Pacman maze

All those colored walls,
Mazes give Pacman the blues,
So teach him to search.


In this assignment, your Pacman agent will find paths through his maze world, both to reach a particular location and to collect food efficiently. You will build general search algorithms and apply them to Pacman scenarios.

As in HW 0, this assignment includes an autograder for you to grade your answers on your machine. This can be run with the command:


See the autograder tutorial in HW 0 for more information about using the autograder.

The code for this assignment consists of several Python files, some of which you will need to read and understand in order to complete the assignment, and some of which you can ignore. You can download all the code and supporting files as a zip archive.

Files you'll edit: Where all of your search algorithms will reside. Where all of your search-based agents will reside.
Files you might want to look at: The main file that runs Pacman games. This file describes a Pacman GameState type, which you use in this assignment. The logic behind how the Pacman world works. This file describes several supporting types like AgentState, Agent, Direction, and Grid. Useful data structures for implementing search algorithms.
Supporting files you can ignore: Graphics for Pacman Support for Pacman graphics ASCII graphics for Pacman Agents to control ghosts Keyboard interfaces to control Pacman Code for reading layout files and storing their contents autograder Parses autograder test and solution files General autograding test classes
test_cases/ Directory containing the test cases for each question HW 1 specific autograding test classes

Files to Edit and Submit: You will fill in portions of and during the assignment. You should submit these files with your code and comments. Please do not change the other files in this distribution or submit any of our original files other than these files. Do not submit code that imports other libraries; these will not be available to the autograder on Gradescope.

Evaluation: Your code will be autograded for technical correctness. Please do not change the names of any provided functions or classes within the code, or you will wreak havoc on the autograder. However, the correctness of your implementation -- not the autograder's judgements -- will be the final judge of your score. If necessary, we will review and grade assignments individually to ensure that you receive due credit for your work.

Academic Dishonesty: We will be checking your code against other submissions, both within and outside of this year's class. If you copy someone else's code and submit it with minor changes, we will know. These cheat detectors are quite hard to fool, so please don't try. We trust you all to submit your own work only; please don't let us down. If you do, we will pursue the strongest consequences available to us.

Getting Help: You are not alone! If you find yourself stuck on something, contact the course staff for help. Office hours, section, and the discussion forum are there for your support; please use them. If you can't make our office hours, let us know and we will try to work something out for you. We want these assignments to be rewarding and instructional, not frustrating and demoralizing. But, we don't know when or how to help unless you ask.

Piazza discussion: Please be careful not to post spoilers. Post anything that reveals answers in private posts. By now, you should have the skills to debug your own code, but if you find it necessary to show us your code, privately post your code. DO NOT POST A PICTURE OF YOUR CODE. That makes it unnecessarily difficult for us to help you.

Welcome to Pacman

After downloading the code (, unzipping it, and changing to the directory, you should be able to play a game of Pacman by typing the following at the command line:


Pacman lives in a shiny blue world of twisting corridors and tasty round treats. Navigating this world efficiently will be Pacman's first step in mastering his domain.

The simplest agent in is called the GoWestAgent, which always goes West (a trivial reflex agent). This agent can occasionally win:

python --layout testMaze --pacman GoWestAgent

But, things get ugly for this agent when turning is required:

python --layout tinyMaze --pacman GoWestAgent

If Pacman gets stuck, you can exit the game by typing CTRL-c into your terminal.

Soon, your agent will solve not only tinyMaze, but any maze you want.

Note that supports a number of options that can each be expressed in a long way (e.g., --layout) or a short way (e.g., -l). You can see the list of all options and their default values via:

python -h

Also, all of the commands that appear in this assignment also appear in commands.txt, for easy copying and pasting. In UNIX/Mac OS X, you can even run all these commands in order with bash commands.txt.

Question 1 (10 points): Finding a Fixed Food Dot using Depth First Search

In, you'll find a fully implemented SearchAgent, which plans out a path through Pacman's world and then executes that path step-by-step. The search algorithms for formulating a plan are not implemented -- that's your job.

First, test that the SearchAgent is working correctly by running:

python -l tinyMaze -p SearchAgent -a fn=tinyMazeSearch

The command above tells the SearchAgent to use tinyMazeSearch as its search algorithm, which is implemented in Pacman should navigate the maze successfully.

Now it's time to write full-fledged generic search functions to help Pacman plan routes! Pseudocode for the search algorithms you'll write can be found in the lecture slides. Remember that a search node must contain not only a state but also the information necessary to reconstruct the path (plan) which gets to that state.

Important note: All of your search functions need to return a list of actions that will lead the agent from the start to the goal. These actions all have to be legal moves (valid directions, no moving through walls).

Important note: Make sure to use the Stack, Queue and PriorityQueue data structures provided to you in! These data structure implementations have particular properties which are required for compatibility with the autograder.

Hint: Each algorithm is very similar. Algorithms for DFS, BFS, UCS, and A* differ only in the details of how the frontier is managed. Concentrate on getting DFS right and the rest should be relatively straightforward. Indeed, one possible implementation requires only a single generic search method which is configured with an algorithm-specific queuing strategy. (Your implementation need not be of this form to receive full credit).

Implement the depth-first search (DFS) algorithm in the depthFirstSearch function in To make your algorithm complete, write the graph search version of DFS, which avoids expanding any already visited states.

Your code should quickly find a solution for:

python -l tinyMaze -p SearchAgent
python -l mediumMaze -p SearchAgent
python -l bigMaze -z .5 -p SearchAgent

The Pacman board will show an overlay of the states explored, and the order in which they were explored (brighter red means earlier exploration). Is the exploration order what you would have expected? Does Pacman actually go to all the explored squares on his way to the goal?

Hint: If you use a Stack as your data structure, the solution found by your DFS algorithm for mediumMaze should have a length of 130 (provided you push successors onto the frontier in the order provided by getSuccessors; you might get 246 if you push them in the reverse order). Is this a least cost solution? If not, think about what depth-first search is doing wrong.

Hint: The autograder can be a bit fiddly for this question, and it will complain if your implementation doesn't do things exactly as it expects. For DFS, add states to the visited when you expand, and be sure to test if a state has been visited before pushing, and to test when you pop.

Question 2 (10 points): Breadth First Search

Implement the breadth-first search (BFS) algorithm in the breadthFirstSearch function in Again, write a graph search algorithm that avoids expanding any already visited states. Test your code the same way you did for depth-first search.

python -l mediumMaze -p SearchAgent -a fn=bfs
python -l bigMaze -p SearchAgent -a fn=bfs -z .5

Does BFS find a least cost solution? If not, check your implementation.

Hint: If Pacman moves too slowly for you, try the option --frameTime 0.

Note: If you've written your search code generically, your code should work equally well for the eight-puzzle search problem without any changes.


Hint: The autograder can also be a bit fiddly for this question, and it will complain if your implementation doesn't do things exactly as it expects. As with DFS, add states to the visited when you expand them, and be sure to test if a state has been visited before pushing, and to test when you pop.

Question 3 (10 points): Varying the Cost Function

While BFS will find a fewest-actions path to the goal, we might want to find paths that are "best" in other senses. Consider mediumDottedMaze and mediumScaryMaze.

By changing the cost function, we can encourage Pacman to find different paths. For example, we can charge more for dangerous steps in ghost-ridden areas or less for steps in food-rich areas, and a rational Pacman agent should adjust its behavior in response.

Implement the uniform-cost graph search algorithm in the uniformCostSearch function in We encourage you to look through for some data structures that may be useful in your implementation. You should now observe successful behavior in all three of the following layouts, where the agents below are all UCS agents that differ only in the cost function they use (the agents and cost functions are written for you):

python -l mediumMaze -p SearchAgent -a fn=ucs
python -l mediumDottedMaze -p StayEastSearchAgent
python -l mediumScaryMaze -p StayWestSearchAgent

Note: You should get very low and very high path costs for the StayEastSearchAgent and StayWestSearchAgent respectively, due to their exponential cost functions (see for details).

Question 4 (10 points): A* search

Implement A* graph search in the empty function aStarSearch in A* takes a heuristic function as an argument. Heuristics take two arguments: a state in the search problem (the main argument), and the problem itself (for reference information). The nullHeuristic heuristic function in is a trivial example.

You can test your A* implementation on the original problem of finding a path through a maze to a fixed position using the Manhattan distance heuristic (implemented already as manhattanHeuristic in

python -l bigMaze -z .5 -p SearchAgent -a fn=astar,heuristic=manhattanHeuristic

You should see that A* finds the optimal solution slightly faster than uniform cost search (about 549 vs. 620 search nodes expanded in our implementation, but ties in priority may make your numbers differ slightly). What happens on openMaze for the various search strategies? (This is for you to ponder; you are not required to submit an expilcit aswer to this question.)

Question 5 (10 points): Finding All the Corners

The real power of A* will only be apparent with a more challenging search problem. Now, it's time to formulate a new problem and design a heuristic for it.

In corner mazes, there are four dots, one in each corner. Our new search problem is to find the shortest path through the maze that touches all four corners (whether the maze actually has food there or not). Note that for some mazes like tinyCorners, the shortest path does not always go to the closest food first! Hint: the shortest path through tinyCorners takes 28 steps.

Note: Make sure to complete Question 2 before working on Question 5, because Question 5 builds upon your answer for Question 2.

Implement the CornersProblem search problem in You will need to choose a state representation that encodes all the information necessary to detect whether all four corners have been reached. Now, your search agent should solve:

python -l tinyCorners -p SearchAgent -a fn=bfs,prob=CornersProblem
python -l mediumCorners -p SearchAgent -a fn=bfs,prob=CornersProblem

To receive full credit, you need to define an abstract state representation that does not encode irrelevant information (like the position of ghosts, where extra food is, etc.). In particular, do not use a Pacman GameState as a search state. Your code will be very, very slow if you do (and also wrong).

Hint: The only parts of the game state you need to reference in your implementation are the starting Pacman position and the location of the four corners.

Our implementation of breadthFirstSearch expands just under 2000 search nodes on mediumCorners. However, heuristics (used with A* search) can reduce the amount of searching required.

Question 6 (10 points): Corners Problem: Heuristic

Note: Make sure to complete Question 4 before working on Question 6, because Question 6 builds upon your answer for Question 4.

Implement a non-trivial, consistent heuristic for the CornersProblem in cornersHeuristic.

python -l mediumCorners -p AStarCornersAgent -z 0.5

Note: AStarCornersAgent is a shortcut for

-p SearchAgent -a fn=aStarSearch,prob=CornersProblem,heuristic=cornersHeuristic

Admissibility vs. Consistency: Remember, heuristics are just functions that take search states and return numbers that estimate the cost to a nearest goal. More effective heuristics will return values closer to the actual goal costs. To be admissible, the heuristic values must be lower bounds on the actual shortest path cost to the nearest goal (and non-negative). To be consistent, it must additionally hold that if an action has cost c, then taking that action can only cause a drop in heuristic of at most c.

Remember that admissibility isn't enough to guarantee correctness in graph search -- you need the stronger condition of consistency. However, admissible heuristics are usually also consistent, especially if they are derived from problem relaxations. Therefore it is usually easiest to start out by brainstorming admissible heuristics. Once you have an admissible heuristic that works well, you can check whether it is indeed consistent, too. The only way to guarantee consistency is with a proof. However, inconsistency can often be detected by verifying that for each node you expand, its successor nodes are equal or higher in in f-value. Moreover, if UCS and A* ever return paths of different lengths, your heuristic is inconsistent. This stuff is tricky!

Non-Trivial Heuristics: The trivial heuristics are the ones that return zero everywhere (UCS) and the heuristic which computes the true completion cost. The former won't save you any time, while the latter will timeout the autograder. You want a heuristic which reduces total compute time, though for this assignment the autograder will only check node counts (aside from enforcing a reasonable time limit).

Grading: Your heuristic must be a non-trivial non-negative consistent heuristic to receive any points. Make sure that your heuristic returns 0 at every goal state and never returns a negative value. Depending on how few nodes your heuristic expands, you'll be graded:

Number of nodes expanded Grade
more than 2000 2/10
at most 2000 6/10
at most 1600 10/10

Remember: If your heuristic is inconsistent, you will receive no credit, so be careful!

Question 7 (10 points): Eating All The Dots

Now we'll solve a hard search problem: eating all the Pacman food in as few steps as possible. For this, we'll need a new search problem definition which formalizes the food-clearing problem: FoodSearchProblem in (implemented for you). A solution is defined to be a path that collects all of the food in the Pacman world. For the present assignment, solutions do not take into account any ghosts or power pellets; solutions only depend on the placement of walls, regular food and Pacman. (Of course ghosts can ruin the execution of a solution! We'll get to that in the adversarial search.) If you have written your general search methods correctly, A* with a null heuristic (equivalent to uniform-cost search) should quickly find an optimal solution to testSearch with no code change on your part (total cost of 7).

python -l testSearch -p AStarFoodSearchAgent

Note: AStarFoodSearchAgent is a shortcut for -p SearchAgent -a fn=astar,prob=FoodSearchProblem,heuristic=foodHeuristic.

You should find that UCS starts to slow down even for the seemingly simple tinySearch. As a reference, our implementation takes 2.5 seconds to find a path of length 27 after expanding 5057 search nodes.

Note: Make sure to complete Question 4 before working on Question 7, because Question 7 builds upon your answer for Question 4.

Fill in foodHeuristic in with a consistent heuristic for the FoodSearchProblem. Try your agent on the trickySearch board:

python -l trickySearch -p AStarFoodSearchAgent

Our UCS agent finds the optimal solution in about 13 seconds, exploring over 16,000 nodes.

Any non-trivial non-negative consistent heuristic will receive 1 point. Make sure that your heuristic returns 0 at every goal state and never returns a negative value. Depending on how few nodes your heuristic expands, you'll get additional points:

Number of nodes expanded Grade
more than 15000 2/10
at most 15000 6/10
at most 12000 10/10

Remember: If your heuristic is inconsistent, you will receive no credit, so be careful! Can you solve mediumSearch in a short time? If so, we're either very, very impressed, or your heuristic is inconsistent.

Question 8 (10 points): A* by hand

This question asks you to verify your understanding of A* by working through a simple example by hand. Consider the 8-puzzled configuration below as your starting state:
8 puzzle configuration
and use the number of misplaced tiles as your heuristic, e.g., h=4 for this state. Run A* by hand for this problem. We suggest you do this on paper or on tablet, writing out the search tree and associated states explicitly. For a more compact representation of the state, you can think of each state as 9 digit number (using 0 for the empty square), by reading the digits in left-to-right, top-to-bottom order. For example, the state shown above would have a compact representation as "023145786". Rather than turning in your hand-drawn tree, you can just turn in a sequence of snapshots of the priority queue, expressed as a list with higher priority items shown to the left. For example, the queue would initially be:


After this node is expanded, you'd have:

((123045786), (203145786))

If you ever encounter a tie in priority, you can break ties by giving the state with lowest numerical representation highest priority. Finally, compare the number of nodes expanded by A* with this heuristic with the worst case number of nodes BFS would expand given the same starting state. You should be able to do this without running or simulating BFS directly.

Question 9 (10 points): Combining Heuristics

Suppose that heuristics are admissible. Prove that h(s)=max(h1(s),...,hn(s)) is also admissible by induction on n. If you need help doing proofs by induction there is a nice little tutorial/refresher here. To answer this question, you should provide three key items: a) A base case where you show that the hypothesis holds for the smallest meaningful case. (You should use n=2). b) An assumption that it holds for n=k. c) A proof that if it holds for k, it holds for k+1.

Question 10 (10 points): Working with Heuristics

You may have heard of bubblesort. It's a sorting algorithm that's rarely used because it scales quadratically in the size of the input array. The bubblesort algorithm walks through the array repeatedly and swaps contiguous elements whenever is sees they're out of order. Have you heard of fancy bubblesort? Here we're going to consider a problem related to bubblesort called "the fancy bubblesort problem." Our task is to sort an array using contiguous swaps as in the bubblesort algorithm. The cost of a fancy bubblesort run is the total number of swaps performed. Is there a benefit, in terms of reduced number of swaps, to deviating from the bubblesort procedure and swapping any two contiguous elements in the array at any time if we have a good heuristic?

Consider the following three heuristics, h1...h3, for fancy bubblesort:

  1. The number of swaps required by bubblesort.
  2. Half the hamming distance to the sorted array, rounded up.
  3. Half the Manhattan distance to the sorted array, rounded up. (As with the 8-puzzle, we interpret the total Manhattan distance to be the sum of the distances each out of place item must move.)

a) Which of these heuristics is admissible, and which is consistent? b) Show that h1 dominates h2 and h3, and c) Show that h2 doesn't dominate h3. For each of these parts, you should give some justification for your answer.

Hint: Note that h1 is a particularly silly heuristic because it takes as much work to compute it as to sort the list, so on this basis alone there would be no sensible reason to use h1 with fancy bubblesort. We might ask, however, whether it would be useful to have an oracle that somehow magically provided us with this information. This touches on a related question: Is it possible to use fewer contiguous swaps than regular bubble sort? To answer this question (and perhaps others) think about the total amount of movement that is required so sort the list and whether bubblesort ever does more work than is needed.


To submit your homework, please upload the following files to HW 1 (code) on Gradescope: and Please do not upload the files in a zip file or a directory as the autograder will not work if you do so.

Separately, please upload a single pdf for quesitons 8-10 to HW 1 on Gradescope.