up the one and nums becomes [**0**,0,0,0].

Constraints:

• 2 <= n <= 105
• 0 <= nums[i] <= 1
• 1 <= k <= 105
• 0 <= maxChanges <= 105
• maxChanges + sum(nums) >= k

Code

Time Complexity: $O()$, Space Complexity: $O()$

Source

• Minimum Moves to Pick K Ones - LeetCode