COMP582 Module 11 Problems

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1. [12 pts] Given input alphabet {A, B, C, D}, construct the DFA for the string
A B C D A B D
NOTE: The spaces in the above string and aphabet are for readability.
The spaces are not part of the pattern or alphabet.
Show a trace of the KMP algorithm and your DFA on the target string
A B C A B C D A B C D A B D
NOTE: Again, the spaces are for readability.
2. [12 pts] Regular expressions (or regexs) are often extended with
convenience operations.
Assuming the alphabet is the lower case letters a,b,c,d,e; write the regular
expression for the following convenience operations:
2.1 [3pts] Wildcard (usually the ‘.’ character): will match any character in the
alphabet
2.2 [3pts] Kleene +: Like *, but the regular expression must have at least 1
instance of the root regex
Example: (ac)+ matches “ac”, “acac”
(ac)+ does not match “” (the empty string)
2.3 [3pts] bounded closure: Match a finite set of concatenations.
Example: (ab){3,5} matches “ababab” or “ababababab”
(ab){3,5} does not match “ab” or “abab” or “abababababab”
2.4 [3pts] character range:
Example: a[b-d] matches ab, ac, or ad
3. [18 pts] Given the regex over the alphabet {A,B,C,D,E,F,G}
((A | B)* | C D* | E F G )*
NOTE: The notation for the regex above uses spaces for readability.
The space character is not part of the pattern
3.1 [9 pts] Construct the NFA state machine and digraph of etranisitons for the given regex
NOTE: The above regex *may not be* fully parenthesized.
If needed, you may add enough parentheses make the regex
fully parenthesized.
3.2 [9 pts] Show the full set of state transitions when your NFA
recognizer is applied to
A B B A C E F G E F G C A A B
NOTE: The spaces in the above string are for readability.
The space character is not part of the source string.
4. [14 pts] A contiguous subsequence of a list S is a subsequence made up of
consecutive elements of S.
For instance, if S is
5, 15, −30, 10, −5, 40, 10
then
15, −30, 10
is a contiguous subsequence.
The subsequence
5, 15, 40
is not contiguous.
Give a linear-time algorithm for the following task:
Input: A list of numbers, a1,a2,…,an
Output: A contiguous subsequence of maximum sum.
NOTE: A subsequence of length 0- has sum 0.
For the preceding example sequence,the answer would be
10, −5, 40, 10
with a sum of 55.
Hint: For each j ∈ {1, 2, . . . , n}, consider contiguous subsequences ending
exactly at position j.
5. [14 pts] You are given a string of n characters s[1 . . . n], which you believe
to be a corrupted text document in which all punctuation and whitespace has
vanished. A sample input is:
itwasthebestoftimesitwastheworst
You wish to reconstruct the document using a dictionary. The dictionary is
available in the form of a Boolean function
dict(w):
for any string w, dict(w) = true if w is a valid word in the dictionary
dict(w) returns false otherwise.
5.1 [9 pts] Give a dynamic programming algorithm that determines whether
the
string s can be reconstituted as a string of valid words.
The running tine should be no worse that O(|s|2), assuming a call to the
dict function takes O(1).
5.2 [5 pts] In the event that the string is valid, make your algorithm
output the corresponding sequence of words.
6. [16 pts] Best Matrix Multiply Order:
Let
A1A2A3A4…An
be a sequence of matrices that must be multiplied together.
Matrix Multiply is associative, so you may parenthesize the multiplication in
whatever way works best.
Some reminders: An m x p matrix can be multiplied by a p x q matrix giving an
m x q matrix. The cost of multiplying the 2 matrices is m * p * q.
Example: Given the sequence
A1A2A3
where A1=10×10, A2=10×10, A3=10×1.
Choosing (A1 * A2) * A3 gives:
cost of A1×A2=10×10×10=1000
cost of(A1×A2)×A3=10×10×1
total cost = (1000) + 100 = 1100
On the other hand, choosing A1×(A2×A3)
cost of A2×A3=10×10×1=100
cost of A1×(A2×A3)=10×10×1=100
total cost = 100 + 100 = 200
So, 2nd choice is clearly best.
Your problem: Given a sequence of matrices and their dimensions, construct
an algorithm that gives the order of matrix multiplies
that yields the lowest number of operations.
7. [14 pts] Minimum Edit Distance:
You have at your disposal 3 character editing operations:
1. d: delete a char
2. i: insert a char
3. c: change 1 char into another char
You are given 2 strings s,t.
Your problem is to always find the minimum # of operations to turn string s
into string t.
Example: s = ‘cast’, t=’cats’
Option 1: c s->t [pos 3]; c t->s [pos 4] has 2 operations
Option 2: d a [pos 2]; c s a [pos 2];i s at the end. This has 3 operations.
Option 1 is clearly better option 2.