README for Longest Increasing Subsequence

Longest_Increasing_Subsequence/README.md

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+# Longest Increasing Subsequence
+
+The Longest Increasing Subsequence (LIS) problem is to find the length of the longest subsequence of a given sequence such that all elements of the subsequence are sorted in increasing order. For example, the length of LIS for `{10, 22, 9, 33, 21, 50, 41, 60, 80}` is `6` and LIS is `{10, 22, 33, 50, 60, 80}`.
+
+![longest-increasing-subsequence](https://media.geeksforgeeks.org/wp-content/cdn-uploads/Longest-Increasing-Subsequence.png)
+
+## Examples
+
+```
+Input  : arr[] = {3, 10, 2, 1, 20}
+Output : Length of LIS = 3
+The longest increasing subsequence is 3, 10, 20
+
+Input  : arr[] = {3, 2}
+Output : Length of LIS = 1
+The longest increasing subsequences are {3} and {2}
+
+Input : arr[] = {50, 3, 10, 7, 40, 80}
+Output : Length of LIS = 4
+The longest increasing subsequence is {3, 7, 40, 80}
+```
+
+## Recursive Approach
+
+```
+Let arr[0..n-1] be the input array and L(i) be the length of the LIS ending at index i such that arr[i] is the last element of the LIS.
+
+Then, L(i) can be recursively written as:
+
+L(i) = 1 + max( L(j) ) where 0 < j < i and arr[j] < arr[i]; or
+L(i) = 1, if no such j exists.
+
+To find the LIS for a given array, we need to return max(L(i)) where 0 < i < n.
+
+Thus, we see the LIS problem satisfies the optimal substructure property as the main problem can be solved using solutions to subproblems.
+```
+
+## Pseudo Code
+
+```
+LIS(A[1..n]):
+    Array L[1..n]
+    (* L[i] = value of LIS ending(A[1..i]) *)
+    for i = 1 to n do
+        L[i] = 1
+        for j = 1 to i − 1 do
+            if (A[j] < A[i]) do
+                L[i] = max(L[i], 1 + L[j])
+    return L
+
+MAIN(A[1..n]):
+    L = LIS(A[1..n])
+        return the maximum value in L
+```
+
+## Overlapping Subproblems:
+
+Considering the above implementation, following is recursion tree for an array of size 4. lis(n) gives us the length of LIS for arr[].
+
+```
+                 lis(4)
+        /          |        \
+      lis(3)     lis(2)   lis(1)
+     /     \       |
+   lis(2) lis(1) lis(1)
+   /
+lis(1)
+```
+
+
+# Time Complexity
+
+O ( n <sup>2</sup> )