264. Ugly Number II

Write a program to find the n-th ugly number.

Ugly numbers are positive numbers whose prime factors only include 2, 3, 5.

Example:

Input: n = 10
Output: 12
Explanation: 1, 2, 3, 4, 5, 6, 8, 9, 10, 12 is the sequence of the first 10 ugly numbers.

Note:

1. 1 is typically treated as an ugly number.
2. n does not exceed 1690.

解法1

暴力搜尋。假設已經知道了前n個醜數\(a_1, a_2, ..., a_n\)，求第n+1個醜數，則有：

\[a_{n+1} = \min\{2*a_i, 3*a_j, 5*a_k\}, \quad \forall i, j, k\in[1, n]\\ s.t\quad a_{n+1} > a_n \]

class Solution {
public:
    int nthUglyNumber(int n) {
        vector<int>u_n{1};
        int prime[3] = {2, 3, 5};
        while(u_n.size() < n){
            int flag = 0;
            int cur_n = INT_MAX;
            for(int i = u_n.size() - 1; i >= 0; --i){
                for(int j = 0; j < 3; ++j){
                    if(u_n[i]*prime[j] > u_n.back()){
                        cur_n = min(cur_n, u_n[i]*prime[j]);
                    }else{
                        flag++;
                    }
                }
                if(flag == 3)break;
            }
            u_n.push_back(cur_n);
        }
        return u_n.back();
    }
};
 

但是提交會超時。。。

解法2 對解法1進行改進。顯然醜數陣列是有序的，可以用二分查詢完成，查詢的問題描述為：

尋找第一次出現的滿足 \(k\times a_i > a_n\)的\(a_i\)

搜尋過程為：對於區間\([l, r]\)

• \(k\times a_{mid} > a_n\)，則滿足條件的肯定在\([l, mid]\)中
• \(k\times a_{mid} \leq a_n\)，則滿足條件的肯定在\([mid+1, r]\)中

typedef long long int LL;
class Solution {
public:
    int nthUglyNumber(int n) {
        vector<LL>u_n{1};
        int prime[3] = {2, 3, 5};
        while(u_n.size() < n){
            LL cur_n = LLONG_MAX;
            for(int j = 0; j < 3; ++j){
                int idx = bin_search(u_n, prime[j]);
                cur_n = min(cur_n, prime[j]*u_n[idx]);
            }
            u_n.push_back(cur_n);
        }
        return u_n.back();
    }
    int bin_search(vector<LL>&nums, int k){
        int last_num = nums.back();
        int l = 0, r = nums.size()-1;
        while(l < r){
            int mid = (l + r) / 2;
            if(nums[mid]*k > last_num)r=mid;
            else l = mid + 1;
        }
        return l;
    }
};
 

解法3 注意到事實：如果\(a_n\)是醜數，則\(2a_n, 3a_n, 5a_n\)也是醜數

typedef long long int LL;
class Solution {
public:
    int nthUglyNumber(int n) {
        priority_queue<LL, vector<LL>, greater<LL>>q;
        set<LL>s;
        int prime[3] = {2, 3, 5};
        
        q.push(1);
        s.insert(1);
        LL ans = q.top();
        for(int i = 0; i < n; ++i){
            ans = q.top();
            q.pop();
            s.erase(ans);
            for(int j = 0; j < 3; ++j){
                if(s.find(ans*prime[j]) == s.end()){
                    q.push(ans*prime[j]);
                    s.insert(ans*prime[j]);
                }
            }
        }
        return ans;
    }
};

解法4 根據解法三種事實，利用動態規劃

class Solution {
public:
    int nthUglyNumber(int n) {
        int pre2 = 0, pre3 = 0, pre5 = 0;
        int nums[1690];
        nums[0] = 1;
        for(int i = 1; i < n; ++i){
            int ugly = min(nums[pre2]*2, min(nums[pre3]*3, nums[pre5]*5));
            nums[i] = ugly;
            if(ugly % 2 == 0)pre2++;
            if(ugly % 3 == 0)pre3++;
            if(ugly % 5 == 0)pre5++;
        }
        return nums[n-1];
    }
};