Skip ListWord文档格式.docx

Skip ListWord文档格式.docx

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Search（x）

while

（1）

whilep->

next->

key<

x

p=p->

next

Ifp->

down==NULL

returnp->

down

thelogiciseasy,goright,ifcan’t,godown.

Insertion

Theprocessneedaktodetermineinsertthenodetowhichlevel,Iwilltalkaboutkafterwards.Pseudocodeislikethis

Node_Insert（x）

recordtheprecursoroftheinsertednodeineachlevelinarrayp[i]

fori=1,2…k

ifp[i]=NULL

createanewlevelandinsertthenode

else

Insertthenodetotheexistedlevel

return

kisdeterminedbytheprocess

Random_K

fori=0…31

ifrand（）

k++

break

Itshouldhavethiskindofdistributiontoguaranteeitstimebound.

delete

deleteisalsoeasy,wejustneedtodeletethenodefromthelists,ifthelevelisempty,theremovethelevel,thepseudocodeislikethis

Note_Delete

p=highestnodeintheskiplist

whilepintheskiplist

ifthelistisempty

removethelist

deletepfromitslist

p=p->

down

Chapter3:

TestingResults

Wefirstinsertalltheoddintotheskiplist,thentesteachoperationfor50timesandcalculatetheaveragetime.Heregivesthetimetable（unit:

millisecond）.

size

500

1000

2000

4000

6000

8000

10000

insert

0.00399

0.005386

0.010896

0.013868

0.024649

0.027711

0.029879

find

0.001199

0.002775

0.008194

0.011241

0.020461

0.025404

0.029748

0.001429

0.003071

0.008369

0.014216

0.024994

0.025379

0.027268

Thecharts

Wecansaythatthetimeofinsertion,deletingandsearchingareroughlyO（lgn）.

Chapter4:

AnalysisandComments

Analysis

Toanalysisthetimeandspaceboundofskiplist,somelemmasarenecessary.

●Boole’sinequality

Pr[E1∪E2∪…∪En]≤Pr[E1]+Pr[E2]+…Pr[En]

●Informaldefinitionof“withhighpossibility”

Aneventoccurswithhighprobabilityif,foranyα≥1,thereisanappropriatechoiceofconstantsforwhichEoccurswithprobabilityatleast1-O（1/nα）.

ItisthedefinitionfromProfessorErikDemaine,MITinthelectureofskiplists.Surelyitisnotastrictdefinition,butwecanuseitforefficiencyinthisproof.Bytheway,theprecisedefinitionislikethis.

●Precisedefinitionof“withhighpossibility”

A（parameterized）eventEαoccurswithhighprobabilityif,foranyα≥1,Eαoccurswithprobabilityatleast1-cα/nα,wherecαisa“constant”dependingonlyonα.

AndwecancallthetermO（1/nα）errorprobability.

●Lemma:

Withhighprobability,skiplistwithnelementshasO（lgn）levels.

Proof:

Wemayaswelldoexactlytheoppositeandconsidertheprobabilityoffall.Thatwillbe

Pr[thelevelsaremorethanclgn]=Pr[anyelementisinmorethanclgnlevels],supposetheyarex1,x2,...xn

ThenaccordingtoBoole’sinequality

Pr[anyelementisinmorethanclgnlevels]≤Pr[x1isinmorethanclgnlevels]+Pr[x2isinmorethanclgnlevels]+…Pr[xnisinmorethanclgnlevels]=n·

（0.5）clgn=1/nc-1

Sofarweseethatα=c-1,andαcanbearbitrarilylarge.

●Claim:

SearchinginskiplistwillbeinO（lgn）timewithhighprobability.

Proofofclaim:

Wealsodoexactlytheopposite,thatis,analyzethesearchprocessbackwards—fromleaftoroot.

WeknowtheheightisinO（lgn）,nowsayitisclgn.Soinasearchprocess,the“up”movecannotbemorethanclgn.Thenextthingistoprovethatinθ（lgn）timesmove,thereareclgntimes“up”movewithhighprobability.

Saywetakeaclgnmoves,accordingtoBernoulliprobabilitymodel

Pr[exactlyclgn“up”moves]=

Pr[atmostclgn“up”moves]=

Accordingtotheinequality

So

Pr[atmostclgn“up”moves]≤

=

=1/nα

Asa→+∞,α=（a-1）-lg（10e）→+∞,q.e.d.

SameanalyzecanbeusedoninsertionanddeletionandtheyhaveO（lgn）timeboundwithhighprobability.

●Spacecomplexity

Fromthepointofexpectation,theithisexpectedtohaven/

nodes,sothenodesintheskiplistshouldbe

n+n/2+n/4+……=2n

sothespacecomplexityisO（n）.

Comments

Theskiplist’sadvantageisthatitiseasytoimplement.Thetimeofinsertion,deletingandsearchingisnotonlyanaveragebound,butalsoa“withhighprobability”bound.Thatistosay,asngrows,thetimewillmoreandmoredistributeonclgn,whichisasignificativeandstrongproposition.

Appendix:

SourceCode（ifrequired）

skiplist.h

#ifndef__SKIPLIST_H__

#define__SKIPLIST_H__

#defineMAXLEVEL32

structsNode;

typedefstructsNode*Node;

structsList;

typedefstructsList*List;

NodeInitializeNode（intlevel,intkey,intvalue）;

voidDeleteNode（Nodenode）;

ListInitializeList（void）;

voidDeleteList（Listlist）;

intRandomLevel（void）;

voidInsert（Listlist,intkey,intvalue）;

voidDelete（Listlist,intkey）;

intFind（Listlist,intkey）;

#endif

skiplist.c

#include<

stdio.h>

stdlib.h>

time.h>

#include"

skiplist.h"

NodeInitializeNode（intlevel,intkey,intvalue）

Nodenode=（Node）malloc（sizeof（structsNode）+level*sizeof（Node））;

if（node==NULL）

returnNULL;

node->

key=key;

value=value;

returnnode;

}

voidDeleteNode（Nodenode）

return;

free（node）;

node=NULL;

ListInitializeList（void）

inti;

Listlist=（List）malloc（sizeof（structsList））;

if（list==NULL）

list->

level=1;

header=InitializeNode（MAXLEVEL,-1,-1）;

if（list->

header==NULL）

for（i=0;

i<

MAXLEVEL;

i++）

list->

header->

next[i]=NULL;

returnlist;

voidDeleteList（Listlist）

Nodetemp1,temp2;

temp1=list->

header;

while（temp1）

{

temp2=temp1->

next[0];

DeleteNode（temp1）;

temp1=temp2;

}

free（list）;

list=NULL;

intRandomLevel（void）

intlevel=1;

srand（（unsigned）time（NULL））;

for（i=1;

=31;

if（rand（）%2）

{

level++;

}

else

break;

returnlevel;

voidInsert（Listlist,intkey,intvalue）

Nodetemp[MAXLEVEL];

Nodet=list->

header,new;

for（i=list->

level-1;

i>

=0;

i--）

while（t->

next[i]&

&

t->

next[i]->

key）

t=t->

next[i];

temp[i]=t;

}

t=t->

if（t&

key==key）

return;

level=RandomLevel（）;

if（level>

level）

for（i=list->

level;

level;

temp[i]=list->

level=level;

new=InitializeNode（level,key,value）;

if（new==NULL）

for（i=0;

new->

next[i]=temp[i]->

temp[i]->

next[i]=new;

voidDelete（Listlist,intkey）

i>

=0;

if（!

t||t->

key!

=key）

i++）

if（temp[i]->

next[i]==t）

temp[i]->

next[i]=t->

while（list->

level>

1&

next[list->

level-1]==NULL）

level--;

DeleteNode（t）;

intFind（Listlist,intkey）

inti;

i--）

while（t->

returnt->

value;

return-1;

References

[1]ErikDemaine,“LectureNotesonSkipLists”,March18,2004

[2]MarkAllenWeiss,“DataStructureandAlgorithmAnalysisinC”,机械工业出版社，2014

[3]sponge,“跳表（SkipList）的介绍以及查找插入删除等操作”,

Declaration

Weherebydeclarethatalltheworkdoneinthisprojecttitled"

SkipList"

isofourindependenteffortasagroup.