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hw3-sol - CS 202 Data Structures and Discrete Mathematics...

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CS 202: Data Structures and Discrete Mathematics II Fall 2010 Homework 3 () Solutions 1. Recurrences (60 points) (a) Find an exact solution to the following recurrence: a n = a n - 1 + 2 a n - 2 + 3 n 2 - 16 n + 17 . 5 , with a 0 = 1 , a 1 = - 0 . 5 Solution: To solve recurrence a n = a n - 1 + 2 a n - 2 + 3 n 2 - 16 n + 17 . 5 The homogeneous annihilator is E 2 - E - 2 = ( E - 2)( E + 1). The residue is h- 6 n 2 + 38 n - 42 i which is annihilated by ( E - 1) 3 . The annihilator is ( E - 2)( E + 1)( E - 1) 3 , and the generic solution is a n = α 2 n + β ( - 1) n + γ + δn + n 2 . The constants α, β, γ, δ, satisfying the equations a 0 = 1 = α + β + γ a 1 = - 0 . 5 = 2 α - β + γ + δ + a 2 = - 1 = 4 α + β + γ + 2 δ + 4 a 3 = - 5 . 5 = 8 α - β + γ + 3 δ + 9 a 4 = - 6 = 16 α + β + γ + 4 δ + 16 are 1 , 3 4 , - 3 4 , 1 2 , - 3 2 respectively, The closed form of the recurrence is a n = 2 n + 3 4 ( - 1) n - 3 2 n 2 + n 2 - 3 4 . (b) Let a n be the number of additions performed in evaluating the determinant of an n × n matrix by the cofactor expansion method (expansion by minors) 1 . i. Write a recursive formula for a n . Justify briefly. Solution: a n = n · a n - 1 + n - 1 a 1 = 0 Explanation: Computing the determinant of an n × n matrix involves computing the de- terminants of n other matrices, each of size ( n - 1) × ( n - 1). Once those are computed we then add them all together. The number of additions performed to compute a determinant of an n × n matrix must be the number of additions to compute the determinant of an 1 See http://mathworld.wolfram.com/DeterminantExpansionbyMinors.html 3-1
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CS 202 Homework 3 ( n - 1) × ( n - 1) matrix multiplied by n because that’s how many ( n - 1) × ( n - 1) matrices there and then plus the n - 1 additions necessary to sum up all the determinants. The base case is set to n = 1 to represent the situation when the matrix is of size 1 × 1 which is really just a single scalar number requiring no computation work at all. ii. The solution to the recursion is a n = n ! - 1. iii. Prove that your formula correctly expresses the number of additions, a n . Solution: We’ll use a Proof by Induction to show that our recurrence relation in part i matches the exact solution in part ii . In other words we’ll show n · a n - 1 + n - 1 = n ! - 1 Base Case: n = 2 . For n = 2, we get: Left Hand Side (LHS) = 2 · 0 + 2 - 1 = 1 Right Hand Side (RHS) = 2! - 1 = 2 - 1 = 1 Both sides are equal which proves the base case. Inductive Hypothesis: Assume that the recurrence relation is true for all a k where 2 k < n . Inductive Step: Prove the recurrence relation is also correct for n . Proof: a n = n · a n - 1 + n - 1 by the definition of a n = n · (( n - 1)! - 1) + n - 1 by the inductive hypothesis = n ! - n + n - 1 = n ! - 1 This proves the recurrence relation from part i is equal to the exact equa- tion given in part ii for a n . End of proof: Since the base case and the inductive step hold, the state- ment is true for all n 2. (c) Weiss, 4th edition, Ch 7, Ex 7.17(e), 7.18(c,h) Solve the recurrences using both the recursion tree and the Master Theorem method.
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