The.Algorithm.Design.Manual.Springer-Verlag.1998

Exercises
whose denominations are , C(5) = 1, C(6) to C(9)=2, C(10)=3, and C(12)=4.
1. How many ways are there to make change of 20 units from ?
2. Give an efficient algorithm to compute C(n), and analyze its complexity. (Hint: think in
terms of computing C(n,d), the number of ways to make change of n units with highest
denomination d. Be careful to avoid overcounting.)
8. (**) Consider the problem of examining a string of characters from an alphabet on
k symbols, and a multiplication table over this alphabet, and deciding whether or not it is possible
to parenthesize x in such a way that the value of the resulting expression is a, where a belongs to
the alphabet. The multiplication table is neither commutative or associative, so the order of
multiplication matters.
a b c
a a c c
b a a b
c c c c
For example, consider the following multiplication table and the string bbbba. Parenthesizing it
(b(bb))(ba) gives a, but ((((bb)b)b)a) gives c.
Give an algorithm, with time polynomial in n and k, to decide whether such a parenthesization
exists for a given string, multiplication table, and goal element.
9. (*) Consider the following data compression technique. We have a table of m text strings, each of
length at most k. We want to encode a data string D of length n using as few text strings as
possible. For example, if our table contains (a,ba,abab,b) and the data string is bababbaababa,
the best way to encode it is (b,abab,ba,abab,a) - a total of five code words. Give an O(nmk)
algorithm to find the length of the best encoding. You may assume that the string has an encoding
in terms of the table.
10. A company database consists of 10,000 sorted names, 40% of whom are known as good
customers and who together account for 60% of the accesses to the data base. There are two data
structure options to consider for representing the database:
❍ Put all the names in a single array and use binary search.
❍ Put the good customers in one array and the rest of them in a second array. Only if we do
not find the query name on a binary search of the first array do we do a binary search of
the second array.
Demonstrate which option gives better expected performance. Does this change if linear search on
an unsorted array is used instead of binary search for both options?
11. Suppose you are given an array A of n sorted numbers that has been circularly shifted k positions
to the right. For example, is a sorted array that has been circularly shifted k=2
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