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3086. Minimum Moves to Pick K Ones

Description

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 [1,0,0,0,0,1,1,0,0,1].
  • 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 == aliceIndex, Alice 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 == aliceIndex, Alice picks up the one and nums becomes [0,0,0,0,0,1,1,0,0,1].

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 [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 == aliceIndex, Alice 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 == aliceIndex, Alice 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

Solutions

We consider enumerating Alice's standing position $i$. For each $i$, we follow the strategy below:

  • First, if the number at position $i$ is $1$, we can directly pick up a $1$ without needing any moves.
  • Then, we pick up the number $1$ from both sides of position $i$, which is action $2$, i.e., move the $1$ from position $i-1$ to position $i$, then pick it up; move the $1$ from position $i+1$ to position $i$, then pick it up. Each pick up of a $1$ requires $1$ move.
  • Next, we maximize the conversion of $0$s at positions $i-1$ or $i+1$ to $1$s using action $1$, then move them to position $i$ using action $2$ to pick them up. This continues until the number of $1$s picked up reaches $k$ or the number of times action $1$ is used reaches $\textit{maxChanges}$. Assuming the number of times action $1$ is used is $c$, then a total of $2c$ moves are needed.
  • After utilizing action $1$, if the number of $1$s picked up has not reached $k$, we need to continue considering moving $1$s to position $i$ from the intervals $[1,..i-2]$ and $[i+2,..n]$ using action $2$ to pick them up. We can use binary search to determine the size of this interval so that the number of $1$s picked up reaches $k$. Specifically, we binary search for an interval size $d$, then within the intervals $[i-d,..i-2]$ and $[i+2,..i+d]$, we perform action $2$ to move $1$s to position $i$ for pickup. If the number of $1$s picked up reaches $k$, we update the answer.

The time complexity is $O(n \times \log n)$, and the space complexity is $O(n)$. Here, $n$ is the length of the array $\textit{nums}$.

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class Solution:
    def minimumMoves(self, nums: List[int], k: int, maxChanges: int) -> int:
        n = len(nums)
        cnt = [0] * (n + 1)
        s = [0] * (n + 1)
        for i, x in enumerate(nums, 1):
            cnt[i] = cnt[i - 1] + x
            s[i] = s[i - 1] + i * x
        ans = inf
        max = lambda x, y: x if x > y else y
        min = lambda x, y: x if x < y else y
        for i, x in enumerate(nums, 1):
            t = 0
            need = k - x
            for j in (i - 1, i + 1):
                if need > 0 and 1 <= j <= n and nums[j - 1] == 1:
                    need -= 1
                    t += 1
            c = min(need, maxChanges)
            need -= c
            t += c * 2
            if need <= 0:
                ans = min(ans, t)
                continue
            l, r = 2, max(i - 1, n - i)
            while l <= r:
                mid = (l + r) >> 1
                l1, r1 = max(1, i - mid), max(0, i - 2)
                l2, r2 = min(n + 1, i + 2), min(n, i + mid)
                c1 = cnt[r1] - cnt[l1 - 1]
                c2 = cnt[r2] - cnt[l2 - 1]
                if c1 + c2 >= need:
                    t1 = c1 * i - (s[r1] - s[l1 - 1])
                    t2 = s[r2] - s[l2 - 1] - c2 * i
                    ans = min(ans, t + t1 + t2)
                    r = mid - 1
                else:
                    l = mid + 1
        return ans

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class Solution {
    public long minimumMoves(int[] nums, int k, int maxChanges) {
        int n = nums.length;
        int[] cnt = new int[n + 1];
        long[] s = new long[n + 1];
        for (int i = 1; i <= n; ++i) {
            cnt[i] = cnt[i - 1] + nums[i - 1];
            s[i] = s[i - 1] + i * nums[i - 1];
        }
        long ans = Long.MAX_VALUE;
        for (int i = 1; i <= n; ++i) {
            long t = 0;
            int need = k - nums[i - 1];
            for (int j = i - 1; j <= i + 1; j += 2) {
                if (need > 0 && 1 <= j && j <= n && nums[j - 1] == 1) {
                    --need;
                    ++t;
                }
            }
            int c = Math.min(need, maxChanges);
            need -= c;
            t += c * 2;
            if (need <= 0) {
                ans = Math.min(ans, t);
                continue;
            }
            int l = 2, r = Math.max(i - 1, n - i);
            while (l <= r) {
                int mid = (l + r) >> 1;
                int l1 = Math.max(1, i - mid), r1 = Math.max(0, i - 2);
                int l2 = Math.min(n + 1, i + 2), r2 = Math.min(n, i + mid);
                int c1 = cnt[r1] - cnt[l1 - 1];
                int c2 = cnt[r2] - cnt[l2 - 1];
                if (c1 + c2 >= need) {
                    long t1 = 1L * c1 * i - (s[r1] - s[l1 - 1]);
                    long t2 = s[r2] - s[l2 - 1] - 1L * c2 * i;
                    ans = Math.min(ans, t + t1 + t2);
                    r = mid - 1;
                } else {
                    l = mid + 1;
                }
            }
        }
        return ans;
    }
}

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class Solution {
public:
    long long minimumMoves(vector<int>& nums, int k, int maxChanges) {
        int n = nums.size();
        vector<int> cnt(n + 1, 0);
        vector<long long> s(n + 1, 0);

        for (int i = 1; i <= n; ++i) {
            cnt[i] = cnt[i - 1] + nums[i - 1];
            s[i] = s[i - 1] + 1LL * i * nums[i - 1];
        }

        long long ans = LLONG_MAX;

        for (int i = 1; i <= n; ++i) {
            long long t = 0;
            int need = k - nums[i - 1];

            for (int j = i - 1; j <= i + 1; j += 2) {
                if (need > 0 && 1 <= j && j <= n && nums[j - 1] == 1) {
                    --need;
                    ++t;
                }
            }

            int c = min(need, maxChanges);
            need -= c;
            t += c * 2;

            if (need <= 0) {
                ans = min(ans, t);
                continue;
            }

            int l = 2, r = max(i - 1, n - i);

            while (l <= r) {
                int mid = (l + r) / 2;
                int l1 = max(1, i - mid), r1 = max(0, i - 2);
                int l2 = min(n + 1, i + 2), r2 = min(n, i + mid);

                int c1 = cnt[r1] - cnt[l1 - 1];
                int c2 = cnt[r2] - cnt[l2 - 1];

                if (c1 + c2 >= need) {
                    long long t1 = 1LL * c1 * i - (s[r1] - s[l1 - 1]);
                    long long t2 = s[r2] - s[l2 - 1] - 1LL * c2 * i;
                    ans = min(ans, t + t1 + t2);
                    r = mid - 1;
                } else {
                    l = mid + 1;
                }
            }
        }

        return ans;
    }
};

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func minimumMoves(nums []int, k int, maxChanges int) int64 {
    n := len(nums)
    cnt := make([]int, n+1)
    s := make([]int, n+1)

    for i := 1; i <= n; i++ {
        cnt[i] = cnt[i-1] + nums[i-1]
        s[i] = s[i-1] + i*nums[i-1]
    }

    ans := math.MaxInt64

    for i := 1; i <= n; i++ {
        t := 0
        need := k - nums[i-1]

        for _, j := range []int{i - 1, i + 1} {
            if need > 0 && 1 <= j && j <= n && nums[j-1] == 1 {
                need--
                t++
            }
        }

        c := min(need, maxChanges)
        need -= c
        t += c * 2

        if need <= 0 {
            ans = min(ans, t)
            continue
        }

        l, r := 2, max(i-1, n-i)

        for l <= r {
            mid := (l + r) >> 1
            l1, r1 := max(1, i-mid), max(0, i-2)
            l2, r2 := min(n+1, i+2), min(n, i+mid)

            c1 := cnt[r1] - cnt[l1-1]
            c2 := cnt[r2] - cnt[l2-1]

            if c1+c2 >= need {
                t1 := c1*i - (s[r1] - s[l1-1])
                t2 := s[r2] - s[l2-1] - c2*i
                ans = min(ans, t+t1+t2)
                r = mid - 1
            } else {
                l = mid + 1
            }
        }
    }

    return int64(ans)
}

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function minimumMoves(nums: number[], k: number, maxChanges: number): number {
    const n = nums.length;
    const cnt = Array(n + 1).fill(0);
    const s = Array(n + 1).fill(0);

    for (let i = 1; i <= n; i++) {
        cnt[i] = cnt[i - 1] + nums[i - 1];
        s[i] = s[i - 1] + i * nums[i - 1];
    }

    let ans = Infinity;
    for (let i = 1; i <= n; i++) {
        let t = 0;
        let need = k - nums[i - 1];

        for (let j of [i - 1, i + 1]) {
            if (need > 0 && 1 <= j && j <= n && nums[j - 1] === 1) {
                need--;
                t++;
            }
        }

        const c = Math.min(need, maxChanges);
        need -= c;
        t += c * 2;

        if (need <= 0) {
            ans = Math.min(ans, t);
            continue;
        }

        let l = 2,
            r = Math.max(i - 1, n - i);

        while (l <= r) {
            const mid = (l + r) >> 1;
            const [l1, r1] = [Math.max(1, i - mid), Math.max(0, i - 2)];
            const [l2, r2] = [Math.min(n + 1, i + 2), Math.min(n, i + mid)];

            const c1 = cnt[r1] - cnt[l1 - 1];
            const c2 = cnt[r2] - cnt[l2 - 1];

            if (c1 + c2 >= need) {
                const t1 = c1 * i - (s[r1] - s[l1 - 1]);
                const t2 = s[r2] - s[l2 - 1] - c2 * i;
                ans = Math.min(ans, t + t1 + t2);
                r = mid - 1;
            } else {
                l = mid + 1;
            }
        }
    }

    return ans;
}

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