Space complexity

Space complexity is a metric that evaluates the total memory space required by an algorithm in terms of the size of its input data. It is analogous to time complexity but focuses on memory usage instead of execution time.

Space Related to Algorithms

Memory usage in an algorithm can be categorized into several types:

• Input space: Memory required to store the input data.
• Temporary space: Memory used to store intermediate data during execution. This includes:
  • Temporary data: Constants, variables, and objects used temporarily.
  • Stack frame space: Memory used to store function call contexts. This is allocated at the top of the stack for each function call and is freed when the function returns.
  • Instruction space: Memory used to store the program's compiled instructions, typically considered negligible for space complexity calculations.
• Output space: Memory used to store the algorithm's output.

Space complexity calculations generally consider the temporary data, stack frame space, and output data.

Calculation Method

Space complexity is calculated similarly to time complexity but measures memory usage instead. It primarily focuses on the worst-case scenario because memory requirements must be met regardless of input conditions. This includes considering the worst input data and the peak memory usage during execution.

Example Analysis

For a given input size $n$, the space complexity might vary based on conditional logic or recursion depth. Here's how different scenarios can affect space complexity:

• Non-recursive scenarios: Space complexity might remain constant if temporary variables do not depend on input size.
• Recursive scenarios: Consideration of the stack space used by recursive calls is crucial, as it can grow linearly or even exponentially with input size.

Common Types of Space Complexity

Space complexity can range from constant to exponential:

• Constant ($O (1)$): Space does not increase with input size. Typical of algorithms with a fixed number and size of variables.
• Logarithmic ($O (\log n)$): Space grows logarithmically in response to the input size. Common in divide-and-conquer algorithms like binary search.
• Linear ($O (n)$): Space grows directly proportionally to the input size. Typical of algorithms that store linear data structures like arrays.
• Quadratic ($O (n^2)$): Space grows as the square of the input size. Often seen in algorithms that handle two-dimensional arrays.
• Exponential ($O (2^n)$): Space doubles with each additional element of input. Common in recursive algorithms that generate a new branch for each call.

Balancing Time and Space

Optimizing an algorithm often requires balancing between minimizing time and space complexity:

• Space-time tradeoff: Increasing space complexity to reduce time complexity. This is common when memory resources are less critical than execution speed.
• Time-space tradeoff: Reducing space complexity at the cost of increased time complexity. This approach is useful in environments where memory is a limiting factor.

The choice of tradeoff depends on the specific requirements and constraints of the application, such as system memory limits and the need for rapid response times.

Reference and Useful Links

• 2.4 Space complexity - Hello Algo