What is Space complexity?

When we design an algorithm to solve a problem, it needs some computer memory to complete its execution. For any algorithm, memory is required for the following purposes…

1. To store program instructions.
2. To store constant values.
3. To store variable values.
4. And for few other things like funcion calls, jumping statements etc,.

Space complexity of an algorithm can be defined as follows…

Total amount of computer memory required by an algorithm to complete its execution is called as space complexity of that algorithm.

Generally, when a program is under execution it uses the computer memory for THREE reasons. They are as follows…

1. Instruction Space: It is the amount of memory used to store compiled version of instructions.
2. Environmental Stack: It is the amount of memory used to store information of partially executed functions at the time of function call.
3. Data Space: It is the amount of memory used to store all the variables and constants.

Note - When we want to perform analysis of an algorithm based on its Space complexity, we consider only Data Space and ignore Instruction Space as well as Environmental Stack. That means we calculate only the memory required to store Variables, Constants, Structures, etc.,

To calculate the space complexity, we must know the memory required to store different datatype values (according to the compiler). For example, the C Programming Language compiler requires the following…

1. 2 bytes to store Integer value.
2. 4 bytes to store Floating Point value.
3. 1 byte to store Character value.
4. 6 (OR) 8 bytes to store double value.

Consider the following piece of code…

Example 1:

int square(int a)
{
	return a*a;
}

In the above piece of code, it requires 2 bytes of memory to store variable ‘a’ and another 2 bytes of memory is used for return value.

That means, totally it requires 4 bytes of memory to complete its execution. And this 4 bytes of memory is fixed for any input value of ‘a’. This space complexity is said to be Constant Space Complexity.

If any algorithm requires a fixed amount of space for all input values then that space complexity is said to be Constant Space Complexity.

Consider the following piece of code…

Example 2:

int sum(int A[ ], int n)
{
   int sum = 0, i;
   for(i = 0; i < n; i++)
      sum = sum + A[i];
   return sum;
}

In the above piece of code it requires ‘n*2’ bytes of memory to store array variable ‘a[ ]’ 2 bytes of memory for integer parameter ‘n’ 4 bytes of memory for local integer variables ‘sum’ and ‘i’ (2 bytes each) 2 bytes of memory for return value.

That means, totally it requires ‘2n+8’ bytes of memory to complete its execution. Here, the total amount of memory required depends on the value of ‘n’. As ‘n’ value increases the space required also increases proportionately. This type of space complexity is said to be Linear Space Complexity.

If the amount of space required by an algorithm is increased with the increase of input value, then that space complexity is said to be Linear Space Complexity.