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What is an Algorithm?

An algorithm is a methodical approach for solving a problem.

For Example:

1) The algorithm to multiply "A" by "B" is to add "A" to itself "B" times.

2) The algorithm to compute the average of "N" number is to add them up and then divide by "N".

What is NP Completeness algorithm?

A decision problem B is NP complete if 

  1. B is in NP
  2. B is NP-hard

Poly time algorithm: 

Here size of the input is n (in some encoding), and in the running time of the worst case is O(n power c) for some constant c. Describing the three types issues, i.e

  1. P - Issue solvable in ploly_time. 
  2. NP - Issue verifiable in poly_time. 
  3. NPC - Issue in NP and as hard as any problem in NP.

Theorem: if A is NPC ( NP-complete ) and (A Є NP), then ( P = NP ).

Example: If the Spanning_tree degree is 4 that is NP complete

  1. Spanning_tree degree is 4 is in NP 
  2. NP-complete problem : Hamiltonian path is called as NP-complete problem
  3. Hamiltonian path is polynomial time reducible to degree of the Spanning tree is 4.

NP Completeness Algorithm

P : the class of problems which can be solved by a deterministic polynomial algorithm. Example of P is shortest path problem

NP: the class of decision problem which can be solved by a non-deterministic polynomial algorithm. Example of NP complete is Vertex cover problem.

NP-hard: the class of problems to which every NP problem reduces. Example of the NP-hard is Turing Halting Problem.

NP-complete (NPC): the class of problems which are NP-hard and belong to NP

Properties of Algorithms

  1. Specified Input
  2. Specified Output
  3. Definiteness
  4. Effectiveness
  5. Finiteness

There are various ways to classify algorithms. Some of the popular methodologies are :

  1. Logical
  2. Serial, parallel or distributed
  3. Deterministic or non-deterministic
  4. Exact or approximate
  5. Divide and conquer method
  6. Search and enumeration
  7. Randomized algorithm
  8. Reduction of complexity
  9. Linear programming
  10. Dynamic programming
  11. The greedy method