# hw5(2) - CS 373 Combinatorial Algorithms Spring 2001...

This preview shows pages 1–3. Sign up to view the full content.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: CS 373: Combinatorial Algorithms, Spring 2001 http://www-courses.cs.uiuc.edu/~cs373 Homework 5 (due Tue. Apr. 17, 2001 at 11:59 pm) Name: Net ID: Alias: U 3 / 4 1 Name: Net ID: Alias: U 3 / 4 1 Name: Net ID: Alias: U 3 / 4 1 Neatly print your name(s), NetID(s), and the alias(es) you used for Homework 0 in the boxes above. Please also tell us whether you are an undergraduate, 3/4-unit grad student, or 1-unit grad student by circling U, 3 / 4 , or 1, respectively. Staple this sheet to the top of your homework. RequiredProblems 1. Prove that finding the second smallest of n elements takes EXACTLY n + ⌈ lg n ⌉ − 2 com- parisons in the worst case. Prove for both upper and lower bounds. Hint: find the (first) smallest using an elimination tournament. 2. Fibonacci strings are defined as follows: F 1 = “b” , F 2 = “a” , and F n = F n- 1 F n- 2 , ( n > 2) where the recursive rule uses concatenation of strings, so F 3 is “ab”, F 4 is “aba”. Note that the length of F n is the n th Fibonacci number. (a) Prove that in any Fibonacci string there are no two b’s adjacent and no three a’s. (b) Give the unoptimized and optimized ‘prefix’ (fail) function for F 7 . (c) Prove that, in searching for the Fibonacci string F k , the unoptimized KMP algorithm can shift ⌈ k/ 2 ⌉ times in a row trying to match the last character of the pattern. In other words, prove that there is a chain of failure links m → fail [ m ] → fail [ fail [ m ]] → ... of length ⌈ k/ 2 ⌉ , and find an example text T that would cause KMP to traverse this entire chain at a single text position. CS 373 Homework 5 (due 4/17/01)...
View Full Document

{[ snackBarMessage ]}

### Page1 / 4

hw5(2) - CS 373 Combinatorial Algorithms Spring 2001...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online