# Graph Traversal: solving the 8-puzzle with basic A.I.

I’ve been working through Peter Norvig and Stuart Russel’s Artificial Intelligence, A Modern Approach (thanks to the Square engineering library) and one of the most helpful chapters involved methodically demonstrating basic graph traversal algorithms for problem solving. If that sounds heady, it’s not — I think you’ll enjoy it.

First, let’s talk about graphs. A graph is any set of points (nodes) and the lines (edges) between those points. A simple kind of graph is a tree structure. This is where there’s a single ‘root’ which has 1 or more branches that then have their own branches, etc. Many things in computer science can be expressed using some kind of tree structure graph.

Here’s how this relates to AI: the root node (the one at the top of the tree) is the state the world is in right now. Each other node represents a different state of the world that’s reachable across an edge by the node immediately above it. The line between them is some kind of action. Moving a car’s wheels, turning on a servo, whatever. The A.I. just needs to know that if you start at node A and do action X you’ll get to node B. Your software will appear to be intelligent if it can start at the root node and find its way to a better state of the world through a series of actions.

## A world where you try to get a big number

Let’s reuse the above image as an example. Imagine it’s a complete map
of all possible states of the world. The only actions that are
available to you are “go left” and “go right” (‘cause you’re always
starting at the root (top) node and heading downward). Imagine,
additionally, that the value of the number of a node is how much we like
that node. From a glance you can easily tell that one of the nodes has a
value of **11** and is therefore the “best” node in the graph; it’s the
“best” state of the world that we could possibly get to. If our software
is intelligent it should start at the root node and find the **11** node
as it’s destination.

Since we can easily see where the **11** node is the right answer is “go
left, then right, then right again.” But how do we teach our code to
find the best node?

## Recursive Depth-first search

Here’s the simplest way we can find the right node on the given tree.

```
best = nil
def walk(node)
best = node if best.nil? || node > best
node.children.each {|child| walk child }
end
walk root
```

Here we “walk” down each branch of the tree all the way to the end using
recursion. This is known as recursive ```
depth first
search
```

and is a great
tool when you think that any path might have a good node really far down
so you just want to get really deep really fast. It’s also the least
code possible to find a solution to our problem. Unfortunately, the
simple implementation of depth-first involves recursion which means
we’re limited to only traversing graphs that have a total depth less
than our runtime stack frame limit. If you’ve ever seen a “stack
overflow error” it’s because there was so much recursion in your program
that the computer assumed you were caught in an infinite loop and gave
up. To demonstrate this try running this simple program in `irb`

:

```
def go(n) # On my machine the last thing printed was 8247 and then I saw:
puts n # SystemStackError: stack level too deep
go n + 1 # which means Ruby let me use 8,247 stack frames
end # before giving up. If I were to try going 9,000 nodes deep
go 0 # in a graph I'd get this error.
```

Now, if you wanted to do a depth-first search without recursion you can but it’s no longer the simplest code so I’ll skip it here. Suffice to say that it would mean you’ll have to manually keep track of which nodes to visit in which order rather than letting your programming language do it for you.

## Breadth-first search in a loop

Another simple way to traverse the graph is to look at the nodes from
left to right on each level rather than going all the way down one
branch and then all the way down another branch. This is called
```
breadth-first
search
```

and is
better if you think there’s a good node close to the root.

This is also a very simple bit of code *and* it frees us
from having to rely on our runtime’s call stack. Rather than recursing
through methods and letting Ruby keep track of our work (as in the above
example) we’ll just run in a loop and store our work in an array. The
advantage here is you can put more than 9,000 items in an array —
it’s only bounded by how much memory is on your machine.

```
queue = [root]
best = -1
begin
current = queue.shift
best = [current, best].max
current.children.each {|child| queue.push child }
end until queue.empty?
```

The name “breadth-first” comes from the fact that it’ll look at all the
nodes at each level from side to side before proceeding down to the next
level. Notice that the array variable is named `queue`

. This is because
in a depth-first implementation you’re always going to have a list that
you put newly-discovered nodes onto the end of and pull nodes to explore
off of the front.

Breadth-first search is easy to reason about, you won’t run out of stack
space like when you used recursion (although it’s possible you’ll run
out of memory), and if your environment supports concurrency primitives
you might be able to run it in parallel quite easily. C# has a ```
Consumer
Queue
```

that can help with this and Clojure has multiple ways to iterate
through a list in parallel. Ruby requires you to do more work
synchronizing when threads get to append their newly-discovered nodes to
the end of the queue but otherwise you get the parallelism cheaply.

## Solving the sliding-block puzzle

We walk along the edges of a graph from the starting node to a better node. With me so far? If this graph represents a set of world states that are reachable from each other via actions then searching the graph is the same thing as figuring out what to do.

Let’s use this technique to try to solve a problem that has a clear
starting state and a clear ending state with many (possibly *very* many)
intermediate states. The sliding-block puzzle (often called an 8-puzzle
or, in it’s larger variant, a 15-puzzle) is a great case for us to
tackle.

In an 8-puzzle you’ve got a bunch of tiles in the wrong places and just one empty space that you can move around until the tiles are in the right order. How do you know which move to make first? How do you know when you’re on the right track? How do you know if you’ve going in loops?

If we model each possible action as edges in a graph and each potential puzzle state as a node then we just start at the beginning and begin exploring the graph. We’ll stop once we’ve found the solution (or, if we built our code poorly, we’ll stop when we run out of memory or time).

Now, there are two ways we can set up our data for this problem. One is
to generate all possible states (nodes) that the puzzle can have and to
then connect the adjacent states. We would then have a complete graph we
could traverse and we’d even be able to mark the solution ahead of time
and know where it was located in the graph. Unfortunately, the number of
states is 9-factorial or about 360,000. Generating that many puzzle
slide orientations and then iterating through each one would take, at
best, `(9!)(9!-1)/2`

node-comparison operations (the formula for how
many edges can exist between nodes in a graph is `(n * n-1)/2`

)

So let’s not do that. Rather, let’s start at the root node (the starting state) and then create branches from each node as we go. We’ll stop when we discover our solution — hopefully long before we examine 360,000 states.

## Defining a puzzle class

We’re going to need a few tools. First, let’s put together a way to represent a puzzle board with tiles in a particular position:

```
class Puzzle
Solution = [0, 1, 2, # Let's put all the tiles in ascending order
3, 4, 5, # the same way you'd see them on a phone keypad.
6, 7, 8] # We use a '0' for the blank cell because
# `nil` doesn't play well with others.
attr_reader :cells
def initialize cells
@cells = cells
end
def solution?
Solution == @cells
end
end
```

What did we just do there? That is a `Puzzle`

class where each instance
knows whether it’s a solution. The cells/tiles of the puzzle are kept in
a list.

Now let’s construct a way to represent a state (a node on the
solution graph). A state isn’t just a representation of puzzle tile
position but *also* the history of how that puzzle arrangement was
reached from the starting point. This is key: if we don’t keep track of
how we got to a solution node on the graph then we’ll never be able to
report how to solve the puzzle. So we need to keep a list as we go of
which actions we’ve taken to arrive at the current node.

```
class Puzzle # We're extending the Puzzle class.
def zero_position # `zero_position` tells us which cell has
@cells.index(0) # the '0'.
end
def swap swap_index # `swap` tells the puzzle:
new_cells = @cells.clone # "give me you, but with the '0' cell
new_cells[zero_position] = new_cells[swap_index] # replaced by the cell at some other
new_cells[swap_index] = 0 # location of my choice."
Puzzle.new new_cells # This is how we'll simulate moving a tile.
end
end
class State # Each `State` instance represents a node
Directions = [:left, :right, :up, :down] # in our solution graph. It keeps track of
# both a puzzle and the list of actions
# required to arrive at there from the
attr_reader :puzzle, :path # starting node.
def initialize puzzle, path = []
@puzzle, @path = puzzle, path # The `path` is a list of actions
end # like 'up', 'down', 'right', right'
def solution? # Each node in our graph should
puzzle.solution? # know if it's a solution.
end
def branches # Returns all adjacent possible states
Directions.map do |dir| # including steps needed to get there.
branch_toward dir # Most nodes will have 2-4 branches based
end.compact.shuffle # on how many directions the blank tile
end # can try to go.
private
def branch_toward direction
blank_position = puzzle.zero_position
blankx = blank_position % 3
blanky = (blank_position / 3).to_i
cell = case direction # The only reason this method is so long
when :left # is because sometimes the blank tile is already
blank_position - 1 unless 0 == blankx # at a wall and that direction isn't possible
when :right
blank_position + 1 unless 2 == blankx
when :up
blank_position - 3 unless 0 == blanky
when :down
blank_position + 3 unless 2 == blanky
end
State.new puzzle.swap(cell), @path + [direction] if cell
end
end
```

This State class knows about one particular arrangement of the puzzle and is able to determine next steps. When we call State#branches we get a list of adjacent puzzle arrangements (anything reachable by moving the empty space over by 1 square) and each of these new states include the full list of steps necessary to reach them.

That’s the setup. Now that we have some problem-specific helpers we can use our breadth-first algorithm from up above to start tackling this.

```
def search state
state.branches.reject do |branch| # Important: don't revisit puzzles
@visited.include? branch.puzzle.cells # you've already seen!
end.each do |branch| # The list of places we need to search
@frontier << branch # is known as the 'frontier'
end
end
require 'set' # We'll remember what we've seen in a set, it has
def solve puzzle # way better lookup times than an array.
@visited = Set.new
@frontier = []
state = State.new puzzle
loop {
@visited << state.puzzle.cells
break if state.solution? # This is the `base` or end condition
search state
state = @frontier.shift # Pull another off the list, keep chugging along
}
state
end
```

If we feed in a solveable puzzle we can see that this code works. Let’s try one where the empty tile was moved right and then down. The solution should be to move it up and then left:

```
p solve(Puzzle.new [1, 4, 2, # `solve` is going to return a State
3, 0, 5, # instance. We care about it's #path
6, 7, 8]).path
## => [:up, :left]
```

So… it works, but it’s just kinda wandering around until it finds a solution. We gave it a problem that was only 2 steps from a solution so if we gave it something harder would it ever finish? And how long would the solution path be?

Here’s our code running with a puzzle who’s optimal solution is 20 steps away:

```
p solve(Puzzle.new [7, 6, 2, # I generated this by creating a solution state and
5, 3, 1, # running `state = state.branches.sample` in a big loop.
0, 4, 8]).path
## => [:up, :up, :right, :down, :right, :down, :left, :left,
## :up, :up, :right, :down, :down, :right, :up, :left,
## :down, :left, :up, :up]
## Time: 27 seconds
```

It works! Eventually. But 27 seconds is a bit slow. What if we were tackling the 15-puzzle instead? Rather than the 9! (360K) options we would be searching through 16! (20 trillion) options. That would take almost literally forever.

## Uniform-cost search

As we walk the graph we’re keeping a `frontier`

— a list of states
we’re hoping to explore in the future. Since we always add to the back
of the list and take (`shift`

) from the front it’s technically a FIFO queue rather than just a list.

What if, rather than picking the next element from the queue to explore
we tried to pick the *best* one? Then we wouldn’t have to explore quite
so many trillions of nodes in our state graph.

Uniform-cost search entails keeping track of the how far any given node
is from the root node and using that as its `cost`

. For our puzzle
example that means that however many steps `n`

it takes to get to state
`s`

then `s`

has a cost of `n`

. In code: ```
s.cost =
steps_to_reach_from_start(s)
```

. A variant of this is called Dijkstra’s
Algorithm.

There’s one missing piece here though: we don’t want to examine every
item in the entire `frontier`

queue every time we want to pick the
next lowest-cost element. What we need is a priority queue that
automatically sorts its members by some value so looking up an element
by cost is cheap and doesn’t slow down the rest of what we’re trying to
do.

```
class PriorityQueue # This is a terrible implementation of a
def initialize &comparator # priority queue. The `#sort!` method iterates
@comparator = comparator # through every item every time.
@elements = [] # What you want is a priority queue backed by a heap data structure.
end # In Ruby you should use the `PriorityQueue` gem
# and on the JVM there's a good Java implementation.
def << element
@elements << element
sort!
end
def pop # `pop` is the typical queue-polling nomenclature.
@elements.shift # Your implementation may call it something else.
end
private
def sort!
@elements = @elements.sort_by &@comparator # This line is why this implementation
# sucks. Don't use this IRL.
end
end
class State
def cost # The cost is pretty simple to calculate here.
path.size # The path contains all the steps, in order that we
end # used to arrive at this state. So the cost
end # is just the number of steps.
require 'set'
def solve puzzle
@visited = Set.new
@frontier = PriorityQueue.new {|s| s.cost }
state = State.new puzzle
loop {
@visited << state.puzzle.cells
break if state.solution?
search state
state = @frontier.pop
}
state
end
```

**Sidebar:** you may be wondering why this gains us any advantage? Sure,
we’re now picking the the best node from the queue rather than whichever
one was added first but we still have to explore all of them, right?
Actually, no. Because we’re sorting by the ‘cost’ of the nodes we can be
guaranteed that whenever we find a solution it’s the best one. There
may be other paths to solutions in our graph but they are all guaranteed
to be of higher cost. So this Uniform-cost search lets us leave a vast
section of the queue unexplored.

What does that do to our performance? Well, if we re-run our above 20-step puzzle the time will drop considerably from 27 seconds to 10 seconds (on my machine).

This is a big speedup and, for larger problems, can shave days off the calculation time. But there’s much more we can do.

## A* Search

The uniform-cost search picks the best next state from the frontier. Let’s enhance the code’s understanding of what makes something “best” by calculating not only the distance from the start to where we are but the distance from where we are to the goal.

Old cost function: `steps_to_get_to(s)`

New cost function: `steps_to_get_to(s) + steps_to_goal_from(s)`

But, uh, how do we know how far we are from the solution? If we knew how far away the solution was we’d probably already have found it, right? Right. So rather than being exact, let’s just pick a healthy estimate of how far we are from a solution. One approximation would be “how many tiles are out of place?” That would at least differentiate almost-solution nodes from not-even-close ones. But we’d like to be a bit more precise.

So let’s say that the distance cost between a given node and the solution node is the number of tile-movements that would be required if tiles could move through each other and go straight to their goal positions. So a near-solution node might have a distance cost of 3 and a not-even-close node might have a distance cost of 26. That should give us decent precision while also being fair. It’s important that our cost-to-get-to-goal function doesn’t accidentaly deprioritize good near-solution states.

To help us we’ll calculate the Manhattan Distance between each tile and where it’s supposed to be. Manhattan Distance is the distance between two places if you have to travel along city blocks. Essentially, you’re adding up the short sides of a right triangle rather than shortcutting across the hypotenuse. The formula is pretty simple:

```
class Puzzle
def distance_to_goal # Here we `zip` the current puzzle with the solution
@cells.zip(Solution).inject(0) do |sum, (a,b)| # and total up the distances between each cell
sum += manhattan_distance a % 3, (a / 3).to_i, # and where that cell should be.
b % 3, (b / 3).to_i # This % and / stuff is just turning an integer
end # into puzzle x,y coordinates
end
private
def manhattan_distance x1, y1, x2, y2 # The manhattan distance of something is just
(x1 - x2).abs + (y1 - y2).abs # the distance between x coordinates
end # plus the distance between y coordinates
end
class State
def cost
steps_from_start + steps_to_goal # Now we have a more informed cost method
end # so our priority queue should be giving
# us better results.
def steps_from_start
path.size
end
def steps_to_goal
puzzle.distance_to_goal
end
end
require 'set'
def solve puzzle
@visited = Set.new
@frontier = PriorityQueue.new {|s| s.cost }
state = State.new puzzle
loop {
break if state.solution?
search state
state = @frontier.pop
}
state
end
```

If you’re following along at home (and using a real priority queue) you might think the code is broken because it exited so fast. With a proper priority queue implementation this latest search took 0.07 seconds.

This A* search is able to quickly pick the best candidate to explore in any situation where the distance from the current state to the goal state is knowable. In real-world pathfinding, e.g., you can use the geospatial distance between two points. It doesn’t work at all, however, in situations where you know the goal when you see it but can’t determine how close you are. A robot trying to find a door in unexplored territory would not be able to use this, it would have to just keep bumbling around.

The full reference code for this is on GitHub including a full implentation in Clojure.

Huge thanks to my reviewer Ashish Dixit without whom this post would have been a typo-filled mess of half-conveyed ideas.

A quick recap of the relative time and memory costs for these search algorithms:

```
uninformed depth-first: {stack overflow error}
breadth-first w/o tracking `visited`: {out of memory error}
uninformed breadth-first: 27 seconds, 47,892 explored states
uniform-cost (Dijkstra's): 10 seconds, 51,963 explored states
A* search: 0.07 seconds, 736 explored states
```