Sale!

# Problem Set 3 Shortest Paths

\$30.00

Category:

CSC 226: Problem Set 3
Problem Set 3, Programming Part
Shortest Paths

1 Programming Assignment
The robotic duo Max and Min are in a bit of a fix. Min has a critically low battery
and cannot move; while they are both at the same charging station, it is broken.
If Min’s battery is not recharged, naturally they will have to miss the robot opera
in the evening. There are a number of charging stations throughout the city, but
because Min cannot move, Min cannot get to any of these. So, the plan is for Max to
visit a charging station, obtain some power, and return to Min to transfer the power
obtained. However, Max can only obtain a small amount of power from each charging
station, because Max lives “off the grid” and is not paying for the power. Also, Max
cannot spend a lot of time in transit because of the imminent robot opera. Max
therefore needs to compute the shortest paths from their current location to each
of the charging stations. Fortunately, Max has a map with the pairwise distances
between all pairs of charging stations. To get Min charged up for the robot opera, Max
needs to compute these (single-source) shortest paths efficiently prior to venturing out.
This problem is described by the following Input and Output.
Input: A weighted graph G of n vertices. Each edge weight corresponds
to the time to travel between two locations.
Output: For each charging station, the shortest path from the starting
location to that charging station and the total distance of the trip.
A Java template has been provided containing two empty functions. The function
ShortestPaths takes a two dimensional integer array G that represents the graph
and an integer representing the source vertex. This function should calculate and
store the single-source shortest paths from the source vertex to all other vertices.
The function PrintPaths takes the source vertex as an argument and prints out the
paths in a specific format. Your task is to write both the functions. The input might
be disconnected, but you can assume that the graphs are undirected.
You must use the provided Java template as the basis of your submission and start
your implementation inside the ShortestPaths and PrintPaths functions in the template. You may not change the name, return type, or parameters of those functions.
The main function in the template contains code to help you test your implementation by reading it from a file or from the console. A sample file is also provided. You
may modify the main function or any other function because your submission will be
tested using a different main function. You can use any helper methods or any helper
classes. You can use any built-in class or write your own classes and data structures.
We advise you to put all the classes you write in the same file, and so no other class
except the provided one should be declared as a public class.
2 Examples
The input format is exactly like that of Problem Set 2. The first line denotes the
number of vertices n in the graph and the subsequent n lines represent the n × n
adjacency matrix. A 0 at row i and column j in the adjacency matrix means that
there is no edge between vertices i and j; otherwise, that number denotes the weight
of the edge between i and j. You can assume that the edge weights are between 1
and 1000.
1
CSC 226: Problem Set 3
Sample input:
3
0 5 6
5 0 7
6 7 0
6
0 7 9 0 0 14
7 0 10 15 0 0
9 10 0 11 0 2
0 15 11 0 6 0
0 0 0 6 0 9
14 0 2 0 9 0
Sample output:
The path from 0 to 0 is: 0 and the total distance is: 0
The path from 0 to 1 is: 0 – – > 1 and the total distance is: 5
The path from 0 to 2 is: 0 – – > 2 and the total distance is: 6
The path from 0 to 0 is: 0 and the total distance is: 0
The path from 0 to 1 is: 0 – – > 1 and the total distance is: 7
The path from 0 to 2 is: 0 – – > 2 and the total distance is: 9
The path from 0 to 3 is: 0 – – > 2 – – > 3 and the total distance is: 20
The path from 0 to 4 is: 0 – – > 2 – – > 5 – – > 4 and the total distance is: 20
The path from 0 to 5 is: 0 – – > 2 – – > 5 and the total distance is: 11
Processed 2 graphs.
Average Time (seconds): 0.00
3 Evaluation Criteria
The programming assignment will be marked out of 40, based on a combination of
automated testing (using large graphs) and human inspection.
You are advised to implement Dijkstra’s algorithm. The running time of your code
should be at most O(V
2 + E log V ), where the V
2
is due to the cost of reading the
adjacency matrix. The mark for your submission will be based on both the asymptotic
worst-case running time and the ability of the algorithm to handle inputs of different
sizes.
Score Description
0 – 15 Submission does not compile or does not conform to the provided
template.
16 – 30 The implemented algorithm is substantially inaccurate on the
tested inputs.
31 – 40 The implemented algorithm is O(V
2 +E log V ) and gives the correct answer on all tested inputs.
To be properly tested, every submission must compile correctly as submitted and
must be based on the provided template. If your submission does not compile for any
reason (including even trivial mistakes like typos) or was not based on the template,
it will receive at most 15 out of 40. The best way to ensure your submission is correct