给定一个 N*M 阶矩阵 M,其中Mij表示 j 在第 i 行中的出现。任务是在应用给定操作后找到给定 k 在最后一行中出现的概率:
- 从第一行开始,我们从整行的任何列中选择一个元素,并将其添加到下一行的同一列中。我们重复这个直到最后一行。
例子:
Input: k = 1, M[][] =
{{0, 1},
{1, 1}}
Output:0.666667
Input: k = 1, M[][] =
{{0, 1},
{1, 1}}
Output: 0.333333
方法 :
- 预先计算一个 sum[] 数组,该数组存储第一行所有元素的总和。
- 用第一行 M[][] 元素填充 dp[][] 的第一行。
- 计算从特定行的每一列中选择一个元素的概率,并将其添加到相应的列中。
- 此外,在更新 dp[][] 矩阵的值的同时更新行的 Sum[]。
- 最后,给定 k 的 dp[n][k] 值是所有迭代后选择元素 k 的概率所需的值。
下面是上述方法的实现:
C++
// C++ implementation of the above approach
#include
#define n 4
#define m 4
using namespace std;
// Function to calculate probability
float calcProbability(int M[][m], int k)
{
// declare dp[][] and sum[]
float dp[m][n], sum[n];
// precalculate the first row
for (int j = 0; j < n; j++) {
dp[0][j] = M[0][j];
sum[0] += dp[0][j];
}
// calculate the probability for
// each element and update dp table
for (int i = 1; i < m; i++) {
for (int j = 0; j < n; j++) {
dp[i][j] += dp[i - 1][j] / sum[i - 1] + M[i][j];
sum[i] += dp[i][j];
}
}
// return result
return dp[n - 1][k - 1] / sum[n - 1];
}
// Driver code
int main()
{
int M[m][n] = { { 1, 1, 0, 3 },
{ 2, 3, 2, 3 },
{ 9, 3, 0, 2 },
{ 2, 3, 2, 2 } };
int k = 3;
cout << calcProbability(M, k);
return 0;
}
Java
// Java implementation of the above approach
class GFG
{
final static int n = 4 ;
final static int m = 4 ;
// Function to calculate probability
static float calcProbability(int M[][], int k)
{
// declare dp[][] and sum[]
float dp[][] = new float[m][n] ;
float sum[] = new float[n];
// precalculate the first row
for (int j = 0; j < n; j++)
{
dp[0][j] = M[0][j];
sum[0] = sum[0] + dp[0][j];
}
// calculate the probability for
// each element and update dp table
for (int i = 1; i < m; i++)
{
for (int j = 0; j < n; j++)
{
dp[i][j] += dp[i - 1][j] / sum[i - 1] +
M[i][j];
sum[i] += dp[i][j];
}
}
// return result
return dp[n - 1][k - 1] / sum[n - 1];
}
// Driver code
public static void main(String []args)
{
int M[][] = { { 1, 1, 0, 3 },
{ 2, 3, 2, 3 },
{ 9, 3, 0, 2 },
{ 2, 3, 2, 2 } };
int k = 3;
System.out.println(calcProbability(M, k));
}
}
// This code is contributed by Ryuga
Python3
# Python3 implementation of the
# above approach
n = 4
m = 4
# Function to calculate probability
def calcProbability(M, k):
# declare dp[][] and sum[]
dp = [[0 for i in range(n)]
for i in range(m)]
Sum = [0 for i in range(n)]
# precalculate the first row
for j in range(n):
dp[0][j] = M[0][j]
Sum[0] += dp[0][j]
# calculate the probability for
# each element and update dp table
for i in range(1, m):
for j in range(n):
dp[i][j] += (dp[i - 1][j] /
Sum[i - 1] + M[i][j])
Sum[i] += dp[i][j]
# return result
return dp[n - 1][k - 1] / Sum[n - 1]
# Driver code
M = [[ 1, 1, 0, 3 ],
[ 2, 3, 2, 3 ],
[ 9, 3, 0, 2 ],
[ 2, 3, 2, 2 ]]
k = 3
print(calcProbability(M, k))
# This code is contributed
# by mohit kumar 29
C#
// C# implementation of the above approach
using System;
class GFG
{
static int n = 4 ;
static int m = 4 ;
// Function to calculate probability
static float calcProbability(int[,] M, int k)
{
// declare dp[][] and sum[]
float[,] dp = new float[m,n] ;
float[] sum = new float[n];
// precalculate the first row
for (int j = 0; j < n; j++)
{
dp[0, j] = M[0, j];
sum[0] = sum[0] + dp[0, j];
}
// calculate the probability for
// each element and update dp table
for (int i = 1; i < m; i++)
{
for (int j = 0; j < n; j++)
{
dp[i, j] += dp[i - 1,j] / sum[i - 1] +
M[i, j];
sum[i] += dp[i, j];
}
}
// return result
return dp[n - 1,k - 1] / sum[n - 1];
}
// Driver code
public static void Main()
{
int[,] M = { { 1, 1, 0, 3 },
{ 2, 3, 2, 3 },
{ 9, 3, 0, 2 },
{ 2, 3, 2, 2 } };
int k = 3;
Console.Write(calcProbability(M, k));
}
}
// This code is contributed by Ita_c.
PHP
Javascript
输出:
0.201212
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