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Discrete Mathematics and its Applications

Kenneth H. Rosen

$Math Studyset Vaia Explanations$ Math

7th Edition · 808 pages

ISBN: 9780073383095

Chapter 8: Q15E (page 567)

How many rounds are in the elimination tournament described in Exercise \(14\) when there are \(32\) teams?

Short Answer

Expert verified

Therefore, the number of rounds required is\(5\).

Step by step solution

01

Recurrence Relation definition

A recurrence relation is an equation that recursively defines a sequence where the next term is a function of the previous terms:\({\rm{f(n) = a f(n / b) + c}}\)

02

Apply Recurrence Relation

Recurrence relation found in the previous exercise:

\(f(n) = f(n/2) + 1\,{\rm{and}}\,f(1) = 0\)

Repeatedly apply the recurrence relation with\(n = 32\):

\(\begin{array}{c}f(32) = f(16) + 1\\f(32) = f(8) + 2\\f(32) = f(4) + 3\\f(32) = f(2) + 4\\f(32) = f(1) + 5\\f(32) = 0 + 5\;\;\;\\f(32) = 5\end{array}\)

Therefore, the number of rounds required is\(5\).

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Most popular questions from this chapter

Find the coefficient of \({x^9}\) in the power series of each of these functions.

a) \({\left( {1 + {x^3} + {x^6} + {x^9} + \cdots } \right)^3}\)

b) \({\left( {{x^2} + {x^3} + {x^4} + {x^5} + {x^6} + \cdots } \right)^3}\)

c) \(\left( {{x^3} + {x^5} + {x^6}} \right)\left( {{x^3} + {x^4}} \right)\left( {x + {x^2} + {x^3} + {x^4} + \cdots } \right)\)

d) \(\left( {x + {x^4} + {x^7} + {x^{10}} + \cdots } \right)\left( {{x^2} + {x^4} + {x^6} + {x^8} + } \right.\)\( \cdots )\)

e) \({\left( {1 + x + {x^2}} \right)^3}\)

(a) Show that if $n$ is a positive integer, then $(\begin{matrix} \frac{- 1}{2} \\ n \end{matrix}) = \frac{(\begin{matrix} 2 n \\ n \end{matrix})}{{(- 4)}^{n}}$

(b) Use the extended binomial theorem and part (a) to show that the coefficient of $x^{n}$ in the expansion of $(1 - 4 x)^{- 1 / 2}$ is $[\begin{matrix} 2 n \\ n \end{matrix}]$ for all nonnegative integers $n$

In this exercise we construct a dynamic programming algorithm for solving the problem of finding a subset S of items chosen from a set of n items where item i has a weight , which is a positive integer, so that the total weight of the items in S is a maximum but does no exceed a fixed weight limit W. Let $M (j, w)$ denote the maximum total weight of the items in a subset of the first j items such that this total weight does not exceed w. This problem is known as the knapsack problem.

a) Show that if $w_{j} > w$ , then $M (j, w) = M (j - 1, w) .$
b) Show that if $w_{j} \leq w$ , then $M (j, w) = m a x (M (j - 1, w), w_{j} + M (j - 1, w - w_{j}))$ .
c) Use (a) and (b) to construct a dynamic programming algorithm for determining the maximum total weight of items so that this total weight does not exceed W. In your algorithm store the values $M (j, w)$ as they are found.
d) Explain how you can use the values $M (j, w)$ computed by the algorithm in part (c) to find a subset of items with maximum total weight not exceeding W.

Show that if $a \neq b^{d}$ and $n$ is a power of $b$ , then $f (n) = C_{1} n^{d} + C_{2} n^{\log_{b^{a}}}$ , where $C_{1 =} b^{d} c / (b^{d} - a)$ and $C_{2} = f (1) + b^{d} / (a - b^{d})$

Use generating functions to solve the recurrence relation $a_{k} = a_{k - 1} + 2 a_{k - 2} + 2^{k}$ with initial conditions $a_{0} = 4$ and $a_{1} = 12$ .

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