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Chapter 2: Q14E (page 85)
You are given an array of elements, and you notice that some of the elements are duplicates; that is, they appear more than once in the array. Show how to remove all duplicates from the array in time .
All duplicates are removed from the array in time:
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A linear, time-invariant system has the following impulse response:
(a) Describe in words the effect of this system.
(b) What is the corresponding polynomial
A way merge operation. Suppose you have sorted arrays, each with elements, and you want to combine them into a single sorted array of elements.
(a)Here’s one strategy: Using the merge procedure from Section 2.3, merge the first two arrays, then merge in the third, then merge in the fourth, and so on. What is the time complexity of this algorithm, in terms of and ?
(b) Give a more efficient solution to this problem, using divide-and-conquer.
Show that for any positive integers n and any base b , there must some power of b lying in the range .
This problem illustrates how to do the Fourier Transform (FT) in modular arithmetic, for example, modulo .(a) There is a number such that all the powers are distinct (modulo ). Find this role="math" localid="1659339882657" , and show that . (Interestingly, for any prime modulus there is such a number.)
(b) Using the matrix form of the FT, produce the transform of the sequence modulo 7; that is, multiply this vector by the matrix , for the value of you found earlier. In the matrix multiplication, all calculations should be performed modulo 7.
(c) Write down the matrix necessary to perform the inverse FT. Show that multiplying by this matrix returns the original sequence. (Again all arithmetic should be performed modulo 7.)
(d) Now show how to multiply the polynomials and using the FT modulo 7.
Suppose you are choosing between the following three algorithms: • Algorithm A solves problems by dividing them into five sub-problems of half the size, recursively solving each sub-problem, and then combining the solutions in linear time. •
Algorithm B solves problems of size n by recursively solving two sub-problems of size and then combining the solutions in constant time. • Algorithm C solves problems of size n by dividing them into nine sub-problems of size , recursively solving each sub-problem, and then combining the solutions in time.
What are the running times of each of these algorithms (in big- O notation), and which would you choose?
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