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API

Graphs and Edges

The file search.py 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. todo

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.

Python advice

#todo copy from old wiki

sorted / .sort -- https://wiki.python.org/moin/HowTo/Sorting

^ in case of a tie, sort paths lexicographically

using lists

Problems

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.

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.

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.)

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 and complete Part 2 by calling generic_search with the arguments 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.

Basic 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):

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. For this part, 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.

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

def best_first(graph, startNode, goalNode):

Beam Search

Beam search is very similar to breadth first search, but there is a modification to what paths are in the agenda. The agenda at any time can have up to k paths of each length n; k is also known as the beam width, and n corresponds to the level of the search graph. You will need to sort your paths by the graph heuristic to ensure that only the best k 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 k is the number of paths to keep at an entire level, not the number of paths to keep from each extended node.

Also remember that beam search sorts paths by just the heuristic value, not the path length (as branch-and-bound does) or the path length + heuristic value (as A* does).

For this part, implement the following function:

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

This function has an additional argument, beam_width.

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

Optimal Search

def branch_and_bound(graph, startNode, goalNode):

def branch_and_bound_with_heuristic(graph, startNode, goalNode):

def branch_and_bound_with_extended_set(graph, startNode, goalNode):

def a_star(graph, startNode, goalNode):

Part 3: Generic Search

generic_search is a generic search algorithm:

generic_search(sort_new_paths_fn, add_paths_to_front_of_agenda, sort_agenda_fn, use_extended_set) :

It takes two functions and two boolean values as arguments. By defining appropriate values for the arguments, you can recreate any of the search algorithms we study in 6.034.

For example, 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 could 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)

For beam search, the search algorithm will take an additional argument, beam_width. For example: my_beam_fn = generic_search(...) my_beam_path = my_beam_fn(graph, startNode, goalNode, beam_width)

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 which 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 which 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 end 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 which simply returns the list of paths without changing them. The variable do_nothing_fn accomplishes this.

Please note that if you do sort the agenda, then the previous two arguments become irrelevant (it doesn't matter whether you added new paths to the front, or 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.

TODO: Provide sample graphs that you can call the algorithm on so as to see visually what it's doing. Show how to call the algorithm with the default arguments: generic_search()(graph, start, goal).

Please implement the following search algorithms (dfs, bfs, hill-climbing, best-first, beam, branch-and-bound, b&b+heuristic, b&b+xset, A*) by designing the right arguments to pass to the generic search algorithm. Your answer should be an ordered list of the appropriate arguments to generic_search.

You will need to write your own helper functions for sorting paths. Each sorting function should take in a graph, goalNode, and list of paths and return a list of paths. (For example, sorted_paths = my_sorting_fn(graph, goalNode, paths)) The only exception is the sort_agenda_fn for beam search, which should take in beam width as a fourth argument. (For example, sorted_beam_agenda = my_beam_sorting_fn(graph, goalNode, paths, beam_width)) Break ties lexicographically. We recommend that you avoid modifying the original list of paths.

Hint for beam search: 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.

As taught in 6.034, dfs and bfs use backtracking but do not use an extended set.

todo: for best-first, hill-climbing, beam: note that quality measure can be something other than est dist to goal; maybe have additional arg for non-generic search

You can call generic_search on a list of arguments (such as generic_dfs) like this: path = generic_search(*generic_dfs)(graph1, 'S', 'G')

Part 4: Counting Extensions

generic_search_counter is identical to the generic_search algorithm in ___. Your task is to slightly modify the search_algorithm code in generic_search_counter so that instead of returning a path to the goal, the search algorithm returns the number of extensions it had to make.

By convention, please include the final extension of the goal node, if any, in your extension count.

Hint: One possible solution involves replacing the extended set with a list.

Part 5: Heuristics

Fill in is_admissible(graph, goalNode) to write a function that returns True if the graph has an admissible heuristic for the given goal node, otherwise False.

Fill in is_consistent(graph, goalNode) to write a function that returns True if the graph's heuristic for the given goal node is consistent, otherwise False. (Hint: it's sufficient to check only neighboring nodes, rather than checking all pairs of nodes in the graph.)

The next questions use the following graph:

   A
(1) (1)

S C-(7)-G

(1) (3)
   B

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.

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).

Search can be applied both to abstract problems like finding recipes, and to concrete problems like finding a route on a map. When you are finding a route on a map, you may have nodes that look like the following, where each node has spatial coordinates:

node_F = {'coordinates' : [2.5, 3.5], 'label' : 'F'}

Whenever you are using spatial coordinates, "Euclidean distance to the goal" can be a useful heuristic. Fill in euclidean_heuristic(startNode, goalNode) to write a function that takes two nodes with spatial coordinates as input, and which returns the Euclidean distance between them.

You can test your heuristic function by uncommenting the line [todo].

Survey

Please answer the following questions:

NAME: What is your name? (string) COLLABORATORS: Other than 6.034 staff, who 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, or empty string if none) WHAT_I_FOUND_BORING = Which parts of this lab, if any, did you find boring or tedious? (string, or empty string if none)

(optional) SUGGESTIONS = What concrete changes would you recommend, if any, to improve this lab for future years? (string)

Bonus API

To create a new graph g, you can instantiate it with: g = UndirectedGraph() To add nodes: g.nodes = ["A","B","C","D","E"] To create an edge between two nodes: g.join("A","B",5)

Lab 1: Search