代写算法作业,用BFS和DFS来解决迷宫寻路问题。
Goals
Practice working with graphs and graph algorithms by designing mazes using
Kruskal’s algorithm, and solving them using either breadth- or depth-first
searches.
In class, we have been discussing various algorithms for working with graphs,
that require the use of several data structures working together. We have
talked about general-purpose maze searching algorithms, like breadth- and
depth-first searches, and we have talked about building minimum spanning trees
on graphs.
To get a visual grasp of how these algorithms work, you are going to be
building and solving mazes, like this one.
The mazes you construct should start in the upper-left corner (shown in green)
and end in the lower-right corner (shown in purple). As you solve the mazes,
you should color in the cells you have explored. Once you have reached the
solution, you must backtrack the path from the end to the start, and draw it
as well. A fully-solved maze might look like this.
Requirements
Your program should support at minimum the following features:
- Construct random mazes using Kruskal’s algorithm and Union/Find (below)
- Display the maze graphically and animate the search for the path.
- Allow the user to choose one of two algorithms for finding the path: Breadth-First Search or Depth-First Search (below).
- Provide an option for designing a new random maze.
- Allow the user to traverse the maze manually - using the keys to select the next move, preventing illegal moves and notifying the user of completion of the game.
- Display the solution path connecting the start and end, once it’s found (either automatically or by the user).
Be sure to submit documentation for your code, so the graders know how to run
and play your game. As always, be sure to test your code thoroughly.
Additionally, you may attempt bells and whistles for extra credit:
Whistles
- Provide an option to toggle the viewing of the visited paths.
- Allow the user the ability to start a new maze without restarting the program.
- Keep the score of wrong moves - for either the automatic solutions or manual ones - and maybe keep statistics on which one of the two algorithms had fewer steps for each maze.
Bells
- In addition to animating the solution of the maze, also animate the construction of the maze: on each tick, show a single wall being knocked down.
- (Tricky) Construct mazes with a bias in a particular direction - a preference for horizontal or vertical corridors. (Hint: you might wish to play tricks with the edge weights here.)
- Hard! (But very cool) Instead of constructing a rectangular maze, try constructing a hexagonal one. You’ll have many of the same problems as you would with hexagonal islands from the Forbidden Island game; refer to that for details.
- Tricky Construct two intertwined mazes, and allow two players to race from their starting points to their ending points. Choosing the start points for both mazes is easy; choosing end points is harder. Even more impressive is ensuring that it’s a fair race, and both mazes have the same length path from start to finish… (Hint: how can you force Kruskal’s algorithm to produce two distinct connected mazes? Talk to an instructor if you choose to do this, and are not sure how to proceed.)
Spend careful thought planning ahead and designing your classes: if your
design is too brittle, you’ll have a very hard time completing the algorithms.
And as always, have fun!
Kruskal’s Algorithm for constructing Minimum Spanning Trees
Here is Kruskal’s algorithm illustrated on a particular example graph.
(The edges are drawn without directional arrows; in your mazes, every maze
cell will be connected to its four neighbors, so edges are effectively
undirected. Edge weights are notated as numbers on the edges.)
Kruskal’s algorithm begins by sorting the list of edges in the graph by edge
weight, from shortest to longest:
(E C 15)
(C D 25)
(A B 30)
(B E 35)
(B C 40)
(F D 50)
(A E 50)
(B F 50)
At each step we remove the shortest edge from the list and add it to the
spanning tree, provided we do not introduce a cycle. In practice, this may
produce many trees during the execution of the algorithm (so in fact, the
algorithm produces a spanning forest while it runs), but they will eventually
merge into a single spanning tree at the completion of the algorithm.
For this particular graph, we add the edges (E C 15), (C D 25), (A B 30) and
(B E 35). When we try to add the edge (B C 40) we see that it would make a
cycle, so this edge is not needed and we discard it. We then add edge (F D
50). This connects the last remaining unconnected node in the graph, and our
spanning tree is complete. In very high-level pseudocode, the algorithm is
quite short and elegant:
while (we do not yet have a complete spanning tree)
find the shortest edge that does not create a cycle
and add it to the spanning tree
—|—
Determining if we have a complete spanning tree is easy: for n nodes, we need
n-1 edges to connect them all.
We can represent the spanning tree itself by a list of edges. Adding an edge
to that list is as easy as Cons’ing it on, or adding it, depending on which
representation of lists you choose to use. Finding the shortest edge is easy,
since we began by sorting the list of edges by their weights. The only tricky
part in this algorithm is figuring out whether a given edge creates a cycle
with the edges we have already selected. For this we use the Union/Find data
structure.
The Union/Find data structure
The goal of the union/find data structure is to allow us to take a set of
items (such as nodes in a graph) and partition them into groups (such as nodes
connected by spanning trees) in such a way that we can easily find whether two
nodes are in the same group, and union two disjoint groups together.
Intuitively, we accomplish this by naming each group by some representative
element, and then two items can be checked for whether they are in the same
group by checking if they have the same representative element.
Example
In class, we represented every node of the graph as a class with a String name
field. (For this assignment, String names will be inconvenient; you will need
to come up with some other uniquely-identifying feature of each cell in a maze
that can serve the same role as a name.) Then the union-find data structure
was a HashMap that mapped (the name of) each node to (the name of) a node that
it is connected to. Initially, every node name is mapped to itself, signifying
that every node is its own representative element, or equivalently, that it is
not connected to anything.
Putting the union/find data structure to work
The full Kruskal’s algorithm needs a union/find data structure to handle
efficiently connecting components, and also needs a list of the edges used by
the algorithm:
HashMap<String, String> representatives;
List
List
initialize every node’s representative to itself
While(there’s more than one tree)
Pick the next cheapest edge of the graph: suppose it connects X and Y.
If find(representatives, X) equals find(representatives, Y):
discard this edge // they’re already connected
Else:
Record this edge in edgesInTree
union(representatives,
find(representatives, X),
find(representatives, Y))
Return the edgesInTree
—|—
To find a representative: if a node name maps to itself, then it is the
representative; otherwise, “follow the links” in the representatives map, and
recursively look up the representative for the current node’s parent.
To union two representatives, simply set the value of one representative’s
representative to the other.
Breadth- and depth-first search
As we worked through in class, breadth- and depth-first searches are very
closely related algorithms. The essential steps of the algorithm are the same;
the only difference is whether to use a queue or a stack.
HashMap<String, Edge> cameFromEdge;
???
initialize the worklist to contain the starting node
While(the worklist is not empty)
Node next = the next item from the worklist
If (next has already been processed)
discard it
Else If (next is the target):
return reconstruct(cameFromEdge, next);
Else:
For each neighbor n of next:
Add n to the worklist
Record the edge (next->n) in the cameFromEdge map
—|—
The cameFromEdge map is used to record which edge of the graph was used to get
from an already-visited node to a not-yet-visited one. This map is used to
reconstruct the path from the source to the given target node, simply by
following the edges backward, from the target node to the node that it came
from, and so on back to the source node. Unlike Kruskal’s algorithm, the
worklist here is a collection of nodes (rather than edges). Like the
union/find algorithm, there is a recursive traversal from one node to a
previous one, using node names as the keys into the auxiliary map that
accumulates the ongoing state of the algorithm.