Question

2557. Maximum Number of Integers to Choose From a Range II

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Description

You are given an integer array banned and two integers n and maxSum. You are choosing some number of integers following the below rules:

• The chosen integers have to be in the range [1, n].
• Each integer can be chosen at most once.
• The chosen integers should not be in the array banned.
• The sum of the chosen integers should not exceed maxSum.

Return _the maximum number of integers you can choose following the mentioned rules_.

 

Example 1:

Input: banned = [1,4,6], n = 6, maxSum = 4
Output: 1
Explanation: You can choose the integer 3.
3 is in the range [1, 6], and do not appear in banned. The sum of the chosen integers is 3, which does not exceed maxSum.

Example 2:

Input: banned = [4,3,5,6], n = 7, maxSum = 18
Output: 3
Explanation: You can choose the integers 1, 2, and 7.
All these integers are in the range [1, 7], all do not appear in banned, and their sum is 18, which does not exceed maxSum.

 

Constraints:

• 1 <= banned.length <= 105
• 1 <= banned[i] <= n <= 109
• 1 <= maxSum <= 1015

Solutions

Solution 1: Deduplication + Sorting + Binary Search

We can add $0$ and $n + 1$ to the array banned, then deduplicate and sort the array banned.

Next, we enumerate every two adjacent elements $i$ and $j$ in the array banned. The range of selectable integers is $[i + 1, j - 1]$. We use binary search to enumerate the number of elements we can select in this range, find the maximum number of selectable elements, and then add it to $ans$. At the same time, we subtract the sum of these elements from maxSum. If maxSum is less than $0$, we break the loop. Return the answer.

The time complexity is $O(n \times \log n)$, and the space complexity is $O(n)$. Where $n$ is the length of the array banned.

class Solution:
  def maxCount(self, banned: List[int], n: int, maxSum: int) -> int:
    banned.extend([0, n + 1])
    ban = sorted(set(banned))
    ans = 0
    for i, j in pairwise(ban):
      left, right = 0, j - i - 1
      while left < right:
        mid = (left + right + 1) >> 1
        if (i + 1 + i + mid) * mid // 2 <= maxSum:
          left = mid
        else:
          right = mid - 1
      ans += left
      maxSum -= (i + 1 + i + left) * left // 2
      if maxSum <= 0:
        break
    return ans

class Solution {
  public int maxCount(int[] banned, int n, long maxSum) {
    Set<Integer> black = new HashSet<>();
    black.add(0);
    black.add(n + 1);
    for (int x : banned) {
      black.add(x);
    }
    List<Integer> ban = new ArrayList<>(black);
    Collections.sort(ban);
    int ans = 0;
    for (int k = 1; k < ban.size(); ++k) {
      int i = ban.get(k - 1), j = ban.get(k);
      int left = 0, right = j - i - 1;
      while (left < right) {
        int mid = (left + right + 1) >>> 1;
        if ((i + 1 + i + mid) * 1L * mid / 2 <= maxSum) {
          left = mid;
        } else {
          right = mid - 1;
        }
      }
      ans += left;
      maxSum -= (i + 1 + i + left) * 1L * left / 2;
      if (maxSum <= 0) {
        break;
      }
    }
    return ans;
  }
}

class Solution {
public:
  int maxCount(vector<int>& banned, int n, long long maxSum) {
    banned.push_back(0);
    banned.push_back(n + 1);
    sort(banned.begin(), banned.end());
    banned.erase(unique(banned.begin(), banned.end()), banned.end());
    int ans = 0;
    for (int k = 1; k < banned.size(); ++k) {
      int i = banned[k - 1], j = banned[k];
      int left = 0, right = j - i - 1;
      while (left < right) {
        int mid = left + ((right - left + 1) / 2);
        if ((i + 1 + i + mid) * 1LL * mid / 2 <= maxSum) {
          left = mid;
        } else {
          right = mid - 1;
        }
      }
      ans += left;
      maxSum -= (i + 1 + i + left) * 1LL * left / 2;
      if (maxSum <= 0) {
        break;
      }
    }
    return ans;
  }
};

func maxCount(banned []int, n int, maxSum int64) (ans int) {
  banned = append(banned, []int{0, n + 1}...)
  sort.Ints(banned)
  ban := []int{}
  for i, x := range banned {
    if i > 0 && x == banned[i-1] {
      continue
    }
    ban = append(ban, x)
  }
  for k := 1; k < len(ban); k++ {
    i, j := ban[k-1], ban[k]
    left, right := 0, j-i-1
    for left < right {
      mid := (left + right + 1) >> 1
      if int64((i+1+i+mid)*mid/2) <= maxSum {
        left = mid
      } else {
        right = mid - 1
      }
    }
    ans += left
    maxSum -= int64((i + 1 + i + left) * left / 2)
    if maxSum <= 0 {
      break
    }
  }
  return
}