Assignment-3 Solution

Description

Question 1. [16 marks]

Given a list L, a contiguous sublist M of L is a sublist of L whose elements occur in immediate succession in L. For instance, [4; 7; 2] is a contiguous sublist of [0; 4; 7; 2; 4] but [4; 7; 2] is not a contiguous sublist of [0; 4; 7; 1; 2; 4].

We consider the problem of computing, for a list of integers L, a contiguous sublist M of L with maximum possible sum.

Algorithm 1 M axSublist(L)

<precondition>: L is a list of integers.

<postcondition>: Return a contiguous sublist of L with maximum possible sum.

Part (1) [5 marks]

Using a divide-and-conquer approach, devise a recursive algorithm which meets the requirements of M axSublist.

Part (2) [8 marks]

Give a complete proof of correctness for your algorithm. If you use an iterative subprocess, prove the correctness of this also.

Part (3) [3 marks]

Analyze the running time of your algorithm.

Question 2. [18 marks]

For a point x 2 Q and a closed interval I = [a; b], a; b 2 Q, we say that I covers x if a x b. Given a set of points S = fx₁; : : : ; x_ng and a set of closed intervals Y = fI₁; : : : ; I_kg we say that Y covers S if every point x_i in S is covered by some interval I_j in Y .

In the “Interval Point Cover” problem, we are given a set of points S and a set of closed intervals Y . The goal is to produce a minimum-size subset Y ^′ Y such that Y ^′ covers S.

Consider the following greedy strategy for the problem.

CSC236: Introduction to the Theory of Computation Due:

Algorithm 2 Cover(S; Y )

<precondition>:

S is a finite collection of points in Q. Y is finite set of closed intervals which covers S.

<postcondition>:

Return a subset Z of Y such that Z is the smallest subset of Y which covers S.

1. L = fx₁; : : : ; x_ng S sorted in nondecreasing order

2. Z ∅

3. i 0

4. while i < n do

5. if x_i+1 is not covered by some interval in Z then

6. I interval [a; b] in Y which maximizes b subject to [a; b] containing x_i+1

7. Z:append(I)

8. i i + 1

9. return Z

Give a complete proof of correctness for Cover subject to its precondition and postcondition.

Question 3. [10 marks]

The first three parts of this question deals with properties of regular expressions (this is question 4 from section 7.7 of Vassos’ textbook). Two regular expressions R and S are equivalent, written R S if their underlying language is the same i.e. L(R) = L(S). Let R; S, and T be arbitrary regular expression. For each assertion, state whether it is true or false and justify your answer.

Part (1) [2 marks]

If RS SR then R S.

Part (2) [2 marks]

If RS RT and R ̸ ∅then S T .

Part (3) [2 marks]

(RS+R) R R(SR+R) :

Part (4) [4 marks]

Prove or disprove the following statement: for every regular expression R, there exists a FA M such that L(R) = L(M ). Note: even if you find the proof of this somewhere else, please try to write up the proof in your own words. Just citing the proof is NOT enough.

Question 4. [16 marks]

Let a block be a maximal sequence of identical characters in a finite string. For example, the string 0010101111 can be broken up into blocks: 00, 1, 0, 1, 0, 1111. Let L₂ = fx 2 f0; 1g : x only contains blocks of length at least threeg.

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