CSC165H1, Fall 2019 Problem Set 4
Please read the following instructions carefully before starting the problem set. They contain important
information about general problem set expectations, problem set submission instructions, and reminders
of course policies.
Your problem sets are graded on both correctness and clarity of communication. Solutions which
are technically correct but poorly written will not receive full marks. Please read over your solutions
carefully before submitting them. Proofs should have headers and bodies in the form described in
the course note.
Each problem set may be completed in groups of up to three. If you are working in a group for this
problem set, please consult https://github.com/MarkUsProject/Markus/wiki/Student_Groups
for a brief explanation of how to create a group on MarkUs.
Exception: Problem Sets 0 and 1 must be completed individually.
Solutions must be typeset electronically, and submitted as a PDF with the correct lename. Handwritten submissions will receive a grade of ZERO.
The required lename for this problem set is problem set4.pdf.
Problem sets must be submitted online through MarkUs. If you haven’t used MarkUs before, give
yourself plenty of time to gure it out, and ask for help if you need it! If you are working with a
partner, you must form a group on MarkUs, and make one submission per group. \I didn’t know
how to use MarkUs” is not a valid excuse for submitting late work.
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processing software like Microsoft Word; if it does, you should look into PDF compression tools to
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The work you submit for credit must be your own; you may not refer to or copy from the work of
other groups, or external sources like websites or textbooks. You may, however, refer to any text
from the Course Notes (or posted lecture notes), except when explicitly asked not to.
For questions in which you’re analysing algorithms, you may use (without proving them) all of the
theorems given in the Properties of Big-Oh, Omega, and Theta handout, as long as you clearly state
which theorems you are using.
1. [12 marks] algorithm analysis
Read over the code for has odd. Assume that number list contains entries from f0; 1; 2; 3; 4g, with duplicates
allowed. Answer the questions below, assuming that n = len(number list)
CSC165H1, Fall 2019 Problem Set 4
1 def has_odd(number_list) -> bool:
2 for i in range(len(number_list)):
3 if number_list[i] % 2 == 1:
4 return True
5 return False
(a) [3 marks] Find a good upper bound, U(n), for W Chas odd(n). Prove that your upper bound is correct.
(b) [3 marks] Find a lower bound, L(n), for W Chas odd(n) that is in the same asymptotic complexity
class as U(n) (that’s what I mean by \good” in the previous part). Prove that your lower bound is
correct, then state and justify a simple big-Theta complexity class for W Chas odd(n)
(c) [3 marks] If has odd returns True after examining k entries in number list, count this as k steps. If an
has odd examines all n entries in number list and proceeds to return False count this as n + 1 steps.
Using these assumptions, show how to calculate the average number of steps for all inputs to has odd
of length 2.
(d) [3 marks] Use the step-counting assumptions in the previous part to devise a formula for the average
number of steps for all inputs to has odd of length n. You may nd it useful to recall (where r is
some positive real number)
Show your work.
2. [18 marks] graph connectivity
Answer the questions below. Assume jV j is nite and positive.
(a) [3 marks] Prove that for all undirected graphs G = (V; E), if C is a cycle in G and e is an edge in
C, then removing e leaves C connected. Notice that this is used in Example 6.8, so you cannot use
Example 6.8 as proof, nor can you use the fact that this is asserted without proof in the paragraph
just before Example 6.8.
(b) [3 marks] Prove or disprove: In every undirected graph G = (V; E) with all vertices having degree at
least bjV j=3c, for every 3 distinct vertices u; v; w 2 V there is a path of length no more than 2 from
u to v, or from v to w, or from w to u.
(c) [3 marks] Prove or disprove: If graph G = (V; E) has an odd number of vertices with even degree,
then jV j is odd.
(d) [3 marks] Prove or disprove: Every undirected graph G = (V; E) with at least 13 vertices, all vertices
having degree at least jV j 7, is connected.
(e) [3 marks] Prove that every undirected graph G = (V; E) with every vertex having degree at least 5,
has a cycle.
(f) [3 marks] Prove or disprove: If G = (V; E) is an undirected graph where every vertex has degree at
least 4 and u 2 V , then there are at least 64 distinct paths in G that start at u.
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