Solving 8 puzzle problem using A* star search

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In this tutorial, we will solve the 8 puzzle problem using A* (star) search algorithm. We will approach the solution by first modelling the problem, then by building the fundamental blocks and finally applying a solver to solve the puzzle.


Part 1 of this tutorial provides the introduction, the background information and the approach towards the solution from the algorithmic point of view.


Part 2 of this tutorial provide an implementation of the algorithm and the solution by using C++ for a console program. Read Part 2 “Solving 8 puzzle problem using A* star search in C++”.


Part 3 of this tutorial implements the solution in C# and creates an 8 puzzle game using Unity.



Download the 8 Puzzle Unlimited App from Google Play.

Part 1 – Introduction

Typically A* (Astar) is used in a grid-based pathfinding problem. However, as a general rule, any pathfinding algorithm (A* included) can be used to solve any graph-based problem. For a very detailed understanding of path-finding, I suggest the brilliant tutorial maintained by Amit in his Stanford’s site. In this tutorial I am not going to go through the theory of A* pathfinding, but rather, I would implement all necessary functions for A* pathfinding to solve the 8 puzzle problem.


The 8 Puzzle Problem

The 8 puzzle problem is a puzzle that was invented and popularized by Noyes Palmer Chapman in the 1870s. The 8-puzzle is a smaller version of the slightly better-known 15-puzzle. It comprises a 3-by-3 grid with 8 square blocks labelled 1 through 8 and a blank square.

The goal is to rearrange the blocks so that they are in order with the blank sometimes at the start or at the end. The diagram above shows one possible initial configuration and the goal. To reach the goal state you are permitted to slide blocks horizontally or vertically into the blank square.

Start and goal graphical representation

Before we can solve the 8 puzzle problem we will need to model the problem. But what is meant by Modelling the Problem?

In generic terms, modelling a problem is the art of formulating the problem at hand in terms of precisely described, well-understood building blocks and logic to reach a solution. In computer science, proper modelling is the key to applying algorithmic design techniques to any real-world problems. Real-world applications involve real-world problems.

You might be working on a system that simulates air traffic in and around an airport, you might be working on optimizing the dispatch of delivery vans for an e-commerce application, or you might be working to search through patterns in a large image set. To solve such problems you will use some sort of modelling techniques to reduce the problem in terms of rigorously defined abstract structures such as graphs, trees, permutations, sets and so on.

For our 8 puzzle problem let’s see how we can model the problem. Let’s take a random state of the 8 puzzle problem as given in the diagram below. From this random state, we can either slide tile 8 up, slide tile 3 right or slide tile 6 left to create three variant states.

Each of these three states will produce subsequent more states (3 for the first, 1 for the second and 1 for the third) and so on. This continues until we find the goal state.

We can see that we are able to transform the various possible states of the 8 puzzle problem into a tree data structure.


The 8 Puzzle Solution Search Space

The 8-puzzle is the largest possible N-puzzle that can be completely solved. It is simple and yet has a large problem space. There are larger variants to the same problem type like the 15-puzzle. But those cannot be solved to completion. This makes the N x N extension of the 8-puzzle an NP-hard problem.

8 puzzle has 9! possible tile permutation states. Out of these, every second permutation state is solvable. Hence, there is a total of 9!/2 = 181,440 solvable problem states.

Alexander Reinefeld from the Paderborn Center for Parallel Computing, Germany, has shown that the average length of all optimal solution paths is about 22 moves for any given random configuration. For the 181440 solvable configurations, there is a total of 500880 optimal solutions. This gives an average solution density of 2.76 per problem, with the minimum and maximum number lying at 1 and 64 solutions.

The solution to the problem lies in creating the possible search tree and then traversing the most optimal branch of the tree that will lead from the start state to the end state.

So, how do we find the most optimal branch of the tree? Enter the Heuristic Search!


Heuristic Search

A Heuristic search is a technique to solve a search problem faster than classical methods. In many cases, it finds an approximate solution when classical methods cannot. It is thus a generalized and approximate strategy for problem-solving.

In layman’s terms, it can be thought of as a rule of thumb, or a common-sense knowledge, where the answer isn’t guaranteed to be exactly correct but helps to reach a decision quickly. It is a kind of a shortcut that we take to trade-off either the optimality, the completeness, the accuracy, or the precision for speed.

Heuristic searches are often associated with heuristic values.

A heuristic value of a node in the construction graph attempts to capture the importance of that node’s value, for example, the cost or the gain. Heuristic search is a type of informed search that uses such a heuristic value for optimizing the search.

The search, at each branching step, evaluates the heuristic value and makes a decision on which branch to follow. It does so by ranking alternatives.

There are many different types of heuristic search algorithms. One of them is the A* search algorithm.

The A* Search

A* search is a computer search algorithm that is widely used for pathfinding and graph traversal. In our case of 8 puzzle problem, we will be using it for optimal graph traversal. A* works by keeping track of all visited nodes and ignoring them for further traversal. At the same time, it also keeps track of all the nodes that are yet to be explored and chooses one with the least cost to be further explored.

This simple mechanism allows us to find the most optimal branch of the tree that will lead us from the start state to the end state.


The Heuristic Value (Cost Function) of an 8 Puzzle State

The heuristic value of an 8 puzzle state is a combination of two values. It is often called the cost function f.

f = h + g

h gives how far the goal node is and g the number of nodes traversed from the start node to the current node. For h we will use the Manhattan distance and for g we will use the depth of the current node.

For our A* search we will use the sum of Manhattan distance and the current depth of the node as the total cost.

Manhattan distance

The Manhattan distance heuristic is used not only for its simplicity but also for its ability to estimate the number of moves required to bring a given puzzle state to the solution state. Manhattan distance is simply computed by the sum of the distances of each tile from where it should belong.

// pseudo code
function manhattan_distance(node, goal) =
    dx = abs(node.x - goal.x)
    dy = abs(node.y - goal.y)
    return dx + dy

For example, the Manhattan distance between “213540678” and “123456780” is 9 and between “647850321” and “123456780” is 21.

The diagram below shows the Manhattan cost for a specific tiles configuration.


Software Design for Solving 8 Puzzle Problem

In the following section, I will start creating the building blocks for the puzzle solution and then finally try to join them together to reach the solution.


State

The first step towards solving the 8 puzzle problem will require a data type to represent the tiles on the puzzle. I will call this the State of the puzzle. A state is a unique combination of the tiles.

During our process of solving we will need to store hundreds of perhaps thousands of tile states. Each combination of tiles in the puzzle will be a unique state. Each unique state of the tiles will represent a Node in the tree data structure.

I will use integer array to represent a state. The indices of the array will refer to a tile location whereas the value in that index will represent the tile number. Look at the diagram below. In this diagram, a unique state of the tile is shown on the left. On the right, an array representation is shown to store the tile information.

Start state graphics representation and index array representation.

Thus, we see that by using a one-dimensional array we can represent any state of the 8 puzzle problem. The indices of the array, which cannot change, represent the fixed location of the tiles. In our case, we have assumed that array index 0 represents the top-left tile, index 1 as top-centre tile, index 2 as top-right tile and so on until index 8 as the bottom-right tile. The value stored in each of these indices will represent the actual number (or picture) on the tile. For example, in the above case, we have index 0 having tile 0 (or the empty tile), index 1 having tile 3 and so on until index 8 with tile 2.

We can thus see that by manipulating the values on the array, with the constraint of where the empty tile slides into for each move, we can arrive at the goal state.

Goal state graphical Representation
Goal state index array representation

Neighbours

After State is defined our next task would be to create the graph of neighbours for the 8 puzzle problem.

We have defined the state in the previous section. Now we will define the neighbours based on where the empty tile is. We will do this for each of the 9 tile indices.

So let’s start with tile index 0. For every index, we will need to store the neighbours (in other words the indices where the empty tile can move) as a collection of indices.

Neighbouring tiles
index = 0, {1, 3}
index = 1, {0, 2, 4};
index = 2, {1, 5};
index = 3, {4, 0, 6};
index = 4, {3, 5, 1, 7};
index = 5, {4, 2, 8};
index = 6, {7, 3};
index = 7, {6, 8, 4};
index = 8, {7, 5};

Looking at the above list we can now access the neighbours for any tile index. For example for tile index 6 the neighbours are tile indices 7 and 3.

For the first figure below we have our empty tile in index 0. So for index 0, the neighbours are index 1 and 3. Please note that I am referring to the index of the array and not the actual value of that element in that index of the array. In the first figure below the value at index 1 is 3 and the value at index 3 is 4. Neighbours, in this case, are index 1 and index 3.

Three randomized states

Similarly, for the state represented by the second figure, the empty tile index is 2. Neighbours, in this case, are 1 and 5. For the third state, represented by the third figure, the empty index is 1 and neighbours are 0, 2 and 4. We can thus form a dictionary (or map) of neighbours for each of the 9 indices.


Node (or the State Tree)

The state tree is the actual tree that comprises all the valid transitions from one state to another state ultimately reaching to the final goal (if the solution exists).

In computer science, a tree is a widely used abstract data type (ADT)—or data structure implementing this ADT—that simulates a hierarchical tree structure, with a root value and subtrees of children with a parent node, represented as a set of linked nodes – source wikipedia.

For this problem, each tree element, we call Node, will comprise one specific state of the tile for an 8 puzzle.


Solver for 8 Puzzle

Finally, we look at the Solver framework. The Solver should be able to solve the puzzle based on the A* algorithm. The solver will be implemented simply as a console main function in C++ and as a Coroutine in the Unity C# version. The solver class will construct the tree, visit the next best node and collect nodes for the solution until the solution is finally found.


Read Part 2 of the tutorial “Solving 8 puzzle problem using A* star search in C++”


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