SUBSTRING SEARCH STUDY GUIDE

Terminology and Basics

• Substring search problem: Find an instance (or all instances) of a query substring of length M in a text string of length N.
• Sustring must be precisely defined (no regular expressions).
• You should be able to manually use a KMP DFA, and you should be able to manually carry out Boyers-Moore and Rabin Karp.

• How do you construct the DFA?
• How much time does it take if you re-simulate every time you have a mismatch? It's ok if you don't fully understand the linear time construction process.
• What is the best-case running time for DFA construction and DFA simulation? The worst-case running time?

• What is the mismatched character heuristic? Why do we use the rightmost character?
• Why is the mismatched character heuristic strictly suboptimal? Why do we use it then -- because the basic idea is very similar to KMP and you'll learn it if you ever really need to.
• What is the best-case running time? The worst-case running time?
• Which inputs result in best and worst case performance?

• If we know mod(ABCDEFG, R), how do we compute mod(BCDEFGH, R) in constant time (where A through H are arbitrary digits of a number from some alphabet of radix R)?
• What are the Las Vegas and Monte Carlo versions of Rabin-Karp?
• How would we extend Rabin-Karp to efficiently search for any one of P possible patterns in a text of length N? How would this technique compare to using KMP or Boyer-Moore for the same task?

Recommended Problems

C level

1. Fall 2012, #10 (Boyer-Moore)
2. (a) Given the following KMP DFA, give the string that this DFA searches for

   j	0	1	2	3	4	5	6
   A	1	1	3	1	5	1	5
   B	0	2	0	4	0	6	7

   (b) Below is a partially-completed KMP DFA for a string sof length 6 over the alphabet {a, B}. State 6 is the accept state. Fill in the missing spots in the table.

   j	0	1	2	3	4	5
   pat.charAt(j)
   A	1	1
   B			3			3

   (c) Given each of the following strings as input, what state would the DFA in (b) end in?
   BABAA
   ABABABA
   BABABABA
   BBAABBABAB Answers
   (a) ABABABB
   (b)

   j	0	1	2	3	4	5
   pat.charAt(j)	A	B	B	A	B	B
   A	1	1	1	4	1	1
   B	0	2	3	0	2	3

   (c) 1,5,5,4

B level

1. (KMP) Below is a partially-completed Knuth-Morris-Pratt DFA for a string s of length 11 over the alphabet { A , B }. Reconstruct the DFA and s in the space below.

         0 1 2 3 4 5 6 7 8 9 10
       A 0 0              10 11
       B       5   2          4
       s                   A
   

   Answers
   Substring search.

     0 1 2 3 4 5 6 7 8  9 10
   A 0 0 3 0 0 3 7 0 0 10 11
   B 1 2 2 4 5 6 2 8 9  6 4
   s B B A B B B A B B  A A
   

   The answer is unique.
2. Spring 2012 Final, #7 (KMP)
3. Fall 2012, #9 (KMP)
4. Give an example of when you might want to use KMP? Boyer Moore? Rabin Karp?

A level

1. For each algorithm (the version discussed in lecture and the textbook), give the worst-case order of growth in terms of M and N.
   ------ brute-force substring search for a query string of size M in a text string of size N
   ------ Knuth-Morris Pratt substring search for a query string of size M in a text string of size N
   ------ Boyer-Moore (with only mismatch heuristic) substring search for a query string of size M in a text string of size N
   ------ Monte Carlo version of Rabin-Karp substring search (that checks only for a hash match) for a query string of size M in a text string of size N
   ------ regular-expression pattern matching for a pattern of size M on a text string of size N
   ------ simulating a DFA with M vertices and 2M edges on a text string of size N
   ------ simulating an NFA with M vertices and 3M edges on a text string of size N Answers
   MN, N or M + N, MN, N or M + N, MN, N, MN
2. Give an example of when you might prefer to use the Monte Carlo version of Rabin Karp over the Las Vegas version.
3. Textbook: 5.3.22
4. Textbook: 5.3.26