I want an Oompa-Loompa! screamed Veruca. Roald Dahl, Charlie and the Chocolate Factory

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1 Greedy and Basic Graph Algorithms CS Algorithms Spring 2019 Apr 5, 2019 Name: Collaborators: I want an Oompa-Loompa! screamed Veruca. Roald Dahl, Charlie and the Chocolate Factory This problem set covers material in Chapter 16 and 22 of the textbook, Introduction to Algorithms, Third Edition (CLRS). To complete your homework, you may ONLY consult the following materials: Lecture slides posted on the class website. Course notes you or others took during lecture. The required text. Websites that may clarify the concepts covered in the material but do not in any way provide solutions to the problems Problem Points Score Total: 25 Remember, your goal is to communicate. Full credit will be given only to correct solutions which are described clearly. Convoluted and obtuse descriptions will receive low marks. Collaboration on the problem sets is not only allowed, it s encouraged. It s proven that students who work on the problem sets in small groups perform better in the course than those that don t. That said, I am well aware that there is material relevant to solving problems on the homework and exams that is readily available. You may not seek out, study from, or otherwise consult these types of materials to solve these problems.

2 1. (5 points) Change-Making Problem. Given a list of coin denominations d 1 > d 2 >... > d m and a specific amount n, list the minimum number of coins of each denomination that add up to n. (a) (4 points) Write a greedy, pseudo-code algorithm that solves the change-making problem. Your algorithm should return the number of each denomination in the solution or the statement no solution if no solution exists. Your algorithm should also work with any set of denominations. What is the time efficiency class of your algorithm? 2

3 (b) (1 point) Give an instance (i.e. a collection of denominations) for which your greedy algorithm will not yield an optimal solution. 3

4 2. (10 points) Huffman Coding. (a) (8 points) Determine a variable-length, prefix-free encoding for the characters in the following string using the Huffman Algorithm: Imagine an imaginary menagerie manager managing an imaginary menagerie. Capitalization does matter, and don t forget to count the spaces and punctuation. Be sure to show all of your work. 4

5 (b) (2 points) What is the maximal possible length of a codeword in a Huffman encoding of an alphabet with n characters? Justify your answer. 5

6 3. (10 points) Basic Graph Search. (a) (5 points) Use the following depth-first search (DFS) algorithm to traverse the above graph starting at vertex D. Assume adjacent vertices are ordered alphabetically. // Uses DFS to traverses given graph G starting at vertex s. // Each vertex has a tri-value attribute called color (white, gray, black) // input: a graph G = <V, E>, a start vertex s DFS(G, s) for all v in V v.color <- white S <- empty stack s.color <- gray S.push(s) while!s.empty u <- S.pop u.color <- black for all w Adj[u] if w.color = white w.color <- gray S.push(w) Show the state of the stack before the first iteration of the while loop, and after each iteration of the loop, and give the order in which the vertices are visited for the first time (i.e. marked as gray and pushed onto the stack), and the order in which they become dead ends (i.e. marked as black and popped off the stack). 6

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8 (b) (5 points) Use the following breadth-first search (BFS) algorithm to traverse the above graph starting at vertex D. Assume adjacent vertices are ordered alphabetically. // Uses BFS to traverse given graph G starting at vertex s. // Each vertex has a tri-value attribute called color (white, gray, black) // input: a graph G = <V, E>, a start vertex s BFS(G, s) for all v in V v.color <- white Q <- empty queue s.color <- gray Q.enqueue(s) while!q.empty u <- Q.dequeue u.color <- black for all w Adj[u] if w.color = white w.color <- gray Q.enqueue(w) Show the state of the queue before the first iteration of the while loop, and after each iteration of the loop, and give the order in which the vertices are removed from the queue (i.e. marked as black and dequeued). 8

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10 4. (5 points) Extra Credit. Optimal Merge Pattern. Suppose you re given n files: f 1, f 2,..., f n with r i records in each file f i. If you can only merge two files at time and each merge has a cost equal to the number of records in the two files, what is the minimum cost to merge all n files? For example, consider three files f 1, f 2, and f 3 containing 10, 15, and 20 records respectively. What s the minimum cost to merge these three files? Order 1: Assume that f 1 and f 2 are merged first, then the resultant file is merged with f 3. The cost of merging f 1 and f 2 is = 25, and the merge cost of this file with f 3 is = 45. So, the total cost of merging the files is = 70. Order 2: In this case, we first merge f 2 and f 3 with a merging cost of = 35. Then, we merge f 1 with the resultant file at a cost of = 45. So the total merging cost of = 80. Order 3: In this case, assume that f 1 and f 3 are merged first, and then the resultant file is merged with f 2. The cost of merging f 1 and f 3 is = 30, and the cost of merging f 2 with the resultant file is = 45. So the total merging cost is = 75. Given these files, Order 1 has the minimum cost. Design a greedy algorithm in pseudo-code that solves the Optimal Merge Pattern problem, and apply it to these five files: f 1, f 2, f 3, f 4, and f 5 with 5, 10, 15, 20, and 25 records respectively. 10

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