给定一个数组A ,该数组A由大小为N的正整数组成。如果数组中索引为i的元素为K,则可以在索引范围(i + 1)到(i + K)之间跳转。
任务是找到使用模块10 9 + 7到达终点的可能方法的数量。
起始位置被视为索引0 。
例子:
Input: A = {5, 3, 1, 4, 3}
Output: 6
Input: A = {2, 3, 1, 1, 2}
Output: 4
天真的方法:我们可以形成一个递归结构来解决该问题。
令F [i]表示从索引i开始的路径数,如果元素A [i]为K,则在每个索引i处执行跳转的方式总数为:
F(i) = F(i+1) + F(i+2) +...+ F(i+k), where i + k <= n,
where F(n) = 1
通过使用此递归公式,我们可以解决问题:
C++
// C++ implementation of
// the above approach
#include
using namespace std;
const int mod = 1e9 + 7;
// Find the number of ways
// to reach the end
int ways(int i, int arr[], int n)
{
// Base case
if (i == n - 1)
return 1;
int sum = 0;
// Recursive structure
for (int j = 1;
j + i < n && j <= arr[i];
j++) {
sum += (ways(i + j,
arr, n))
% mod;
sum %= mod;
}
return sum % mod;
}
// Driver code
int main()
{
int arr[] = { 5, 3, 1, 4, 3 };
int n = sizeof(arr) / sizeof(arr[0]);
cout << ways(0, arr, n) << endl;
return 0;
}
Java
// Java implementation of
// the above approach
import java.io.*;
class GFG
{
static int mod = 1000000000;
// Find the number of ways
// to reach the end
static int ways(int i,
int arr[], int n)
{
// Base case
if (i == n - 1)
return 1;
int sum = 0;
// Recursive structure
for (int j = 1; j + i < n &&
j <= arr[i]; j++)
{
sum += (ways(i + j,
arr, n)) % mod;
sum %= mod;
}
return sum % mod;
}
// Driver code
public static void main (String[] args)
{
int arr[] = { 5, 3, 1, 4, 3 };
int n = arr.length;
System.out.println (ways(0, arr, n));
}
}
// This code is contributed by ajit
Python3
# Python3 implementation of
# the above approach
mod = 1e9 + 7;
# Find the number of ways
# to reach the end
def ways(i, arr, n):
# Base case
if (i == n - 1):
return 1;
sum = 0;
# Recursive structure
for j in range(1, arr[i] + 1):
if(i + j < n):
sum += (ways(i + j, arr, n)) % mod;
sum %= mod;
return int(sum % mod);
# Driver code
if __name__ == '__main__':
arr = [5, 3, 1, 4, 3];
n = len(arr);
print(ways(0, arr, n));
# This code is contributed by PrinciRaj1992
C#
// C# implementation of
// the above approach
using System;
class GFG
{
static int mod = 1000000000;
// Find the number of ways
// to reach the end
static int ways(int i,
int []arr, int n)
{
// Base case
if (i == n - 1)
return 1;
int sum = 0;
// Recursive structure
for (int j = 1; j + i < n &&
j <= arr[i]; j++)
{
sum += (ways(i + j,
arr, n)) % mod;
sum %= mod;
}
return sum % mod;
}
// Driver code
public static void Main (String[] args)
{
int []arr = { 5, 3, 1, 4, 3 };
int n = arr.Length;
Console.WriteLine(ways(0, arr, n));
}
}
// This code is contributed by 29AjayKumar
C++
// C++ implementation
#include
using namespace std;
const int mod = 1e9 + 7;
// find the number of ways to reach the end
int ways(int arr[], int n)
{
// dp to store value
int dp[n + 1];
// base case
dp[n - 1] = 1;
// Bottom up dp structure
for (int i = n - 2; i >= 0; i--) {
dp[i] = 0;
// F[i] is dependent of
// F[i+1] to F[i+k]
for (int j = 1; ((j + i) < n
&& j <= arr[i]);
j++) {
dp[i] += dp[i + j];
dp[i] %= mod;
}
}
// Return value of dp[0]
return dp[0] % mod;
}
// Driver code
int main()
{
int arr[] = { 5, 3, 1, 4, 3 };
int n = sizeof(arr) / sizeof(arr[0]);
cout << ways(arr, n) % mod << endl;
return 0;
}
Java
// Java implementation of above approach
class GFG
{
static final int mod = (int)(1e9 + 7);
// find the number of ways to reach the end
static int ways(int arr[], int n)
{
// dp to store value
int dp[] = new int[n + 1];
// base case
dp[n - 1] = 1;
// Bottom up dp structure
for (int i = n - 2; i >= 0; i--)
{
dp[i] = 0;
// F[i] is dependent of
// F[i+1] to F[i+k]
for (int j = 1; ((j + i) < n &&
j <= arr[i]); j++)
{
dp[i] += dp[i + j];
dp[i] %= mod;
}
}
// Return value of dp[0]
return dp[0] % mod;
}
// Driver code
public static void main (String[] args)
{
int arr[] = { 5, 3, 1, 4, 3 };
int n = arr.length;
System.out.println(ways(arr, n) % mod);
}
}
// This code is contributed by AnkitRai01
Python3
# Python3 implementation of above approach
mod = 10**9 + 7
# find the number of ways to reach the end
def ways(arr, n):
# dp to store value
dp = [0] * (n + 1)
# base case
dp[n - 1] = 1
# Bottom up dp structure
for i in range(n - 2, -1, -1):
dp[i] = 0
# F[i] is dependent of
# F[i + 1] to F[i + k]
j = 1
while((j + i) < n and j <= arr[i]):
dp[i] += dp[i + j]
dp[i] %= mod
j += 1
# Return value of dp[0]
return dp[0] % mod
# Driver code
arr = [5, 3, 1, 4, 3 ]
n = len(arr)
print(ways(arr, n) % mod)
# This code is contributed by SHUBHAMSINGH10
C#
// C# implementation of above approach
using System;
class GFG
{
static readonly int mod = (int)(1e9 + 7);
// find the number of ways to reach the end
static int ways(int []arr, int n)
{
// dp to store value
int []dp = new int[n + 1];
// base case
dp[n - 1] = 1;
// Bottom up dp structure
for (int i = n - 2; i >= 0; i--)
{
dp[i] = 0;
// F[i] is dependent of
// F[i+1] to F[i+k]
for (int j = 1; ((j + i) < n &&
j <= arr[i]); j++)
{
dp[i] += dp[i + j];
dp[i] %= mod;
}
}
// Return value of dp[0]
return dp[0] % mod;
}
// Driver code
public static void Main (String[] args)
{
int []arr = { 5, 3, 1, 4, 3 };
int n = arr.Length;
Console.WriteLine(ways(arr, n) % mod);
}
}
// This code is contributed by Rajput-Ji
输出:
6
高效方法:在以前的方法中,有些计算不只一次进行。最好将这些值存储在dp数组中,并且dp [i]将存储从索引i开始到数组末尾的路径数。
因此,dp [0]将是该问题的解决方案。
下面是该方法的实现:
C++
// C++ implementation
#include
using namespace std;
const int mod = 1e9 + 7;
// find the number of ways to reach the end
int ways(int arr[], int n)
{
// dp to store value
int dp[n + 1];
// base case
dp[n - 1] = 1;
// Bottom up dp structure
for (int i = n - 2; i >= 0; i--) {
dp[i] = 0;
// F[i] is dependent of
// F[i+1] to F[i+k]
for (int j = 1; ((j + i) < n
&& j <= arr[i]);
j++) {
dp[i] += dp[i + j];
dp[i] %= mod;
}
}
// Return value of dp[0]
return dp[0] % mod;
}
// Driver code
int main()
{
int arr[] = { 5, 3, 1, 4, 3 };
int n = sizeof(arr) / sizeof(arr[0]);
cout << ways(arr, n) % mod << endl;
return 0;
}
Java
// Java implementation of above approach
class GFG
{
static final int mod = (int)(1e9 + 7);
// find the number of ways to reach the end
static int ways(int arr[], int n)
{
// dp to store value
int dp[] = new int[n + 1];
// base case
dp[n - 1] = 1;
// Bottom up dp structure
for (int i = n - 2; i >= 0; i--)
{
dp[i] = 0;
// F[i] is dependent of
// F[i+1] to F[i+k]
for (int j = 1; ((j + i) < n &&
j <= arr[i]); j++)
{
dp[i] += dp[i + j];
dp[i] %= mod;
}
}
// Return value of dp[0]
return dp[0] % mod;
}
// Driver code
public static void main (String[] args)
{
int arr[] = { 5, 3, 1, 4, 3 };
int n = arr.length;
System.out.println(ways(arr, n) % mod);
}
}
// This code is contributed by AnkitRai01
Python3
# Python3 implementation of above approach
mod = 10**9 + 7
# find the number of ways to reach the end
def ways(arr, n):
# dp to store value
dp = [0] * (n + 1)
# base case
dp[n - 1] = 1
# Bottom up dp structure
for i in range(n - 2, -1, -1):
dp[i] = 0
# F[i] is dependent of
# F[i + 1] to F[i + k]
j = 1
while((j + i) < n and j <= arr[i]):
dp[i] += dp[i + j]
dp[i] %= mod
j += 1
# Return value of dp[0]
return dp[0] % mod
# Driver code
arr = [5, 3, 1, 4, 3 ]
n = len(arr)
print(ways(arr, n) % mod)
# This code is contributed by SHUBHAMSINGH10
C#
// C# implementation of above approach
using System;
class GFG
{
static readonly int mod = (int)(1e9 + 7);
// find the number of ways to reach the end
static int ways(int []arr, int n)
{
// dp to store value
int []dp = new int[n + 1];
// base case
dp[n - 1] = 1;
// Bottom up dp structure
for (int i = n - 2; i >= 0; i--)
{
dp[i] = 0;
// F[i] is dependent of
// F[i+1] to F[i+k]
for (int j = 1; ((j + i) < n &&
j <= arr[i]); j++)
{
dp[i] += dp[i + j];
dp[i] %= mod;
}
}
// Return value of dp[0]
return dp[0] % mod;
}
// Driver code
public static void Main (String[] args)
{
int []arr = { 5, 3, 1, 4, 3 };
int n = arr.Length;
Console.WriteLine(ways(arr, n) % mod);
}
}
// This code is contributed by Rajput-Ji
输出:
6
时间复杂度: O(K)