Naive Solution

We can try all possible values of h from 0to n. For each h, loop through citations to see if h is a possible h-index, using the condition we are given:

A scientist has an index h if h of their n papers have at least h citations each, and the other n-h papers have no more than h citations each.

The answer is the highest h for which this is true.

This takes $\mathcal{O}(n^2)$ time because for each of the n+1 possible h values, we have to loop through n citations.

Binary Search Solution

Create a function hasAtLeastHPapersWithHCitations with a parameter h to check if there are at least h papers with >= h citations. When hasAtLeastHPapersWithHCitations(x) is true, hasAtLeastHPapersWithHCitations(x-1) is also true. This means that hasAtLeastHPapersWithHCitations is a monotonic function, so we can binary search for the highest h for which it return true. This h is our h-index.

Time Complexity

Each call to hasAtLeastHPapersWithHCitations checks all $n$ papers, taking $\mathcal{O}(n)$.

Binary searching the range $[0, n]$ takes $\mathcal{O}(\log n)$.

Multiplying these together, we take $\mathcal{O}(n \log n)$.

Space Complexity

citations is passed by reference, so we aren't allocating any memory for it. We allocate a constant amount of memory for a couple of variables, so the space complexity is $\mathcal{O}(1)$.

C++ Solution

class Solution {
public:
    bool hasAtLeastHPapersWithHCitations(int h, vector<int>& citations) {
        int count = 0;
        for (int cite_count : citations) {
            if (cite_count >= h)
                count++;
        }
        return count >= h;
    }
    int hIndex(vector<int>& citations) {
        int low = 0, high = citations.size();
        while (low <= high) {
            int mid = (low + high) / 2;
            if (hasAtLeastHPapersWithHCitations(mid, citations))
                low = mid + 1;
            else
                high = mid - 1;
        }
        return high;
    }
};

Java Solution

class Solution {
    static boolean hasAtLeastHPapersWithHCitations(int h, int[] citations) {
        int count = 0;
        for (int cite_count : citations) {
            if (cite_count >= h)
                count++;
        }
        return count >= h;
    }
    public int hIndex(int[] citations) {
        int low = 0, high = citations.length;
        while (low <= high) {
            int mid = (low + high) / 2;
            if (hasAtLeastHPapersWithHCitations(mid, citations))
                low = mid + 1;
            else
                high = mid - 1;
        }
        return high;
    }
}

Python Solution

class Solution:
    def hIndex(self, citations: List[int]) -> int:

        def hasAtLeastHPapersWithHCitations(h, citations):
            return sum(cite_count >= h for cite_count in citations) >= h

        low = 0
        high = len(citations)
        while low <= high:
            mid = (low + high) // 2;
            if hasAtLeastHPapersWithHCitations(mid, citations):
                low = mid + 1;
            else:
                high = mid - 1;
        return high

Sort and Loop Solution

First we sort the papers by decreasing # of citations. Imagine a histogram where each bar represents a paper and its height is the # of citations it has.

<img src={require('@site/static/img/274/1.png').default} style={{ height: 450 }}/>

If the h-index were h, we'd need exactly h bars with height as least h. That is to say, we'd need the green square covered by bars. To find the h index, first set h = 0. Then keep increasing h by 1 as long as the next tallest bar is >= h+1. When we can no longer increase h, we have our answer.

In the diagram above, if we continued to increase h, the next added bar would not be tall enough for the new h.

Time Complexity

Sorting is $\mathcal{O}(n \log n)$. Looping h is $\mathcal{O}(n)$. So the time complexity is $\mathcal{O}(n \log n)$.

Space Complexity

The only memory we allocate is the integer h, so the space complexity is $\mathcal{O}(1)$.

C++ Solution

class Solution {
public:
    int hIndex(vector<int>& citations) {
        sort(citations.rbegin(), citations.rend());
        int h = 0;
        while (h < citations.size() and citations[h] >= h+1) {
            h++;
        }
        return h;
    }
};

Java Solution

class Solution {
    public int hIndex(int[] citations) {
        // Sorting an int[] in reverse in Java is annoying
        // We first sort normally then reverse the array
        Arrays.sort(citations);
        for (int i = 0; i < citations.length/2; i++) {
            int tmp = citations[i];
            citations[i] = citations[citations.length-1-i];
            citations[citations.length-1-i] = tmp;
        }

        int h = 0;
        while (h < citations.length && citations[h] >= h+1) {
            h++;
        }
        return h;
    }
}

Python Solution

class Solution:
    def hIndex(self, citations: List[int]) -> int:
        citations.sort(reverse=True)
        h = 0
        while h < len(citations) and citations[h] >= h+1:
            h += 1
        return h