Sponsored
Sponsored
This approach leverages dynamic programming to find the number of ways to stay at index 0 after a given number of steps. We define a 2D table dp[i][j] where i represents the number of steps remaining, and j represents the current position of the pointer.
To optimize computation, we can limit the table size to the minimum of steps and arrLen since going beyond these positions is unnecessary.
Time Complexity: O(steps * min(steps, arrLen))
Space Complexity: O(steps * min(steps, arrLen))
1const MOD = 1000000007;
2
3function numWays(steps, arrLen) {
4 const maxPos = Math.min(steps, arrLen -
This JavaScript implementation follows the dynamic programming strategy to compute the required count of ways. It controls the array size using generic JavaScript functionality such that the number of computations remains efficient and manageable.
This approach utilizes recursion combined with memoization to optimize the recursive calls. Here, recursion is used to explore all possible paths dynamically adjusting by staying at, moving left, or moving right from each position in every step.
The results of the recursive calls are stored in a memoization table to avoid redundant calculations.
Time Complexity: O(steps * min(steps, arrLen))
Space Complexity: O(steps * min(steps, arrLen))
#include <vector>
#define MOD 1000000007
using namespace std;
vector<vector<int>> cache;
int numWaysHelper(int steps, int pos, int maxPos) {
if (pos < 0 || pos > maxPos) return 0;
if (steps == 0) return pos == 0 ? 1 : 0;
if (cache[steps][pos] != -1) return cache[steps][pos];
int stay = numWaysHelper(steps - 1, pos, maxPos);
int left = numWaysHelper(steps - 1, pos - 1, maxPos);
int right = numWaysHelper(steps - 1, pos + 1, maxPos);
return cache[steps][pos] = ((stay + left) % MOD + right) % MOD;
}
int numWays(int steps, int arrLen) {
int maxPos = min(steps, arrLen - 1);
cache.assign(steps + 1, vector<int>(maxPos + 1, -1));
return numWaysHelper(steps, 0, maxPos);
}
int main() {
cout << numWays(3, 2) << endl; // Output: 4
return 0;
}A C++ implementation of recursion with memoization, using vectors to dynamically manage memory and cache recursive results, considerably speeding up the process.