Some new dp problems added

Dynamic_Programming/Longest_Increasing_Subsequence.py

@@ -0,0 +1,21 @@
+'''
+Task : Given an integer array nums, return the length of the longest strictly increasing subsequence
+Leetcode problem 300.
+'''
+
+def solve(nums) :
+    dp = [1]*len(nums)
+    for i in range(len(nums)) :
+        for j in range(i) :
+            if nums[j]<nums[i] : dp[i] = max(dp[i], dp[j]+1)
+    return max(dp)
+
+def inlt():
+    return(list(map(int,input().split())))
+
+if __name__=='__main__' :
+    print('Longest Increasing Subsequence Subroutine')
+    print('Input Space Separated Integers')
+    a = inlt()
+    ans = solve(a)
+    print(ans)

Dynamic_Programming/Max_Product_Subarray

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+'''
+Task : Given an integer array nums, find a  subarray that has the largest product, and return the product.
+Leetcode problem 152.
+'''
+
+def maxProduct(nums) :
+    if len(nums)==1 : return nums[0]
+    if len(nums)==2 : return max(nums[0]*nums[1], max(nums))
+
+    # store the largest and smallest pdt ending at each index
+    mx = nums[0]
+    mn = nums[0]
+    ans = nums[0]
+    for num in nums[1:] :
+        temp = max(num, mx*num, mn*num) # 6
+        mn = min(num, mx*num, mn*num) # -3
+        mx = temp
+        ans = max(ans, mx) # 6
+    return ans
+
+def inlt():
+    return(list(map(int,input().split())))
+
+if __name__=='__main__' : 
+    print('Max Product Subarray Subroutine')
+    print('Input Space Separated Integers')
+    a = inlt()
+    ans = maxProduct(a)
+    print(ans)

Dynamic_Programming/README.md

@@ -1,50 +1 @@
-# Longest Common Subsequence
-
-The longest common subsequence problem is finding the longest sequence 
-which exists in both the given strings.
-
-Subsequence
-Let us consider a sequence S = <s1, s2, s3, s4, …,sn>.
-
-A sequence Z = <z1, z2, z3, z4, …,zm> over S is called a subsequence of S, 
-if and only if it can be derived from S deletion of some elements.
-
-Common Subsequence
-Suppose, X and Y are two sequences over a finite set of elements. 
-We can say that Z is a common subsequence of X and Y, if Z is a subsequence of both X and Y.
-
-Longest Common Subsequence
-If a set of sequences are given, the longest common subsequence problem is to 
-find a common subsequence of all the sequences that is of maximal length.
-
-The longest common subsequence problem is a classic computer science problem, 
-the basis of data comparison programs such as the diff-utility, and has applications 
-in bioinformatics. It is also widely used by revision control systems, such as SVN 
-and Git, for reconciling multiple changes made to a revision-controlled collection of files.
-
-Naïve Method
-Let X be a sequence of length m and Y a sequence of length n. Check for every subsequence 
-of X whether it is a subsequence of Y, and return the longest common subsequence found.
-
-There are 2m subsequences of X. Testing sequences whether or not it is a subsequence of Y 
-takes O(n) time. Thus, the naïve algorithm would take O(n2m) time.
-
-Dynamic Programming
-Let X = < x1, x2, x3,…, xm > and Y = < y1, y2, y3,…, yn > be the sequences. 
-To compute the length of an element the following algorithm is used.
-
-In this procedure, table C[m, n] is computed in row major order and 
-another table B[m,n] is computed to construct optimal solution.
-
-# Example
-In this example, we have two strings X = BACDB and Y = BDCB to find the longest common subsequence.
-
-Following the algorithm LCS-Length-Table-Formulation (as stated above), we have calculated table C 
-(shown on the left hand side) and table B (shown on the right hand side).
-
-In table B, we are using ‘D’, ‘L’ and ‘U’ for diagonal, left and up
-respectively. After generating table B, the LCS is determined by function LCS-Print. The result is BCB.
-
-As can be seen in the image below on the left, we are recurring from bottom to top to get the desired subsequence
-
-![alt text](https://www.tutorialspoint.com/design_and_analysis_of_algorithms/images/lcs.jpg)
+Files containing solutions to some basic problems on Dynamic Programming.