Algorithm Complexity 편집하기

'''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.
==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.
==Time Complexity==
Time complexity is classified based on how the runtime increases with input size:
{| class="wikitable"
!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.
|}
==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.
{| class="wikitable"
!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.
|}
==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.
==Complexity Classes==
Computational complexity theory categorizes problems into different classes:
{| class="wikitable"
!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.
|}
==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.
==See Also==
*[[Big O Notation]]
*[[Computational Complexity]]
*[[Time Complexity]]
*[[Space Complexity]]
*[[Sorting Algorithms]]
*[[NP-Complete Problems]]
*[[Divide-and-Conquer Algorithm]]
[[Category:Algorithm]]
설명	입력하는 내용	문서에 나오는 결과
기울임꼴	''기울인 글씨''	기울인 글씨
굵게	'''굵은 글씨'''	굵은 글씨
굵고 기울인 글씨	'''''굵고 기울인 글씨'''''	*굵고 기울인 글씨*