Problem:
You are given K
eggs, and you have access to a building with N
floors from 1
to N
.
Each egg is identical in function, and if an egg breaks, you cannot drop it again.
You know that there exists a floor F
with 0 <= F <= N
such that any egg dropped at a floor higher than F
will break, and any egg dropped at or below floor F
will not break.
Each move, you may take an egg (if you have an unbroken one) and drop it from any floor X
(with 1 <= X <= N
).
Your goal is to know with certainty what the value of F
is.
What is the minimum number of moves that you need to know with certainty what F
is, regardless of the initial value of F
?
Example 1:
Input: K = 1, N = 2 Output: 2 Explanation: Drop the egg from floor 1. If it breaks, we know with certainty that F = 0. Otherwise, drop the egg from floor 2. If it breaks, we know with certainty that F = 1. If it didn‘t break, then we know with certainty F = 2. Hence, we needed 2 moves in the worst case to know what F is with certainty.
Example 2:
Input: K = 2, N = 6 Output: 3
Example 3:
Input: K = 3, N = 14 Output: 4
Note:
1 <= K <= 100
1 <= N <= 10000
Solution:
说道扔鸡蛋问题,就不得不提一下那个经典的dp问题。给m个鸡蛋尝试n次,最坏情况下可以在多少楼层内确定会让鸡蛋破裂的最低层数。
对于这道题,我们可以创建一个二维数组dp,dp[i][j]表示用i个鸡蛋尝试j次可以确定的最高楼层数。这个最高楼层可以由两部分组成,假设我们在x楼扔下一个鸡蛋,它可能会破也可能不会破,如果鸡蛋破了,那么我们需要用剩下的i-1个鸡蛋和就-1次机会在x楼之下寻找这个最高楼层,即dp[i-1][j-1],若鸡蛋破了,则我们需要在x+1层之上用i个鸡蛋尝试j-1次找到那个最高楼层,即dp[i][j-1],因此可以得到状态转移方程:dp[i][j] = dp[i-1][j-1]+1+dp[i][j-1],对于i=1,dp[1][i] = i。代码部分如下:
1 class Solution { 2 public: 3 int superEggDrop(int K, int N) { 4 vector<vector<int>> dp(K+1,vector<int>(N+1,0)); 5 for(int j = 0;j <= N;++j) 6 dp[1][j] = j; 7 for(int i = 2;i <= K;++i){ 8 for(int j = 1;j <= N;++j){ 9 dp[i][j] = dp[i-1][j-1]+1+dp[i][j-1]; 10 } 11 } 12 return dp[K][N]; 13 } 14 };
现在来看这道题有什么不同呢,现在他告诉我们最高楼层,要我们找到最小的尝试次数,即告诉我们i和dp[i][j]要我们计算j的值。和上面那题的思路类似,我们令x为测试的楼层,在这一楼层扔鸡蛋有两种情况,如果蛋碎了,则我们需要在x-1层里用i-1个鸡蛋找到最小的尝试次数,如果蛋没碎,则要在x+1到j层用i个鸡蛋找到最小尝试次数,所以max(dp[i-1][x],dp[i][j-x])即在x层扔下第一个鸡蛋的情况下,用i个鸡蛋在j层内的最小尝试次数。所以,我们需要在0-j层内找到合适的x,使得这个最小尝试次数最少,所以这就成了一个极小化极大值的问题。在这里其实我们可以观察到dp[i-1][x]是递增的,而dp[i][j-x]是递减的,所以我们在0-j内遍历x,当dp[i-1][x] == dp[i][j-x]时,可以断定此时的x即第一个鸡蛋应该扔的楼层,所以dp[i][j]就等于dp[i-1][x]+1。
对于这个解法,其时间复杂度为O(kN2),我们可以观察到在确定x的过程中,我们通过遍历的方法寻找x,在这个地方我们其实可以通过二分法进行优化,将时间复杂度降低为O(kN*logN)。(虽然通过二分法时间复杂度降低了,但程序的运行时间却增加了,所以算法复杂度和程序运行时间真是没必然联系啊)
由于这道题的状态转移方程只需要i-1层的数据,其空间复杂度也可以降低为O(N),感兴趣的读者可以继续优化算法。
Code:
1 class Solution { 2 public: 3 int superEggDrop(int K, int N) { 4 vector<vector<int>> dp(K+1,vector<int>(N+1,0)); 5 for(int j = 0;j <= N;++j) 6 dp[1][j] = j; 7 for(int i = 2;i <= K;++i){ 8 int x = 1; 9 for(int j = 1;j <= N;++j){ 10 while(x < j && dp[i][j-x] > dp[i-1][x]) 11 x++; 12 dp[i][j] = 1+dp[i][j-x]; 13 } 14 } 15 return dp[K][N]; 16 } 17 };
1 class Solution { 2 public: 3 int superEggDrop(int K, int N) { 4 vector<vector<int>> dp(K+1,vector<int>(N+1,0)); 5 for(int j = 0;j <= N;++j) 6 dp[1][j] = j; 7 for(int i = 2;i <= K;++i){ 8 for(int j = 1;j <= N;++j){ 9 int start = 1; 10 int end = j; 11 while(start < end){ 12 int pivot = start+(end-start)/2; 13 if(dp[i][j-pivot] > dp[i-1][pivot]) 14 start = pivot+1; 15 else 16 end = pivot; 17 } 18 dp[i][j] = 1+dp[i][j-start]; 19 } 20 } 21 return dp[K][N]; 22 } 23 };
原文地址:https://www.cnblogs.com/haoweizh/p/10182908.html