# hw-solns - Homework 1 - Solutions Data Structures and...

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Data Structures and Algorithms, EECS 281 January 23, 2008 Problem 1: a) To Prove: 3 n < n ! using Induction for n > 6 Basis Step: For n = 7, 3 7 < 7! Hypothesis step: Assume 3 k < k ! for k < n Induction step: Prove 3 k +1 < ( k + 1)! Proof: 3 k +1 = 3 k * 3 But since 3 k < k ! from the hypothesis and 3 < ( k + 1) for all k > 6 Since both factors are lesser 3 k +1 < ( k + 1)! Therefore by the principle of mathematical induction, 3 n > n ! for n > 6 b) 2 n o ( n !) since for any constant C, 2 n < Cn ! because there clearly exists a n 0 such that for all n > n 0 2 n n ! < C because 2 n n ! is a strictly decreasing function. Problem 2: a) Suppose f ( n ) = 18 log 2 n + 3 n , then f ( n ) Θ( n ) b) If g ( n ) = 2 n 2 - 5, then g ( n ) f ( n ) θ ( n ) Problem 3: The tail pointer could be of signiﬁcant value for a singly linked circular linked list if we want to insert elements at the end of the list or delete elements at the end. In this case

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## This note was uploaded on 04/11/2008 for the course EECS 281 taught by Professor Jag during the Winter '08 term at University of Michigan.

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hw-solns - Homework 1 - Solutions Data Structures and...

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