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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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