Algorithm Complexity

Algorithm Complexity is a measure of the efficiency of an algorithm in terms of time and space usage as the input size grows. It helps in comparing different algorithms and understanding their performance characteristics.

1 Key Concepts

• Time Complexity: Measures the amount of time an algorithm takes to complete as a function of input size.
• Space Complexity: Measures the amount of memory an algorithm requires during execution.
• Asymptotic Notation: Describes the growth rate of complexity functions, ignoring constant factors.

2 Time Complexity

Time complexity is classified based on how the runtime increases with input size:

Complexity	Notation	Description	Example
Constant	O(1)	Execution time is independent of input size.	Accessing an array element.
Logarithmic	O(log n)	Execution time increases logarithmically.	Binary search.
Linear	O(n)	Execution time grows proportionally to input size.	Iterating through an array.
Linearithmic	O(n log n)	Grows slightly faster than linear.	Merge Sort, Quick Sort (average case).
Quadratic	O(n²)	Execution time grows quadratically.	Bubble Sort, Insertion Sort.
Exponential	O(2ⁿ)	Grows exponentially with input size.	Solving the traveling salesman problem using brute force.

3 Space Complexity

Space complexity considers both:

• Auxiliary Space: Extra memory used by the algorithm beyond input storage.
• Recursive Stack Space: Memory consumed by recursive function calls.

Algorithm	Space Complexity	Description
Merge Sort	O(n)	Requires additional space for merging.
Quick Sort	O(log n)	Uses recursive stack space.
Bubble Sort	O(1)	Uses minimal extra memory.

4 Best, Worst, and Average Case Complexity

An algorithm’s complexity can be analyzed in different scenarios:

• Best Case: The input requires the fewest steps (e.g., sorted input in insertion sort).
• Worst Case: The input requires the maximum number of steps (e.g., reversed input in insertion sort).
• Average Case: The expected runtime over all possible inputs.

5 Complexity Classes

Computational complexity theory categorizes problems into different classes:

Complexity Class	Description	Example Problems
P	Problems solvable in polynomial time.	Sorting, shortest path.
NP	Problems verifiable in polynomial time.	Traveling Salesman, Boolean Satisfiability (SAT).
NP-complete	Hardest problems in NP; if one is solved in polynomial time, all NP problems can be solved in polynomial time.	3-SAT, Hamiltonian Cycle.
NP-hard	At least as hard as NP-complete problems but not necessarily in NP.	Halting Problem, Generalized TSP.

6 Practical Considerations

• Trade-offs: Algorithms with lower time complexity may use more space and vice versa.
• Optimization: Some problems allow for better solutions through heuristics or approximation.
• Parallelization: Some algorithms can be optimized using parallel computing.

7 See Also

• Big O Notation
• Computational Complexity
• Time Complexity
• Space Complexity
• Sorting Algorithms
• NP-Complete Problems
• Divide-and-Conquer Algorithm