Backtracking

1. Kamalesh Karmakar,
Assistant Professor,
Dept. of C.S.E.
Meghnad Saha Institute of Technology

3. Adynamic-programmingalgorithmsolveseverysubsubproblemjustonceandthensavesitsanswerinatable,therebyavoidingtheworkofrecomputingtheanswereverytimethesubsubproblemisencountered.
Dynamicprogrammingistypicallyappliedtooptimizationproblems. Insuchproblemstherecanbemanypossiblesolutions.Eachsolutionhasavalue,andwewishtofindasolutionwiththeoptimal(minimumormaximum)value.Wecallsuchasolutionanoptimalsolutiontotheproblem,asopposedtotheoptimalsolution, sincetheremaybeseveralsolutionsthatachievetheoptimalvalue.

4. AlgorithmPlace(k,i)
//Returnstrueifaqueencanbeplacedat[k,i]position.
//x[]isaglobalarraywhosefirst(k-1)valueshavebeenset.
//Abs(r)returnstheabsolutevalueofr.
{
forj:=1tok-1do
if((x[j]=i)or(Abs(x[j]-i)=Abs(j-k)))
returnfalse;
returntrue;
}

5. AlgorithmNQueens(k,n)
{
fori:=1tondo
ifPlace(k,i)then
{
x[k]:=i;
if(k=n)thenwrite(x[1:n])
elseNQueens(k+1,n);
}
returntrue;
}

6. Thealgorithmwasformedusingtherecursivebacktrackingschema.ThegraphisrepresentedbyitsbooleanadjacencymatrixG[1:n,1:n].Allassignmentof1,2,3,….,mtotheverticesofthegraphsuchthatadjacentverticesareassigneddistinctintegersareprinted.Kistheindexofthenextvertextocolor.
mcoloringmeansmaximummcolorscanbeused.
AlgorithmmColoring(k)
{
repeat
{//Generatealllegalassignmentofx[k].
NextValue(k);//Assigntox[k]alegalcolor.
if(x[k]=0)thenreturn;//Nonewcolorpossible.
if(k=n)//Atmostmolorshavebeenusedtocolornvertices
write(x[1:n]);
elsemColoring(k+1);
}untill(false);
}

7. AlgorithmNextValue(k)
{
repeat
{
x[k]:=(x[k]+1)mod(m+1);//Nexthighestcolor
if(x[k]=0)thenreturn;//Allcolorshavebeenused.
forj:=1tondo
{//checkifthiscolorisdistantfromadjacentcolors.
if((G[k,j]!=0)and(x[k]=x[j]))
//if(k,j)isandedgeandifadj.verticeshavethesamecolor.
thenbreak;
}
if(j=n+1)thenreturn;//Newcolorfound
}untill(false);//Trytofindanothercolor.
}