Data Structures and Algorithms: CHAPTER 9: Algorithm Analysis Techniques
Solve the following recurrences, where T(1) = 1 and T(n) for n ³ 2 satisfies:
a. T(n) = 4T(n/3) + n
9.3
b. T(n) = 4T(n/3) + n2 c. T(n) = 9T(n/3) + n2
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Give tight big-oh and big-omega bounds on T(n) defined by the |
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following recurrences. Assume T(1) = 1. |
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a. T(n) = T(n/2) + 1 |
9.4 |
b. T(n) = 2T(n/2) + logn |
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c. T(n) = 2T(n/2) + n |
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d. T(n) = 2T(n/2) + n2 |
Solve the following recurrences by guessing a solution and checking your answer.
a. |
T(1) |
= 2 |
*9.5 |
T(n) |
= 2T(n-1) + 1 for n ³ 2 |
b. T(1) |
= 1 |
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T(n) |
= 2T(n-1) + n for n ³ 2 |
9.6
Check your answers to Exercise 9.5 by solving the recurrences by repeated substitution.
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Data Structures and Algorithms: CHAPTER 9: Algorithm Analysis Techniques
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Generalize Exercise 9.6 by solving all recurrences of the form |
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9.7 |
T(1) = 1 |
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T(n) = aT(n-1) + d(n) |
for n ³ 1 |
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in terms of a and d(n). |
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*9.8 |
Suppose in Exercise 9.7 that d(n) = cn for some constant c ³ 1. How |
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does the solution to T(n) depend on the relationship between a and c. |
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What is T(n)? |
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Solve for T(n): |
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**9.9 |
T(1) = 1 |
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T(n) = Ö`nT(Ö`n) + n |
for n ³ 2 |
Find closed form expressions for the following sums.
9.10
Show that the number of different orders in which to multiply a sequence of n matrices is given by the recurrence
*9.11
Show that 
. The T(n)'s are called Catalan numbers.
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Data Structures and Algorithms: CHAPTER 9: Algorithm Analysis Techniques
Show that the number of comparisons required to sort n elements using mergesort is given by
T(1) = 0
T(n) = T([n/2]) + T([n/2]) + n - 1
**9.12
where [x] denotes the integer part of x and [x] denotes the smallest integer ³ x. Show that the solution to this recurrence is
T(n) = n[logn] - 2[logn] + 1
Show that the number of Boolean functions of n variables is given by the recurrence
9.13T(1) = 4
T(n) = (T(n - 1))2
Solve for T(n).
Show that the number of binary trees of height £ n is given by the recurrence
**9.14 |
T(1) = 1 |
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T(n) = (T(n - 1))2 + 1 |
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Show that T(n) = [k2n] for some constant k. What is the value of k? |
Bibliographic Notes
Bentley, Haken, and Saxe [1978], Greene and Knuth [1983], Liu [1968], and Lueker [1980] contain additional material on the solution of recurrences. Aho and Sloane [1973] show that many nonlinear recurrences of the form T(n) = (T(n-1))2 + g(n) have a solution of the form T(n) = [k2n] where k is a constant, as in Exercise 9.14.
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Data Structures and Algorithms: CHAPTER 9: Algorithm Analysis Techniques
† But don't hold out much hope for discovering a way to merge two sorted lists of n/2 elements in less than linear time; we couldn't even look at all the elements on the list in that case.
Table of Contents Go to Chapter 10
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