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Guidelines and Grading Criteria for Project 3 and Homework Assignments in Graph Theory

This document outlines the grading criteria for Project 3 and Homework 4 in a Graph Theory course. It details expectations for program efficiency, correctness, and the handling of Dimacs graph files. Grading scales are described, including additional points for reasonable answers and deductions for inefficiencies. Homework tasks include constructing algorithms for minimum spanning trees and dynamic programming challenges. Students are encouraged to follow examples provided in class while implementing their solutions.

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Guidelines and Grading Criteria for Project 3 and Homework Assignments in Graph Theory

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  1. Lab 1012/1

  2. Project 2 Grades • Grades rescaled • +5 points • To a maximum of 105/100.

  3. Project 3 Grade Scale • 80: A program that compiles, runs, halts and returns a sensical answer. • Sensical: 0 < k < |V|+1 • +10 if your wrong answer is reasonable. • A 30-color coloring of a 20-colorable graph is reasonable. • A 50-color coloring might be too, if there are around 500 vertices. • A 100-color coloring probably wouldn’t be. • +10 if your answer is very close. • Consistantly off by 1-2 • But never *under* the correct answer. • Prof. Cheng’s program gives “very close” answers. • +10 if your answer is correct. • 110/100 • The minimum coloring • Not readily achievable with the assigned algorithm. • -5 points if your program is inefficient. • Takes an extremely long time to halt. • ANDI can find some particularly inefficient code to complain about.

  4. Project 3—Advice for Reading Dimacs Graph Files • Use Adjacency Matrix representation for the graph. • Read each line of the file... • Use a string tokenizer to break up the line into tokens. • Delimited by space ‘ ‘. • If the first token is “c”, ignore the line. • Otherwise if the first token is ‘p’ then the next token is |V| and the third token is |E|. • So make a |V| x |V| matrix gAdj of booleans. • Initialize all values to false.

  5. Project 3—Advice for Reading Dimacs Graph Files • If the first token is “e” then the line represents an edge. • The second token is u and the third token is v. • gAdj[u][v]=true; gAdj[v][u]=true;

  6. Homework 4 • Problem 1 • Hint: Given a set X={x1…xn} and a set of subsets Y={{xa…},{xb…},{xc…}} of that set… • Suppose {xa…} is the minimum member of Y. • Suppose we construct X1 and Y1 by squaring every xi. • Does that change the minimum member of Y? • The Shortest Path case is equivalent to the “Hint” problem. • Why? • The Minimum Spanning Tree case is not. • Why?

  7. Homework 4 • Problem 2 • Given G=(V,E) and T a minimum spanning tree of G. • Consider H=(V,E+{x}) • One edge ({x}) has been added. • Two steps: • Make an O(|E|) or O(|V|) algorithm to determine if T is also a minimum spanning tree of H. • Make an O(|E|) algorithm to construct a minimum spanning tree of H (in the case where T is not such a tree).

  8. Homework 4 • Problem 3 • Dynamic programming • You are given the recursion relation. • C(n,k) = C(n-1,k-1) + C(n-1, k) • Follow the examples given in class.

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