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Nonrecursive Algorithm Analysis *

Dr. Ying Luylu@cse.unl.edu

September 6, 2012

RAIK 283: Data Structures & RAIK 283: Data Structures & AlgorithmsAlgorithms

*slides refrred to http://www.aw-bc.com/info/levitinhttp://www.aw-bc.com/info/levitin

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Time efficiency of nonrecursive Time efficiency of nonrecursive algorithmsalgorithms

Steps in mathematical analysis of nonrecursive Steps in mathematical analysis of nonrecursive algorithms:algorithms:

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Time efficiency of nonrecursive Time efficiency of nonrecursive algorithmsalgorithms

Steps in mathematical analysis of nonrecursive Steps in mathematical analysis of nonrecursive algorithms:algorithms: • Decide on parameter Decide on parameter nn indicating indicating input’s sizeinput’s size• Identify algorithm’s Identify algorithm’s basic operationbasic operation• Determine Determine worstworst, , averageaverage, & , & bestbest case for inputs of size case for inputs of size nn• Set up summation for Set up summation for C(n) C(n) reflecting algorithm’s loop reflecting algorithm’s loop

structurestructure• Simplify summation using standard formulas (see Simplify summation using standard formulas (see

Appendix A)Appendix A)

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SeriesSeries

Proof by Gauss when 9 years old (!):Proof by Gauss when 9 years old (!):

2

)1(

1

NNiS

N

i

)1(2

123...)2()1(

)1()2(...321

NNS

NNNS

NNNS

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General rules for sumsGeneral rules for sums

n

mi

c ?

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General rules for sumsGeneral rules for sums

i

ii

kki

ii

kn

kmii

n

miki

ii

ii

ii

iii

ii

n

mi

n

mi

xaxxa

aa

acca

baba

mnccc

)(

)1(1

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Examples:Examples:

Matrix multiplicationMatrix multiplication

• Section 2.3Section 2.3

Selection sortSelection sort• Section 3.1Section 3.1

Insertion sortInsertion sort• Section 4.1Section 4.1

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Matrix multiplicationMatrix multiplication

MatrixMultiplication(A[0..n-1, 0..n-1], B[0..n-1, 0..n-1])MatrixMultiplication(A[0..n-1, 0..n-1], B[0..n-1, 0..n-1]) Input: two n-by-n matrices A and BInput: two n-by-n matrices A and B Output: C = A * BOutput: C = A * B

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=

Sorting problemSorting problem

Given a list of n orderable items, rearrange them Given a list of n orderable items, rearrange them in a non-decreasing orderin a non-decreasing order

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Sorting problemSorting problem

Given a list of n orderable items, rearrange Given a list of n orderable items, rearrange them in a non-decreasing orderthem in a non-decreasing order• Selection SortSelection Sort

– E.g. “3, 7, 8, 2”E.g. “3, 7, 8, 2”

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Selection sortSelection sort

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Sorting problemSorting problem

Given a list of n orderable items, rearrange Given a list of n orderable items, rearrange them in a non-decreasing orderthem in a non-decreasing order• Insertion SortInsertion Sort

– E.g. “5, 2, 9, 1”E.g. “5, 2, 9, 1”

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Insertion sortInsertion sort

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In-class exercisesIn-class exercises

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P67 2.3.1 (c) & (d)P67 2.3.1 (c) & (d)

P68 2.3.11 (a) & (b)P68 2.3.11 (a) & (b)

In-Class ExercisesIn-Class Exercises

Problem 12: Door in a wall You are facing a wall that Problem 12: Door in a wall You are facing a wall that stretches infinitely in both directions. There is a door in the stretches infinitely in both directions. There is a door in the wall, but you know neither how far away nor in which wall, but you know neither how far away nor in which direction. You can see the door only when you are right direction. You can see the door only when you are right next to it. Design an algorithm that enables you to reach the next to it. Design an algorithm that enables you to reach the door. How many steps will it require? door. How many steps will it require?

Please analyze the following two solutions of the problem. Please analyze the following two solutions of the problem.

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Solution 1Solution 1

Walk right and left going each time one step farther from the Walk right and left going each time one step farther from the initial position. A simple implementation of this idea is to do initial position. A simple implementation of this idea is to do the following until the door is reached: For i = 0, 1, ..., make i the following until the door is reached: For i = 0, 1, ..., make i steps to the right, return to the initial position, make i steps to steps to the right, return to the initial position, make i steps to the left, and return to the initial position again. the left, and return to the initial position again.

How many steps will this algorithm require to find the door?How many steps will this algorithm require to find the door? Does it require walking at most O(n) steps where n is the Does it require walking at most O(n) steps where n is the

(unknown to you) number of steps between your initial (unknown to you) number of steps between your initial position and the door.position and the door.

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Solution 2Solution 2

Walk intermittently right and left going each time Walk intermittently right and left going each time exponentially farther from the initial position. A simple exponentially farther from the initial position. A simple implementation of this idea is to do the following until the door implementation of this idea is to do the following until the door is reached: For i = 0, 1, ..., make 2is reached: For i = 0, 1, ..., make 2ii steps to the right, return to steps to the right, return to the initial position, make 2the initial position, make 2ii steps to the left, and return to the steps to the left, and return to the initial position again. Let 2initial position again. Let 2k−1 k−1 < n ≤ 2< n ≤ 2kk. .

How many steps will this algorithm require to find the door?How many steps will this algorithm require to find the door? Does it require walking at most O(n) steps where n is the Does it require walking at most O(n) steps where n is the

(unknown to you) number of steps between your initial (unknown to you) number of steps between your initial position and the door.position and the door.

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