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2555. Maximize Win From Two Segments
Description
There are some prizes on the Xaxis. You are given an integer array prizePositions
that is sorted in nondecreasing order, where prizePositions[i]
is the position of the i^{th}
prize. There could be different prizes at the same position on the line. You are also given an integer k
.
You are allowed to select two segments with integer endpoints. The length of each segment must be k
. You will collect all prizes whose position falls within at least one of the two selected segments (including the endpoints of the segments). The two selected segments may intersect.
 For example if
k = 2
, you can choose segments[1, 3]
and[2, 4]
, and you will win any prize i that satisfies1 <= prizePositions[i] <= 3
or2 <= prizePositions[i] <= 4
.
Return the maximum number of prizes you can win if you choose the two segments optimally.
Example 1:
Input: prizePositions = [1,1,2,2,3,3,5], k = 2 Output: 7 Explanation: In this example, you can win all 7 prizes by selecting two segments [1, 3] and [3, 5].
Example 2:
Input: prizePositions = [1,2,3,4], k = 0 Output: 2 Explanation: For this example, one choice for the segments is[3, 3]
and[4, 4],
and you will be able to get2
prizes.
Constraints:
1 <= prizePositions.length <= 10^{5}
1 <= prizePositions[i] <= 10^{9}
0 <= k <= 10^{9}
prizePositions
is sorted in nondecreasing order.
Solutions
Solution 1: Dynamic Programming + Binary Search
We define $f[i]$ as the maximum number of prizes that can be obtained by selecting a segment of length $k$ from the first $i$ prizes. Initially, $f[0] = 0$. We define the answer variable as $ans = 0$.
Next, we enumerate the position $x$ of each prize, and use binary search to find the leftmost prize index $j$ such that $prizePositions[j] \geq x  k$. At this point, we update the answer $ans = \max(ans, f[j] + i  j)$, and update $f[i] = \max(f[i  1], i  j)$.
Finally, we return $ans$.
The time complexity is $O(n \times \log n)$, and the space complexity is $O(n)$. Where $n$ is the number of prizes.

class Solution { public int maximizeWin(int[] prizePositions, int k) { int n = prizePositions.length; int[] f = new int[n + 1]; int ans = 0; for (int i = 1; i <= n; ++i) { int x = prizePositions[i  1]; int j = search(prizePositions, x  k); ans = Math.max(ans, f[j] + i  j); f[i] = Math.max(f[i  1], i  j); } return ans; } private int search(int[] nums, int x) { int left = 0, right = nums.length; while (left < right) { int mid = (left + right) >> 1; if (nums[mid] >= x) { right = mid; } else { left = mid + 1; } } return left; } }

class Solution { public: int maximizeWin(vector<int>& prizePositions, int k) { int n = prizePositions.size(); vector<int> f(n + 1); int ans = 0; for (int i = 1; i <= n; ++i) { int x = prizePositions[i  1]; int j = lower_bound(prizePositions.begin(), prizePositions.end(), x  k)  prizePositions.begin(); ans = max(ans, f[j] + i  j); f[i] = max(f[i  1], i  j); } return ans; } };

class Solution: def maximizeWin(self, prizePositions: List[int], k: int) > int: n = len(prizePositions) f = [0] * (n + 1) ans = 0 for i, x in enumerate(prizePositions, 1): j = bisect_left(prizePositions, x  k) ans = max(ans, f[j] + i  j) f[i] = max(f[i  1], i  j) return ans

func maximizeWin(prizePositions []int, k int) (ans int) { n := len(prizePositions) f := make([]int, n+1) for i, x := range prizePositions { j := sort.Search(n, func(h int) bool { return prizePositions[h] >= xk }) ans = max(ans, f[j]+ij+1) f[i+1] = max(f[i], ij+1) } return }

function maximizeWin(prizePositions: number[], k: number): number { const n = prizePositions.length; const f: number[] = Array(n + 1).fill(0); let ans = 0; const search = (x: number): number => { let left = 0; let right = n; while (left < right) { const mid = (left + right) >> 1; if (prizePositions[mid] >= x) { right = mid; } else { left = mid + 1; } } return left; }; for (let i = 1; i <= n; ++i) { const x = prizePositions[i  1]; const j = search(x  k); ans = Math.max(ans, f[j] + i  j); f[i] = Math.max(f[i  1], i  j); } return ans; }