## Fun with Genetic Algorithms and the N Queens Problem

Genetic Algorithms are cool!

I was recently skimming through Russel and Norvig’s AI: A Modern Approach and came to the section on Genetic Algorithms (GA). Briefly, they’re a type of algorithm inspired by genetics and evolution, in which you have a problem you’d like to solve and some initial attempts at solutions to the problem, and you combine those solutions (and randomly alter them slightly) to hopefully produce better solutions. It’s cool for several reasons, but one really cool one is that they’re often used to “evolve” to an optimal solution in things like design of objects (see the antenna in the Wikipedia article). So, that’s kind of doing evolution on objects rather than living things. Just take a look at the applications they’re used for.

A lot of it makes more sense when you look at it in the context of evolution, because it’s a pretty decent analogy. A GA “solution attempt” I mentioned above is called an “individual”, like the individual of a species. It could be in many different formats (a string, a 2D array, etc), depending on the problem. You call the group of individuals you currently have the “population”. For species in nature, the “problem” they’re trying to solve is finding the best way to survive and pass on their genes, given the environment. For real evolution and genetics, the success of an individual is somewhat binary: does it pass on its genes, or not? (I guess you could actually also consider that there are grades of passing on your genes; i.e., it might be better to have 10 offspring than 1.) For GA, the success is measured by a “fitness function”, which is a heuristic that you have to choose, depending on the problem.

For each generation, you have a population of different individuals. To continue the analogy, real species mate, and create offspring with combined/mixed genes. In GA we do something called “crossover”, in which the attributes of two individuals are mixed to produce another individual. Similarly, we introduce “mutations” to this new individual, where we slightly change some attribute of them. This is actually very important (see evidence below!), because it allows new qualities to be introduced to the population, which you wouldn’t necessarily get if you just mixed together the current population repeatedly (exactly analogous with real evolution).

So, that’s the rough idea: you have a population, you have them “mate” in some aspect to produce new individuals, you mutate those new ones a little, and now apply your fitness function (i.e., how good is this individual?), and keep some subset of this new population. You could keep the whole population if you wanted to — but the number would quickly grow so large that you’d basically just be doing the same thing as a brute force search over all possible individuals.

I was aware of GA already, but had never actually implemented one. The example they use in AIMA was the 8 Queens problem (8QP), which is a fun little puzzle. Embarrassingly, for a chess player, I had never actually solved it! So I thought I’d dip my toe into GA and do it, and also maybe characterize it a little.

So, let’s set it up! An “individual” here is obviously going to be a board state (i.e., where all the queens are). Most naively, you might think that means a board state is an 8×8 2D array, with 0’s for “no queen on this spot” vs 1 for “queen here”. But, if you look at the 8QP for a second, you’ll quickly see that they each have to be on a different row, and different column. Therefore, you can really specify a board by an 8-long array of integers, where each index/entry represents a row, and the value of that entry is the column number than queen is in. So it’s automatically constraining them to be in different rows, and it makes the coding a lot simpler.

What’s the fitness function (FF)? You want some way of deciding how good a board is. A pretty obvious one for this problem is the number of pairs of queens that are attacking each other, for a given board state. If you solve the problem, FF = 0. So for this problem, you want a lower FF.

Here, crossover is combining two boards. To do this, we just choose a row number, and then split the two parents at that index, and create two new boards by swapping the sides. It’s probably more easily explained in the code:

def mate(self,board1,board2): board1 = deepcopy(board1) board2 = deepcopy(board2) crossover = randint(1,board1.board_size-1) temp = board1.board_state[:crossover] board1.board_state[:crossover] = board2.board_state[:crossover] board2.board_state[:crossover] = temp return(board1,board2)