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# hw4 - (a A simple path in T is one where all the nodes are...

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Fall 10: CSci 5421—Advanced Algorithms and Data Structures Out 11/3 Homework 4 Due 11/17 Please do all problems; we will grade a subset of four problems. Any Exercise/Problem numbers refer to the 3rd edition of the text. Please follow all of the instructions given in the handout for Homework 1. 1. (14 points) Let M = ( S, I ) be a matroid. Let H be any subset of S and define I = { C | C ∈ I and C H = ∅} . Prove that M = ( S, I ) is a matroid. 2. (14 points) Prove that the set system M = ( S , I ) defined in Lemma 16.10 is indeed a matroid. (The book assumes this to be the case implicitly.) 3. (14 points) Problem 12-1, p. 303–304. 4. (14 points) In class, we proved inductively that the height, h , of a red-black tree, T , with n internal nodes is at most 2 log( n + 1). We now explore a different approach to proving this.
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Unformatted text preview: (a) A simple path in T is one where all the nodes are distinct and its length is the number of edges on it. The height , h , of T is the length of a longest simple path from the root to a descendant external node (leaf). Let s be the length of a shortest simple path from the root to a descendant external node. Prove that h/s ≤ 2. (b) Let n ′ be the number of internal nodes in T that are at distance less than s from the root (i.e., at most s-1 edges away from the root). Derive an expression for n ′ . (c) Use the results in parts (a) and (b) to obtain the desired upper bound of 2log( n + 1) for h . 5. (14 points) Ex. 13.2-4, p. 314. 6. (16 points) Problem 13-2, p. 332-333....
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