Avalg homework 2 fall 2002

This homework is due December 10 at 10.15. It can be delivered to Stefan or Isaac personally at their offices or put in their 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 and required. The size of a group of collaboration should be two people. 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 Thurday and Friday December 12-13. All members of the group must be present when the homework is returned. If your group talked to Stefan for homework 1 you should talk to Isaac this time and vice versa.

1. Suffix array implementation (100p)
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 for a text t. (40p)
  • Find the number of substrings of t that match a given string. (10p)
  • Find the length of the longest repeated substring in t. (20p)
  • Find the number of substrings of t that match a string which may contain one or more '*'-symbols. A '*' matches any string. (30p)
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.

Questions and answers

Q: When counting the number of strings matching a pattern, should we count the number of MATCHES or the number of MATCHING STRINGS?

Consider the string "abbc" and the pattern "a*b*c". There are two matches, b can match either the first or the second b, but there is only one string that matches, namely "abbc".

A: You should count the number of matching strings.

Q: Look at the pattern "ab" and the string "abab". What are the number of matching substrings?

A: 2.

Q: Look at the pattern "a*" and the string "abc". What are the number of matching substrings?

A: 3.

Q: I've found a solution with worst-case time complexity Omega(n*n), where n is the length of the string. Is that good enough?

A: For a full score you will need to find a better algorithm. It's not likely that a quadratic algorithm will meet the time bounds for strings containing 1,000,000 characters.

Q: We have a question about the "longest repeated substring". Does this mean the longest substring that occurs at least two times or does it mean the longest substring that repeats itself, as in "abcabc".

A: It's the longest substring that occurs at least twice.

Q: Look at the string "AABB" and the pattern "A*B". Can the string that matches * contain the characters 'A' and 'B'? Is the answer 1 or 4?

A: The '*' character matches _any_ string. There are 4 matching substrings in your example.

2. Fast sorting (100p)
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. (25p)
  • If the elements are small you may also sort in linear time. Give an algorithm and discuss how small the elements need to be. (25p)
  • You may sort faster if there are many repeated elements. How many distinct elements can you handle and how fast can you sort? (25p)
  • 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. (25p)

Stefan Nilsson