Merge pull request #173 from abhisheks008/main

Subset Sum Problem using Dynamic Programming

Author: prathimacode-hub

Dynamic Programming/Subset Sum Problem/Images/subset1.png

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+# Subset Sum Problem using Dynamic Programming
+Language used : **Python 3**
+
+## 🎯 Aim
+The aim of this script is to find out if there is a subset of the given set with sum equal to given sum.
+
+## 👉 Purpose
+The main purpose of this script is to show the implementation of Dynamic Programming to find out if there is a subset of the given set with sum equal to given sum.
+
+## 📄 Description
+Given a set of non-negative integers, and a value sum, determine if there is a subset of the given set with sum equal to given sum. 
+```
+Example: 
+
+Input: set[] = {3, 34, 4, 12, 5, 2}, sum = 9
+Output: True  
+There is a subset (4, 5) with sum 9.
+
+Input: set[] = {3, 34, 4, 12, 5, 2}, sum = 30
+Output: False
+There is no subset that add up to 30.
+```
+
+## 📈 Workflow of the script
+- `isSubsetSum` - Returns true if there is a subset of set[] with sum equal to given sum.
+- `main` - This is the driver code for this python script.
+
+## 📃 Explanation
+We will create a 2D array of `size (arr.size() + 1) * (target + 1)` of type boolean. The state `DP[i][j]` will be true if there exists a subset of elements from `A[0….i]` with sum value = `‘j’`. The approach for the problem is: 
+```
+if (A[i-1] > j)
+DP[i][j] = DP[i-1][j]
+else 
+DP[i][j] = DP[i-1][j] OR DP[i-1][j-A[i-1]]
+```
+1. This means that if current element has value greater than ‘current sum value’ we will copy the answer for previous cases
+2. And if the current sum value is greater than the `‘ith’` element we will see if any of previous states have already experienced the `sum=’j’` OR any previous states experienced a value `‘j – A[i]’` which will solve our purpose.
+
+## 🧮 Algorithm
+The below simulation will clarify the above approach: 
+```
+set[]={3, 4, 5, 2}
+target=6
+ 
+    0    1    2    3    4    5    6
+
+0   T    F    F    F    F    F    F
+
+3   T    F    F    T    F    F    F
+     
+4   T    F    F    T    T    F    F   
+      
+5   T    F    F    T    T    T    F
+
+2   T    F    T    T    T    T    T
+```
+
+## 💻 Input and Output 
+- **Test Case 1 :**
+
+![](https://github.com/abhisheks008/PyAlgo-Tree/blob/main/Dynamic%20Programming/Subset%20Sum%20Problem/Images/subset1.png)
+
+- **Test Case 2 :**
+
+![](https://github.com/abhisheks008/PyAlgo-Tree/blob/main/Dynamic%20Programming/Subset%20Sum%20Problem/Images/subset2.png)
+
+- **Test Case 3 :**
+
+![](https://github.com/abhisheks008/PyAlgo-Tree/blob/main/Dynamic%20Programming/Subset%20Sum%20Problem/Images/subset3.png)
+
+- **Test Case 3 :**
+
+![](https://github.com/abhisheks008/PyAlgo-Tree/blob/main/Dynamic%20Programming/Subset%20Sum%20Problem/Images/subset4.png)
+
+
+## ⏰ Time and Space complexity
+- **Time Complexity:** `O(sum*n)`, where sum is the ‘target sum’ and ‘n’ is the size of array. 
+- **Space Complexity:** `O(sum*n)`, as the size of 2-D array is `sum*n`.
+
+---------------------------------------------------------------
+## 🖋️ Author
+**Code contributed by, _Abhishek Sharma_, 2021 [@abhisheks008](github.com/abhisheks008)**
+
+[![forthebadge made-with-python](http://ForTheBadge.com/images/badges/made-with-python.svg)](https://www.python.org/)

Dynamic Programming/Subset Sum Problem/Images/subset2.png

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+# Subset Sum Problem using Dynamic Programming
+# Directory : Dynamic Programming
+# Language used : Python 3
+# Abhishek S, 2021
+
+# ------------------------------------------------------------------
+
+# Problem Statement : A Dynamic Programming solution for subset
+#                     sum problem Returns true if there is a subset of
+#                     set[] with sun equal to given sum
+
+# ------------------------------------------------------------------
+
+# Solution : Creating the Python script to find out the required output.
+
+
+ 
+# Returns true if there is a subset of set[]
+# with sum equal to given sum
+def isSubsetSum(set, n, sum):
+     
+    # The value of subset[i][j] will be
+    # true if there is a
+    # subset of set[0..j-1] with sum equal to i
+    subset =([[False for i in range(sum + 1)]
+            for i in range(n + 1)])
+     
+    # If sum is 0, then answer is true
+    for i in range(n + 1):
+        subset[i][0] = True
+         
+    # If sum is not 0 and set is empty,
+    # then answer is false
+    for i in range(1, sum + 1):
+         subset[0][i]= False
+             
+    # Fill the subset table in bottom up manner
+    for i in range(1, n + 1):
+        for j in range(1, sum + 1):
+            if j<set[i-1]:
+                subset[i][j] = subset[i-1][j]
+            if j>= set[i-1]:
+                subset[i][j] = (subset[i-1][j] or
+                                subset[i - 1][j-set[i-1]])
+     
+    # uncomment this code to print table
+    # for i in range(n + 1):
+    # for j in range(sum + 1):
+    # print (subset[i][j], end =" ")
+    # print()
+    return subset[n][sum]
+         
+# Driver code
+if __name__=='__main__':
+    print (" - Subset Sum Problem using Dynamic Programming - ")
+    print ("--------------------------------------------------")
+    print ()
+    set = list(map(int, input("Enter the set of numbers : ").split(" ")))
+    print ()
+    sum = int(input("Enter the subset sum : "))
+    print ()
+    print ("--------------------------------------------------")
+    print ()
+    print ("Calculating the result...")
+    print ()
+    print ("The Output is : ")
+    n = len(set)
+    if (isSubsetSum(set, n, sum) == True):
+        print("Found a subset with given sum")
+    else:
+        print("No subset with given sum")
+        
+        
+# ------------------------------------------------------------------
+# Input Cases :
+# Enter the set of numbers : 1 5 3 7 4
+# Enter the subset sum : 12
+
+# ------------------------------------------------------------------
+# Output :
+# Calculating the result...
+
+# The Output is : 
+# Found a subset with given sum
+
+# ------------------------------------------------------------------
+
+# Code contributed by, Abhishek Sharma, 2021