Homework 1: Search


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Artificial Intelligence  Homework 1: Search
In this assignment you will create an agent to solve the N­puzzle game. You will implement and
compare several search algorithms, and collect some statistics related to their performances.
Visit for the game’s rules. Please read all sections carefully:
I. Introduction
II. Algorithm Review
III. What You Need To Submit
IV. What Your Program Outputs
V. Implementation and Testing
VI. Before You Finish
I. Introduction
The N­puzzle game consists of a board holding N = m^2 − 1 distinct movable tiles, plus one
empty space. There is one tile for each number in the set {1, …, m^2 − 1}. In this assignment,
we will represent the blank space with the number 0 and focus on the m = 3 case (8­puzzle).
In this combinatorial search problem, the aim is to get from any initial board state to the
configuration with all tiles arranged in ascending order ⟨0, 1,…, m^2 − 1⟩ ­­ this is your goal
state. The search space is the set of all possible states reachable from the initial state. Each
move consists of swapping the empty space with a component in one of the four directions
{‘Up’, ‘Down’, ‘Left’, ‘Right’}. Give each move a cost of one. Thus, the total cost of a path will be
equal to the number of moves made.
II. Algorithm Review
Recall from lecture that search begins by visiting the root node of the search tree, given by the
initial state. Three main events occur when visiting a node:
First, we remove a node from the frontier set.
Second, we check if this node matches the goal state.
If not, we then expand the node. To expand a node, we generate all of its immediate
successors and add them to the frontier, if they (i) are not yet already in the frontier, and
(ii) have not been visited yet.
This describes the life cycle of a visit, and is the basic order of operations for search agents in
this assignment—(1) remove, (2) check, and (3) expand. We will implement the assignment
algorithms as described here. Please refer to lecture notes for further details, and review the
lecture pseudo­code before you begin.
IMPORTANT: You may encounter implementations that attempt to short­circuit this order
by performing the goal­check on successor nodes immediately upon expansion of a
parent node. For example, Russell & Norvig’s implementation of BFS does precisely this.
Doing so may lead to edge­case gains in efficiency, but do not alter the general
characteristics of complexity and optimality for each method. For simplicity and grading
purposes in this assignment, do not make such modifications to algorithms learned in
III. What You Need To Submit
Your job in this assignment is to write, which solves any 8­puzzle board when given an
arbitrary starting configuration. The program will be executed as follows:
$ python <method <board
The method argument will be one of the following. You must implement all three of them:
bfs (Breadth­First Search)
dfs (Depth­First Search)
ast (A­Star Search)
The board argument will be a comma­separated list of integers containing no spaces. For
example, to use the bread­first search strategy to solve the input board given by the starting
configuration {0,8,7,6,5,4,3,2,1}, the program will be executed like so (with no spaces between
$ python bfs 0,8,7,6,5,4,3,2,1
IMPORTANT: If you are using Python 3, please name your file, so we use the
correct version while grading. If you name your file, the default version for our
box is Python 2.
IV. What Your Program Outputs
Your program will create and/or write to a file called output.txt, containing the following statistics:
path_to_goal: the sequence of moves taken to reach the goal
cost_of_path: the number of moves taken to reach the goal
nodes_expanded: the number of nodes that have been expanded
search_depth: the depth within the search tree when the goal node is found
max_search_depth: the maximum depth of the search tree in the lifetime of the algorithm
running_time: the total running time of the search instance, reported in seconds
max_ram_usage: the maximum RAM usage in the lifetime of the process as measured by
the ru_maxrss attribute in the resource module, reported in megabytes
Example #1: Breadth­First Search
Suppose the program is executed for breadth­first search as follows:
$ python bfs 1,2,5,3,4,0,6,7,8
This should result in the solution path:
.The output file will contain exactly the following lines:
path_to_goal: [‘Up’, ‘Left’, ‘Left’]
cost_of_path: 3
nodes_expanded: 10
search_depth: 3
max_search_depth: 4
running_time: 0.00188088
max_ram_usage: 0.07812500
Example #2: Depth­First Search
Suppose the program is executed for depth­first search as follows:
$ python dfs 1,2,5,3,4,0,6,7,8
This should result in the solution path:
The output file will contain exactly the following lines:
path_to_goal: [‘Up’, ‘Left’, ‘Left’]
cost_of_path: 3
nodes_expanded: 181437
search_depth: 3
max_search_depth: 66125
running_time: 5.01608433
max_ram_usage: 4.23940217
More test cases are provided in the FAQs.
Note on Correctness
All variables, except running_time and max_ram_usage, have one and only one correct
answer when running BFS and DFS. A* nodes_expanded might vary depending on
implementation details. You’ll be fine as long as your algorithm follows all specifications listed in
these instructions.
As running_time and max_ram_usage values vary greatly depending on your machine and
implementation details, there is no “correct” value to look for. They are for you to monitor time
and space complexity of your code, which we highly recommend. A good way to check the
correctness of your program is to walk through small examples by hand, like the ones above.
V. Implementation and Testing
For your first programming project, we are providing hints and explicit instructions. Before
posting a question on the discussion board, make sure your question is not already answered
here or in the FAQs.
1. Implementation
You will implement the following three algorithms as demonstrated in lecture. In particular:
● Breadth­First Search. Use an explicit queue, as shown in lecture.
● Depth­First Search. Use an explicit stack, as shown in lecture.
● A­Star Search. Use a priority queue, as shown in lecture. For the choice of heuristic,
use the Manhattan priority function; that is, the sum of the distances of the tiles from their
goal positions. Note that the blanks space is not considered an actual tile here.
2. Order of Visits
In this assignment, where an arbitrary choice must be made, we always visit child nodes in the
“UDLR” order; that is, [‘Up’, ‘Down’, ‘Left’, ‘Right’] in that exact order. Specifically:
● Breadth­First Search. Enqueue in UDLR order; de­queuing results in UDLR order.
● Depth­First Search. Push onto the stack in reverse­UDLR order; popping off results in
UDLR order.
● A­Star Search. Since you are using a priority queue, what happens with duplicate keys?
How do you ensure nodes are retrieved from the priority queue in the desired order?
3. Submission Test Cases
Run all three of your algorithms on the following test cases:
Test Case #1
python bfs 3,1,2,0,4,5,6,7,8
python dfs 3,1,2,0,4,5,6,7,8
python ast 3,1,2,0,4,5,6,7,8
Test Case #2
python bfs 1,2,5,3,4,0,6,7,8
python dfs 1,2,5,3,4,0,6,7,8
python ast 1,2,5,3,4,0,6,7,8
Make sure your code passes at least these test cases and follows our formatting exactly.
The results of each test are assessed by 8 items: 7 are listed in Section IV. What Your
Program Outputs. The last point is for code that executes and produces any output at all. Each
item is worth 0.75 point.
4. Grading and Stress Tests
We will grade your project by running additional test cases on your code. In particular, there
will be five test cases in total, each tested on all three of your algorithms, for a total of 15 distinct
tests. Similar to the submission test cases, each test will be graded by 8 items, for a total of
90 points. Plus, we give 10 points for code completing all 15 test cases within 10 minutes.
If you implement your code with reasonable designs of data structures, your code will solve all
15 test cases within a minute in total. We will be using a wide variety of inputs to stress­test your
algorithms to check for correctness of implementation. So, we recommend that you test your
own code extensively.
Don’t worry about checking for malformed input boards, including boards of non­square
dimensions, other sizes, or unsolvable boards.
You will not be graded on the absolute values of your running time or RAM usage
statistics. The values of these statistics can vary widely depending on the machine. However,
we recommend that you take advantage of them in testing your code. Try batch­running
your algorithms on various inputs, and plotting your results on a graph to learn more about
the space and time complexity characteristics of your code. Just because an algorithm provides
the correct path to goal does not mean it has been implemented correctly.
5. Tips on Getting Started
Begin by writing a class to represent the state of the game at a given turn, including parent and
child nodes. We suggest writing a separate solver class to work with the state class. Feel free
to experiment with your design, for example including a board class to represent the low­level
physical configuration of the tiles, delegating the high­level functionality to the state class.
You will not be graded on your design, so you are at a liberty to choose among your favorite
programming paradigms. Students have successfully completed this project using an entirely
object­oriented approach, and others have done so with a purely functional approach. Your
submission will receive full credit as long as your driver program outputs the correct information.
VI. Before You Finish
● Make sure your code passes at least the submission test cases.
● Make sure your algorithms generate the correct solution for an arbitrary solvable
problem instance of 8­puzzle.
● Make sure your program always terminates without error, and in a reasonable amount of
time. You will receive zero points from the grader if your program fails to
terminate. Running times of more than a minute or two may indicate a problem
with your implementation. If your implementation exceeds the time limit allocated (20
minutes for all test cases), your grade may be incomplete.
● Make sure your program output follows the specified format exactly. In particular, for the
path to goal, use square brackets to surround the list of items, use single quotes
around each item, and capitalize the first letter of each item. Round floating­point
numbers to 8 places after the decimal. You will not receive proper credit from the grader
if your format differs from the provided examples above.

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