Homework 1- twist the problem of path planning


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CSCI-561 – – Foundations of Artificial Intelligence
Homework 1

Image  The Original Oregon Trail Game
This is a programming assignment. You will be provided sample inputs and outputs (see below).
Please understand that the goal of the samples is to check that you can correctly parse the
problem definitions and generate a correctly formatted output. The samples are very simple and
it should not be assumed that if your program works on the samples it will work on all test cases.
There will be more complex test cases and it is your task to make sure that your program will
work correctly on any valid input. You are encouraged to try your own test cases to check how
your program would behave in some complex special case that you might think of. Since each
homework is checked via an automated A.I. script, your output should match the specified
format exactly. Failure to do so will most certainly cost some points. The output format is simple
and examples are provided below. You should upload and test your code on, and
you will also submit it there. You may use any of the programming languages provided by
Your code will be tested as follows: Your program should not require any command-line
argument. It should read a text file called “input.txt” in the current directory that contains a
problem definition. It should write a file “output.txt” with your solution to the same current
directory. Format for input.txt and output.txt is specified below. End-of-line character is LF (since
vocareum is a Unix system and follows the Unix convention).
The grading A.I. script will, 50 times:
– Create an input.txt file, delete any old output.txt file.
– Run your code.
– Check correctness of your program’s output.txt file.
– If your outputs for all 50 test cases are correct, you get 100 points.
– If one or more test case fails, you get 100 – 4xN points where N is the number of failed
test cases, or 0 if your code fails more than 25 test cases.
Note that if your code does not compile, or somehow fails to load and parse input.txt, or writes
an incorrectly formatted output.txt, or no output.txt at all, or OuTpUt.TxT, you will get zero
points. Anything you write to stdout or stderr will be ignored and is ok to leave in the code you
submit (but it will likely slow you down). Please test your program with the provided sample files
to avoid any problem.

Project description
In this project, we twist the problem of path planning a little bit just to give you the opportunity
to deepen your understanding of search algorithms by modifying search techniques to fit the
criteria of a realistic problem. To give you a context for expanding your ideas about search
algorithms, we invite you to take part in a simplified version of the computer game The Oregon
Trail. In this educational computer game, the player takes the role of leader to a group of
settlers in 1848, traveling with a wagon from Independence, Missouri, to Oregon’s Willamette
Valley. The goal is to reach Oregon without running out of provisions or falling victim to nature.
We are invited to develop an algorithm to find the optimal path for navigation of the wagon
based on a particular objective.
The input of our program includes a topographical map of the land, plus some information
about where our party starts their journey, the intended site our party wants to settle and
some other quantities that control the quality of the solution. The land can be imagined as a
surface in a 3-dimensional space, and a popular way to represent it is by using a mesh-grid. The
M value assigned to each cell will represent how muddy the patch of land is or whether it
contains a rock. At each cell, the wagon can move to each of 8 possible neighbor cells: North,
North-East, East, South-East, South, South-West, West, and North-West. Actions are assumed
to be deterministic and error-free (the wagon will always end up at the intended neighbor cell).
The wagon cannot go over rocks that are too high, and the wheels are such that, as the land
gets muddier, the wagon slows down. Therefore, the value M in each cell can advise us on
whether we can take that route (in case of rocks) or how much moving into that cell will cost
the settler party in terms of time if they move into it (in case of mud).
Search for the optimal paths
Our task is to lead the party of settlers from their start position to the land they aim to reach. If
we had the ideal vehicle that can go across any land without a slow-down, usually the shortest
geometrical path is defined as the optimal path; however, since our wagon is far from ideal, our
objective is to avoid rocks we can’t cross over as well as really muddy areas. Thus, we want to
minimize the path from A to B under those constraints. Our goal is, roughly, finding the shortest
path among the safe paths. What defines the safety of a path is whether there are rocks we
can’t cross and the muddiness of the cells along that path.
Problem definition details
You will write a program that will take an input file that describes the land, the starting point,
potential settling sites for our party of settlers, and some other characteristics for our wagon. For
each settling site, you should find the optimal (shortest) safe path from the starting point to that
target site. A path is composed of a sequence of elementary moves. Each elementary move
consists of moving the wagon party to one of its 8 neighbors. To find the solution you will use the
following algorithms:
– Breadth-first search (BFS)
– Uniform-cost search (UCS)
– A* search (A*).
Your algorithm should return an optimal path, that is, with shortest possible journey cost.
Journey cost is further described below and is not equal to geometric path length. If an optimal
path cannot be found, your algorithm should return “FAIL” as further described below.
Terrain map
We assume a terrain map that is specified as follows:
A matrix with H rows (where H is a strictly positive integer) and W columns (W is also a strictly
positive integer) will be given, with a value M (an integer number, to avoid rounding problems)
specified in every cell of the WxH map. If M is a negative integer, this means there is a rock of
height |M| in that cell. If M is a positive integer, the value represents the level of muddiness of
that cell. For example:
10 20 -30
12 13 40
is a map with W=3 columns and H=2 rows, and each cell contains an M value (in arbitrary units).
By convention, we will use North (N), East (E), South (S), West (W) as shown above to describe
motions from one cell to another. In the above example, mud level M in the North West corner
of the map is 10, and it is 40 in the South East corner, which means our wagon will take more
time to move into the SE corner than the NW corner. There is a rock of height 30 in the NE corner.
To help us distinguish between your three algorithm implementations, you must follow the
following conventions for computing operational path length:
– Breadth-first search (BFS)
In BFS, each move from one cell to any of its 8 neighbors counts for a unit path cost of 1. You do
not need to worry about the muddiness levels or about the fact that moving diagonally (e.g.,
North-East) actually is a bit longer than moving along the North to South or East to West
directions, but you still need to make sure the move is allowed by checking how steep the move
is. Therefore, any allowed move from one cell to an adjacent cell costs 1.
– Uniform-cost search (UCS)
When running UCS, you should compute unit path costs in 2D. Assume that cells’ center
coordinates projected to the 2D ground plane are spaced by a 2D distance of 10 North-South and
East-West. That is, a North or South or East or West move from a cell to one of its 4-connected
neighbors incurs a unit path cost of 10, while a diagonal move to a neighbor incurs a unit path
cost of 14 as an approximation to 10√� when running UCS. You still need to make sure the
move is allowed if a cell with a rock is involved.
– A* search (A*).
When running A*, you should compute an approximate integer unit path cost of each move by
also considering the muddiness levels of the land, by summing the horizontal move distance as
in the UCS case (unit cost of 10 when moving North to South or East to West, and unit cost of 14
when moving diagonally), plus the muddiness level in the cell we are trying to move in to, plus
the absolute height we have to traverse from our current cell to the next (cells where M >= 0
have height 0). For example, moving diagonally from the current position with M=-2 to an
adjacent North-East cell with mud level M=18 would cost 14 (diagonal move) + 18 (mud level) +
|0-2| (height change) = 34. Moving from a cell with M=1 to an adjacent cell with M=5 to the West
would cost 10+5+0=15. You need to design an admissible heuristic for A* for this problem.
Input: The file input.txt in the current directory of your program will be formatted as follows:
First line: Instruction of which algorithm to use, as a string: BFS, UCS or A*
Second line: Two strictly positive 32-bit integers separated by one space character, for
“W H” the number of columns (width) and rows (height), in cells, of the land map.
Third line: Two positive 32-bit integers separated by one space character, for
“X Y” the coordinates (in cells) of the starting position for our party. 0 £ X £ W-1
and 0 £ Y £ H-1 (that is, we use 0-based indexing into the map; X increases when
moving East and Y increases when moving South; (0,0) is the North West corner
of the map). Starting point remains the same for each of the N target sites below.
Fourth line: Positive 32-bit integer number for the maximum rock height that the wagon can
climb between two cells. The difference of heights between two adjacent cells
must be smaller than or equal (£ ) to this value for the wagon to be able to travel
from one cell to the other.
Fifth line: Strictly positive 32-bit integer N, the number of settling sites.
Next N lines: Two positive 32-bit integers separated by one space character, for
“X Y” the coordinates (in cells) of each target settling site. 0 £ X £ W-1 and 0 £ Y £
H-1 (that is, we again use 0-based indexing into the map). These N target settling
sites are not related to each other, so you will run your search algorithm from
the starting point to each target site in turn, and write each result to the output
as specified below. We will never give you a target settling site that is the same
as the starting point.
Next H lines: W 32-bit integer numbers separated by any numbers of spaces for the M values
of each of the W cells in each row of the map. Each number can represent the
following cases:
§ M >= 0, muddy land with height 0 and mud-level M
§ M < 0, rock of height |M| with mud-level 0
For example:
8 6
4 4
1 1
6 3
-10 40 34 21 42 37 18 7
-20 10 6 27 -6 5 2 0
-30 8 17 -3 -4 -1 0 4
-25 -4 12 14 -1 9 6 9
-15 -9 46 6 5 11 31 -21
-5 -6 -3 -7 0 25 53 -42
In this example, on an 8-cells-wide by 6-cells-high grid, we start at location (4, 4) highlighted in
green above, where (0, 0) is the North West corner of the map. The maximum elevation change
that the wagon can travel is 3 (in arbitrary units which are the same as for the M values of the
map). We have 2 possible targets sites, at locations (1, 1) and (6, 3), both highlighted in red
above. The map of the land is then given as six lines in the file, with eight M values in each line,
separated by spaces.
Output: The file output.txt which your program creates in the current directory should be
formatted as follows:
N lines: Report the paths in the same order as the target sites were given in the input.txt
file. Write out one line per target. Each line should contain a sequence of X,Y pairs
of coordinates of cells visited by the wagon party to travel from the starting point
to the corresponding settling site for that line. Only use a single comma and no
space to separate X,Y and a single space to separate successive X,Y entries.
If no solution was found (settling site unreachable by the wagon from the given
starting point), write a single word FAIL in the corresponding line.
For example, output.txt may contain:
4,4 4,3 3,2 2,1 1,1
4,4 5,3 6,3
Here the first line is a sequence of five X,Y locations which trace the path from the starting
point (4,4) to the first settling site (1,1). Note how both the starting location and the settling
site location are included in the path. The second line is a sequence of three X,Y locations which
trace the path from the starting point (4,4) to the second possible settling site (6,3).
The first path looks like this:
-10 40 34 21 42 37 18 7
-20 10 6 27 -6 5 2 0
-30 8 17 -3 -4 -1 0 4
-25 -4 12 14 -1 9 6 9
-15 -9 46 6 5 11 31 -21
-5 -6 -3 -7 0 25 53 -42
With the starting point shown in green, the settling sites in red, and each traversed cell in
between in yellow. Note how one could have thought of a perhaps shorter path: 4,4 3,3 2,2
1,1 (straight diagonal from landing site to target site). But this was not as good as this path has
much higher mud levels and therefore costs more for A*.
And the second path looks like this:
-10 40 34 21 42 37 18 7
-20 10 6 27 -6 5 2 0
-30 8 17 -3 -4 -1 0 4
-25 -4 12 14 -1 9 6 9
-15 -9 46 6 5 11 31 -21
-5 -6 -3 -7 0 25 53 -42
Notes and hints:
– Please name your program “” where ‘xxx’ is the extension for the
programming language you choose (“py” for python, “cpp” for C++, and “java” for Java).
If you are using C++11, then the name of your file should be “homework11.cpp” and if
you are using python3 then the name of your file should be “”.
– Likely (but no guarantee) we will create 15 BFS, 15 UCS, and 20 A* text cases.
– Your program will be killed after some time if it appears stuck on a given test case, to
allow us to grade the whole class in a reasonable amount of time. We will make sure that
the time limit for a given test case is at least 10x longer than it takes for the reference
algorithm written by the TA to solve that test case correctly.
– There is no limit on input size, number of targets, etc. other than specified above (32-bit
integers, etc.). However, you can assume that all test cases will take < 30 secs to run on a
regular laptop.
– If several optimal solutions exist, any of them will count as correct.
Example 1:
For this input.txt:
2 2
0 0
1 1
0 -10
-10 -20
the only possible correct output.txt is:
Example 2:
For this input.txt:
5 3
0 0
4 1
1 5 1 -1 -2
6 2 4 10 3
9 8 -10 -20 40
one possible correct output.txt is:
0,0 1,0 2,0 3,0 4,1
Example 3:
For this input.txt:
5 4
1 0
4 3
20 2 1 -2 -10
-8 1 10 2 -20
9 -1 4 15 11
6 -5 1 1 -1
one possible correct output.txt is:
1,0 1,1 2,2 3,3 4,3

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