Chapter 8: Q30E (page 536)
Use Exercise 29 to show that if \(a = {b^d}\), then \(f(n)\) is \(O\left( {{n^d}\log n} \right)\).
The expression\(f(n) = O\left( {{n^d}\log n} \right)\)is proved.
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Let denote the length of a longest common subsequence ofand, whereand.
Use partsof Exercise 15to show thatsatisfies the recurrence relation,
If both andare nonzero and,
and
If both andare nonzero and, and the initial condition,
If
or
.
Give a big-O estimate for the size of f in Exercise \(1{20}\) if f is an increasing function.
Determine a value for the constant in Example 4 and use it to estimate the number of bit operations needed to multiply two 64-bit integers using the fast multiplication algorithm.
Find a closed form for the generating function for each of these sequences. (For each sequence, use the most obvious choice of a sequence that follows the pattern of the initial terms listed.)
a) \(0,2,2,2,2,2,2,0,0,0,0,0, \ldots \)
b) \(0,0,0,1,1,1,1,1,1, \ldots \)
c) \(0,1,0,0,1,0,0,1,0,0,1, \ldots \)
d) \(2,4,8,16,32,64,128,256, \ldots \)
e) \(\left( {\begin{array}{*{20}{l}}7\\0\end{array}} \right),\left( {\begin{array}{*{20}{l}}7\\1\end{array}} \right),\left( {\begin{array}{*{20}{l}}7\\2\end{array}} \right), \ldots ,\left( {\begin{array}{*{20}{l}}7\\7\end{array}} \right),0,0,0,0,0, \ldots \)
f) \(2, - 2,2, - 2,2, - 2,2, - 2, \ldots \)
g) \(1,1,0,1,1,1,1,1,1,1, \ldots \)
h) \(0,0,0,1,2,3,4, \ldots \)
Find the solutions of the simultaneous system of recurrence relations,
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