Cycle Detection in a Directed Graph (Using DFS)

Graph Algorithms/Cycle detection using dfs/Readme.md

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+# Longest Increasing Subsequence in C
+
+This program calculates the **Longest Increasing Subsequence (LIS)** in an array of integers, using dynamic programming. It allows users to input the array size and its elements, then outputs the length of the longest increasing subsequence.
+
+## How It Works
+
+The program uses a dynamic programming approach where:
+- An auxiliary array, `lis`, is maintained to store the LIS length at each index.
+- For each element, it calculates the maximum LIS that ends at that position by considering all previous elements.
+- Finally, it finds the maximum value in the `lis` array to determine the length of the longest increasing subsequence.
+
+### Time Complexity
+The time complexity of this approach is **O(n²)** due to the nested loops.
+
+### Space Complexity
+The space complexity is **O(n)**, as it uses an auxiliary array `lis` of the same length as the input array.
+
+## Getting Started
+
+### Prerequisites
+
+- A C compiler (like GCC).
+
+### Running the Program
+
+1. **Clone the Repository** (optional if using version control):
+   ```bash
+   git clone https://github.com/your-username/longest-increasing-subsequence
+   cd longest-increasing-subsequence
+2. **Compile the Code **:
+
+```bash
+    Copy code
+    gcc lis.c -o lis
+
+3. **Run the Program**:
+
+```bash
+    Copy code
+    ./lis
+
+**Example Usage**
+```bash
+Enter the number of elements in the array: 9
+Enter the elements of the array:
+10 22 9 33 21 50 41 60 80
+Length of LIS is 6
+
+In this example, the longest increasing subsequence is [10, 22, 33, 50, 60, 80] with a length of 6.

Graph Algorithms/Cycle detection using dfs/code.c

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+#include <stdio.h>
+
+int longestIncreasingSubsequence(int arr[], int n)
+{
+    int lis[n];
+    for (int i = 0; i < n; i++)
+        lis[i] = 1; // Initialize LIS values for all indexes as 1
+
+    for (int i = 1; i < n; i++)
+    {
+        for (int j = 0; j < i; j++)
+        {
+            if (arr[i] > arr[j] && lis[i] < lis[j] + 1)
+            {
+                lis[i] = lis[j] + 1;
+            }
+        }
+    }
+
+    int max = 0;
+    for (int i = 0; i < n; i++)
+    {
+        if (max < lis[i])
+            max = lis[i];
+    }
+    return max;
+}
+
+int main()
+{
+    int n;
+
+    // Take the number of elements as input
+    printf("Enter the number of elements in the array: ");
+    scanf("%d", &n);
+
+    int arr[n];
+
+    // Take array elements as input
+    printf("Enter the elements of the array:\n");
+    for (int i = 0; i < n; i++)
+    {
+        scanf("%d", &arr[i]);
+    }
+
+    // Calculate and print the length of LIS
+    printf("Length of LIS is %d\n", longestIncreasingSubsequence(arr, n));
+    return 0;
+}