668. Kth Smallest Number in Multiplication Table

668. Kth Smallest Number in Multiplication Table (Hard)

Nearly every one have used the Multiplication Table. But could you find out the k-th smallest number quickly from the multiplication table?

Given the height m and the length n of a m * n Multiplication Table, and a positive integer k, you need to return the k-th smallest number in this table.

Example 1:

Input: m = 3, n = 3, k = 5
Output: 
Explanation: 
The Multiplication Table:
1	2	3
2	4	6
3	6	9
The 5-th smallest number is 3 (1, 2, 2, 3, 3).

Example 2:

Input: m = 2, n = 3, k = 6
Output: 
Explanation: 
The Multiplication Table:
1	2	3
2	4	6
The 6-th smallest number is 6 (1, 2, 2, 3, 4, 6).

Note:

1. The m and n will be in the range [1, 30000].
2. The k will be in the range [1, m * n]

Related Topics:
Binary Search

Similar Questions:

• Kth Smallest Element in a Sorted Matrix (Medium)
• Find K-th Smallest Pair Distance (Hard)
• K-th Smallest Prime Fraction (Hard)

TLE version: Next Heap

// OJ: https://leetcode.com/problems/kth-smallest-number-in-multiplication-table/
// Author: github.com/lzl124631x
// Time: O(k * MlogM) = O(M^2 * NlogM)
// Space: O(M)
class Solution {
public:
    int findKthNumber(int m, int n, int k) {
        auto cmp = [](pair<int, int> &a, pair<int, int> &b) { return a.first * a.second > b.first * b.second; };
        priority_queue<pair<int, int>, vector<pair<int, int>>, decltype(cmp)> q(cmp);
        for (int i = 1; i <= m; ++i) q.emplace(i, 1);
        int ans = 0;
        while (k--) {
            auto p = q.top();
            q.pop();
            ans = p.first * p.second;
            if (p.second < n) q.emplace(p.first, p.second + 1);
        }
        return ans;
    }
};

Solution 1. Binary Answer

The range of the answer is [1, m * n]. We can use binary answer to find it.

Let L = 1, R = m * n, M = (L + R) / 2.

Define cnt(M) as the count of numbers less than or equal to M.

For the answer ans, the corresponding cnt(ans) could be exactly k (when there is only one ans in the table) or greater than k (when there are multiple ans in the table).

The goal is to find the first element M whose cnt(M) is greater than or equal to k.

So let the left part of the array be those elements whose cnt < k, and the right part be cnt >= k.

In the end, L will point to the first element whose cnt >= k and it is the answer.

// OJ: https://leetcode.com/problems/kth-smallest-number-in-multiplication-table/
// Author: github.com/lzl124631x
// Time: O(Mlog(MN))
// Space: O(1)
class Solution {
public:
    int findKthNumber(int m, int n, int k) {
        long long L = 1, R = m * n;
        while (L <= R) {
            long long M = (L + R) / 2, cnt = 0;
            for (int i = 1; i <= m; ++i) cnt += min(M / i, (long long) n);
            if (cnt < k) L = M + 1;
            else R = M - 1;
        }
        return L;
    }
};