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2554. Maximum Number of Integers to Choose From a Range I

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 = [1,6,5], n = 5, maxSum = 6
Output: 2
Explanation: You can choose the integers 2 and 4.
2 and 4 are from the range [1, 5], both did not appear in banned, and their sum is 6, which did not exceed maxSum.

Example 2:

Input: banned = [1,2,3,4,5,6,7], n = 8, maxSum = 1
Output: 0
Explanation: You cannot choose any integer while following the mentioned conditions.

Example 3:

Input: banned = [11], n = 7, maxSum = 50
Output: 7
Explanation: You can choose the integers 1, 2, 3, 4, 5, 6, and 7.
They are from the range [1, 7], all did not appear in banned, and their sum is 28, which did not exceed maxSum.

 

Constraints:

  • 1 <= banned.length <= 104
  • 1 <= banned[i], n <= 104
  • 1 <= maxSum <= 109

Solutions

Solution 1: Greedy + Enumeration

We use the variable $s$ to represent the sum of the currently selected integers, and the variable $ans$ to represent the number of currently selected integers. We convert the array banned into a hash table for easy determination of whether a certain integer is not selectable.

Next, we start enumerating the integer $i$ from $1$. If $s + i \leq maxSum$ and $i$ is not in banned, then we can select the integer $i$, and add $i$ and $1$ to $s$ and $ans$ respectively.

Finally, we return $ans$.

The time complexity is $O(n)$, and the space complexity is $O(n)$. Where $n$ is the given integer.

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class Solution:
    def maxCount(self, banned: List[int], n: int, maxSum: int) -> int:
        ans = s = 0
        ban = set(banned)
        for i in range(1, n + 1):
            if s + i > maxSum:
                break
            if i not in ban:
                ans += 1
                s += i
        return ans

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class Solution {
    public int maxCount(int[] banned, int n, int maxSum) {
        Set<Integer> ban = new HashSet<>(banned.length);
        for (int x : banned) {
            ban.add(x);
        }
        int ans = 0, s = 0;
        for (int i = 1; i <= n && s + i <= maxSum; ++i) {
            if (!ban.contains(i)) {
                ++ans;
                s += i;
            }
        }
        return ans;
    }
}

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class Solution {
public:
    int maxCount(vector<int>& banned, int n, int maxSum) {
        unordered_set<int> ban(banned.begin(), banned.end());
        int ans = 0, s = 0;
        for (int i = 1; i <= n && s + i <= maxSum; ++i) {
            if (!ban.count(i)) {
                ++ans;
                s += i;
            }
        }
        return ans;
    }
};

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func maxCount(banned []int, n int, maxSum int) (ans int) {
    ban := map[int]bool{}
    for _, x := range banned {
        ban[x] = true
    }
    s := 0
    for i := 1; i <= n && s+i <= maxSum; i++ {
        if !ban[i] {
            ans++
            s += i
        }
    }
    return
}

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function maxCount(banned: number[], n: number, maxSum: number): number {
    const set = new Set(banned);
    let sum = 0;
    let ans = 0;
    for (let i = 1; i <= n; i++) {
        if (i + sum > maxSum) {
            break;
        }
        if (set.has(i)) {
            continue;
        }
        sum += i;
        ans++;
    }
    return ans;
}

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use std::collections::HashSet;
impl Solution {
    pub fn max_count(banned: Vec<i32>, n: i32, max_sum: i32) -> i32 {
        let mut set = banned.into_iter().collect::<HashSet<i32>>();
        let mut sum = 0;
        let mut ans = 0;
        for i in 1..=n {
            if sum + i > max_sum {
                break;
            }
            if set.contains(&i) {
                continue;
            }
            sum += i;
            ans += 1;
        }
        ans
    }
}

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int cmp(const void* a, const void* b) {
    return *(int*) a - *(int*) b;
}

int maxCount(int* banned, int bannedSize, int n, int maxSum) {
    qsort(banned, bannedSize, sizeof(int), cmp);
    int sum = 0;
    int ans = 0;
    for (int i = 1, j = 0; i <= n; i++) {
        if (sum + i > maxSum) {
            break;
        }
        if (j < bannedSize && i == banned[j]) {
            while (j < bannedSize && i == banned[j]) {
                j++;
            }
        } else {
            sum += i;
            ans++;
        }
    }
    return ans;
}

If $n$ is very large, the enumeration in Method One will time out.

We can add $0$ and $n + 1$ to the array banned, deduplicate the array banned, remove elements greater than $n+1$, and then sort it.

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.

Similar problems:

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class Solution:
    def maxCount(self, banned: List[int], n: int, maxSum: int) -> int:
        banned.extend([0, n + 1])
        ban = sorted(x for x in set(banned) if x < n + 2)
        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

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class Solution {
    public int maxCount(int[] banned, int n, int maxSum) {
        Set<Integer> black = new HashSet<>();
        black.add(0);
        black.add(n + 1);
        for (int x : banned) {
            if (x < n + 2) {
                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;
    }
}

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class Solution {
public:
    int maxCount(vector<int>& banned, int n, int maxSum) {
        banned.push_back(0);
        banned.push_back(n + 1);
        sort(banned.begin(), banned.end());
        banned.erase(unique(banned.begin(), banned.end()), banned.end());
        banned.erase(remove_if(banned.begin(), banned.end(), [&](int x) { return x > n + 1; }), 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;
    }
};

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func maxCount(banned []int, n int, maxSum int) (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]) || x > n+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 (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
}

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