I’m working on the classic N-Queens puzzle using a specific approach where I place queens column by column, making sure each new queen doesn’t attack any previously placed ones.
My algorithm works like this: for each empty column starting from the left, I try placing a queen in each valid row position where it won’t be attacked by existing queens.
I’ve read that this method results in a state space of around 2057 possible states for the 8x8 board, but I’m not sure how this number is calculated.
Since I’m planning to use Depth First Search to explore the solution tree, I need to understand the computational complexity. What would be the time and space complexity for this approach?
I’m confused about the branching factor because it seems like the number of valid positions decreases significantly as we go deeper into the search tree. Would O(8^8) be accurate for the worst-case time complexity, or is there a better way to analyze this?
The branching factor isn’t constant throughout the search - that’s key to understanding the complexity. Sure, you could theoretically hit O(N^N) states if every position worked, but reality’s way more constrained. Each queen you place eliminates roughly 3N-2 squares for future placements. This cuts down your branching factor massively as you go deeper. That 2057 figure? That’s what you actually get when you run the algorithm and count recursive calls or state visits. It accounts for all the natural pruning from problem constraints. I’ve implemented this myself - the average branching factor starts around 8 in the first column but drops to just 1-2 viable options by the middle columns. Space complexity stays linear at O(N) since you’re only tracking the current partial solution and recursion depth. The exponential worst-case bound exists on paper, but N-Queens’ inherent constraints make practical performance way better than naive exponential estimates suggest.
The figure of 2057 is derived from actual counts of the search tree nodes during backtracking rather than from theoretical estimates. The N-Queens problem is known for its exponential worst-case complexity, but stating it as O(8^8) is an overestimation of the actual computational effort involved. As you delve deeper into the search tree, the branching factor declines significantly due to the constraints imposed by previous placements. For instance, the first queen may have 8 potential positions, but the second queen typically has 6 or 7 valid options, and by the time you reach the fourth or fifth column, you might be left with only 1 to 3 feasible placements. This pruning effect results in a far lower number of explored states than 8^8 suggests. Regarding space complexity, it remains O(N) since the depth-first search backtracking merely requires storage for the current board configuration and the recursion stack. The key takeaway is that efficient constraint checking during placement improves practical performance considerably beyond what the worst-case scenario implies.