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Lecture10:Dijkstra'sShortestPath

Algorithm

CLRS24.3

OutlineofthisLecture

problemforunweighted(di)graphs. ?Dijkstra'salgorithm. undirectedgraphs. 1

Recall:ShortestPathProblemforGraphs

Let

Callthisthelink-distance.

est(link-distance)pathsfroma singlesourcever- tex toallothervertices. distancefromtherootvertex. ?BFShasrunningtime? 2

ShortestPathProblemforWeightedGraphs

Let function ????mappingedgestoreal-valued weights.If ??????for ?Thelengthofapath? sumoftheweightsofitsconstituentedges: length

Thedistancefrom

?to ?,denoted? ??????,isthe lengthofthe minimumlengthpathifthereisa pathfrom ?to ?;andis?otherwise. distancefrom?to?is 3

Single-SourceShortest-PathsProblem

weights andadistinguishedsourcevertex,?? determinethe distanceandashortestpathfromthe sourcevertextoeveryvertexinthedigraph. forthisproblem? 4

Single-SourceShortest-PathsProblem

pathmustalsobeashortestpath.Why?

Example:Inthefollowingdigraph,

?isashort- estpath.Thesubpath ?isalsoashortestpath. distancefrom?to?is istenceofa shortestpathtreeinwhichdistancefrom sourcetovertex ?islengthofshortestpathfromsource tovertexinoriginaltree. 5

IntuitionbehindDijkstra'sAlgorithm

tancefromthesourcevertex. eachstepaddingonenewedge,corresponding vertex. 6

TheRoughIdeaofDijkstra'sAlgorithm

?Maintainanestimate ?????ofthelength? ???of theshortestpathforeachvertex ?Always ???and ?????equalsthelength ofaknownpath ???????ifwehavenopathssofar). ?Initially ???andalltheother ?????valuesare setto ?.Thealgorithmwillthenprocessthever- ticesonebyonein someorder. asbeingrealshortestdistance, i.e.

Here"processingavertex

?"meansfindingnew pathsand updating ?????forall ??ifnec- essary.Theprocessbywhichanestimateisup- datediscalled relaxation.

Whenallverticeshavebeenprocessed,

???forall 7

TheRoughIdeaofDijkstra'sAlgorithm

anddothe relaxation?

Question2:Inwhichorderdoesthealgorithm

pro- cess theverticesonebyone? 8

AnswertoQuestion1

?Findingnewpaths.Whenprocessingavertex thealgorithmwillexamineallvertices

Foreachvertex

??,anewpathfrom?to ?isfound(pathfrom?to ?+newedge). ?Relaxation.Ifthelengthofthenewpathfrom? to ?isshorterthan ?????,thenupdate ?????tothe lengthofthisnewpath.

Remark:Wheneverweset

?????toafinitevalue,there existsapathofthatlength.Therefore (Note:If ??,thenfurtherrelaxationscannotchange itsvalue.) 9

ImplementingtheIdeaofRelaxation

Consideranedgefromavertex?to

?whoseweightis?

Supposethatwehavealreadyprocessed

?sothatweknow Then ?Thereisa(shortest)pathfrom ?to?withlength ?Thereisapathfrom ?to ?withlength

Combiningthispathfrom

?to?withtheedge ??,weobtain anotherpathfrom ?to ?withlength If ?????,thenwereplacetheoldpath? withthenewshorterpath? ??.Henceweupdate (originally, s u vd[v] d[u]w 10

TheAlgorithmforRelaxinganEdge

Relax(u,v)?

if(

Remark:Thepredecessorpointer????

?isfordeter- miningtheshortestpaths. 11 setofvertices, ?,forwhichweknowthetrue distance,thatis ?Initially? ??,theemptyset,andweset and ???????forallothersvertices ?.Oneby onewe selectverticesfrom ???toaddto?. ?Theset?canbeimplementedusinganarrayof weset ???blacktoindicatethat 12

TheSelectioninDijkstra'sAlgorithm

teedtoconvergetothetruedistances.

Thatis,howdoesthealgorithm

selectwhichvertex amongtheverticesof ???toprocessnext?

Answer:Weusea

greedyalgorithm.Foreachver- texin ???,wehavecomputedadistancees- timate ?????.Thenextvertexprocessedisalwaysa vertex ???forwhich ????isminimum,thatis, estimate)to tices efficiently? 13

TheSelectioninDijkstra'sAlgorithm

Answer:Westoretheverticesof

???inapriority queue,wherethekeyvalueofeachvertex ?is

Min(),

andDecrease

Key(),eachin?

14

ReviewofPriorityQueues

ations: insert( ???????):Insert ?withthekeyvalue ????in?. u=extractMin():

Extracttheitemwiththeminimum

keyvaluein decreaseKey( ????):Decrease ?'skeyvalueto thateachoperationtakestime ???.SeeCLRS! 15

DescriptionofDijkstra'sAlgorithm

Dijkstra(G,w,s)

?%Initialize for(each???) ?white; ??NIL; ??(queuewithallvertices); while(Non-Empty( ?))%Processallvertices

Extract-Min

??;%Findnewvertex for(each if( ???????)%Ifestimateimproves ;relax

Decrease-Key

black; 16

Dijkstra'sAlgorithm

Example:

s abc d7 2321
8

5450inf

infinf inf

Step0:Initialization.

?sabcd ?????0???? ??nilnilnilnilnil ??WWWWW

PriorityQueue:

?sabcd ?????0???? 17

Dijkstra'sAlgorithm

Example:

s abc d7 2321
8

545inf

inf7 2 0

Step1:As?

??,workon ?and ?and updateinformation. ?sabcd ?????0 ??nilssnilnil ??BWWWW

PriorityQueue:

?abcd 18

Dijkstra'sAlgorithm

Example:

s abc d7 2321
8 545
0 2 7510

Step2:AfterStep1,

?hastheminimumkeyinthe priorityqueue.As ??,workon andupdateinformation. ?sabcd ?????0 ??nilsaaa ??BBWWW

PriorityQueue:

?bcd 19

Dijkstra'sAlgorithm

Example:

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