Algorithm Complexity By: Ashish Patel and Alex Golebiewski

Outline Complexity of Algorithms Complexity Classes Tractable and Intractable Traveling Salesman Problem Knapsack Problem

Complexity of Algorithms The complexity of an algorithms deals with the problem of how much time is required for completion and the amount of space it uses. Measurement is in terms of input size.

Why Complexity is Important log n1 sec3 sec5 sec6 sec n2 sec8 sec32 sec64 sec nlogn2 sec24 sec2.67 min6.4 min n2n2 4 sec64 sec17.07 min1.14 hrs n3n3 8 sec8.53 min9.1 hrs3.03 days 2n2n 4 sec4.27 min1.36 cent5x10 9 cent n!2 sec4.2 hrs8x10 25 cent 4x10 79 cent

Complexity Classes Set of computational problems with the same bounds on time and space. Tractable (solvable in polynomial time) –P Intractable (not solvable in polynomial time) –NP –NP-Complete –NP-Hard Unsolvable (cannot be solved by a computer)

Complexity Classes P Intractable Not solvable NP NP-CompleteNP-Hard

P The set of concrete decision problems that are solvable in polynomial time Problems that include those with reasonably efficient algorithms If a problem is not in P, it will be extremely expensive and probably impossible to solve in practice

NP NP stands for “non-deterministic polynomial” Class of decision problems solved by a non-deterministic polynomial algorithm Non-deterministic polynomial Algorithm –Guess potential solution –Verify solution in polynomial time

NP - Complete NP–complete is the term used to describe decision problems that are the hardest ones in NP in the sense that, if there were a polynomial algorithm for an NP- complete problem, then there would be a polynomial algorithm for every problem in NP. A problem is NP-complete if: –It is in NP and all other problems in NP can be reduced to it in polynomial time An examples of NP-complete problems are the Traveling Salesman, the Knapsack problem, Hamiltonian Circuit, CNF-Satisfiability, etc.

NP - Hard Class of decision problems harder than those that can be solved by a non- deterministic polynomial algorithm If decision version of optimization problem is in NP-Complete, then optimization is NP-Hard

Unsolvable Any problem that cannot be solved by any means through the use of a computer. Example: The Halting problem

Approximation Algorithms An algorithm that seems to work well for the inputs that usually occur. The strategies, or heuristics, used by many of the approximation algorithms are simple and straightforward, yet for some problems they provide surprisingly good results.

Hamiltonian Circuit A Hamiltonian Circuit, in a graph or digraph, is a path (cycle) that starts and ends ate the same vertex and passes through every vertex exactly once Hamiltonian Circuit Problem: Determine whether a given graph has a Hamiltonian Circuit.

Traveling Salesman Problem (TSP) Optimization Problem: Given a weighted graph, find a minimum weighted Hamiltonian Circuit. (NP- Hard) Decision Problem: Given a weighted graph and an integer k, is there a Hamiltonian Circuit with total weight at most k? (NP-complete) This problem is widely known as the Traveling Salesman problem; the salesman wants to minimize total traveling while visiting all the cities in a territory. Other applications include routing trucks for garbage pickup and package deliver. General Solution is of Factorial Complexity

 2  4  3  1 = = 18 1  4  3  2  1 = = 24 1  3  2  4  1 = = 28 Can also be reversed with same cost *** optimal ***

Knapsack Problem Let C = capacity of knapsack (positive) and S = set of objects with assigned “sizes/weights” and “values” (all positive) Optimization Problem: Find the largest valued subset of S that fits in the knapsack of capacity C Decision Problem: Given value K, is there a subset of S that fits in the knapsack of capacity C and has a total value at least K? This problem is mostly exemplified by a robber with a limited size bag stealing items from someone. He wants to get the most value he can inside the bag.

ItemValueWeight TV$ lbs Jewelry$ lbs Clothes$ lbs Piano$ lbs Couch$ lbs ItemValueWeight Piano$ lbs Total=$ lbs T weight = 300lbs ItemValueWeight Couch$ lbs TV$ lbs Jewelry$ lbs Total=$ lbs ItemValueWeight Couch$ lbs TV$ lbs Clothes$ Total=$ lbs *** OPTIMAL***

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