Merge branch 'AlgoGenesis:main' into dice-roll

Dynamic Programming/CLIMBING STAIR/README.md

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+Climbing Stairs Problem
+This program provides a solution to the "Climbing Stairs" problem, a classic example of dynamic programming. The problem can be stated as follows:
+
+Problem Statement:
+A person is at the bottom of a staircase with n steps and wants to reach the top. They can take either 1 or 2 steps at a time. The task is to determine the number of distinct ways the person can reach the top of the staircase.
+
+Approach:s
+Dynamic Programming
+
+This problem has overlapping subproblems, making it a suitable candidate for dynamic programming.
+Define dp[i] as the number of ways to reach the i-th step.
+The number of ways to reach step i is the sum of ways to reach the previous step i-1 and the step before that, i-2. This is because the person can arrive at step i by taking a single step from i-1 or a double step from i-2.
+Thus, the recurrence relation is:
+
+𝑑𝑝[𝑖]=𝑑𝑝[𝑖−1]+𝑑𝑝[𝑖−2]dp[i]=dp[i−1]+dp[i−2]
+Base Cases:
+
+If there are no steps (n = 0), there is 1 way (doing nothing).
+If there is one step (n = 1), there is also 1 way to reach it.
+Building the Solution:
+
+Create an array dp of size n+1 to store the number of ways to reach each step up to n.
+Initialize dp[0] = 1 and dp[1] = 1 as per the base cases.
+Use a loop to fill the array from dp[2] up to dp[n] using the recurrence relation.
+Finally, dp[n] will contain the number of distinct ways to reach the top of the staircase with n steps.
+Example:
+For n = 5, the program calculates the ways as follows:
+
+dp[0] = 1
+dp[1] = 1
+dp[2] = dp[1] + dp[0] = 2
+dp[3] = dp[2] + dp[1] = 3
+dp[4] = dp[3] + dp[2] = 5
+dp[5] = dp[4] + dp[3] = 8
+So, there are 8 distinct ways to reach the 5th step.
+
+Complexity Analysis:
+Time Complexity: O(n) because we only need to compute each value in dp from 0 to n.
+Space Complexity: O(n) for storing the dp array.
+This dynamic programming approach efficiently computes the number of ways to climb the staircase, demonstrating how overlapping subproblems and optimal substructure can be leveraged to solve problems effectively.

Dynamic Programming/CLIMBING STAIR/program.c

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+#include <stdio.h>
+
+int climbStairs(int n) 
+{
+    if (n == 0 || n == 1) 
+    {
+        return 1;
+    }
+
+    int dp[n + 1];
+    dp[0] = dp[1] = 1;
+
+    for (int i = 2; i <= n; i++) 
+    {
+        dp[i] = dp[i - 1] + dp[i - 2];
+    }
+
+    return dp[n];
+}
+
+int main() 
+{
+    int n = 5; // Example input
+    printf("Ways to climb %d stairs: %d\n", n, climbStairs(n));
+    return 0;
+}