Backtracking optimization - algorithm

Question

recently I was trying to solve famous little bishops algorithmic problem. In one of the websites I read that I should divide chessboard into black and white parts to optimize the execution. After that I should use backtracking to count number of possible ways to put bishops on black squares and white squares separetely.
In the following code I try to put 6 bishops ONLY ON WHITE squares of an 8 by 8 chessboard. I do it only to verify that technique is really working.
//inside main function
int k = 6; //number of bishops
int n = 8; //length of one side of chessboard
Integer[] positions = new Integer[k];
long result = backtrack(positions, 0, n);
//find how many times we double counting each possible combination of bishops
int factor = 1;
for(int i = k; i>0; i--) {
factor = factor * i;
}
System.out.println("The result is " + result/factor);
//implementation of other functions
public long backtrack(Integer[] prevPositions, int k, int n) {
if(k == 6) {
return 1;
}
long sum = 0;
Integer[] candidates = new Integer[n*n];
int length = getCandidates(prevPositions, k, candidates, n);
for(int i=0 ; i<length ; i++) {
prevPositions[k] = candidates[i];
sum += backtrack(prevPositions,k+1,n);
}
return sum;
}
public Integer getCandidates(Integer[] prevPositions, int k, Integer[] candidates, int n) {
int length = 0;
//only white squares are considered as candidates, hence i+=2
for (int i = 0; i < n*n; i+=2) {
boolean isGood = true;
int iRow = i / n;
int iCol = i % n;
for (int j = 0; j < k; j++) {
int prev = prevPositions[j];
if (i == prev) {
isGood = false;
break;
} else {
int prevRow = prev / n;
int prevCol = prev % n;
if (Math.abs(iRow - prevRow) == Math.abs(iCol - prevCol)) {
isGood = false;
break;
}
}
}
if(isGood) {
candidates[length] = new Integer(i);
length++;
}
}
return length;
}
Even though I can see why dividing chessboard into white and black squares optimizes the problem, it is still takes around 11 seconds to count number of possible ways to put all bishops ONLY ON WHITE SQUARES. Can you help me pls? What am I doing wrong?

Answer 1

here are a few ways to improve your search.
(1) Instead of generate-and-test, you could consider finite domain search, where every bishop has a "domain" of possible places. Whenever you place a bishop, you prune the domains of the remaining bishops. If a bishop's domain becomes empty, you must backtrack.
(2) As a refinement, if you have n bishops to place and m < n places left, you must backtrack.
(3) Use dynamic programming/memoization, where you store solutions for 1 bishop, 2 bishops, ..., and compute the set of n + 1 bishop solutions from the set of n bishop solutions.
(4) Exploit symmetry to reduce your search space. In this case there is (at least) black/white symmetry and rotational/reflective symmetry.
(5) Try to find a better representation. For example, bit patterns.
(6) If you use a different representation, look into using a "trail" (cf. Prolog) to track the operations you need to undo on backtracking.
Cheers!