Avalg homework 2 fall 2000

This homework is due Monday December 4 at 15.00. It can be delivered to Stefan personally at his office or put in his mail slot at the department. It can also be given to Stefan in connection with one of the lectures. Solutions handed in late are not accepted and will not be graded.

Some forms of collaboration are allowed. The size of a group of collaboration is limited to three. The group should hand in only one solution and for each problem it should be clearly marked which of the members have contributed. The homework will be returned on Wednesday December 6. I will be in my room from 9-12 and 13-15 and the groups are expected to arrive at suitable moments in time. To minimize the waiting time, you should arrive when the queue is short. (You may, for example, use a randomized algorithm.) All members of the group must be present when the homework is returned.

I do not consider the set of problems below easy. Thus a performance of getting half the total score on this set is at least equivalent to the grade 4 on this subset of the course. Credit may be given for partially solved problems.

1. Suffix array implementation (90p)
A suffix array is a simple data structure that facilitates searching in a text. It consists of a sorted array of pointers. There is one pointer for each position in the text. The pointer array is sorted according to the suffixes to which the pointers refer. For example, the text "abba0" (0 is a special end of string character) has the suffix array [3, 0, 2, 1], where the pointers refer to the suffixes "a0", abba0", "ba0", and "bba0", respectively.

  • Build a suffix array. (40p)
  • Find the number of occurrences of a particular string. (10p)
  • Find the length of the longest repeated substring. (20p)
  • Find the number of occurrences of a string which may contain one or more *-symbols. A * matches any string. (20p)
You should give a clear description of the algorithms and an estimate of the asymptotic worst-case performance for each of the four subtasks.
     This is a programming exercise and the same quality standards as in Homework 1 apply. You may assume that the text is stored on file and consists of ASCII characters (excluding 0 and '*'). The program should contain a simple user interface for the queries and the time to execute each command should be reported to the user. The text contains at most 1,000,000 characters. Your goal is to build the suffix array within one minute and the searches should take no more than 10 seconds. You do not know what the text files and queries will look like.

2. Fast sorting (80p)
In the lectures you've seen how to sort n word-sized integers on a unit-cost RAM in O(n log log n) time. In this exercise you will study some special cases where it's possible to find easier algorithms or better time bounds.

  • If there are many elements - e.g. if n is close to 2^w, where w is the word length - you may sort in linear time. Give an algorithm and discuss how large n needs to be. (20p)
  • If the elements are small you may also sort in linear time. Give an algorithm and discuss how small the elements need to be. (20p)
  • You may sort faster if there are many repeated elements. How many distinct elements can you handle and how fast can you sort? (20p)
  • The algorithm discussed in class uses large amounts of memory. How much? By doing the radix sorting phase in more than two steps you may reduce the memory requirements while increasing the running time. Explain how to do this and give formulas describing the time-space tradeoff. (20p)

3. Linear programming (30p)
Give an example of a problem that can be solved with linear programming using a handful of inequalities. Show how to solve your problem using the simplex method.

Stefan Nilsson