I think the complexity class of Sudoko is NP-Complete (a quick google confirms it's not exactly NP but very close: http://11011110.livejournal.com/23221.html?thread=19381 -- complexity is the same as solving SAT problems which have only one unique solution)
> The solver uses depth first and/or breadth first with constraint propagation to prune the search
That would be backtracking.
Further, the purpose of many the constraint techniques seem to be aimed at creating puzzles that are amenable to humans, not solving them. The article itself states that for solving, backtracking with fewer constraints performs better.
You are right that "the purpose of many the constraint techniques seem to be aimed at creating puzzles that are amenable to humans, not solving them"
I shared the link because I found his sudoku generator and the constraint methods interesting.
> That would be backtracking.
Let me quote the first para in full:
The solver uses depth first and/or breadth first tree search
with constraint propagation to prune the search for the next
best move (forms of forward checking.) There are space/time
tradeoffs between depth/breadth first search and the constraints
used; sudoku(1) has options to control the combinations.
The common characteristic for all constraints, here and elsewhere,
is that they avoid trial and error. Its fine for a computer
to guess and backtrack but a definite breach of puzzle manners
to require a human to do so.
I could be wrong but, yes, it does use backtracking to find the constraint that can give a number for an empty cell but it never has to change a number it has put in a cell. That differs from the trial and error approach that moves forward by guessing values and checking if it leads to a valid solution.
I usually solve them by hand without guess and check, though occasionally, you get down to where there are pairs of values where you have no choice but to try one and see what works (or maybe I just need to figure out more constraints to use).
Well, an ambiguous Sudoku is like a 12 line sonnet...
It isn't computationally relevant but even if trial and error were a simpler approach I feel like that is not in the spirit of the game. Sudoku is a number maze, the goal is to backtrack.
Also I refuse to attempt something that is supposedly too difficult for humans to solve, I'm incapable of quitting puzzles and don't relish spending the next X hours proving it can be done :)
9x9 grid of digits with an ambiguous solution is not a Sudoku. It is part of the definition.
Within the set of initial grids that have unambiguous solutions there are ones which require guessing or backtracking, these are generally not considered proper Sudokus and are not presented to humans to solve.
If a Sudoku has one and only one solution, then it is always solvable without guessing or backtracking. It just might not be within the scope of the human brain. There exist (many) configurations where every cell is influenced by every other, so to solve any one cell essentially requires a simultaneous solve of the entire puzzle. There's no theoretical reason a human couldn't do that too; it's a problem simply of computational capacity not fundamental approach.
If you do the constraint propagation right, the number of trials and errors will be very small (and it will be only one trial and no errors for the easy ones).
I'd be very interested to see what an algorithm that solves any valid sudoku puzzle looks like. (apart from pathologic solutions like querying a database of all valid 9x9 sudoku combinations).
Edit: using the symmetries described in the same Wikipedia article, the size of the database could be reduced significantly to roughly 5*10^9, which is quite manageable.
The code in the post is based on depth-first search, which basically is a trial and error approach. A well implemented and effective one thanks to constraint propagation, but it is still trial and error. And there are cases, in which backtracking (trial and error) is necessary - at least one of them is documented in the post (the puzzle "hard1" that took 188 seconds).
