给定n个鸡蛋和k个地板,找到在最坏情况下找到所有地板都安全的地板所需的最少试验次数。如果从地板上扔鸡蛋不会破坏鸡蛋,则地板是安全的。请查看n个鸡蛋和k层地板。完整的陈述
例子
Input : n = 2, k = 10
Output : 4
We first try from 4-th floor. Two cases arise,
(1) If egg breaks, we have one egg left so we
need three more trials.
(2) If egg does not break, we try next from 7-th
floor. Again two cases arise.
We can notice that if we choose 4th floor as first
floor, 7-th as next floor and 9 as next of next floor,
we never exceed more than 4 trials.
Input : n = 2. k = 100
Output : 14
我们已经讨论了2个鸡蛋和k层地板的问题。我们还讨论了动态编程解决方案以找到该解决方案。动态编程解决方案基于问题的以下递归性质。让我们从不同的角度看待所讨论的递归公式。
我们可以用x次试验覆盖几层楼?
当我们放一个鸡蛋时,就会出现两种情况。
- 如果鸡蛋破裂,那么我们将进行x-1次试验和n-1个鸡蛋。
- 如果鸡蛋没有破裂,那么我们将进行x-1次试验和n个鸡蛋
Let maxFloors(x, n) be the maximum number of floors
that we can cover with x trials and n eggs. From above
two cases, we can write.
maxFloors(x, n) = maxFloors(x-1, n-1) + maxFloors(x-1, n) + 1
For all x >= 1 and n >= 1
Base cases :
We can't cover any floor with 0 trials or 0 eggs
maxFloors(0, n) = 0
maxFloors(x, 0) = 0
Since we need to cover k floors,
maxFloors(x, n) >= k ----------(1)
The above recurrence simplifies to following,
Refer this for proof.
maxFloors(x, n) = ∑xCi
1 <= i <= n ----------(2)
Here C represents Binomial Coefficient.
From above two equations, we can say.
∑xCj >= k
1 <= i <= n
Basically we need to find minimum value of x
that satisfies above inequality. We can find
such x using Binary Search.
C++
// C++ program to find minimum
// number of trials in worst case.
#include
using namespace std;
// Find sum of binomial coefficients xCi
// (where i varies from 1 to n).
int binomialCoeff(int x, int n)
{
int sum = 0, term = 1;
for (int i = 1; i <= n; ++i) {
term *= x - i + 1;
term /= i;
sum += term;
}
return sum;
}
// Do binary search to find minimum
// number of trials in worst case.
int minTrials(int n, int k)
{
// Initialize low and high as 1st
// and last floors
int low = 1, high = k;
// Do binary search, for every mid,
// find sum of binomial coefficients and
// check if the sum is greater than k or not.
while (low < high) {
int mid = (low + high) / 2;
if (binomialCoeff(mid, n) < k)
low = mid + 1;
else
high = mid;
}
return low;
}
/* Driver code*/
int main()
{
cout << minTrials(2, 10);
return 0;
}
Java
// Java program to find minimum
// number of trials in worst case.
class Geeks {
// Find sum of binomial coefficients xCi
// (where i varies from 1 to n). If the sum
// becomes more than K
static int binomialCoeff(int x, int n, int k)
{
int sum = 0, term = 1;
for (int i = 1; i <= n && sum < k; ++i) {
term *= x - i + 1;
term /= i;
sum += term;
}
return sum;
}
// Do binary search to find minimum
// number of trials in worst case.
static int minTrials(int n, int k)
{
// Initialize low and high as 1st
//and last floors
int low = 1, high = k;
// Do binary search, for every mid,
// find sum of binomial coefficients and
// check if the sum is greater than k or not.
while (low < high) {
int mid = (low + high) / 2;
if (binomialCoeff(mid, n, k) < k)
low = mid + 1;
else
high = mid;
}
return low;
}
/* Driver code*/
public static void main(String args[])
{
System.out.println(minTrials(2, 10));
}
}
// This code is contributed by ankita_saini
Python3
# Python3 program to find minimum
# number of trials in worst case.
# Find sum of binomial coefficients
# xCi (where i varies from 1 to n).
# If the sum becomes more than K
def binomialCoeff(x, n, k):
sum = 0;
term = 1;
i = 1;
while(i <= n and sum < k):
term *= x - i + 1;
term /= i;
sum += term;
i += 1;
return sum;
# Do binary search to find minimum
# number of trials in worst case.
def minTrials(n, k):
# Initialize low and high as
# 1st and last floors
low = 1;
high = k;
# Do binary search, for every
# mid, find sum of binomial
# coefficients and check if
# the sum is greater than k or not.
while (low < high):
mid = int((low + high) / 2);
if (binomialCoeff(mid, n, k) < k):
low = mid + 1;
else:
high = mid;
return int(low);
# Driver Code
print(minTrials(2, 10));
# This code is contributed
# by mits
C#
// C# program to find minimum
// number of trials in worst case.
using System;
class GFG
{
// Find sum of binomial coefficients
// xCi (where i varies from 1 to n).
// If the sum becomes more than K
static int binomialCoeff(int x,
int n, int k)
{
int sum = 0, term = 1;
for (int i = 1;
i <= n && sum < k; ++i)
{
term *= x - i + 1;
term /= i;
sum += term;
}
return sum;
}
// Do binary search to find minimum
// number of trials in worst case.
static int minTrials(int n, int k)
{
// Initialize low and high
// as 1st and last floors
int low = 1, high = k;
// Do binary search, for every
// mid, find sum of binomial
// coefficients and check if the
// sum is greater than k or not.
while (low < high)
{
int mid = (low + high) / 2;
if (binomialCoeff(mid, n, k) < k)
low = mid + 1;
else
high = mid;
}
return low;
}
// Driver Code
public static void Main()
{
Console.WriteLine(minTrials(2, 10));
}
}
// This code is contributed
// by Akanksha Rai(Abby_akku)
PHP
输出
4
时间复杂度: O(n Log k)