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COMP20003 Algorithms and Data Structures
Second (Spring) Semester 2018
[Assignment 2]
Solving Puzzle Games Optimally:
Graph Search
Handed out: Wednesday, 3 of October
Due: 12:00 Noon, Wednesday, 17 of October
Purpose
The purpose of this assignment is for you to:
Increase your proficiency in C programming, your dexterity with efficient memory allocation and
your algorithmic thinking, through programming a search algorithm over Graphs.
Practice your ability to understand new algorithms based on those already taught in class.
Gain experience with applications of graphs and graph algorithms to create an AI solver for a
game.
Foster your capability to understand a scientific paper and implement its ideas (optional).
Assignment description
In this programming assignment you’ll be expected to build a solver for the 15–puzzle https://en.
wikipedia.org/wiki/15_puzzle.
The 15–puzzle
These puzzles consist in assembling an image or some geometrical patterns, which has been decomposed
into a number of sliding tiles. We can consider each tile to be identified by a number, so we can
represent the initial state of the puzzle with the following diagram:
1 2
3 4 8 5
14 15 13
9 10 11 12
7 6
B
where B is a blank space. A puzzle solution is a sequence of moves which achieves the following state
B 1 3 2
4 5 6 7
8 9 10 11
12 13 14 15
subject to the constraint that we can only swap positions of the blank tile with some adjacent one.
For instance, in the initial state depicted above, the only valid moves would be:
1. Swap positions of tile 9 and blank B.
2. Swap positions of tile 14 and blank B.
Each possible configuration of the puzzle is called a state. The 15-puzzle Graph G = hV, Ei is implicitly
defined. The vertex set V is defined as all the possible puzzle configurations (states), and the edges
E connecting two vertexes are defined by the legal movements, also called actions or operators.
Algorithm
Your solver should contain the implementation of the following search algorithm:
1. Iterative Deepening A(IDA)
The algorithm follows the Depth-First Search search strategy and can be easily implemented as a
recursive function.
IDA? has two parts:
1. The main loop, initializes the thresholds B and B0 first. If no solution is found the search is
triggered again but with an updated B = B0
threshold.
2. The recursive function that implements the bounded Depth-First Search.
In this assignment IDA?
is guaranteed to find the optimal solution. In general, IDA?
returns optimal
solutions as long as the heuristic function is a lower bound on the true optimal solution length (see
next section for more information on the heuristic used in this assignment).
Definitions and Comments about the Pseudocode
The pseudocode assumes that you have a Node data structure – for variables n and n
0 –containing the fields s for the state, g(n) for the cost of the path up to s from the initial state,
and f(n) for the evaluation function. Note that the notation n0.g stands for the value of g
function of node n0.
Node - The data structure representing a possible path in the search. A node may have children
nodes (corresponding to its possible successor states), a parent node (corresponding to the node
from where the current node has been generated), and an operator index (recording which
operator or action has been applied to generate the node, e.x. move blank left, right, up or
down)
Successor State - A successor state s0 of state s is the resulting state of applying an action a.
Generated Nodes - Nodes that have been created within the search. If you imagine the search
algorithm building a tree, these are all the nodes of the tree.
Expanded Nodes - Nodes for which its children have been generated. Here, it corresponds to
the nodes that haven’t been pruned by the threshold, or in terms of the search tree would be
the internal nodes of the tree.
Pruning - Evaluation of the path quality (node) and deciding whether we should not explore
it further given a function f(n) to evaluate and a threshold.
A(s) is a function that returns the actions applicable in a given state s. In 15-puzzle it may be
between 2 and 4 actions.
The function f(a, s) returns the state resulting of applying action a in state s.
To print the cost of the solution found, you should look at g(n) for node n = r in the
pseudocode.
? h(s) implements the heuristic function, taking as an argument the state s.
Note that a node n is pruned only if f(n) is strictly bigger than the threshold B.
Note that in 15-puzzle the cost function of applying any action a in any state s is c(a, s) = 1.
IDA-Control-Loop()
1 B ← h(s0)
2 repeat
3 B0 ← ∞
4 n.s ← s0
5 n.g ← 0
6 r ← IDA?
(n, B, B0
)
7 if r = nil
8 then B ← B0
9 until r 6= nil
10 return r.g
Figure 1: Main search loop. s0 stands for the initial configuration.
IDA?
(n, B, B0
)
1 for a ∈ A(n.s)
2 do n
0
.s ← f(a, n.s)
3 n
0
.g ← n.g + 1
4 n
0
.f ← n
0
.g + h(n
0
.s)
5 if n
0
.f > B
6 then B0 ← min(n
0
.f, B0
)
7 else
8 if h(n
0
.s) = 0
9 then return n
0
10 r ← IDA?
(n
0
, B, B0
)
11 if r 6= nil
12 then return r
13 return nil
Figure 2: Recursive function implementing IDA?
. If a solution is found, it returns the node r,
otherwise returns NIL.
Heuristic
The heuristic that you will be using to prune the search space is the Sum of Manhattan Distances,
which is defined as follows:
h(s) = Xi
d(ti, ti) (1)
where ti
is the position of the ith tile and ti
is the position where this tile should be. Positions are
given by 2–component vectors ti = (x, y). Manhattan distance is easily computed
d(ti, ti) = |x x| + |y y| (2)
For the initial state depicted above, the heuristic computation would be, starting from the upper left
corner
h(s) = 1 + 1 + 2 + 2 + 4 + 1 + 1 + 4 + 1 + 1 + 1 + 4 + 1 + 1 + 2 = 27 (3)
note that we don’t take into account the (mis)placement of the blank space.
This heuristic is proved to be admissible, i.e. h(s) is always lower or equal to the optimal distance
(s) to the objective configuration from a any state s, for the 15–puzzle, being a useful heuristic to
search for optimal solutions.
Deliverables, evaluation and delivery rules
Deliverable 1 – Solver source code
Just look for the base code in LMS. Understand the code and write your own code in the section
where you see the comment: FILL WITH YOUR CODE.
You are expected to hand in the source code for your solver, written in C. Obviously, your source code
is expected to compile and execute flawlessly using the following makefile command: make generating
an executable called 15puzzle. Remember to compile using the optimization flag gcc -O3 for doing
your experiments, it will run twice faster than compiling with the debugging flag gcc -g.
In order to test your solver, you can use the following 15–puzzle instances
N Initial State s0 h(s0) Cost Nodes
1 14 13 15 7 11 12 9 5 6 0 2 1 4 8 10 3 41 57 276M
2 13 5 4 10 9 12 8 14 2 3 7 1 0 15 11 6 43 55 15M
3 14 7 8 2 13 11 10 4 9 12 5 0 3 6 1 15 41 59 565M
4 5 12 10 7 15 11 14 0 8 2 1 13 3 4 9 6 42 56 62M
14 7 6 8 1 11 5 14 10 3 4 9 13 15 2 0 12 41 59 1369M
88 15 2 12 11 14 13 9 5 1 3 8 7 0 10 6 4 43 65 6009M
where N is a unique ID from a test set of 100 initial states, s0 is the initial state (first number
corresponds with the tile placed in the upper left corner and zero represents the blank space), h(s0)
is the value of the heuristic for the initial state, Cost is the optimal cost of the problem, and Nodes
is the number of nodes for which their children are generated (it’s a rough figure, your mileage might
vary slightly depending on your implementation).
Your IDA?
implementation will be required to find solutions of the same quality. With the data of
the table you’ll be able to asses the correctness of your implementation: it has to compute the exact
same values for h(s0) and cost, with roughly a similar amount of Nodes.
Input
Your solver has to read the initial configuration from a file with the same format as it appears
in the table, that is, a single line, containing a sorted list of indexes separated by a blank space. For
example, instance N = 1 from the table would be:
14 13 15 7 11 12 9 5 6 0 2 1 4 8 10 3
output
Assuming we have a file called “1.puzzle” containting a single line with the configuration of instance
N = 1 from the table, we will call our solver by running the following command:
./15puzzle 1.puzzle
and it will print into the stdout the following information:
1. Initial state of the puzzle.
2. h(s0) heuristic estimate for the initial state.
3. Thresholds the search have used to solve the problem.
4. Number of moves of the optimal solution.
5. Number of generated nodes.
6. Number of expanded nodes.
7. Number of expanded nodes per second.
8. Total Search Time, in seconds.
For example, the output of our solver ./15puzzle 1.puzzle for instance N = 1 is:
Initial State:
14 13 15 7
11 12 9 5
6 0 2 1
4 8 10 3
Initial Estimate = 41
Threshold = 41 43 45 47 49 51 53 55 57
Solution = 57
Generated = 499,911,606
Expanded = 253,079,561
Time (seconds) = 5.12
Expanded/Second = 49,082,100
Deliverable 2 – Experimentation
Besides handing in the solver source code, you’re required to provide a table, for your implementation
of the algorithm and each of the problems in the table above, which accounts for
1. ID N of the Initial state of the puzzle.
2. h(s0) heuristic estimate for the initial state.
3. Thresholds the search have used to solve the problem.
4. Number of moves of the optimal solution.
5. Number of generated nodes.
6. Number of expanded nodes.
7. Number of expanded nodes per second.
8. Total Search Time, in seconds.
(optional) Explain concisely any improvements you made to speed up the algorithm in terms of number
of expanded nodes per second, as well as any improvements you implemented suggested by the paper
to reduce the number of expanded nodes.
Evaluation
Assignment marks will be divided into four different components.
1. IDA?
(10)
2. Manhattan heuristic (2)
3. Experimentation (2)
4. Code Style (1)
Please note that you should be ready to answer any question we might have on the details of your
assignment solution by e–mail, or even attending a brief interview with me, in order to clarify any
doubts we might have.
Extra Mark
An extra mark can be earned if you implement any (1 at least) of the optimizations described in the
the section Improved Admissible Heuristics of the article http://www.aaai.org/Papers/
AAAI/1996/AAAI96-178.pdf written by the founder of IDA? Richard Korf.
If you do any of the optimizations for the extra marks, please report and discuss it in your experimentation.
Delivery rules
You will need to make two submissions for this assignment:
Your C code files (including your Makefile) will be submitted through the LMS page for this
subject: Assignments → Assignment 2 → Assignment 2: Code.
Your experiments report file will be submitted through the LMS page for this subject: Assignments
→ Assignment 2 → Assignment 2: Experimentation. This file can be of any format, e.g.
.pdf, text or other.
This double submissions process is being used because the program files need to be tested under UNIX,
and the presence of non-text experiment files can interfere with the submit and verify process.
Program files submitted (Code)
Submit the program files for your assignment and your Makefile.
If you wish to submit any scripts or code used to generate input data, you may, although this is not
required. Just be sure to submit all your files at the same time.
Your programs must compile and run correctly on the CIS machines. You may have developed your
program in another environment, but it still must run on the department machines at submission time.
For this reason, and because there are often small, but significant, differences between compilers, it is
suggested that if you are working in a different environment, you upload and test your code on the
department machines at reasonably frequent intervals.
A common reason for programs not to compile is that a file has been inadvertently omitted from the
submission. Please check your submission, and resubmit all files if necessary. .
Advice and Hints
1. Create an initial puzzle that you know can be solved with few steps, e.g. take the goal state and
just move the blank two positions. Your algorithm should be able to solve this problem quickly.
2. If you allocate memory for new nodes (Line 2 Fig 2), make sure to delete them as memory
leaks might take all the space of your RAM.
3. Once you’ve got a working version, if you want to speed up your code think carefully where
most of the computation goes. It’s a good excuse to practice the art of profiling your code
(understanding where most of the computation goes). Valgrind can be used for this purpose as
well. Check documentation about it in this link. Most likely the most expensive operations will
be computing the heuristic and generating new nodes. And just in case, we won’t give extra
marks or remove marks depending on your speed, but feel free to create a Piazza post with a
ladder of the fastest programs. Of course be honest and feel free to share your programming
tricks after the due date of the assignment.
4. Have in mind that some these problems might need – depending on hardware and implementation
– more than 30 minutes to be solved. In order to test your code, just come up with a couple
simple instances (simple as in “solutions requiring few moves”). You should be able to check by
hand the solutions for those problems and assess the soundness of your implementation.
Plagiarism
This is an individual assignment. The work must be your own.
While you may discuss your program development, coding problems and experimentation with your
classmates, you must not share files, as this is considered plagiarism.
If you refer to published work in the discussion of your experiments, be sure to include
a citation to the publication or the web link.
“Borrowing” of someone else’s code without acknowledgement is plagiarism, e.x. taking
code from a book without acknowledgement. Plagiarism is considered a serious offense at the
University of Melbourne. You should read the University code on Academic Honesty and details on
plagiarism. Make sure you are not plagiarizing, intentionally or unintentionally.
You are also advised that there will be a C programming component (on paper, not on a computer) on
the final examination. Students who do not program their own assignments will be at a disadvantage
for this part of the examination.
Administrative issues
When is late? What do I do if I am late? The due date and time are printed on the front
of this document. The lateness policy is on the handout provided at the first lecture and also available
on the subject LMS page. If you decide to make a late submission, you should send an email
directly to the lecturer as soon as possible and he will provide instructions for making a late submission.
What are the marks and the marking criteria Recall that this project is worth 15% of your
final score. There is also a hurdle requirement: you must earn at least 15 marks out of a subtotal of
30 for the projects to pass this subject.
Finally Despite all these stern words, we are here to help! There is information about getting help
in this subject on the LMS pages. Frequently asked questions about the project will be answered in
the LMS discussion group.

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