Tuesday, February 28, 2006

Problems on Algorithms

While 99.9% of Sudoku puzzles can be solved using relatively simple techniques, certain very difficult Sudoku puzzles require more advanced methods. One of these techniques consists of searching the puzzle for hidden sets, i.e. couples, triples or quadruples.
Again, puzzles which can only be resolved by searching for hidden sets occur very infrequently. Nevertheless, any self-respecting Sudoku solver must implement these more advanced techniques. Sudoku-Grok is no exception to this rule.

After half a day work, I managed to devise an algorithm to detect hidden couples. I thought that in principle the same algorithm could be used to detect hidden triples and quadruples. After all, it worked perfectly for couples, and the set size was just a parameter supplied to the algorithm. It was not hard to change the set size to three or four instead of two. Unfortunately, my approach assumed that each set was complete.

When the set size is two, the problems consist of detecting two sets of size two. My approach could detect three sets of size three, or four sets of size four. It so happens that when the set size is larger, say 3, hidden sets occur in three sets of size three but also two. Similarly hidden quadruples occur in four sets of size four but also three or two.

The problem was evidently more complicated than I thought initially. After several hours of unfruitful toil trying to find the proper recursive algorithm, it occurred to me that the problem could be approached as a choice of r elements among 9 elements, where r is 2, 3 or 4, the size of the desired hidden set.

Once I had realized that the problem could be expressed in terms of simple combinatorics, I turned to my copy of the book "Problems on Algorithms" by Ian Parberry. Lo and behold, page 120 of the book had an outline of a recursive algorithm generating all the combinations of r elements chosen among n. It was expressed on four (4) lines of pseudo-code which was just what I needed to get my hidden set problem solved. As it turns out, many sophisticated Sudoku techniques (but not all) can be implemented in terms of a choice of r elements in a set of n, where n is usually 9.

I can hear you reaching your keyboard to order a copy of "Problems on Algorithms". The bad news is that this book is out of print. Now for the good news: you can get it from Ian Parberry's web-site for free. Enjoy.

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