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Assignment 3: Shortest Paths
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 betweenallpairsofchargingstations. TogetMinchargedupfortherobotopera, Max needstocomputethese(single-source)shortestpathsefficientlypriortoventuringout. 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 yourimplementationinsidetheShortestPathsandPrintPathsfunctionsinthetemplate. 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: Reading graph 1 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 Reading graph 2 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 logV ), where the V 2 is due to the cost of reading the adjacencymatrix. Themarkforyoursubmissionwillbebasedonboththeasymptotic 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 logV ) and gives the correct answer on all tested inputs.
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 betweenallpairsofchargingstations. TogetMinchargedupfortherobotopera, Max needstocomputethese(single-source)shortestpathsefficientlypriortoventuringout. 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 yourimplementationinsidetheShortestPathsandPrintPathsfunctionsinthetemplate. 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: Reading graph 1 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 Reading graph 2 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 logV ), where the V 2 is due to the cost of reading the adjacencymatrix. Themarkforyoursubmissionwillbebasedonboththeasymptotic 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 logV ) and gives the correct answer on all tested inputs.
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