Lab 2

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This lab is due by Thursday, September 24 at 10:00pm.

To work on this lab, you will need to get the code, much like you did for the first two labs.

Your answers for this lab belong in the main file


Graphs and Edges

The file contains definitions for graphs and edges. The relevant information is described here, so you probably won't need to read the file.

A graph is an object of type UndirectedGraph.

For an UndirectedGraph g, you can use the following attributes:

  • You can get a list of all nodes in the graph via the .nodes attribute.
  • You can get a list of all edges in the graph via the .edges attribute.

For an UndirectedGraph g, you can use the following methods:

  • g.get_neighbors(node1): Given a node node1, returns a list of all nodes that are directly connected to node1.
  • g.get_neighboring_edges(node1): Given a node node1, returns a list of all edges that have node1 as their startNode.
  • g.get_edge(node1, node2): Given two nodes, returns the edge that directly connects node1 to node2 (or None if there is no such edge)
  • g.is_neighbor(node1, node2): Given two nodes, returns True if there is an edge that connects them, or False if there is no such edge
  • g.get_heuristic_value(node1, goalNode): Given two nodes node1 and goalNode, returns the heuristic estimate of the distance from node1 to goalNode.

An edge is an object of type Edge. It has the following attributes:

  • .startNode: a node
  • .endNode: a node
  • .length: a positive number, or None if the edge has no length specified

For this lab, you may assume that edge weights will always be positive numbers (or None).

You can check whether two edges e1 and e2 are the same with e1 == e2.

Graphs for Testing

Here are some sample graphs that are used in the tester and which you can use to test and debug your functions. The graphs have been imported into by their names -- that is, if you want to access GRAPH_2, it's stored in the variable GRAPH_2.





The graphs come from the files graphs.txt and, which you should not need to read.

Python advice

  • You will undoubtably need to sort Python lists during this lab (using either the in-place .sort method or sorted). Python has built-in sorting functionality, documented at <>.
  • You will need to know how to access elements in lists and dictionaries. For some portions of this lab, you may want to treat lists like either stacks or queues, as documented at <>. However, you should not import other modules (such as deque) for this purpose because they will confuse the tester.


The Agenda

Different search techniques explore nodes in different orders, and we will keep track of the nodes remaining to explore in a list we will call the agenda (sometimes called the queue). Some search techniques will add paths to the front (or top) of the agenda, treating it like a stack, while others will add to the back (or end) of the agenda, treating it like a queue. Some agendas are organized by heuristic value, others are ordered by path distance, and others by depth in the search tree. Your job will be to show your knowledge of search techniques by implementing different types of search and making slight modifications to how the agenda is accessed and updated.

Extending a path

At a high level, you search for a path through a graph by starting with shorter paths then extending them into progressively longer paths until (hopefully) you discover a path that terminates at the goal node. When we talk about extending a partial path, we mean that you consider every possible node that the path could visit next (i.e. all of the nodes which are neighbors of the last node in the path) and create a new, incrementally longer path for each possibility. Algorithmically, you'll collect those longer paths into a list then add them to the agenda in some order.

For example, in the following graph:


If your partial path is sxw, then the possible one-node extensions of sxw are sxwy, sxwz, and sxwg. Note that we exclude the possibility sxwx because we want our paths to be as short as possible, and so we always want to ignore any paths with loops.

Hint: when writing the function extensions(graph, path), you should ensure that the paths you create are new objects. If you try to extend a path by modifying (or "mutating") the existing path -- for example by using list .append() or .pop() -- you may find yourself in a quagmire.

Extended sets make optimal search more efficient

When you want to find not just any path but the shortest path between two nodes, it is often useful to keep track of which nodes in the graph you've already extended (i.e. which nodes have already been at the end of paths you've extended.) This is because of a fact about paths: if the shortest route from the start to the goal passes through some intermediate node X, then the first part of that route must be shortest path between the start node and X.

Now consider how you can use this principle to avoid unnecessary searching when you use an algorithm such as branch and bound, which is guaranteed to look at shorter paths before longer paths. If you extend a node X once, you know that you've found the shortest path from the start to X. Then, if you later find a second path that ends in X, you can safely reject the second path: The second path takes longer to reach X than the first path, and so it cannot be part of the shortest, most direct path to the goal.

Algorithmically, you employ an extended set by maintaining a set/list of the nodes you've extended so far (i.e. nodes that have been at the end of paths you've extended.) Then, whenever you remove a path from the agenda, you check the last node in the path and see whether you've already extended it. If you have extended it already, you reject the path and continue with the next iteration of search. If you haven't extended it already, you add the node to your extended set and proceed as usual.

When to Exit a Search

A non-optimal search such as DFS, BFS, hill climbing and beam may exit either:

  • when it finds a path-to-goal in the agenda
  • when a path-to-goal is first removed from the agenda.

Optimal searches such as branch and bound and A* must always exit:

  • when a path-to-goal is first removed from the agenda.

(This is because the agenda is ordered by path length (or heuristic path length), so a path-to-goal is not necessarily the best when it is added to the agenda, but when it is removed, it is guaranteed to have the shortest path length (or heuristic path length).)

For the sake of consistency, you should implement all your searches to exit:

  • when a path-to-goal is first removed from the agenda.


Part 1: Helper Functions

Your first task is to write four helper functions which may be useful in Part 2:

def path_length(graph, path):

def has_loops(path):

def extensions(graph, path):

def sort_by_heuristic(graph, goalNode, nodes):

A path is a list of nodes where each neighboring pair is connected by an edge in the graph. By convention, the start of the path is at index [0] and the end of the path is at index [-1].

path_length(graph, path) should return the length of a path. You may assume that the path is a valid sequence of edge-connected nodes in the graph. If there is only one node in the path, your function should return 0.

has_loops(path) should return True if the path contains a loop, otherwise False. (Hint: One efficient solution involves using a set.)

extensions(graph, path) should take a path and return a list of possible one-node extensions of that path. That is, your function should return a list of all possible paths that can be created by adding one more node to the given path (as described in Extending a Path, above).

sort_by_heuristic(graph, goalNode, nodes) should takes a list of any nodes in the graph and sort them based on their heuristic value to the goalNode, from smallest to largest. The function should return the sorted list. (Hint: Use the Python keyword sorted with the "key" argument.) In case of a tie, sort nodes lexicographically.

Part 2: Search Algorithms

In this section, you will implement each of the search algorithms we study in 6.034.

Alternatively, you may skip to Part 3: Generic Search in which you code all of the algorithms based on a generic template. Then you can complete Part 2 using your answers from Part 3.

The inputs to each search function are:

  • graph: The graph, as an UndirectedGraph object
  • startNode: The node that you want to start traversing from
  • goalNode: The node that you want to reach

Each algorithm should return an appropriate path from startNode to goalNode, or None if no such path is found. A path should consist of a list of nodes, beginning with the start node and following existing edges to the goal node.

Blind Search

The first two types of search to implement are depth-first search and breadth-first search. You should allow backtracking.

def dfs(graph, startNode, goalNode):

def bfs(graph, startNode, goalNode):

For an explanation of DFS and BFS, refer to Chapter 4 of the textbook (pages 4-6 of the pdf, pages 65-67 of the textbook).

Heuristic Search

Next, you will implement three types of heuristic search.

Hill Climbing

Hill climbing is very similar to depth-first search. There is only a slight modification to the ordering of paths that are added to the agenda. Implement the following function:

def hill_climbing(graph, startNode, goalNode):

The hill-climbing procedure you define here should use backtracking, for consistency with the other methods, even though hill climbing typically is not implemented with backtracking. Hill climbing does not use an extended set.

For an explanation of hill climbing, refer to Chapter 4 of the textbook (page 8 of the pdf, page 70 of the textbook).

Best-First Search

Best-first search is similar to hill climbing in that it uses a heuristic, but best-first search always extends the "best" path, as determined by heuristic values to the goal. Like hill climbing and beam search, best-first search sorts paths by only the heuristic value, not the path length (as branch and bound does) or the path length + heuristic value (as A* does). Implement the following function:

def best_first(graph, startNode, goalNode):

For a brief explanation of best-first search, refer to Chapter 4 of the textbook (page 13 of the pdf, page 75 of the textbook).

Beam Search

Beam search is very similar to breadth-first search, but there is a modification to which paths are in the agenda. The agenda at any time can have up to w paths of each length n; w is also known as the beam width, and n corresponds to the level of the search graph. Beam search can be useful in situations where you want to reduce memory usage, especially when working with a large graph.

You will need to sort your paths by the graph's heuristic to ensure that only the best w paths at each level are in your agenda. You may want to use an array or dictionary to keep track of paths of different lengths. Beam search does NOT use an extended set, and does NOT use backtracking to the paths that are eliminated at each level.

Remember that w is the number of paths to keep at an entire level, not the number of paths to keep from each extended node.

Implement the following function:

def beam(graph, startNode, goalNode, beam_width):

Unlike the other search functions, beam has an additional argument, beam_width, which is the same as w, the number of paths to keep at each level.

For an explanation and an example of beam search, refer to Chapter 4 of the textbook (pages 13-14 of the pdf, pages 75-76 of the textbook). However, note that this example is incorrect according to the way we do beam search: At the third level, the node B (with heuristic value 6.7, at the end of path S-D-E-B) should not be included -- only C and F should be included (because the beam width is 2) -- so at the fourth level, B should not be extended to A and C.

Optimal Search

The search techniques you have implemented so far have not taken into account the edge lengths. Instead, we were just trying to find one possible solution of many. Optimal searches try to find the path with the shortest distance from the start node to the goal node. The search types that guarantee optimal solutions are branch and bound and A* (under certain conditions).

Branch and Bound

Using the path_length helper function, this part should be easy to complete. You might find the list procedure remove and/or the Python del keyword useful (though not necessary). Implement basic branch and bound with no extended set:

def branch_and_bound(graph, startNode, goalNode):

For an explanation of branch-and-bound, refer to Chapter 5 of the textbook (page 3 of the pdf).

Branch and Bound Variations

Now you've used heuristic estimates to speed up search, and you've used edge lengths to compute optimal paths. If you add a heuristic to branch and bound, the path with the least {heuristic estimate + path length} is taken from the agenda to extend. Implement this variation of branch and bound (using a heuristic but no extended set):

def branch_and_bound_with_heuristic(graph, startNode, goalNode):

Next, implement a variation on branch and bound that uses an extended set (but no heuristic):

def branch_and_bound_with_extended_set(graph, startNode, goalNode):


A* is simply branch and bound using both a heuristic and an extended set. (Note: If the heuristic is not consistent, then using an extended-set can sometimes prevent A* from finding an optimal solution.) Implement A*:

def a_star(graph, startNode, goalNode):

Part 3: Generic Search

generic_search is a generic search algorithm that has already been written for you:

def generic_search(sort_new_paths_fn, add_paths_to_front_of_agenda, sort_agenda_fn, use_extended_set):
    return search_algorithm

Arguments for generic_search

generic_search takes in two functions and two boolean values as arguments, and it returns a function representing a search algorithm. By defining appropriate values for the arguments, you can recreate any of the search algorithms we study in 6.034.

Here's an explanation of each of the arguments in generic_search:

  • When you extend a node, you create a list of new paths. Some algorithms will sort these paths before adding them to the agenda. The argument sort_new_paths_fn is a function that takes the list of new paths and performs the appropriate sorting routine; it returns the sorted list of new paths. If you don't want any sorting to happen, you can pass a function that simply returns the list of paths without changing it. The function do_nothing_fn accomplishes this.
  • Some algorithms will add new paths to the front of the agenda (like a stack). Others will add new paths to the back of the agenda (like a queue). When the argument add_paths_to_front_of_agenda is True, paths will get added to the front of the agenda. Otherwise, paths will get added to the back.
  • After new paths are added to the agenda, some algorithms will then sort the entire agenda. The argument sort_agenda_fn takes the agenda as an argument, and performs the appropriate sorting routine; it returns a sorted agenda. If you don't want any sorting to happen, you can pass a function that simply returns the list of paths without changing them (do_nothing_fn). Note that if you do sort the agenda, then the previous two arguments become irrelevant (it no longer matters whether you added new paths to the front, or whether you sorted them before adding them to the agenda).
  • Some algorithms use an extended set. Set use_extended_set to True to maintain an extended set, else set it to False.

Examples for using generic_search

To create a search algorithm that does not sort new paths, adds paths to the front of the agenda, does not sort the agenda, and does not use an extended set, you can call generic_search like this:

my_search_fn = generic_search(do_nothing_fn, True, do_nothing_fn, False)

You can then call your search algorithm on a graph:

my_path = my_search_fn(graph, startNode, goalNode)

You can also call generic_search directly on a list of arguments (such as generic_dfs) like this:

path = generic_search(*generic_dfs)(graph, startNode, goalNode)

If you want to test your search algorithm on a provided graph, you can use one of the graphs shown in Graphs for Testing, above. For example:

path = generic_search(*generic_dfs)(graph1, 'S', 'G')

Your Task

Please implement the following search algorithms by designing the right arguments to pass to the generic search algorithm. Your answer to each should be an ordered list of the appropriate four arguments to generic_search. No argument should be None.

generic_dfs = [None, None, None, None]
generic_bfs = [None, None, None, None]
generic_hill_climbing = [None, None, None, None]
generic_best_first = [None, None, None, None]
generic_branch_and_bound = [None, None, None, None]
generic_branch_and_bound_with_heuristic = [None, None, None, None]
generic_branch_and_bound_with_extended_set = [None, None, None, None]
generic_a_star = [None, None, None, None]

You will need to write your own path-sorting functions. Each sorting function should take in a graph, goalNode, and list of paths, and then return a list of paths. For example: sorted_paths = my_sorting_fn(graph, goalNode, paths) We recommend that you avoid modifying the original list of paths in your sorting function.

Break ties lexicographically. (For example, the path ['a', 'c', 'f'] comes before the path ['a', 'd', 'b'], because acf comes before adb in the English dictionary.) For your convenience, here is tie-breaking function that you can add to to use in your sorting, extensions, and sort_by_heuristic functions:

def break_ties(paths):
    return sorted(paths)

As taught in 6.034, depth-first search and breadth-first search use backtracking but do not use an extended set. Refer to Part 2 above for more details about individual search methods.

Optional: Generic Beam Search

Beam search is trickier to implement with generic_search, so it's optional. If you want to run local tests to test your generic_beam, change the boolean TEST_GENERIC_BEAM to True in There are no online tests for generic_beam.

To implement generic_beam, fill in the four arguments:

generic_beam = [None, None, None, None]  # OPTIONAL (see below)

The sort_agenda_fn for beam search takes a fourth argument, beam_width. For example:

sorted_beam_agenda = my_beam_sorting_fn(graph, goalNode, paths, beam_width)

Similarly, the search algorithm produced by generic_search for beam search will take an additional argument, beam_width. For example:

my_beam_fn = generic_search(*generic_beam)

my_beam_path = my_beam_fn(graph, startNode, goalNode, beam_width)

Hint: You can implement beam search in a way almost identical to bfs. The only difference is that beam search will need to limit the number of paths whenever an entire level has been expanded. You can do this by defining a sort_agenda_fn function that checks whether an entire level has been extended. If it has, the function should limit the number of paths in the agenda. If it hasn't, the function should do nothing to the agenda. To tell whether an entire level has been expanded, check whether all paths in the agenda have the same number of nodes. Use the variable beam_width to refer to the limiting size of the agenda.

Part 4: Heuristics

Admissibility and Consistency

A heuristic value gives an approximation from a node to a goal. You've learned that in order for the heuristic to be admissible, the heuristic value for every node in a graph must be less than or equal to the distance of the shortest path from the goal to that node. In order for a heuristic to be consistent, for each edge in the graph, the edge length must be greater than or equal to the absolute value of the difference between the two heuristic values of its nodes.

For this part, complete the following functions, which return True iff the heuristics for the given goal node are admissible or consistent, respectively, and False otherwise:

def is_admissible(graph, goalNode):

def is_consistent(graph, goalNode):

Hint: For is_consistent, it's sufficient to check only neighboring nodes, rather than checking all pairs of nodes in the graph.

Optional: Picking Heuristics

This final section is intended to help you test your understanding of heuristics. To check your heuristics, you can run local tests by changing the boolean TEST_HEURISTICS to True. There are no online tests for this section.

The questions in this section use the following graph:


For heuristic_1, pick heuristic values so that, for the goal node G, the heuristic is both admissible and consistent. Fill in the values by replacing the list of five None's with five numbers:

[h1_S, h1_A, h1_B, h1_C, h1_G] = [None, None, None, None, None]

For heuristic_2, pick heuristic values so that, for the goal node G, the heuristic is admissible but NOT consistent.

For heuristic_3, pick heuristic values so that, for the goal node G, the heuristic is admissible but A* returns a non-optimal (not shortest) path to the goal (S-A-C-G).

For heuristic_4, pick heuristic values so that, for the goal node G, the heuristic is admissible but NOT consistent, yet A* still finds the optimal (shortest) path to the goal (S-B-C-G).


Please answer these questions at the bottom of your file:

  • NAME: What is your name? (string)
  • COLLABORATORS: Other than 6.034 staff, whom did you work with on this lab? (string, or empty string if you worked alone)
  • HOW_MANY_HOURS_THIS_LAB_TOOK: Approximately how many hours did you spend on this lab? (number)
  • WHAT_I_FOUND_INTERESTING: Which parts of this lab, if any, did you find interesting? (string)
  • WHAT_I_FOUND_BORING: Which parts of this lab, if any, did you find boring or tedious? (string)
  • (optional) SUGGESTIONS: What specific changes would you recommend, if any, to improve this lab for future years? (string)

(We'd ask which parts you find confusing, but if you're confused you should really ask a TA.)

When you're done, run the online tester to submit your code.


Q: I implemented dfs (or any other search method) by calling generic_dfs. When I try to test my dfs function by calling it in, I get NameError: global name 'generic_dfs' is not defined.

A: Try testing dfs below PART 3 of Because generic_dfs is defined after dfs, all calls to dfs need to be made after the generic search arguments have been defined.

Q: How should A* behave if given a bad (inadmissible or inconsistent) heuristic?

A: The A* algorithm should still search for and return a path; however, the path returned is not guaranteed to be optimal. In practice, you can patch A* so that it will return the shortest path even if the heuristic is bad (see Chapter 5 of the textbook) --- but for this lab, we ask you to implement naive A*, which optimistically assumes that the first path it finds to each node is the shortest.

Q: How should I break ties if two paths have the same length?

A: See tie-breaking, above: Break ties lexicographically. (For example, the path ['a', 'c', 'f'] comes before the path ['a', 'd', 'b'], because acf comes before adb in the English dictionary.)

Q: My code passes all the offline tests, but when I run the online tests, it prints a bunch of errors such as "AttributeError: 'module' object has no attribute 'generic_hill_climbing_sort_new_paths_fn'".

A: If you downloaded before 4pm on Sept 19, you will need to add a patch to in order to run the online tests. Simply copy all the lines from patch.txt and paste them at the bottom of your file.

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