6.006IntroductiontoAlgorithms

Lecture14:ShortestPathsIProf.ErikDemaine

Today• Shortestpaths• Negative‐weightcycles• Triangleinequality• Relaxationalgorithm• Optimalsubstructure

ShortestPaths

ShortestPaths

HowLongIsYourPath?• Directedgraph• Edge‐weightfunction• Path• Weight of ,denoted ,is

v1v2

v3v4

v54 −2 −5 1Example:

MyPathIsShorterThanYours

• Ashortestpathfrom to isapath ofminimumpossibleweight from to

• Theshortest‐pathweight from toistheweightofanysuchshortestpath:

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Example:1

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72BA

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72 ,6

YouCan’tGetThereFromHere

YouCan’tGetThereFromHere

• Ifthereisnopathfrom to ,thenneitheristhereashortest pathfrom to

• Define inthiscase

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Example: 1

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TheMoreIWalk,TheLessItTakes

• Ashortestpathfrom to mightnotexist,eventhoughthereisapathfrom to

• Negative‐weightcycle

has

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Example: 1

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TheMoreIWalk,TheLessItTakes

• Define ifthere’sapathfromto thatvisitsanegative‐weightcycle

•

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Example:1

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Ds t2

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Single‐SourceShortestPaths• Problem: Givenadirectedgraphwithedge‐weightfunction ,andasource vertex ,compute forall

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Example:1

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Ds t2

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0 1

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Shortest‐PathTree• Ideallyalsocomputeashortest‐pathtreecontainingashortestpathfromsource toevery (assumingshortestpathsexist)– Representbystoringparent foreach

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Example:1

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Ds t2

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Shortest‐PathTrees

• Ideallyalsocomputetheunionofoneshortestpathfromsource toevery ,whichformstheshortest‐pathtree

bicycletravelinSanFranciscoarea

bicycletravelinSeattlearea

BrandonMartin‐Anderson&NinoWalker

http://graphserver.sourceforge.net/gallery.html

Single‐SourceShortest‐PathAlgorithms

• Relaxationalgorithm (TODAY)– Frameworkformostshortest‐pathalgorithms– Notnecessarilyefficient

• Bellman‐Fordalgorithm (LECTURE15)– Dealswithnegativeweights– Slowbutpolynomial

• Dijkstra’s algorithm (LECTURE16)– Fast(nearlylineartime)– Requiresnonnegativeweights

Brute‐ForceAlgorithm

• Numberofpathscanbeinfinite:

return encountered with smallest

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Ds t2

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Brute‐ForceAlgorithm

• Numberofpathscanbeexponential:

##assumenonegative‐weightcycles

return encountered with smallest

s vnv1 v2 …=t

pathsfrom to ; verticesandedges

Relaxation• Ingeneral,referstolettingasolution(temporarily)violateaconstraint,andtryingtofixtheseviolations

Magic Geekhttp://www.youtube.com/

watch?v=Y12daEZTUYo

TriangleInequality• Theorem: Forall ,wehave

• Proof: Shortestpathfrom to isatmostanyparticularpath,e.g.,thebluechain.

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RelaxationApproach• Maintaindistanceestimateforeach

• Goal: forall• Invariant:• Initialization:

• Repeatedlyimproveestimatestowardgoal,byaimingtoachievetriangleinequality

in :

EdgeRelaxation• Consideranedge

• [triangleineq.][candidatepath]

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:

RelaxationAlgorithmfor in :

whilesomeedge has :picksuchanedgerelax :

if :u

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RelaxationAlgorithmwithShortest‐PathTree

for in :

whilesomeedge has :picksuchanedgerelax :

if :u

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RelaxingIsSafe• Lemma: Therelaxationalgorithmmaintainstheinvariantthat forall .

• Proof: Byinductiononthenumberofsteps.– Considerrelax– Byinduction,– Bytriangleinequality,

– Sosetting is“safe”

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InfiniteRelaxation• Ifanegative‐weightcycleisreachablefrom ,thenrelaxationcanneverterminate

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∞ ∞

∞ ∞

∞

LongRelaxation

v1 v2 v3 v4 v5 v7v6

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4 4 2 2

2 1

1…

0 ∞ ∞ ∞ ∞ ∞ ∞

LongRelaxation

• Analysis:– relax– relax– recurseon– relax– recurseon

v1 v2 v3 v4 v5 v7v6

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88

4

4 4 2 2

2 1

1…

0 ∞ ∞ ∞ ∞ ∞ ∞

/

AreYouSureThisIsaGoodIdea?

• Bellman‐Fordalgorithm: (LECTURE15)– Relaxalloftheedges– Repeat times– Polynomialtime!

• Dijkstra’s algorithm: (LECTURE16)– Relaxedgesinagrowingballaround– Nearlylineartime!– (butdoesn’tworkwithnegativeedgeweights)

OptimalSubstructure• Lemma: Asubpath ofashortestpathisashortestpath(betweenitsendpoints).

• Proof: Bycontradiction.– Iftherewereashorterpathfrom to ,thenwecouldshortcut thepathfrom to ,contradictingthatwehadashortestpath.

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