6.006IntroductiontoAlgorithms

Lecture2:PeakFindingProf.ErikDemaine

Today• Peakfinding(newproblem)

– 1Dalgorithms– 2Dalgorithms

• Divide&conquer(newtechnique)

FindingWater…IN SPACE

• YouareGeordi LaForge• Trappedonalienmountainrange• Needtofindapoolwherewateraccumulates

• Canteleport,butcan’tsee http://en.wikipedia.org/wiki/File:GeordiLaForge.jpg

photo by Erik Demaine Banff, Canada

FindingWater…IN SPACE

• Problem: Findalocalminimumormaximuminaterrainbysampling

photo by Erik Demaine Banff, Canada

1DPeakFinding• Givenanarray :

• isapeak ifitisnotsmallerthanitsneighbor(s):

whereweimagine

• Goal: Findany peak

1 2 6 5 3 7 40 1 2 3 4 5 6

:

“BruteForce”Algorithm• Testallelementsforpeakyness

for in :if :

return

1 2 6 5 3 7 40 1 2 3 4 5 6

:

Algorithm1½•

– Globalmaximumisa localmaximum

for in :if :

return

1 2 6 5 3 7 40 1 2 3 4 5 6

:

ClevererIdea• Lookatanyelement anditsneighbors &– Ifpeak:return– Otherwise:locallyrisingonsomeside

• Mustbeapeakinthatdirection• Socanthrowawayrestofarray,leaving : or 1:

1 2 6 5 3 7 40 1 2 3 4 5 6

:

WheretoSample?• Wanttominimizetheworst‐caseremainingelementsinarray– Balance oflengthwith oflength

–– :middleelement– Reduce to

1 2 6 5 3 7 40 1 2 3 4 5 6

:

Algorithm

if :return

elif :return peak1d

elifreturn peak1d

1 2 6 5 3 7 40 1 2 3 4 5 6

:

Divide&Conquer• Generaldesigntechnique:1. Divideinputintopart(s)2. Conquereachpart

recursively3. Combineresult(s)tosolve

originalproblem

• 1Dpeak:1. Onehalf2. Recurse3. Return

Divide&ConquerAnalysis• Recurrence fortimetakenbyproblemsize

1. Divideinputintopart(s):

2. Conquereachpartrecursively

3. Combineresult(s)tosolveoriginalproblem

combinecost

dividecost

1DPeakFindingAnalysis• Divide probleminto1problemofsize

• Dividecost:• Combinecost:• Recurrence:

SolvingRecurrence

2DPeakFinding• Given matrixofnumbers

• Wantanentrynotsmallerthanits(upto)4neighbors:

2 1 2 1 1 1 18 9 8 0 5 3 09 0 6 0 4 6 47 6 3 1 3 2 39 8 9 3 2 4 87 2 5 1 4 0 39 3 5 2 4 9 8

Divide&Conquer#0• Lookingatcenterelementdoesn’tsplittheproblemintopieces…

2 1 2 1 1 1 18 9 8 0 5 3 09 0 6 0 4 6 47 6 3 1 3 2 39 8 9 3 2 4 87 2 5 1 4 0 39 3 5 2 4 9 8

Divide&Conquer#½• Considermaxelementineachcolumn

• 1Dalgorithmwouldsolvemaxarrayin

time• But timetocomputemaxarray 2 1 2 1 1 1 1

8 9 8 0 5 3 09 0 6 0 4 6 47 6 3 1 3 2 39 8 9 3 2 4 87 2 5 1 4 0 39 3 5 2 4 9 8

9 9 9 3 5 9 8

Divide&Conquer#1• Lookatcentercolumn• Findglobalmaxwithin• Ifpeak:returnit• Else:

– Largerleft/rightneighbor– Largermaxinthatcolumn– Recurseinleft/righthalf

• Basecase: 1column– Returnglobalmaxwithin

2 1 2 1 1 1 18 9 8 0 5 3 09 0 6 0 4 6 47 6 3 1 3 2 39 8 9 3 2 4 87 2 5 1 4 0 39 3 5 2 4 9 8

9 9 9 3 5 9 8

Analysis#1• timetofindmaxincolumn

• iterations(likebinarysearch)

• timetotal

• Canwedobetter?2 1 2 1 1 1 18 9 8 0 5 3 09 0 6 0 4 6 47 6 3 1 3 2 39 8 9 3 2 4 87 2 5 1 4 0 39 3 5 2 4 9 8

Divide&Conquer#2• Lookatboundary,centerrow,andcentercolumn(window)

• Findglobalmaxwithin• Ifit’sapeak:returnit• Else:

– Findlargerneighbor– Can’tbeinwindow– Recurseinquadrant,includinggreenboundary

22 11 22 11 11 11 1188 99 88 00 55 33 0099 00 66 00 44 66 4477 66 33 11 33 22 3399 88 99 33 22 44 8877 22 55 11 44 00 3399 33 55 22 44 99 88

00 00 00 00 00 00 00

000000000000000000

00 00 00 00 00 00 00 000000000000000000

Correctness• Lemma: Ifyouenteraquadrant,itcontainsapeakoftheoverallarray[climbup]

• Invariant: Maximumelementofwindowneverdecreasesaswedescendinrecursion

• Theorem: Peakinvisitedquadrantisalsopeakinoverallarray

22 11 22 11 11 11 1188 99 88 00 55 33 0099 00 66 00 44 66 4477 66 33 11 33 22 3399 88 99 33 22 44 8877 22 55 11 44 00 3399 33 55 22 44 99 88

00 00 00 00 00 00 00

000000000000000000

00 00 00 00 00 00 00 000000000000000000

Analysis#2• Reduce matrixto

submatrixintime(|window|)

22 11 22 11 11 11 1188 99 88 00 55 33 0099 00 66 00 44 66 4477 66 33 11 33 22 3399 88 99 33 22 44 8877 22 55 11 44 00 3399 33 55 22 44 99 88

00 00 00 00 00 00 00

000000000000000000

00 00 00 00 00 00 00 000000000000000000

Divide&ConquerWrapup• Leadstosurprisinglyefficientalgorithms• Notterriblygeneral,butstillquiteuseful• We’lluseitagainin

– Module4(sorting)– Module8(geometry)

http://en.wikipedia.org/wiki/File:CaesarTusculum.jpg