题目描述:
我们把只包含质因子 2、3 和 5 的数称作丑数(Ugly Number)。求按从小到大的顺序的第 n 个丑数。
示例:
输入: n = 10
输出: 12
解释: 1, 2, 3, 4, 5, 6, 8, 9, 10, 12 是前 10 个丑数。
说明:
1 是丑数。
n 不超过1690。
来源:力扣(LeetCode)
链接:https://leetcode-cn.com/problems/chou-shu-lcof
著作权归领扣网络所有。商业转载请联系官方授权,非商业转载请注明出处。
解题思路:
方法一: 暴力打表法。我们可以将1~1690个丑数计算出来,然后打表存到一个数组中,每次只需要返回ans[n-1]即可。PS:讲解一下怎么判断该数字是丑数,由题目要求已知,丑数只包含质因子2、3、5,这就说明丑数一定能除一个或多个2、3、5,最终得到1。即丑数 = 2 * x + 3 * y + 5 * z;(x、y、z全部大于等于0),所以我们可以利用三个循环来判断该数是不是丑数:
首先,若该数num能整除2,即num%2 ==0,我们就循环的让num/2,直到不能整除为止。
然后若该数num能整除3,即num%3 ==0,我们就循环的让num/3,直到不能整除为止。
然后若该数num能整除5,即num%5 ==0,我们就循环的让num/5,直到不能整除为止。
最后的num如果变成1,则说明该数是丑数。
注:打表的时候,因为不知道上界即不知道n=1690时,该数字是多少,所以我们可以先计算一下n=1690时的结果。
方法二: 动态规划,我只能说dp永远滴神!看了评论区大佬的解法,瞬间茅塞顿开。
根据题意,丑数只包含2、3、5三个质因子。
所以, 丑数 = 某较小丑数 * 某质因子
进而得到转移方程:
***dp[n] = min(dp[a]2,dp[b]3,dp[c]5),其中a、b、c为三个比 n小的索引
此时的a、b、c要满足以下条件:
dp[a-1]*2<dp[n-1]≤dp[a]*2,即dp[a]为首个乘以2后大于dp[n]的丑数
dp[b-1]*3<dp[n-1]≤dp[b]*3,即dp[b]为首个乘以3后大于dp[n]的丑数
dp[c-1]*5<dp[n-1]≤dp[c]*5,即dp[c]为首个乘以5后大于dp[n]的丑数
例如,当dp[a]2==dp[i]时,我们就让a+1,说明a也已经2并存到数组中了,此时a乘没乘过3、5并不知道。
这里a、b、c除了索引的意思,还代表着前a-1个数都已经乘过2并且保存到dp数组中了,只有这样才能保证一个丑数也不丢。
对于某个状态下的丑数序列,我们知道此时第a个数还没有乘2(有没有乘3或者乘5不知道), 第b个数还没有乘3(有没有乘2或者乘5不知道),第c个数还没有乘5(有没有乘2或者乘3不知道), 下一个丑数一定是从第a丑数乘2, 第b个数乘3, 第c个数乘5中获得,他们三者最小的那个就是下个丑数。
注意: 可能存在这样的情况:dp[a]*2== dp[b]*3==dp[c]*5,此时我们需要将a、b、c都+1,所以代码中用的是并列的if语句,而不是if-else if-else。两个相等且是最小值的情况也是如此,需要将他们的索引+1。
这一块的图解可以详见Krathets的题解。
初始状态: dp[0] = 1,a、b、c均为0.
结束状态: 返回dp[n-1]
时间复杂度:O(N) 空间复杂度:O(N)
详细代码:
public class jianzhi_Offer_49 {
//方法一:暴力打表,每个数都去判断是不是丑数,如果是则让计数器+1,直到找到第n个
public int nthUglyNumber(int n) {
//用来判断打表的上届,即判断n=1690时,丑数是什么?---->2123366400
// int cnt = 0;
// int ans = 0;
// int i = 1;
// while (cnt < n){
// if(isUglyNumber(i) == true){
// cnt++;
// }
// i++;
// }
// return i-1;
// run();
int[] ans = new int[]{
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};
return ans[n-1];
}
//丑数的质因子只有2,3,5,则将其不断地除2,3,5后最后得到的值一定为1
public boolean isUglyNumber(long num){
while (num % 2 == 0){
num /= 2;
}
while (num % 3 == 0){
num /= 3;
}
while (num % 5 == 0){
num /= 5;
}
if(num == 1)
return true;
else
return false;
}
//打表
public void run(){
for (int i = 1; i <= 2123366400; i++){
if(isUglyNumber(i)){
System.out.print(i+",");
}
}
}
//方法二:dp。因为丑数只包含2、3、5三个质因子。
//所以,丑数 = 某较小丑数 * 某质因子
//得到转移方程 dp[i] = min(dp[a] * 2, dp[b] * 3, dp[c] *5)
//其中a,b,c初始化为0,在循环时,若dp[i]等于他们所指的数*所表示的质因子时,让它们+1.
public int nthUglyNumber1(int n) {
int[] dp = new int[n];
dp[0] = 1;
int a = 0,b = 0,c = 0;
for (int i = 1; i < n; i++){
dp[i] = Math.min(dp[a]*2,Math.min(dp[b]*3,dp[c]*5));
if(dp[i] == dp[a] * 2)
a++;
if(dp[i] == dp[b] * 3)
b++;
if(dp[i] == dp[c] * 5)
c++;
}
return dp[n-1];
}
public static void main(String[] args) {
jianzhi_Offer_49 obj = new jianzhi_Offer_49();
System.out.println(obj.nthUglyNumber1(1690));
}
}