Computational Problem is a problem that can be solved using an algorithm executed on a computational model, such as a Turing machine. It involves defining an input, processing it through a set of rules or algorithms, and obtaining an output.
Key Concepts
- Input: A well-defined set of data that the problem operates on.
- Output: The expected result derived from the given input.
- Algorithm: A finite sequence of steps that transforms the input into the output.
- Complexity: The amount of computational resources (time and space) required to solve the problem.
Types of Computational Problems
| Type | Description | Example |
|---|---|---|
| Decision Problem | A problem with a yes/no answer. | "Is a given number prime?" |
| Function Problem | A problem where the output is a computed value. | "Find the greatest common divisor of two numbers." |
| Optimization Problem | A problem where the goal is to find the best solution among many. | "Find the shortest path between two cities." |
Examples
- Sorting Problem
- Input: A list of numbers.
- Output: The list sorted in ascending order.
- Example Algorithm: Merge Sort, Quick Sort.
- Complexity: O(n log n).
- Graph Traversal Problem
- Input: A graph and a starting node.
- Output: A traversal order of nodes.
- Example Algorithm: Breadth-First Search (BFS), Depth-First Search (DFS).
- Complexity: O(V + E) (where V is the number of vertices and E is the number of edges).
- Pathfinding Problem
- Input: A weighted graph and two nodes.
- Output: The shortest path between the nodes.
- Example Algorithm: Dijkstra’s Algorithm, A* Algorithm.
- Complexity: O((V + E) log V).
Complexity Classes
Computational problems are categorized based on the resources required to solve them.
| Complexity Class | Description | Example Problems |
|---|---|---|
| P | Problems solvable in polynomial time. | Sorting, shortest path (Dijkstra’s algorithm). |
| NP | Problems verifiable in polynomial time. | Traveling Salesman Problem, Boolean Satisfiability Problem (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 Problem. |
| NP-hard | At least as hard as NP-complete problems, but not necessarily in NP. | Halting Problem, TSP with arbitrary constraints. |
Applications
Computational problems are fundamental in:
- Cryptography: Prime factorization, hash functions.
- Artificial Intelligence: Machine learning optimization, heuristic search.
- Operations Research: Linear programming, supply chain optimization.
- Computer Vision: Object recognition, image segmentation.
Open Problems
Some computational problems remain unsolved or require better solutions:
- P vs NP Problem: One of the biggest open questions in computer science, asking whether every problem verifiable in polynomial time (NP) can also be solved in polynomial time (P).
- Graph Isomorphism Problem: Finding an efficient algorithm to determine whether two graphs are structurally identical.
- Integer Factorization: Used in cryptography, with RSA security relying on its difficulty.