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## Chapter 11 Limitations of Algorithm Power Copyright © 2007 Pearson Addison-Wesle

Cruz, Donna, Midday On-Air Personality has reference to this Academic Journal, PHwiki organized this Journal Chapter 11 Limitations of Algorithm Power Copyright © 2007 Pearson Addison-Wesley. All rights reserved. Lower Bounds Lower bound: an estimate on a minimum amount of work needed to solve a given problem Examples: number of comparisons needed to find the largest element in a set of n numbers number of comparisons needed to sort an array of size n number of comparisons necessary as long as searching in a sorted array number of multiplications needed to multiply two n-by-n matrices Lower Bounds (cont.) Lower bound can be an exact count an efficiency class () Tight lower bound: there exists an algorithm with the same efficiency as the lower bound Problem Lower bound Tightness sorting (comparison-based) (nlog n) yes searching in a sorted array (log n) yes element uniqueness (nlog n) yes n-digit integer multiplication (n) unknown multiplication of n-by-n matrices (n2) unknown

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Methods as long as Establishing Lower Bounds trivial lower bounds in as long as mation-theoretic arguments (decision trees) adversary arguments problem reduction Trivial Lower Bounds Trivial lower bounds: based on counting the number of items that must be processed in input in addition to generated as output Examples finding max element – n steps or n/2 comparisons polynomial evaluation sorting element uniqueness Hamiltonian circuit existence Conclusions may in addition to may not be useful be careful in deciding how many elements must be processed Decision Trees Decision tree a convenient model of algorithms involving comparisons in which: internal nodes represent comparisons leaves represent outcomes (or input cases) Decision tree as long as 3-element insertion sort

Decision Trees in addition to Sorting Algorithms Any comparison-based sorting algorithm can be represented by a decision tree ( as long as each fixed n) Number of leaves (outcomes) n! Height of binary tree with n! leaves log2n! Minimum number of comparisons in the worst case log2n! as long as any comparison-based sorting algorithm, since the longest path represents the worst case in addition to its length is the height log2n! n log2n (by Sterling approximation) This lower bound is tight (mergesort or heapsort) Ex. Prove that 5 (or 7) comparisons are necessary in addition to sufficient as long as sorting 4 keys (or 5 keys, respectively). Adversary Arguments Adversary argument: Its a game between the adversary in addition to the (unknown) algorithm. The adversary has the input in addition to the algorithm asks questions to the adversary about the input. The adversary tries to make the algorithm work the hardest by adjusting the input (consistently). It wins the game after the lower bound time (lower bound proven) if it is able to come up with two different inputs. Example 1: Guessing a number between 1 in addition to n using yes/no questions (Is it larger than x) Adversary: Puts the number in a larger of the two subsets generated by last question Example 2: Merging two sorted lists of size n a1 < a2 < < an in addition to b1 < b2 < < bn Adversary: Keep the ordering b1 < a1 < b2 < a2 < < bn < an in mind in addition to answer comparisons consistently Claim: Any algorithm requires at least 2n-1 comparisons to output the above ordering (because it has to compare each pair of adjacent elements in the ordering) Ex: Design an adversary to prove that finding the smallest element in a set of n elements requires at least n-1 comparisons. Lower Bounds by Problem Reduction Fact: If problem Q can be reduced to problem P, then Q is at least as easy as P, or equivalent, P is at least as hard as Q. Reduction from Q to P: Design an algorithm as long as Q using an algorithm as long as P as a subroutine. Idea: If problem P is at least as hard as problem Q, then a lower bound as long as Q is also a lower bound as long as P. Hence, find problem Q with a known lower bound that can be reduced to problem P in question. Example: P is finding MST as long as n points in Cartesian plane, in addition to Q is element uniqueness problem (known to be in (nlogn)) Reduction from Q to P: Given a set X = {x1, , xn} of numbers (i.e. an instance of the uniqueness problem), we as long as m an instance of MST in the Cartesian plane: Y = {(0,x1), , (0,xn)}. Then, from an MST as long as Y we can easily (i.e. in linear time) determine if the elements in X are unique. Classifying Problem Complexity Is the problem tractable, i.e., is there a polynomial-time (O(p(n)) algorithm that solves it Possible answers: yes (give example polynomial time algorithms) no because its been proved that no algorithm exists at all (e.g., Turings halting problem) because its been be proved that any algorithm as long as it would require exponential time unknown. How to classify their (relative) complexity using reduction Problem Types: Optimization in addition to Decision Optimization problem: find a solution that maximizes or minimizes some objective function Decision problem: answer yes/no to a question Many problems have decision in addition to optimization versions. E.g.: traveling salesman problem optimization: find Hamiltonian cycle of minimum length decision: find Hamiltonian cycle of length L Decision problems are more convenient as long as as long as mal investigation of their complexity. Class P P: the class of decision problems that are solvable in O(p(n)) time, where p(n) is a polynomial of problems input size n Examples: searching element uniqueness graph connectivity graph acyclicity primality testing (finally proved in 2002) Class NP NP (nondeterministic polynomial): class of decision problems whose proposed solutions can be verified in polynomial time = solvable by a nondeterministic polynomial algorithm A nondeterministic polynomial algorithm is an abstract two-stage procedure that: generates a solution of the problem (on some input) by guessing checks whether this solution is correct in polynomial time By definition, it solves the problem if its capable of generating in addition to verifying a solution on one of its tries Why this definition led to development of the rich theory called computational complexity Example: CNF satisfiability Problem: Is a boolean expression in its conjunctive normal as long as m (CNF) satisfiable, i.e., are there values of its variables that make it true This problem is in NP. Nondeterministic algorithm: Guess a truth assignment Substitute the values into the CNF as long as mula to see if it evaluates to true Example: (A ¬B ¬C) & (A B) & (¬B ¬D E) & (¬D ¬E) Truth assignments: A B C D E 0 0 0 0 0 1 1 1 1 1 Checking phase: O(n) What other problems are in NP Hamiltonian circuit existence Partition problem: Is it possible to partition a set of n integers into two disjoint subsets with the same sum Decision versions of TSP, knapsack problem, graph coloring, in addition to many other combinatorial optimization problems. All the problems in P can also be solved in this manner (but no guessing is necessary), so we have: P NP Big (million dollar) question: P = NP NP-Complete Problems A decision problem D is NP-complete if it is as hard as any problem in NP, i.e., D is in NP every problem in NP is polynomial-time reducible to D Cooks theorem (1971): CNF-sat is NP-complete NP-Complete Problems (cont.) Other NP-complete problems obtained through polynomial- time reductions from a known NP-complete problem Examples: TSP, knapsack, partition, graph-coloring in addition to hundreds of other problems of combinatorial nature P = NP Dilemma Revisited P = NP would imply that every problem in NP, including all NP-complete problems, could be solved in polynomial time If a polynomial-time algorithm as long as just one NP-complete problem is discovered, then every problem in NP can be solved in polynomial time, i.e. P = NP Most but not all researchers believe that P NP , i.e. P is a proper subset of NP. If P NP, then the NP-complete problems are not in P, although many of them are very useful in practice.

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