How to Optimize Auxiliary Space Of a DP Solution.

Space Optimization in Dynamic Programming Problems:

This is a way to optimize the Auxlillary space of a solution which also produces the same correct result as before optimization.

How to Optimize Space in the DP Solution:

The idea is to store only the values that are necessary to generate the result for the current state of the DP solution.

Avoid storing the states that do not contribute to the current state.

Follow the below steps to optimize the space of any DP solution:

• Identify the transition of states and observe clearly what states are needed to calculate the current state.
• Store all the previous states that are needed to calculate the current state separately in variables or arrays.
• After calculating the current state, update the previous states for the next state that is to be calculated.
• The result is stored in the first previous state if there are multiple previous states.

Let’s see these steps with examples,

Space optimization from O(N)Linear to O(1)Constant:

Calculate the nth Fibonacci number:

State: f[i] –> Denotes the ith Fibonacci number.

Transition: f[i] = f[i-1] + f[i-2]

Code:

C++

// C++ program for Fibonacci Series
// using Dynamic Programming
#include <bits/stdc++.h>
using namespace std;
class GFG {
public:
    int fib(int n)
    {
        // Declare an array to store
        // Fibonacci numbers.
        // 1 extra to handle
        // case, n = 0
        int f[n + 2];
        int i;

        // 0th and 1st number of the
        // series are 0 and 1
        f[0] = 0;
        f[1] = 1;

        for (i = 2; i <= n; i++) {

            // Add the previous 2 numbers
            // in the series and store it
            f[i] = f[i - 1] + f[i - 2];
        }
        return f[n];
    }
};
// Driver code
int main()
{
    GFG g;
    int n = 9;
    cout << g.fib(n);
    return 0;
}

Java

public class Fibonacci {
    public int fib(int n) {
        // Declare an array to store Fibonacci numbers.
        // 1 extra to handle case, n = 0
        int[] f = new int[n + 2];
        int i;

        // 0th and 1st number of the series are 0 and 1
        f[0] = 0;
        f[1] = 1;

        for (i = 2; i <= n; i++) {
            // Add the previous 2 numbers in the series and store it
            f[i] = f[i - 1] + f[i - 2];
        }
        return f[n];
    }

    // Driver code
    public static void main(String[] args) {
        Fibonacci fibonacci = new Fibonacci();
        int n = 9;
        System.out.println(fibonacci.fib(n));
    }
}

C#

using System;

public class Fibonacci
{
    public int Fib(int n)
    {
        // Declare an array to store Fibonacci numbers.
        // 1 extra to handle case, n = 0
        int[] f = new int[n + 2];
        int i;

        // 0th and 1st number of the series are 0 and 1
        f[0] = 0;
        f[1] = 1;

        for (i = 2; i <= n; i++)
        {
            // Add the previous 2 numbers in the series and store it
            f[i] = f[i - 1] + f[i - 2];
        }
        return f[n];
    }

    // Driver code
    public static void Main(string[] args)
    {
        Fibonacci fibonacci = new Fibonacci();
        int n = 9;
        Console.WriteLine(fibonacci.Fib(n));
    }
}

Javascript

class Fibonacci {
    fib(n) {
        // Declare an array to store Fibonacci numbers.
        // 1 extra to handle case, n = 0
        const f = new Array(n + 2);
        let i;

        // 0th and 1st number of the series are 0 and 1
        f[0] = 0;
        f[1] = 1;

        for (i = 2; i <= n; i++) {
            // Add the previous 2 numbers in the series and store it
            f[i] = f[i - 1] + f[i - 2];
        }
        return f[n];
    }
}

// Driver code
const fibonacci = new Fibonacci();
const n = 9;
console.log(fibonacci.fib(n));

Python3

class GFG:
    def fib(self, n):
        # Declare a list to store Fibonacci numbers.
        # 1 extra to handle the case, n = 0
        f = [0] * (n + 2)

        # 0th and 1st number of the series are 0 and 1
        f[0] = 0
        f[1] = 1

        for i in range(2, n + 1):
            # Add the previous 2 numbers
            # in the series and store it
            f[i] = f[i - 1] + f[i - 2]

        return f[n]

# Driver code
if __name__ == "__main__":
    g = GFG()
    n = 9
    print(g.fib(n))

Output

34

Time Complexity: O(N)
Auxiliary space: O(N)

Steps To Optimise the Space in action for above problem:

• If we observe clearly, To generate the f[i] we need only two previous states that is f[i-1] and f[i-2].
• Store previous two states in prev1 and prev2 and current in curr.
• After calculating the curr, update the prev2 as prev1 and prev1 as curr.
• At the end, result is stored in the prev1.

Space Optimised Code:

C++

// Fibonacci Series using Space Optimized Method
#include <bits/stdc++.h>
using namespace std;

int fib(int n)
{
    int prev2 = 0;
    if (n == 0)
        return prev2;
    int prev1 = 1;
    int curr;
    for (int i = 2; i <= n; i++) {
        curr = prev1 + prev2;
        prev2 = prev1;
        prev1 = curr;
    }
    return prev1;
}

// Driver code
int main()
{
    int n = 9;
    cout << fib(n);
    return 0;
}

Java

public class FibonacciSeries {
    public static int fib(int n) {
        int prev2 = 0;
        if (n == 0)
            return prev2;
        int prev1 = 1;
        int curr;
        for (int i = 2; i <= n; i++) {
            curr = prev1 + prev2;
            prev2 = prev1;
            prev1 = curr;
        }
        return prev1;
    }

    public static void main(String[] args) {
        int n = 9;
        System.out.println(fib(n));
    }
}

C#

using System;

class Program {
    // Function to calculate the Fibonacci number at index n
    static int Fibonacci(int n)
    {
        int prev2 = 0;
        if (n == 0)
            return prev2;
        int prev1 = 1;
        int curr;
        for (int i = 2; i <= n; i++) {
            curr = prev1 + prev2;
            prev2 = prev1;
            prev1 = curr;
        }
        return prev1;
    }

    // Main method
    static void Main(string[] args)
    {
        int n = 9; // Index of the Fibonacci number to find
        Console.WriteLine(Fibonacci(
            n)); // Print the Fibonacci number at index n
    }
}

Javascript

// Function to calculate the Fibonacci series using space-optimized method
function fib(n) {
    let prev2 = 0; // Initialize the first Fibonacci number (0)
    if (n === 0) return prev2; // If n is 0, return the first Fibonacci number
    let prev1 = 1; // Initialize the second Fibonacci number (1)
    let curr; // Initialize the current Fibonacci number
    for (let i = 2; i <= n; i++) { // Loop through from the 3rd Fibonacci number to the nth Fibonacci number
        curr = prev1 + prev2; // Calculate the current Fibonacci number by adding the previous two Fibonacci numbers
        prev2 = prev1; // Update the previous two Fibonacci numbers for the next iteration
        prev1 = curr;
    }
    return prev1;  
}

// Driver code
let n = 9; 
console.log(fib(n));  

Python3

def fib(n):
    prev2, prev1 = 0, 1
    if n == 0:
        return prev2
    for i in range(2, n + 1):
        curr = prev1 + prev2
        prev2, prev1 = prev1, curr
    return prev1

# Driver code
if __name__ == "__main__":
    n = 9
    print(fib(n))

Output

34

Time Complexity: O(N)
Auxillary space: O(1)

Space optimization from O(N*Sum)Quadratic to O(Sum)Linear In Subset Sum Problem:

Subset Sum problem Using Dynamic programming:

In approach of dynamic programming we have derived the relation between states as given below:

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]]

Steps To Optimize Space in action for above problem:

• If we observe that for calculating current dp[i][j] state we only need previous row dp[i-1][j] or dp[i-1][j-A[i-1]].There is no need to store all the previous states just one previous state is used to compute result.
• Define two arrays prev and curr of size Sum+1 to store the just previous row result and current row result respectively.
• Once curr array is calculated then curr becomes our prev for the next row.
• When all rows are processed the answer is stored in prev array.

Space optimised Implementation:

C++

// A Dynamic Programming solution
// for subset sum problem with space optimization
#include <bits/stdc++.h>
using namespace std;

// Returns true if there is a subset of set[]
// with sum equal to given sum
bool isSubsetSum(int set[], int n, int sum)
{

    vector<bool> prev(sum + 1);

    // If sum is 0, then answer is true
    for (int i = 0; i <= n; i++)
        prev[0] = true;

    // If sum is not 0 and set is empty,
    // then answer is false
    for (int i = 1; i <= sum; i++)
        prev[i] = false;

    // curr array to store the current row result generated
    // with help of prev array
    vector<bool> curr(sum + 1);

    for (int i = 1; i <= n; i++) {
        for (int j = 1; j <= sum; j++) {
            if (j < set[i - 1])
                curr[j] = prev[j];
            if (j >= set[i - 1])
                curr[j] = prev[j] || prev[j - set[i - 1]];
        }
        // now curr becomes prev for i+1 th element
        prev = curr;
    }

    return prev[sum];
}

// Driver code
int main()
{
    int set[] = { 3, 34, 4, 12, 5, 2 };
    int sum = 9;
    int n = sizeof(set) / sizeof(set[0]);
    if (isSubsetSum(set, n, sum) == true)
        cout << "Found a subset with given sum";
    else
        cout << "No subset with given sum";
    return 0;
}

Java

import java.util.*;

class SubsetSum {
    // Returns true if there is a subset of set[]
    // with sum equal to given sum
    static boolean isSubsetSum(int[] set, int n, int sum) {
        boolean[] prev = new boolean[sum + 1];

        // If sum is 0, then answer is true
        for (int i = 0; i <= n; i++)
            prev[0] = true;

        // If sum is not 0 and set is empty,
        // then answer is false
        for (int i = 1; i <= sum; i++)
            prev[i] = false;

        // curr array to store the current row result generated
        // with help of prev array
        boolean[] curr = new boolean[sum + 1];

        for (int i = 1; i <= n; i++) {
            for (int j = 1; j <= sum; j++) {
                if (j < set[i - 1])
                    curr[j] = prev[j];
                if (j >= set[i - 1])
                    curr[j] = prev[j] || prev[j - set[i - 1]];
            }
            // now curr becomes prev for i+1 th element
            prev = curr.clone();
        }

        return prev[sum];
    }

    // Driver code
    public static void main(String args[]) {
        int[] set = { 3, 34, 4, 12, 5, 2 };
        int sum = 9;
        int n = set.length;
        if (isSubsetSum(set, n, sum))
            System.out.println("Found a subset with given sum");
        else
            System.out.println("No subset with given sum");
    }
}
//This code is contributed by Utkarsh

C#

using System;

public class SubsetSum {
    // Returns true if there is a subset of set[]
    // with sum equal to given sum
    public static bool IsSubsetSum(int[] set, int n,
                                   int sum)
    {
        // Create a boolean array to store the results
        // of subproblems. The value dp[i, j] will be
        // true if there is a subset of set[0..j-1]
        // with sum equal to i.
        bool[, ] dp = new bool[sum + 1, n + 1];

        // If sum is 0, then answer is true
        for (int i = 0; i <= n; i++)
            dp[0, i] = true;

        // If sum is not 0 and set is empty,
        // then answer is false
        for (int i = 1; i <= sum; i++)
            dp[i, 0] = false;

        // Fill dp[][] in bottom up manner
        for (int i = 1; i <= sum; i++) {
            for (int j = 1; j <= n; j++) {
                dp[i, j] = dp[i, j - 1];
                if (i >= set[j - 1])
                    dp[i, j] = dp[i, j]
                               || dp[i - set[j - 1], j - 1];
            }
        }

        return dp[sum, n];
    }

    // Driver code
    public static void Main(string[] args)
    {
        int[] set = { 3, 34, 4, 12, 5, 2 };
        int sum = 9;
        int n = set.Length;
        if (IsSubsetSum(set, n, sum))
            Console.WriteLine(
                "Found a subset with given sum");
        else
            Console.WriteLine("No subset with given sum");
    }
}

Javascript

// Returns true if there is a subset of set[]
// with sum equal to given sum
function isSubsetSum(set, n, sum) {
    let prev = new Array(sum + 1);

    // If sum is 0, then answer is true
    for (let i = 0; i <= n; i++)
        prev[0] = true;

    // If sum is not 0 and set is empty,
    // then answer is false
    for (let i = 1; i <= sum; i++)
        prev[i] = false;

    // curr array to store the current row result generated
    // with help of prev array
    let curr = new Array(sum + 1);

    for (let i = 1; i <= n; i++) {
        for (let j = 1; j <= sum; j++) {
            if (j < set[i - 1])
                curr[j] = prev[j];
            if (j >= set[i - 1])
                curr[j] = prev[j] || prev[j - set[i - 1]];
        }
        // now curr becomes prev for i+1 th element
        prev = curr.slice(); // Copying array to avoid reference sharing
    }

    return prev[sum];
}

// Driver code
function main() {
    let set = [3, 34, 4, 12, 5, 2];
    let sum = 9;
    let n = set.length;
    if (isSubsetSum(set, n, sum))
        console.log("Found a subset with given sum");
    else
        console.log("No subset with given sum");
}

// Call the main function
main();

Python3

def isSubsetSum(set, n, sum):
    prev = [False] * (sum + 1)

    # If sum is 0, then answer is true
    for i in range(n + 1):
        prev[0] = True

    # If sum is not 0 and set is empty,
    # then answer is false
    for i in range(1, sum + 1):
        prev[i] = False

    # curr array to store the current row result generated
    # with help of prev array
    curr = [False] * (sum + 1)

    for i in range(1, n + 1):
        for j in range(1, sum + 1):
            if j < set[i - 1]:
                curr[j] = prev[j]
            if j >= set[i - 1]:
                curr[j] = prev[j] or prev[j - set[i - 1]]
        # now curr becomes prev for i+1 th element
        prev = curr[:]

    return prev[sum]


# Driver code
if __name__ == "__main__":
    set = [3, 34, 4, 12, 5, 2]
    sum = 9
    n = len(set)
    if isSubsetSum(set, n, sum):
        print("Found a subset with given sum")
    else:
        print("No subset with given sum")

Output

Found a subset with given sum

Complexity Analysis:

• Time Complexity: O(sum * n), where n is the size of the array.
• Auxiliary Space: O(sum), as the size of the 1-D array is sum+1.