Algorithms: Union-Find, Huffman Coding and Minimum Spanning Tree, Assignments of Algorithms and Programming

Download Algorithms: Union-Find, Huffman Coding and Minimum Spanning Tree and more Assignments Algorithms and Programming in PDF only on Docsity! Head-Banging Session 5 Fall 2008 CS 473: Algorithms Problem 1. [Saving with Union-Find] Show the Union-Find forest that results from the following sequence of operations on the ten singleton elements a, b, . . . , j. Do this exercise two ways: with and without path compression. On which steps does path compression save running time? UNION(FIND(a), FIND(e)) UNION(FIND(c), FIND(b)) UNION(FIND(b), FIND(h)) FIND(e) UNION(FIND(f), FIND(e)) UNION(FIND(j), FIND(g)) UNION(FIND(g), FIND(b)) UNION(FIND(i), FIND(c)) FIND(j) UNION(FIND(e), FIND(d)) UNION(FIND(a), FIND(c)) FIND(g) Problem 2. [Practice With Huffman] The following table contains Huffman code for the given nine-letter alphabet. Letter — Code Word O — 1111100 D — 1111101 M — 11110 R — 11101 S — 11100 A — 100 I — 110 T — 101 E — 0 Build the Huffman coding tree that corresponds to the above code. Is the given encoding an optimal encoding? If not, how could you improve it? What are possible frequencies for the letters in this alphabet that would result in the optimal tree that you have drawn? 1