Fast Nearest Neighbor Algorithms on a Linear Array with a 线性阵列上的快速最近邻算法

FastNearestNeighborAlgorithmsonaLinearArraywithaReconfigurablePipelinedBusSystem

YiPan,KeqinLi,andSi-QingZheng

ParallelAlgorithmsandApplication,Vol.13,pp.1-25(1998)

Introduction

LetA=(ai,j),0≦i,j≦n-1,beablackandwhiten*nbinaryimageintheformofatwo-dimensionalarray,witheachelementai,jbeingapixel.Pixelai,jisblackifai,j=1,andwhiteifai,j=0.Inthispaper,weconsidersolvingthenearestneighborproblemonalineararraywithareconfigurablepipelinedbussystem(LARPBS).

TheLARPBSModel

BasicOperations

One-to-OneCommunication

tset(j)=tref+(N–j–1)ω.

BroadcastingandMulticasting

NearestNeighborAlgorithms

AnAlgorithmFramework

Eq.(1):dist(ai,j,ai’,j')=((i-j’)^2+(j-j’)^2)^(1/2)

Step1:ForeachpixelA(i,j),findj`suchthatA(i,j’)istheblackpixelclosesttoA(i,j)toitsrightinrowi,andstoreindexpair(i,j’)inRIGHT_NEAREST(i,j).Ifsuchapixeldoesnotexist,letj’=∞.

Step2:ForeachpixelA(i,j),findi’suchthatA(i’,j)istheblackpixelclosesttoA(i,j)fromaboveincolumnj,andstoreindexpair(i’,j)inUP_NEAREST(i,j).Ifsuchapixeldoesnotexist,leti’=∞.

Step3:ForeachpixelA(i,j),findi”suchthatA(i”,j)istheblackpixelclosesttoA(i,j)frombelowincolumnj,andstoreindexpair(i”,j)inDOWN_NEAREST(i,j).Ifsuchapixeldoesnotexist,leti”=∞.

Step4:ForeachPE(i,j)thatoriginallystoresawhitepixelA(i,j),computeDISTup(i,j)andDISTdown(i,j)usingEq.(1),whereDISTup(i,j)isthedistancebetweenpixelUP_NEAREST(i,j)andpixelRIGHT_NEAREST(i,j),andDISTdown(i,j)isthedistancebetweenpixelDOWN_NEAREST(i,j)andpixelRIGHT_NEAREST(i,j).LetDIST(i,j):=min{DISTup(i,j),DISTdown(i,j)}.

Step5:ForeachPE(i,j)thatoriginallystoresablackpixelA(i,j),computeDISTup(i,j),DISTdown(i,j),DISTright(i,j)usingEq.(1),whereDISTup(i,j)isthedistancebetweenpixelUP_NEAREST(i,j)andpixelA(i,j),DISTdown(i,j)isthedistancebetweenpixelDOWN_NEAREST(i,j)andpixelA(i,j),andDISTright(i,j)isthedistancebetweenpixelRIGHT_NEAREST(i,j)andpixelA(i,j).

Step6:ForeachPE(i,j)thatoriginallystoresablackpixelA(i,j),sendA(i,j)toprocessorPE(i,j’),wherej”=RIGHT_NEAREST(i,j).ForeachPE(i,j’),ifitreceivesablackpixelA(i,j),itstoresthepair(i,j)inLEFT_NEAREST(i,j’);otherwise,itstores(i,∞)inLEFT_NEAREST(i,j’).

Step7:ForeachPE(i,j)thatoriginallystoresablackpixelA(i,j),letDIST(i,j):=min{DISTup(i,j),DISTdown(i,j),DISTleft(i,j)},whereDISTleft(i,j)isthedistancebetweenA(i,j)andLEFT_NEAREST(i,j).

Step8:Considereachrowoftheinputimageasseveralsegmentsseparatedbyblackpixels.Eachsegmentconsistsofablackpixelandthemaximalnumberofconsecutivewhitepixelsimmediatelytoitsleft.LetthePEthatoriginallystoresablackpixelbetheleaderofthesegment.Foreachsegment,findtheminimumofthevaluesDISTstoredintheprocessorsofthesegment,andsendthepixelwiththisDISTvaluetotheleaderofthesegment.

AnO(loglogn)TimeImplementation

ThetaskoffindingtheminimumofasetofNelementscanbecarriedoutinO(loglogN)timeonanLARPBSofNprocessors.

AnO(1)TimeImplementation

Foreveryfixedε>0,theminimumofNelementscanbecomputedinO(1)timeusinganLARPBSofN^(1+ε)processors.

ARandomizedImplementation

Theneare