CS 371 - Introduction to Artificial Intelligence
Homework #3 |

1. **Rook Jumping Maze Generation: **

**Rook Jumping Maze Instructions**** :**
Starting at the circled cell in the upper-left corner, find a path to the goal
cell marked “G”. From each numbered cell, one may move that exact
number of cells horizontally or vertically in a straight line. How many
moves does the shortest path have?

For this exercise, you will generate random *n*-by-*n* Rook Jumping Mazes (RJMs)
(5 £ *n*
£ 10) where there is a legal move (jump) from each non-goal state.
You will also use stochastic local search to optimize for (1) solvability and
(2) maximum length of the shortest maze solution path.

Let array row and
column indices both be numbered (0, ..., *n *
- 1). The RJM is represented by a 2D-array of jump numbers. A
cell's *jump
number* is the number of array cells one must move in a straight line
horizontally or vertically from that cell. The start cell is located at
(0, 0). For the goal cell, located at (*n *
- 1, *n *
- 1), let the jump number be 0. For all non-goal cells, the randomly
generated jump number must allow a legal move. In the example 5-by-5 maze
above, legal jump numbers for the start cell are {1, 2, 3, 4}, whereas legal
jump numbers for the center cell are {1, 2}. In general, the minimum legal
jump number for a non-goal cell is 1, and the maximum legal jump number for a
non-goal cell (*r*, *c*) is the maximum of *n* - 1 - *r*,
*r*, *n* - 1 - *c*, and *c*. This
defines a directed graph
where vertices are cells, edges are legal jumps, and the outdegree is 0 for the
goal cell vertex and positive otherwise.

There are many features of a good RJM. One obvious feature is that the maze has a solution, i.e. one can reach the goal from the start. One simple measure of maze quality is the minimum number of moves from the start to the goal. For this exercise, we will limit our attention to these two measures of maze quality.

Using breadth-first search, or some other suitable graph algorithm, compute
the minimum distance (i.e. depth, number of moves) to each cell from the start
cell. Create an *objective function* (a.k.a. *energy function*)
that returns the negated distance from start to goal, or a large positive number
(use 1,000,000) if no path from start to
goal exists. Then the task of maze generation can be reformulated as a
search through changes in the maze configuration so as to minimize this
objective function.

Implement stochastic local search state RookJumpingMaze according to this specification. Note that the specification of implemented interface methods is listed towards the top of the RookJumpingMaze documentation.

Once you have RookJumpingMaze implemented, complete the implementation of RJMGenerator.java. For a size 5 RJM, you should be able to implement and tune the parameters of a stochastic local search algorithm of your choice to achieve a median energy of less than or equal to -18.0 in 5000 iterations of search or less. That is, your median-energy maze generated should require a minimum of 18 moves to solve.

*Hint: When computing the energy() method, you
should not use your previous uniformed search code. Instead, coding a simple breadth-first
search within the RookJumpingMaze class will allow you to easily do
repeated-state elimination. Initialize the depth of the initial
location to 0, and all others to a negative constant representing
"unreached" status. Replace "unreached" values with search depths (one
greater than current depth) as child locations are being added to the queue,
and not adding children to the queue that have already been visited (have
non-negative depth).*

2. **Distant Sampling: **In many different problem, it
is useful to be able to select a *biased *sample from a data set such
that the items are very different from each other. Examples include
initialization of clustering algorithms, choosing a color palette where colors
are well distinguished from one another, and choosing sites to geographically
distribute resources.

For this problem, you will create a stochastic local search State implemented
according to this specification. Note
that the specification of implemented interface methods is listed towards the
top of the DistantSamplerState documentation. To summarize, the state is
constructed with a 2D array where each row is an n-dimensional double
point/vector, and a given number of distant samples from among these is the
objective. Thus, to compute the energy, one computes the *inverse
squared Euclidean distance *for each pair of points and sums these to get
the "energy" of the State. To compute the Euclidean distance, compute the
sum of the squared differences in each dimension between two points/vectors and
then take the square root of this sum. To get the inverse squared
Euclidean distance, you divide one by the distance squared.
(Alternatively, *don't * take the square root when computing the
distance and directly compute the inverse (i.e. reciprocal) without squaring.
The square root and squaring are inverse operations and cancel.) It is as
if we're simulating
electrostatic
rupulsion forces between our selected sample data points.

A stochastic local search step here is switching from one unique sample choice (no repetitions are allowed) to a different unique sample choice. It is convenient to internally represent our sample choices as a list of row indices from the original given data. More details to more methods are given in this specification.

Once you have DistantSamplerState implemented, complete the implementation of DistantSampler.java. For the default dataset given therein, you should be able to implement and tune the parameters of a stochastic local search algorithm of your choice to achieve a median energy of less than 132.0 in 10000 iterations of search or less.

Hopefully, in looking at these problems, you'll see the wide applicability of such optimization algorithms. Often, the most challenging aspect of such optimization is devising a good, efficient next state generator that gives beneficial structure to the state space, rather than merely bouncing randomly from one state to another. Good next state generators both allow one to traverse the entire state space, and immediately sample states that are similar in quality to the current state.