Data Structure - City University of New...
Transcript of Data Structure - City University of New...
![Page 1: Data Structure - City University of New Yorkvisionlab.engr.ccny.cuny.edu/~fhu/Lecture20-Hashing.pdf · · 2016-11-28Data Structure -Section FG Lecture 20 Hashing Instructor: Feng](https://reader031.fdocuments.net/reader031/viewer/2022030710/5af987f27f8b9a19548cc017/html5/thumbnails/1.jpg)
CSC212 Data Structure
- Section FG
Lecture20Hashing
Instructor:FengHUDepartmentofComputerScience
CityCollegeofNewYork
![Page 2: Data Structure - City University of New Yorkvisionlab.engr.ccny.cuny.edu/~fhu/Lecture20-Hashing.pdf · · 2016-11-28Data Structure -Section FG Lecture 20 Hashing Instructor: Feng](https://reader031.fdocuments.net/reader031/viewer/2022030710/5af987f27f8b9a19548cc017/html5/thumbnails/2.jpg)
HashTables
•Chapter12discussesseveralwaysofstoringinformationinanarray,andlatersearchingfortheinformation.
•Hashtables areacommonapproachtothestoring/searchingproblem.
• Thispresentationintroduceshashtables.Data Structures
and Other ObjectsUsing C++
![Page 3: Data Structure - City University of New Yorkvisionlab.engr.ccny.cuny.edu/~fhu/Lecture20-Hashing.pdf · · 2016-11-28Data Structure -Section FG Lecture 20 Hashing Instructor: Feng](https://reader031.fdocuments.net/reader031/viewer/2022030710/5af987f27f8b9a19548cc017/html5/thumbnails/3.jpg)
WhatisaHashTable?
• Thesimplestkindofhashtableisanarrayofrecords.
• Thisexamplehas701records.
[ 0 ] [ 1 ] [ 2 ] [ 3 ] [ 4 ] [ 5 ]
An array of records
. . .
[ 700]
![Page 4: Data Structure - City University of New Yorkvisionlab.engr.ccny.cuny.edu/~fhu/Lecture20-Hashing.pdf · · 2016-11-28Data Structure -Section FG Lecture 20 Hashing Instructor: Feng](https://reader031.fdocuments.net/reader031/viewer/2022030710/5af987f27f8b9a19548cc017/html5/thumbnails/4.jpg)
WhatisaHashTable?
• Eachrecordhasaspecialfield,calleditskey.
• Inthisexample,thekeyisalongintegerfieldcalledNumber.
[ 0 ] [ 1 ] [ 2 ] [ 3 ] [ 4 ] [ 5 ]
. . .
[ 700]
[ 4 ]
Number 506643548
![Page 5: Data Structure - City University of New Yorkvisionlab.engr.ccny.cuny.edu/~fhu/Lecture20-Hashing.pdf · · 2016-11-28Data Structure -Section FG Lecture 20 Hashing Instructor: Feng](https://reader031.fdocuments.net/reader031/viewer/2022030710/5af987f27f8b9a19548cc017/html5/thumbnails/5.jpg)
WhatisaHashTable?
• Thenumbermightbeaperson'sidentificationnumber,andtherestoftherecordhasinformationabouttheperson.
[ 0 ] [ 1 ] [ 2 ] [ 3 ] [ 4 ] [ 5 ]
. . .
[ 700]
[ 4 ]
Number 506643548
![Page 6: Data Structure - City University of New Yorkvisionlab.engr.ccny.cuny.edu/~fhu/Lecture20-Hashing.pdf · · 2016-11-28Data Structure -Section FG Lecture 20 Hashing Instructor: Feng](https://reader031.fdocuments.net/reader031/viewer/2022030710/5af987f27f8b9a19548cc017/html5/thumbnails/6.jpg)
WhatisaHashTable?
• Whenahashtableisinuse,somespotscontainvalidrecords,andotherspotsare"empty".
[ 0 ] [ 1 ] [ 2 ] [ 3 ] [ 4 ] [ 5 ] [ 700]Number 506643548Number 233667136Number 281942902 Number 155778322
. . .
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InsertingaNewRecord
• Inordertoinsertanewrecord,thekeymustsomehowbeconvertedto anarrayindex.
• Theindexiscalledthehashvalue ofthekey.
[ 0 ] [ 1 ] [ 2 ] [ 3 ] [ 4 ] [ 5 ] [ 700]Number 506643548Number 233667136Number 281942902 Number 155778322
. . .
Number 580625685
![Page 8: Data Structure - City University of New Yorkvisionlab.engr.ccny.cuny.edu/~fhu/Lecture20-Hashing.pdf · · 2016-11-28Data Structure -Section FG Lecture 20 Hashing Instructor: Feng](https://reader031.fdocuments.net/reader031/viewer/2022030710/5af987f27f8b9a19548cc017/html5/thumbnails/8.jpg)
InsertingaNewRecord
• Typicalwaytocreateahashvalue:
[ 0 ] [ 1 ] [ 2 ] [ 3 ] [ 4 ] [ 5 ] [ 700]Number 506643548Number 233667136Number 281942902 Number 155778322
. . .
Number 580625685
(Number mod 701)
What is (580625685 mod 701) ?
![Page 9: Data Structure - City University of New Yorkvisionlab.engr.ccny.cuny.edu/~fhu/Lecture20-Hashing.pdf · · 2016-11-28Data Structure -Section FG Lecture 20 Hashing Instructor: Feng](https://reader031.fdocuments.net/reader031/viewer/2022030710/5af987f27f8b9a19548cc017/html5/thumbnails/9.jpg)
InsertingaNewRecord
• Typicalwaytocreateahashvalue:
[ 0 ] [ 1 ] [ 2 ] [ 3 ] [ 4 ] [ 5 ] [ 700]Number 506643548Number 233667136Number 281942902 Number 155778322
. . .
Number 580625685
(Number mod 701)
What is (580625685 mod 701) ?3
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InsertingaNewRecord
• Thehashvalueisusedforthelocationofthenewrecord.
Number 580625685
[ 0 ] [ 1 ] [ 2 ] [ 3 ] [ 4 ] [ 5 ] [ 700]Number 506643548Number 233667136Number 281942902 Number 155778322
. . .
[3]
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InsertingaNewRecord
• Thehashvalueisusedforthelocationofthenewrecord.
[ 0 ] [ 1 ] [ 2 ] [ 3 ] [ 4 ] [ 5 ] [ 700]Number 506643548Number 233667136Number 281942902 Number 155778322
. . .Number 580625685
![Page 12: Data Structure - City University of New Yorkvisionlab.engr.ccny.cuny.edu/~fhu/Lecture20-Hashing.pdf · · 2016-11-28Data Structure -Section FG Lecture 20 Hashing Instructor: Feng](https://reader031.fdocuments.net/reader031/viewer/2022030710/5af987f27f8b9a19548cc017/html5/thumbnails/12.jpg)
Collisions
• Hereisanothernewrecordtoinsert,withahashvalueof2.
[ 0 ] [ 1 ] [ 2 ] [ 3 ] [ 4 ] [ 5 ] [ 700]Number 506643548Number 233667136Number 281942902 Number 155778322
. . .Number 580625685
Number 701466868
My hashvalue is [2].
![Page 13: Data Structure - City University of New Yorkvisionlab.engr.ccny.cuny.edu/~fhu/Lecture20-Hashing.pdf · · 2016-11-28Data Structure -Section FG Lecture 20 Hashing Instructor: Feng](https://reader031.fdocuments.net/reader031/viewer/2022030710/5af987f27f8b9a19548cc017/html5/thumbnails/13.jpg)
Collisions
• Hereisanothernewrecordtoinsert,withahashvalueof2.
[ 0 ] [ 1 ] [ 2 ] [ 3 ] [ 4 ] [ 5 ] [ 700]Number 506643548Number 233667136Number 281942902 Number 155778322
. . .Number 580625685
Number 701466868
My hashvalue is [2].When a collision occurs,
move forward until youfind an empty spot.
![Page 14: Data Structure - City University of New Yorkvisionlab.engr.ccny.cuny.edu/~fhu/Lecture20-Hashing.pdf · · 2016-11-28Data Structure -Section FG Lecture 20 Hashing Instructor: Feng](https://reader031.fdocuments.net/reader031/viewer/2022030710/5af987f27f8b9a19548cc017/html5/thumbnails/14.jpg)
Collisions
• Thisiscalledacollision,becausethereisalreadyanothervalidrecordat[2].
[ 0 ] [ 1 ] [ 2 ] [ 3 ] [ 4 ] [ 5 ] [ 700]Number 506643548Number 233667136Number 281942902 Number 155778322
. . .Number 580625685
Number 701466868
When a collision occurs,move forward until you
find an empty spot.
![Page 15: Data Structure - City University of New Yorkvisionlab.engr.ccny.cuny.edu/~fhu/Lecture20-Hashing.pdf · · 2016-11-28Data Structure -Section FG Lecture 20 Hashing Instructor: Feng](https://reader031.fdocuments.net/reader031/viewer/2022030710/5af987f27f8b9a19548cc017/html5/thumbnails/15.jpg)
Collisions
• Thisiscalledacollision,becausethereisalreadyanothervalidrecordat[2].
[ 0 ] [ 1 ] [ 2 ] [ 3 ] [ 4 ] [ 5 ] [ 700]Number 506643548Number 233667136Number 281942902 Number 155778322
. . .Number 580625685
Number 701466868
When a collision occurs,move forward until you
find an empty spot.
![Page 16: Data Structure - City University of New Yorkvisionlab.engr.ccny.cuny.edu/~fhu/Lecture20-Hashing.pdf · · 2016-11-28Data Structure -Section FG Lecture 20 Hashing Instructor: Feng](https://reader031.fdocuments.net/reader031/viewer/2022030710/5af987f27f8b9a19548cc017/html5/thumbnails/16.jpg)
Collisions
• Thisiscalledacollision,becausethereisalreadyanothervalidrecordat[2].
[ 0 ] [ 1 ] [ 2 ] [ 3 ] [ 4 ] [ 5 ] [ 700]Number 506643548Number 233667136Number 281942902 Number 155778322
. . .Number 580625685
Number 701466868
When a collision occurs,move forward until you
find an empty spot.
![Page 17: Data Structure - City University of New Yorkvisionlab.engr.ccny.cuny.edu/~fhu/Lecture20-Hashing.pdf · · 2016-11-28Data Structure -Section FG Lecture 20 Hashing Instructor: Feng](https://reader031.fdocuments.net/reader031/viewer/2022030710/5af987f27f8b9a19548cc017/html5/thumbnails/17.jpg)
Collisions
• Thisiscalledacollision,becausethereisalreadyanothervalidrecordat[2].
[ 0 ] [ 1 ] [ 2 ] [ 3 ] [ 4 ] [ 5 ] [ 700]Number 506643548Number 233667136Number 281942902 Number 155778322
. . .Number 580625685 Number 701466868
The new record goesin the empty spot.
![Page 18: Data Structure - City University of New Yorkvisionlab.engr.ccny.cuny.edu/~fhu/Lecture20-Hashing.pdf · · 2016-11-28Data Structure -Section FG Lecture 20 Hashing Instructor: Feng](https://reader031.fdocuments.net/reader031/viewer/2022030710/5af987f27f8b9a19548cc017/html5/thumbnails/18.jpg)
AQuiz
Where would you be placed in this table, if there is no collision? Use your social security number or some other favorite number.
[ 0 ] [ 1 ] [ 2 ] [ 3 ] [ 4 ] [ 5 ] [ 700]Number 506643548Number 233667136Number 281942902 Number 155778322Number 580625685 Number 701466868
. . .
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AnotherKindofCollision
Where would you be placed in this table, if there is no collision? Use your social security number or some other favorite number.
[ 0 ] [ 1 ] [ 2 ] [ 3 ] [ 4 ] [ 5 ] [ 700]Number 506643548Number 233667136Number 281942902 Number 155778322Number 580625685 Number 701466868
. . .
Number 155779023
My hashvalue is [700].
![Page 20: Data Structure - City University of New Yorkvisionlab.engr.ccny.cuny.edu/~fhu/Lecture20-Hashing.pdf · · 2016-11-28Data Structure -Section FG Lecture 20 Hashing Instructor: Feng](https://reader031.fdocuments.net/reader031/viewer/2022030710/5af987f27f8b9a19548cc017/html5/thumbnails/20.jpg)
AnotherKindofCollision
Where would you be placed in this table, if there is no collision? Use your social security number or some other favorite number.
[ 0 ] [ 1 ] [ 2 ] [ 3 ] [ 4 ] [ 5 ] [ 700]Number 506643548Number 233667136Number 281942902 Number 155778322Number 580625685 Number 701466868
. . .
Number 155779023
My hashvalue is [700].
![Page 21: Data Structure - City University of New Yorkvisionlab.engr.ccny.cuny.edu/~fhu/Lecture20-Hashing.pdf · · 2016-11-28Data Structure -Section FG Lecture 20 Hashing Instructor: Feng](https://reader031.fdocuments.net/reader031/viewer/2022030710/5af987f27f8b9a19548cc017/html5/thumbnails/21.jpg)
SearchingforaKey
• Thedatathat'sattachedtoakeycanbefoundfairlyquickly.
[ 0 ] [ 1 ] [ 2 ] [ 3 ] [ 4 ] [ 5 ] [ 700]Number 506643548Number 233667136Number 281942902 Number 155778322
. . .Number 580625685 Number 701466868
Number 701466868
![Page 22: Data Structure - City University of New Yorkvisionlab.engr.ccny.cuny.edu/~fhu/Lecture20-Hashing.pdf · · 2016-11-28Data Structure -Section FG Lecture 20 Hashing Instructor: Feng](https://reader031.fdocuments.net/reader031/viewer/2022030710/5af987f27f8b9a19548cc017/html5/thumbnails/22.jpg)
SearchingforaKey
• Calculatethehashvalue.• Checkthatlocationofthearrayforthekey.
[ 0 ] [ 1 ] [ 2 ] [ 3 ] [ 4 ] [ 5 ] [ 700]Number 506643548Number 233667136Number 281942902 Number 155778322
. . .Number 580625685 Number 701466868
Number 701466868
My hashvalue is [2].
Not me.
![Page 23: Data Structure - City University of New Yorkvisionlab.engr.ccny.cuny.edu/~fhu/Lecture20-Hashing.pdf · · 2016-11-28Data Structure -Section FG Lecture 20 Hashing Instructor: Feng](https://reader031.fdocuments.net/reader031/viewer/2022030710/5af987f27f8b9a19548cc017/html5/thumbnails/23.jpg)
SearchingforaKey
• Keepmovingforwarduntilyoufindthekey,oryoureachanemptyspot.
[ 0 ] [ 1 ] [ 2 ] [ 3 ] [ 4 ] [ 5 ] [ 700]Number 506643548Number 233667136Number 281942902 Number 155778322
. . .Number 580625685 Number 701466868
Number 701466868
My hashvalue is [2].
Not me.
![Page 24: Data Structure - City University of New Yorkvisionlab.engr.ccny.cuny.edu/~fhu/Lecture20-Hashing.pdf · · 2016-11-28Data Structure -Section FG Lecture 20 Hashing Instructor: Feng](https://reader031.fdocuments.net/reader031/viewer/2022030710/5af987f27f8b9a19548cc017/html5/thumbnails/24.jpg)
SearchingforaKey
• Keepmovingforwarduntilyoufindthekey,oryoureachanemptyspot.
[ 0 ] [ 1 ] [ 2 ] [ 3 ] [ 4 ] [ 5 ] [ 700]Number 506643548Number 233667136Number 281942902 Number 155778322
. . .Number 580625685 Number 701466868
Number 701466868
My hashvalue is [2].
Not me.
![Page 25: Data Structure - City University of New Yorkvisionlab.engr.ccny.cuny.edu/~fhu/Lecture20-Hashing.pdf · · 2016-11-28Data Structure -Section FG Lecture 20 Hashing Instructor: Feng](https://reader031.fdocuments.net/reader031/viewer/2022030710/5af987f27f8b9a19548cc017/html5/thumbnails/25.jpg)
SearchingforaKey
• Keepmovingforwarduntilyoufindthekey,oryoureachanemptyspot.
[ 0 ] [ 1 ] [ 2 ] [ 3 ] [ 4 ] [ 5 ] [ 700]Number 506643548Number 233667136Number 281942902 Number 155778322
. . .Number 580625685 Number 701466868
Number 701466868
My hashvalue is [2].
Yes!
![Page 26: Data Structure - City University of New Yorkvisionlab.engr.ccny.cuny.edu/~fhu/Lecture20-Hashing.pdf · · 2016-11-28Data Structure -Section FG Lecture 20 Hashing Instructor: Feng](https://reader031.fdocuments.net/reader031/viewer/2022030710/5af987f27f8b9a19548cc017/html5/thumbnails/26.jpg)
SearchingforaKey
• Whentheitemisfound,theinformationcanbecopiedtothenecessarylocation.
[ 0 ] [ 1 ] [ 2 ] [ 3 ] [ 4 ] [ 5 ] [ 700]Number 506643548Number 233667136Number 281942902 Number 155778322
. . .Number 580625685 Number 701466868
Number 701466868
My hashvalue is [2].
Yes!
![Page 27: Data Structure - City University of New Yorkvisionlab.engr.ccny.cuny.edu/~fhu/Lecture20-Hashing.pdf · · 2016-11-28Data Structure -Section FG Lecture 20 Hashing Instructor: Feng](https://reader031.fdocuments.net/reader031/viewer/2022030710/5af987f27f8b9a19548cc017/html5/thumbnails/27.jpg)
DeletingaRecord
• Recordsmayalsobedeletedfromahashtable.
[ 0 ] [ 1 ] [ 2 ] [ 3 ] [ 4 ] [ 5 ] [ 700]Number 506643548Number 233667136Number 281942902 Number 155778322
. . .Number 580625685 Number 701466868
Pleasedelete me.
![Page 28: Data Structure - City University of New Yorkvisionlab.engr.ccny.cuny.edu/~fhu/Lecture20-Hashing.pdf · · 2016-11-28Data Structure -Section FG Lecture 20 Hashing Instructor: Feng](https://reader031.fdocuments.net/reader031/viewer/2022030710/5af987f27f8b9a19548cc017/html5/thumbnails/28.jpg)
DeletingaRecord
• Recordsmayalsobedeletedfromahashtable.• Butthelocationmustnotbeleftasanordinary"emptyspot"sincethatcouldinterferewithsearches.
[ 0 ] [ 1 ] [ 2 ] [ 3 ] [ 4 ] [ 5 ] [ 700]Number 233667136Number 281942902 Number 155778322
. . .Number 580625685 Number 701466868
![Page 29: Data Structure - City University of New Yorkvisionlab.engr.ccny.cuny.edu/~fhu/Lecture20-Hashing.pdf · · 2016-11-28Data Structure -Section FG Lecture 20 Hashing Instructor: Feng](https://reader031.fdocuments.net/reader031/viewer/2022030710/5af987f27f8b9a19548cc017/html5/thumbnails/29.jpg)
DeletingaRecord
• Recordsmayalsobedeletedfromahashtable.• Butthelocationmustnotbeleftasanordinary"emptyspot"sincethatcouldinterferewithsearches.
• Thelocationmustbemarkedinsomespecialwaysothatasearchcantellthatthespotusedtohavesomethinginit.
[ 0 ] [ 1 ] [ 2 ] [ 3 ] [ 4 ] [ 5 ] [ 700]Number 233667136Number 281942902 Number 155778322
. . .Number 580625685 Number 701466868
![Page 30: Data Structure - City University of New Yorkvisionlab.engr.ccny.cuny.edu/~fhu/Lecture20-Hashing.pdf · · 2016-11-28Data Structure -Section FG Lecture 20 Hashing Instructor: Feng](https://reader031.fdocuments.net/reader031/viewer/2022030710/5af987f27f8b9a19548cc017/html5/thumbnails/30.jpg)
TimeAnalysis
• Withoutanycollisions• constant
• Withcollisions• O(k)wherekistheaveragecollisions foritems• k<<n, sizeof theproblem
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ImprovingHashing
• Sizeofthehashingtablewhenusingdivisionhashfunction• primenumberintheformof4k+3
•Otherhashingfunctions• mid-square,multiplicative
•Doublehashing(insteadoflinearprobing)• the2nd hashfunctionforsteppingthroughthearray
•Chainedhashing• usingalinkedlistforeachcomponentofthehashtable
![Page 32: Data Structure - City University of New Yorkvisionlab.engr.ccny.cuny.edu/~fhu/Lecture20-Hashing.pdf · · 2016-11-28Data Structure -Section FG Lecture 20 Hashing Instructor: Feng](https://reader031.fdocuments.net/reader031/viewer/2022030710/5af987f27f8b9a19548cc017/html5/thumbnails/32.jpg)
Summary
•Hashtablesstoreacollectionofrecordswithkeys.• Thelocationofarecorddependsonthehashvalueoftherecord'skey.
•Whenacollisionoccurs,thenextavailablelocationisused.
• Searchingforaparticularkeyisgenerallyquick.•Whenanitemisdeleted,thelocationmustbemarkedinaspecialway,sothatthesearchesknowthatthespotusedtobeused.
![Page 33: Data Structure - City University of New Yorkvisionlab.engr.ccny.cuny.edu/~fhu/Lecture20-Hashing.pdf · · 2016-11-28Data Structure -Section FG Lecture 20 Hashing Instructor: Feng](https://reader031.fdocuments.net/reader031/viewer/2022030710/5af987f27f8b9a19548cc017/html5/thumbnails/33.jpg)
Hash Table Exercise
Five records of past students
- Create a small hash table with size 5 (indexes 0 to 4).
- Insert the five items
- Remove Bill Clinton
- Do three searches (for Will Smith, Bill Clinton, and Elizabeth).
![Page 34: Data Structure - City University of New Yorkvisionlab.engr.ccny.cuny.edu/~fhu/Lecture20-Hashing.pdf · · 2016-11-28Data Structure -Section FG Lecture 20 Hashing Instructor: Feng](https://reader031.fdocuments.net/reader031/viewer/2022030710/5af987f27f8b9a19548cc017/html5/thumbnails/34.jpg)
Kathy Martin817339024
Took Data Structures in Fall 1993.Grade A.
Hard worker. Always gets things doneon time.
Currently working for ABCin New York City.
![Page 35: Data Structure - City University of New Yorkvisionlab.engr.ccny.cuny.edu/~fhu/Lecture20-Hashing.pdf · · 2016-11-28Data Structure -Section FG Lecture 20 Hashing Instructor: Feng](https://reader031.fdocuments.net/reader031/viewer/2022030710/5af987f27f8b9a19548cc017/html5/thumbnails/35.jpg)
Will Smith506643973
Took Data Structures in Fall 1995.Grade A.
A bit of a goof-off, but he comes throughin a pinch.
Currently saving the world from alieninvasion.
![Page 36: Data Structure - City University of New Yorkvisionlab.engr.ccny.cuny.edu/~fhu/Lecture20-Hashing.pdf · · 2016-11-28Data Structure -Section FG Lecture 20 Hashing Instructor: Feng](https://reader031.fdocuments.net/reader031/viewer/2022030710/5af987f27f8b9a19548cc017/html5/thumbnails/36.jpg)
William “Bill” Clinton330220393
Took Data Structures in Fall 1995.Grade B-.
Gets along with mostpeople well.
Been laid off even before the slowdown of the economy.
![Page 37: Data Structure - City University of New Yorkvisionlab.engr.ccny.cuny.edu/~fhu/Lecture20-Hashing.pdf · · 2016-11-28Data Structure -Section FG Lecture 20 Hashing Instructor: Feng](https://reader031.fdocuments.net/reader031/viewer/2022030710/5af987f27f8b9a19548cc017/html5/thumbnails/37.jpg)
Elizabeth Windsor092223340
Took Data Structures in Fall 1995.Grade B-.
Prefers to be called “Elizabeth II” or “HerMajesty.” Has some family problems.
Currently working in public relationsnear London.
![Page 38: Data Structure - City University of New Yorkvisionlab.engr.ccny.cuny.edu/~fhu/Lecture20-Hashing.pdf · · 2016-11-28Data Structure -Section FG Lecture 20 Hashing Instructor: Feng](https://reader031.fdocuments.net/reader031/viewer/2022030710/5af987f27f8b9a19548cc017/html5/thumbnails/38.jpg)
Al Einstein699200102
Took CSCI 2270 in Fall 1995.Grade F.
In spite of poor grade, I think there isgood academic ability in Al.
Currently a well-known advocate forpeace.
![Page 39: Data Structure - City University of New Yorkvisionlab.engr.ccny.cuny.edu/~fhu/Lecture20-Hashing.pdf · · 2016-11-28Data Structure -Section FG Lecture 20 Hashing Instructor: Feng](https://reader031.fdocuments.net/reader031/viewer/2022030710/5af987f27f8b9a19548cc017/html5/thumbnails/39.jpg)
THE END
Presentation copyright 1997 Addison Wesley Longman,For use with Data Structures and Other Objects Using C++by Michael Main and Walter Savitch.
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Students and instructors who use Data Structures and Other Objects Using C++ are welcometo use this presentation however they see fit, so long as this copyright notice remainsintact.