2018 Pummill Relay problem statement

Transcription

1 2018 Pummill Relays CS Problem: Minimum Spanning Tree Missouri State University For information about the Pummill Relays CS Problem, please contact: Suppose there are N locations (such as cities), and we wish to build some physical pathway that connects all N locations for instance, a new high-speed internet cable that connects every city. For example, the diagram shows a set of N = 10 cities and the pathway that connects them. (source) The pathway, shown in thick lines, consists of a set of line segments. Each segment connects two cities using a straight line. The new pathway will not directly connect every pair of cities. Instead, a set of line segments will be chosen so that at least one connection exists at every city. There must be only one path between any two cities, but that path may require several line segments. The goal is to minimize the total length of the pathways while connecting all N locations. The numbers shown in the diagram are the length of each segment, totalling 38 units, which is the minimum for this example. This Pummill Relays Problem is to find the length of the Minimum Spanning Tree, and to give one set of line segments that give that length. (There might be ties of MST length: multiple sets of line segments that all give a valid tree of the same length.) This Pummill Relays Problem is provided in three parts of increasing level of challenge, and scoring will be based on the number of parts completed. You need not complete all three parts to enter and score. All teams are encouraged to consider this problem and solve as many parts as possible. Prim s Algorithm. See an animation of Prim s Algorithm. 1. Sort the edges by length, shortest to longest. 2. Choose the shortest edge. (If there is a tie for the shortest edge, select any one at random.) That edge is one of the segments of the Minimum Spanning Tree. Add the edge to a set of edges that are known to be in the MST. Name that set E. (An edge can be identified and stored by using the two endpoints.) 3. Add the two endpoints of that selected edge into a set of points, named P. The set P contains the points that are already known to be part of the Minimum Spanning Tree. 4. If the set P contains every point (location), you have found the complete Minimum Spanning Tree, contained in the set E. You re finished! 5. If you re not yet finished, select the next-shortest edge that has exactly one endpoint in P. (The edge you choose cannot have both endpoints in P.) With that edge, go back to step 3. The order that the edges are found will not necessarily also be in order of increasing edge length. The Minimum Spanning Tree for N points will always have N 1 edges. Page 1

2 Part 1. Find the Minimum Spanning Tree using a spreadsheet, and display the tree Use spreadsheet operations of sorting and plotting to display your Minimum Spanning Tree. Example. Each point (location) in the example graph has been labelled with a red identifying number. There are N = 10 points in the example graph. Refer to the provided spreadsheet, Provided Part 1 Example to observe the steps in finding a Minimum Spanning Tree a. b. 2 c. d. 4 e. f. g. 3 6 h. i. j. k. l. 5 m. n. a. The first spreadsheet tab, Edge Weight, has 21 data rows that correspond to the 21 edges in the test graph. Sort these rows (edges) by length, shortest to longest. Use Prim s Algorithm (previous page) to decide which edges are in the MST. b. The second spreadsheet tab, Point (x,y) is an (x, y) Cartesian coordinate for the location of each red-numbered point. c. The third spreadsheet tab, Plot of MSTree, plots the edges that you found for the Minimum Spanning Tree. i. This plot uses MS Excel Scatter Plot function, which can plot either single data points or a sequence of data points connected by line segments. Separate data with a blank line so that Excel Scatter Plot doesn t connect those points with a line segment. ii. The first line segment you found for the MST is the edge between point 8 and point 9, length 1. The Excel cells that plot that line segment are highlighted in blue in tab The Part 1 task is to find the Minimum Spanning Tree for the points and edges shown in the Excel spreadsheet Part 1 Problem two tabs As the filename indicates, the data in the Part 1 file has two tabs containing the data. Use Excel to plot the Minimum Spanning Tree, as demonstrated in the provided spreadsheet Provided Part 1 Example Turn in: a Word document, with the following clearly stated: o team members and school name o list of the edges in the MST o length of the MST o diagram of the MST (cut-and-paste from Excel Page 2

3 Part 2. Write a program to find the Minimum Spanning Tree. Write a computer program in either Java, C++, or Python to read problem inputs from a data file and produce output for the solution of finding a Minimum Spanning Tree. All data of this problem is numeric integers, and you may use arrays or lists to contain that data. Input to your program: A data file is used to specify the characteristics of this Minimum Spanning Tree problem. Example and test input files are provided on the Pummill Relays website. The input data file contains only a series of integer values described by the following form: N, the number of points for which an MST will be found. Those points will be numbered 1...N. E, the number of edges that will follow this... (until E edges have been listed) Sample input file TestMST.txt (see meanings of each number above): Output of your program: text output. Shown below is the expected form of the output of the example used in part 1. NOTES: There may be more than one valid, equal-length MST, with different sets of edges whose sum length is the same, because you may break ties of equal-length next-selected edges differently. White-space formatting in the output is not critical. Page 3

4 The identifying keyword OUTPUT should not appear in any other printed output lines. OUTPUT Minimum Spanning Tree length is 38 OUTPUT Edge 8-9, length 1 OUTPUT Edge 8-10, length 3 OUTPUT Edge 7-9, length 4 OUTPUT Edge 6-7, length 7 OUTPUT Edge 4-6, length 8 OUTPUT Edge 3-4, length 2 OUTPUT Edge 2-4, length 2 OUTPUT Edge 1-3, length 3 OUTPUT Edge 5-6, length 8 Notes on output: The identifying keyword OUTPUT will help to identify any genuine output from other print statements. Please use that keyword on no other printed output statement. In fact, we would appreciate a minimum of print statements in your submitted code. Turn in: your program source code Part 3. Write a program to find the Minimum Spanning Tree with specific conditions. Similarly to Part 2, write a computer program in either Java, C++, or Python to read problem inputs from a data file and produce output for the solution of finding a Minimum Spanning Tree --- but in this Part 3, the MST has specific conditions. Find an MST for the points of the input file such that: 1. The first edge chosen is the shortest edge between an odd-numbered point and an evennumbered point. 2. All other edges selected are between two odd-odd points or two even-even points. That is, the MST you find will be the union of the MST on the odd-numbered points, the MST on the even-numbered points, and the shortest edge having one odd-numbered endpoint and one evennumbered endpoint. All other instructions are the same as were shown in Part 2. Turn in: your program source code (Contest rules and guidelines are on next page) Page 4

5 Program rules and guidelines (see also submission rules below): 1. Your program must read from a text input file the data for the problems. The form of the input file is described above. 2. Your program must create output with the identifying keyword OUTPUT. Example output is shown above. Pummill relay programming contest hints: 1. Start small and work up. That is, code and test one small activity at a time, then move to the next activity. For instance, for an initial implementation of a recursive function, simply pass a parameter, demonstrate within the recursive routine that the parameter was correctly received, and return. 2. Partial credit will be awarded where it makes sense. 3. A snippet of code to read numeric values from a file is provided in two languages, Java and C Your program will be tested on other, larger, input files than those provided. Important: Of course you may use print statements to debug your code. However, please do not turn in code that causes massive amounts of output to the screen. Rules for submitting entries to the Pummill Relay Programming Contest 1. A team consists of at most three students from the same high school. There may be up to two teams per school. No student can be a member of more than one team. Each team must work independently. Each team may have only one entry (latest-dated entry will be used). 2. Submission of entries should be by of some generally legible format, with contents described above. If you have questions about formatting, please contact KenVollmar@missouristate.edu, There are separate parts to the 2018 Pummill Relays problem. Name your documents or programs to indicate the problem part that program solves. 4. All entries are welcome by , prior to 8:00am Monday Apr. 9, 2018 (the Monday prior to the Wednesday Pummill Relay competition). No entries can be accepted after that time. 5. Important: the form of the contest entry submission: Send entries by to KenVollmar@missouristate.edu, with a subject line that clearly indicates Pummill Relay. You should expect your s to be acknowledged within a few business hours, including a statement that the entry is in a legible format. Clearly show in the the school name, the team members' names, and the supervising teacher's name. Attach to the your source code for solving the Pummill Relay problem. Your source code will be compiled and tested on additional problems of the same format. The latest-dated submitted by a team will be the one used in the competition. 6. On the day of competition, the results of all entries will be displayed in a public area at Missouri State University. 7. Scoring and tiebreakers: number of parts correctly solved, earlier date, quality of source code and solution. Page 5