Computation, Complexity, and Algorithm IST 501 Fall 2014 Dongwon Lee, Ph.D

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  • Computation, Complexity, and Algorithm

    IST 501Fall 2014

    Dongwon Lee, Ph.D.

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  • Learning ObjectivesUnderstand what computation and algorithm areBe able to compare alternative algorithms for a problem w. r. t their complexityGiven an algorithm, be able to describe its big O complexityBe able to explain NP-complete

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    *Computational ComplexityWhy complexity?Primarily for evaluating difficulty in scaling up a problemHow will the problem grow as resources increase?Knowing if a claimed solution to a problem is optimal (best)Optimal (best) in what sense?

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    *Why do we have to deal with this?Moores lawHwangs lawGrowth of information and information resourcesStoring stuffFinding stuff

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    *ImpactThe efficiency of algorithms/methods The inherent "difficulty" of problems of practical and/or theoretical importanceA major discovery in the science was that computational problems can vary tremendously in the effort required to solve them precisely. The technical term for a hard problem is _________________" which essentially means: "abandon all hope of finding an efficient algorithm for the exact (and sometimes approximate) solution of this problem".

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    *OptimalityA solution to a problem is sometimes stated as optimalOptimal in what sense?Empirically?Theoretically? (the only real definition)

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    *Youre given a problemData informatics: How will our system scale if we store everybodys email every day and we ask everyone to send a picture of themselves and to everyone else. We will also hire a new employ each week for a year

    Social informatics: We want to know how the number of social connections in a growing community increase over time.

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  • We will use algorithmsIST 501

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  • Eg. Sorting AlgorithmsProblem: Sort a set of numbers in ascending order[1 7 3 5 9]Many variations of sorting algorithmsBubble SortInsertion SortMerge SortQuick SortIST 501

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  • ScenariosIve got two algorithms that accomplish the same taskWhich is better?I want to store some dataHow do my storage needs scale as more data is storedGiven an algorithm, can I determine how long it will take to run?Input is unknownDont want to trace all possible paths of executionFor different input, can I determine how system cost will change?IST 501

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    *Measuring the Growth of WorkWhile it is possible to measure the work done by an algorithm for a given set of input, we need a way to:Measure

    Compare

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    *Basic unit of the growth of workInput NDescribing the relationship as a function f(n)f(N): the number of steps required by an algorithm

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  • Why Input Size N?IST 501

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  • Comparing the growth of workSize of input NWork doneIST 501

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    *Time vs. SpaceVery often, we can trade space for time:

    For example: maintain a collection of students with SSN information.Use an array of a billion elements and have immediate access (better time)Use an array of number of students and have to search (better space)

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    *Introducing Big O NotationUsed in a sense to put algorithms and problems into families

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    *Size of Input (measure of work)In analyzing rate of growth based upon size of input, well use a variableWhy?For each factor in the size, use a new variableN is most common

    Examples:A linked list of N elementsA 2D array of N x M elementsA Binary Search Tree of P elements

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    *Formal Definition of Big-OFor a given function g(n), O(g(n)) is defined to be the set of functions

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    *Visual O( ) Meaningf(n)n0f(n) = O(g(n))Size of inputWork doneOur Algorithm

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    *Simplifying O( ) AnswersWe say Big O complexity of 3n2 + 2 = O(n2) drop constants!

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  • Correct but MeaninglessIST 501

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    *Comparing AlgorithmsNow that we know the formal definition of O( ) notation (and what it means)If we can determine the O( ) of algorithms and problemsThis establishes the worst they perform.

    Thus now we can compare them and see which has the better performance.

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    Correctly Interpreting O( )O(1) or Order OneO(N) or Order N

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    *Complex/Combined FactorsAlgorithms and problems typically consist of a sequence of logical steps/sections

    We need a way to analyze these more complex algorithms

    Its easy analyze the sections and then combine them!

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    *Eg: Insert into Linked ListTask: Insert an element into an ordered listFind the right locationDo the steps to create the node and add it to the list1738142head//Inserting 75

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    *Eg: Insert into Linked ListInsert an element into an ordered listFind the right locationDo the steps to create the node and add it to the list1738142head//75

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  • Combine the AnalysisIST 501

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    *Eg: Search a 2D ArrayTask: Search an unsorted 2D array (row, then column)Traverse all rowsFor each row, examine all the cells (changing columns)

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    *Search an unsorted 2D array (row, then column)Traverse all rowsFor each row, examine all the cells (changing columns)RowColumn 123451 2 3 4 5 6 7 8 9 10O(M)Eg: Search a 2D Array

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  • Combine the AnalysisIST 501

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    *Sequential StepsIf steps appear sequentially (one after another), then add their respective O().

    loop. . .endlooploop. . .endloopNM

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    *Embedded Steps

    If steps appear embedded (one inside another), then multiply their respective O().

    loop loop . . . endloopendloopMN

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    *Correctly Determining O( )Can have multiple factors:O(N*M)O(logP + N2)But keep only the dominant factors:O(N + NlogN) O(N*M + P) O(V2 + VlogV) Drop constants:O(2N + 3N2)

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  • SummaryIST 501

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  • Poly-Time vs. Expo-TimeSuch algorithms with running times of orders O(log n), O(n ), O(n log n), O(n2), O(n3) etc. Are called polynomial-time algorithms.

    On the other hand, algorithms with complexities which cannot be bounded by polynomial functions are called exponential-time algorithms. These include "exploding-growth" orders which do not contain exponential factors, like n!.

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    *The Towers of HanoiA B CGoal: Move stack of rings to another pegRule 1: May move only 1 ring at a timeRule 2: May never have larger ring on top of smaller ring

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    *Original StateMove 1Move 2Move 3Move 4Move 5Move 6Move 7Towers of Hanoi: Solution

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    *Towers of Hanoi - ComplexityFor 3 rings we have 7 operations.

    In general, the cost is 2N 1 = O(2N)

    Each time we increment N, we double the amount of work.

    This grows incredibly fast!

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    *Towers of Hanoi (2N) RuntimeFor N = 642N = 264 = 18,450,000,000,000,000,000

    If we had a computer that could execute a billion instructions per second

    It would take 584 years to complete

    But it could get worse

    The case of 4 or more pegs: Open Problem

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    *The Traveling Salesman ProblemA traveling salesman wants to visit a number of cities and then return to the starting point. Of course he wants to save time and energy, so he wants to determine the shortest path for his trip.We can represent the cities and the distances between them by a weighted, complete, undirected graph.The problem then is to find the round-trip of minimum total weight that visits each vertex exactly once.

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    *The Traveling Salesman ProblemExample: What path would the traveling salesman take to visit the following cities?

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  • Traveling Salesman Problem (TSP): one of the best-known computational problems that is known to be extremely hard to solveNP-hardA brute force solution takes O(n!)A dynamic programming solution takes O(cn), c>2Harder to solve than Tower of Hanoi problem

    The Traveling Salesman ProblemIST 501

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    *Where Does this Leave Us?Clearly algorithms have varying runtimes or storage costs.

    Wed like a way to categorize them:

    Reasonable, so it may be useful Unreasonable, so why bother running

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    *Performance Categories

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    *Two Categories of Algorithms2 4 8 16 32 64 128 256 512 1024Size of Input (N)10351030102510201015trillionbillionmillion100010010Runtime

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  • *Reasonable vs. UnreasonableReasonable algorithms feature polynomial factors in their O( ) and may be usable depending upon input size.O (Log N), O (N), O (NK) where K is a constantUnreasonable algorithms feature exponential factors in their O( ) and have little practical utilityO (2N), O (N!), O (NN)IST 501

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    *Eg: Computational ComplexityBig O complexity in