csc212 dtst tdata structurewolberg/cs212/pdf/csc212-19-searching.pdfwhen does the best-case time...

CSC212 D t St tCSC212 D t St tData Structure Data Structure

Lecture 18Lecture 18SearchingSearching

Instructor: George Instructor: George WolbergWolbergDepartment of Computer Science Department of Computer Science

City College of New YorkCity College of New York

@ @ George Wolberg, 2016George Wolberg, 2016 11

TopicsTopicsTopicsTopics

ApplicationsApplicationsMost Common MethodsMost Common MethodsSerial Search Serial Search Binary SearchBinary SearchyySearch by Hashing (next lecture)Search by Hashing (next lecture)

RunRun--Time AnalysisTime AnalysisyyAverageAverage--time analysistime analysisTime analysis of recursive algorithmsTime analysis of recursive algorithms


y gy g

ApplicationsApplicationsApplicationsApplications

Searching a list of values is a common Searching a list of values is a common computational taskcomputational taskpp

ExamplesExamplesdatabase: student record bank account recorddatabase: student record bank account recorddatabase: student record, bank account record, database: student record, bank account record,

credit record...credit record... InternetInternet –– information retrieval: Yahoo Googleinformation retrieval: Yahoo Google Internet Internet information retrieval: Yahoo, Googleinformation retrieval: Yahoo, GoogleBiometrics Biometrics ––face/ fingerprint/ iris IDsface/ fingerprint/ iris IDs


Most Common MethodsMost Common MethodsMost Common MethodsMost Common Methods

Serial Search Serial Search simplest, O(n)simplest, O(n) simplest, O(n)simplest, O(n)

Binary SearchBinary Search averageaverage case O(log n)case O(log n) averageaverage--case O(log n)case O(log n)

Search by Hashing (the next lecture)Search by Hashing (the next lecture)bb ffbetter averagebetter average--case performancecase performance


Serial SearchSerial SearchSerial SearchSerial SearchPseudocode for Serial SearchPseudocode for Serial Search// search for a desired item in an array a of size n// search for a desired item in an array a of size n

A serial search A serial search algorithm steps algorithm steps

set i to 0 and set found to false;set i to 0 and set found to false;

while (i<n && ! found)while (i<n && ! found)through (part through (part of ) an array of ) an array one item aone item a

while (i<n && ! found)while (i<n && ! found){{

if (a[i] is the desired item) if (a[i] is the desired item) found = true;found = true;one item a one item a

time, looking time, looking for a “desiredfor a “desired

elseelse++i;++i;

}}for a desired for a desired item”item” if (found) if (found)

return i; // indicating the location of the desired itemreturn i; // indicating the location of the desired itemelseelse


elseelsereturn return ––1; // indicating “not found”1; // indicating “not found”

Serial Search -AnalysisSerial Search -AnalysisSerial Search -AnalysisSerial Search -Analysis

Size of array: nSize of array: nBestBest--Case: O(1)Case: O(1)

33 22 44 66 55 11 88 77

( )( ) item in [0]item in [0]

WorstWorst--Case: O(n)Case: O(n)3 6 7 9

Wo sWo s C se: O( )C se: O( ) item in [nitem in [n--1] or not found1] or not found

AverageAverage--CaseCaseAverageAverage CaseCaseusually requires fewer than n array accessesusually requires fewer than n array accessesBut, what are the average accesses?But, what are the average accesses?


, g, g

Average-Case Time for Serial SearchAverage-Case Time for Serial SearchAverage-Case Time for Serial SearchAverage-Case Time for Serial Search

A more accurate computation:A more accurate computation:Assume the target to be searched is in the arrayAssume the target to be searched is in the arrayAssume the target to be searched is in the arrayAssume the target to be searched is in the array and the probability of the item being in any and the probability of the item being in any

array location is the samearray location is the sameyyThe average accessesThe average accesses

2)1(2/)1(...321

n

nnn

nn


When does the best-case time When does the best-case time make more sense?make more sense?

For an array of For an array of nn elements, the bestelements, the best--case case time for serial search is just one array time for serial search is just one array j yj yaccess.access.

The bestThe best--case time is more useful if thecase time is more useful if theThe bestThe best case time is more useful if the case time is more useful if the probability of the target being in the [0] probability of the target being in the [0] location is the highestlocation is the highestlocation is the highest. location is the highest. or loosely if the target is most likely in the front or loosely if the target is most likely in the front

part of the arraypart of the array


part of the arraypart of the array

Binary SearchBinary SearchBinary SearchBinary Search

If If nn is huge, and the item to be searched can be in is huge, and the item to be searched can be in any locations, serial search is slow on averageany locations, serial search is slow on average

But if the items in an array are sorted, we can But if the items in an array are sorted, we can somehow know a target’s location earliersomehow know a target’s location earlier Array of integers from smallest to largestArray of integers from smallest to largest Array of strings sorted alphabetically (e.g. dictionary)Array of strings sorted alphabetically (e.g. dictionary)

A f t d t d t d b ID bA f t d t d t d b ID b Array of students records sorted by ID numbersArray of students records sorted by ID numbers


Binary Search in an Integer ArrayBinary Search in an Integer ArrayBinary Search in an Integer ArrayBinary Search in an Integer Arrayif target is in the arrayif target is in the array

Items are sortedItems are sorted target = 16target = 16 n = 8n = 8

22 33 66 77 1010 1212 1616 1818

[0] [1] [2] [3] [4] [5] [6] [7][0] [1] [2] [3] [4] [5] [6] [7] n 8n 8

Go to the middle Go to the middle location i = n/2location i = n/2

[0] [1] [2] [3] [4] [5] [6] [7][0] [1] [2] [3] [4] [5] [6] [7]

location i = n/2location i = n/2 if (a[i] is target)if (a[i] is target)

done!done! else if (target <a[i]) else if (target <a[i])

go to the first halfgo to the first half else if (target >a[i])else if (target >a[i])


else if (target a[i])else if (target a[i]) go to the second halfgo to the second half



22 33 66 77 1010 1212 1616 1818

[0] [1] [2] [3] [4] [5] [6] [7][0] [1] [2] [3] [4] [5] [6] [7] n 8n 8


[0] [1] [2] [3] [4] [5] [6] [7][0] [1] [2] [3] [4] [5] [6] [7]

[0] [1] [2] [3] [0] [1] [2][0] [1] [2] [3] [0] [1] [2]location i = n/2location i = n/2 if (a[i] is target)if (a[i] is target)

done!done!

[0] [1] [2] [3] [0] [1] [2][0] [1] [2] [3] [0] [1] [2]

else if (target <a[i]) else if (target <a[i]) go to the first halfgo to the first half

else if (target >a[i])else if (target >a[i])





22 33 66 77 1010 1212 1616 1818

[0] [1] [2] [3] [4] [5] [6] [7][0] [1] [2] [3] [4] [5] [6] [7] n 8n 8


[0] [1] [2] [3] [4] [5] [6] [7][0] [1] [2] [3] [4] [5] [6] [7]


done!done!

[0] [1] [2] [3] [0] [1] [2][0] [1] [2] [3] [0] [1] [2]

DONEDONE else if (target <a[i]) else if (target <a[i]) go to the first halfgo to the first half


DONEDONE





22 33 66 77 1010 1212 1616 1818

[0] [1] [2] [3] [4] [5] [6] [7][0] [1] [2] [3] [4] [5] [6] [7] n 8n 8


[0] [1] [2] [3] [4] [5] [6] [7][0] [1] [2] [3] [4] [5] [6] [7]


done!done!

[0] [1] [2] [3] [0] [1] [2][0] [1] [2] [3] [0] [1] [2]



DONEDONE

recursive calls: what recursive calls: what are the parameters?are the parameters?



are the parameters?are the parameters?



22 33 66 77 1010 1212 1616 1818

[0] [1] [2] [3] [4] [5] [6] [7][0] [1] [2] [3] [4] [5] [6] [7] n 8n 8


[0] [1] [2] [3] [4] [5] [6] [7][0] [1] [2] [3] [4] [5] [6] [7]


done!done!

[0] [1] [2] [3] [0] [1] [2][0] [1] [2] [3] [0] [1] [2]



DONEDONE

recursive calls with parameters:recursive calls with parameters:


else if (target a[i])else if (target a[i]) go to the second halfgo to the second half array, start, size, targetarray, start, size, target

found, location // referencefound, location // reference

Binary Search in an Integer ArrayBinary Search in an Integer ArrayBinary Search in an Integer ArrayBinary Search in an Integer Arrayif target is not in the arrayif target is not in the array

Items are sortedItems are sorted target = target = 1717 n = 8n = 8

22 33 66 77 1010 1212 1616 1818

[0] [1] [2] [3] [4] [5] [6] [7][0] [1] [2] [3] [4] [5] [6] [7] n 8n 8


[0] [1] [2] [3] [4] [5] [6] [7][0] [1] [2] [3] [4] [5] [6] [7]


done!done!

[0] [1] [2] [3] [0] [1] [2][0] [1] [2] [3] [0] [1] [2]

else if (target <a[i]) else if (target <a[i]) go to the first halfgo to the first half






22 33 66 77 1010 1212 1616 1818

[0] [1] [2] [3] [4] [5] [6] [7][0] [1] [2] [3] [4] [5] [6] [7] n 8n 8


[0] [1] [2] [3] [4] [5] [6] [7][0] [1] [2] [3] [4] [5] [6] [7]


done!done!

[0] [1] [2] [3] [0] [1] [2][0] [1] [2] [3] [0] [1] [2]

[0] [0][0] [0] else if (target <a[i]) else if (target <a[i])


[ ] [ ][ ] [ ]





22 33 66 77 1010 1212 1616 1818

[0] [1] [2] [3] [4] [5] [6] [7][0] [1] [2] [3] [4] [5] [6] [7] n 8n 8


[0] [1] [2] [3] [4] [5] [6] [7][0] [1] [2] [3] [4] [5] [6] [7]


done!done!

[0] [1] [2] [3] [0] [1] [2][0] [1] [2] [3] [0] [1] [2]



[ ] [ ][ ] [ ]





22 33 66 77 1010 1212 1616 1818

[0] [1] [2] [3] [4] [5] [6] [7][0] [1] [2] [3] [4] [5] [6] [7] n 8n 8


[0] [1] [2] [3] [4] [5] [6] [7][0] [1] [2] [3] [4] [5] [6] [7]


done!done!

[0] [1] [2] [3] [0] [1] [2][0] [1] [2] [3] [0] [1] [2]



[ ] [ ][ ] [ ]

the size of the first half is 0!the size of the first half is 0!





target = target = 1717

If (n == 0 )If (n == 0 )

22 33 66 77 1010 1212 1616 1818

[0] [1] [2] [3] [4] [5] [6] [7][0] [1] [2] [3] [4] [5] [6] [7] If (n 0 )If (n 0 ) not found!not found!


[0] [1] [2] [3] [4] [5] [6] [7][0] [1] [2] [3] [4] [5] [6] [7]


done!done!

[0] [1] [2] [3] [0] [1] [2][0] [1] [2] [3] [0] [1] [2]



[ ] [ ][ ] [ ]



else if (target >a[i])else if (target >a[i]) go to the second halfgo to the second half


Binary Search CodeBinary Search CodeBinary Search CodeBinary Search Codevoid search (const int a[ ], size_t first, size_t size,

int target,bool& found, size_t& location)

6 parameters6 parameters 2 stopping 2 stopping

{size_t middle;

if (size == 0) // stopping case if not foundcasescases

2 recursive 2 recursive call casescall cases

found = false;else{

middle = first + size/2;call casescall cases ;if (target == a[middle]) // stopping case if found{

location = middle;found = true;found = true;

}else if (target < a[middle]) // search the first half

search(a, first, size/2, target, found, location);


else //search the second halfsearch(a, middle+1, (size-1)/2, target, found, location);

} }

Binary Search - AnalysisBinary Search - AnalysisBinary Search - AnalysisBinary Search - Analysisvoid search (const int a[ ], size_t first, size_t size,


Analysis of Analysis of recursive recursive algorithmsalgorithms

{size_t middle;

if (size == 0) // stopping case if not foundalgorithmsalgorithms Analyze the Analyze the

worstworst--casecase A i thA i th

found = false;else{

middle = first + size/2; Assuming the Assuming the

target is in the target is in the array array

;if (target == a[middle]) // stopping case if found{

location = middle;found = true;

and we and we always go to always go to the second the second

found = true;}else if (target < a[middle]) // search the first half



halfhalf else //search the second halfsearch(a, middle+1, (size-1)/2, target, found, location);

} }

Binary Search - AnalysisBinary Search - AnalysisBinary Search - AnalysisBinary Search - Analysisvoid search (const int a[ ], size_t first, size_t size,


Analysis of Analysis of recursive recursive algorithmsalgorithms

{size_t middle;

if (size == 0) // 1 operation

Define T(n) is Define T(n) is the total the total operationsoperations

found = false;else{

middle = first + size/2; // 1 operationoperations operations when size=nwhen size=n

T( ) 6+T( /2)T( ) 6+T( /2)

; pif (target == a[middle]) // 2 operations{

location = middle; // 1 operationfound = true; // 1 operationT(n) = 6+T(n/2)T(n) = 6+T(n/2)

T(1) = 6 T(1) = 6

found = true; // 1 operation}else if (target < a[middle]) // 2 operations



( )( )else // T(n/2) operations for the recursive call

search(a, middle+1, (size-1)/2, target, found, location);} // ignore the operations in parameter passing

}

Binary Search - AnalysisBinary Search - AnalysisBinary Search - AnalysisBinary Search - Analysis

How many recursive calls for the longest chain?How many recursive calls for the longest chain?)(nT original calloriginal call

)2/(66

)2/(6

)(

2

1

nT

nT

nT original calloriginal call1st recursion, 1 six1st recursion, 1 six2nd recursion, 2 six2nd recursion, 2 six

)2/(6...66

...)2/(66

nT

nT

m

2nd recursion, 2 six2nd recursion, 2 six

mmth recursion, m sixth recursion, m six

)1(666...66

)2/(6...66

m

nT mmth recursion, m sixth recursion, m six

and n/2and n/2mm = 1 = 1 –– target foundtarget found

d th f th i lld th f th i ll


6log6 2 n depth of the recursive call depth of the recursive call m = logm = log22nn

Worst-Case Time for Binary SearchWorst-Case Time for Binary SearchWorst-Case Time for Binary SearchWorst-Case Time for Binary Search

For an array of n elements, the worstFor an array of n elements, the worst--case time for case time for binary search is logarithmicbinary search is logarithmic We have given a rigorous proofWe have given a rigorous proof The binary search algorithm is very efficientThe binary search algorithm is very efficient

What is the average running time?What is the average running time?Th i ti f t ll fi di bTh i ti f t ll fi di b The average running time for actually finding a number The average running time for actually finding a number is O(log n) is O(log n)

Can we do a rigorous analysis????Can we do a rigorous analysis????


Can we do a rigorous analysis????Can we do a rigorous analysis????

SummarySummarySummarySummary

Most Common Search MethodsMost Common Search MethodsSerial Search Serial Search –– O(n)O(n)Binary Search Binary Search –– O (log n ) O (log n ) Search by Hashing (*) Search by Hashing (*) –– better averagebetter average--case case

performance ( next lecture)performance ( next lecture)RunRun--Time AnalysisTime AnalysisAverageAverage--time analysistime analysisTime analysis of recursive algorithmsTime analysis of recursive algorithms


HomeworkHomeworkHomeworkHomework

Review Chapters 10 & 11 (Trees), and Review Chapters 10 & 11 (Trees), and do the do the self_testself_test exercises exercises

Read Chapters 12 & 13, and Read Chapters 12 & 13, and do the do the self testself test exercises exercises __

Homework/Quiz (on Searching):Homework/Quiz (on Searching):SelfSelf--Test 12.7, p 590 (binary search reTest 12.7, p 590 (binary search re--coding)coding), p ( y, p ( y g)g)


csc212 dtst tdata structurewolberg/cs212/pdf/csc212-19-searching.pdfwhen does the best-case time...

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