## Description

School of Computer Science

CompSci 220

There are three problems listed below. To get full credit for this assignment you need to complete all of them!

If you are stuck or confused by any of the problems, feel free to post on Piazza!

Show all working; if you do not show your work the mark will be reduced. You should submit via Canvas

a SINGLE PDF file containing the answers to the written questions. A scanned handwritten submission

is acceptable if and only if it is very neatly written. If typing the assignment, do the best you can with

mathematical symbols. For exponents, write something like

2^n

if using plain text. Use LaTeX if you really want it to look good.

Answers to programming questions must be submitted via the automated marker system:

automarker.cs.auckland.ac.nz

Best of luck, and enjoy the problems!

1. Apply DFS to the following graph and give the order of the vertices in which they get visited. If there are

several choices, choose the vertex with the smallest number.

(1 mark)

Apply BFS to the following graph and give the order of the vertices in which they get visited. If there are

several choices, choose the vertex with the smallest number.

(a) (1 mark)

CompSci 220 (2020 S2) Assignment 2 Page 1 of 4

(b) Draw two distinct graphs G1 and G2 on 7 vertices {a, b, c, d, e, f, g} such that (i) a DFS starting at a of

G1 visits the vertices of G1 in the same order as a BFS starting at a of G1, AND (ii) a DFS starting at a

of G2 visits the vertices of G2 in the same order as a BFS starting at a of G2.

Note. If DFS has several choices, it chooses the vertex that comes first in the alphabetical ordering. For

example, if c and e can both be visited next, DFS chooses c.

(2 marks)

(c) Consider the following digraph G.

Karlsruher Institut für Technologie Prof. Dr. Dennis Hofheinz

Institut für Theoretische Informatik Lukas Barth, Lisa Kohl

8. Übungsblatt zu Algorithmen I im SoSe 2016

https://crypto.iti.kit.edu/index.php?id=algo-sose16

{lisa.kohl,lukas.barth}@kit.edu

Musterlösungen

Aufgabe 1 (Breitensuche, 4 Punkte)

Gegeben sei folgender gerichteter Graph mit der Knotenmenge {A, B, C, D, E, F, G}:

B C

D E F

A

G

In diesem Graph werde nun eine Breitensuche durchgeführt, und zwar ausgehend vom Knoten A.

Wählen Sie die Knoten bei mehreren Möglichkeiten jeweils in alphabetischer Reihenfolge aus. (Am

Ende des Übungsblattes finden Sie weitere Kopien dieses Graphs zum anmalen. Wenn Sie diese nutzen,

kennzeichnen Sie bitte eindeutig, welche Kopie die Lösung zu welcher Teilaufgabe enthält.)

a) Zählen Sie die Knoten des Graphen in einer Reihenfolge auf, in der diese von einer Breitensuche

jeweils zum ersten Mal berührt werden. Heben Sie im Graph außerdem alle Kanten (z. B. farbig)

hervor, die bezüglich dieser Breitensuche tree-Kanten sind.

b) Sind im Graph bzgl. dieser Breitensuche irgendwelche backward-Kanten vorhanden? Wenn ja,

heben Sie diese hervor.

c) Sind im Graph bzgl. dieser Breitensuche irgendwelche cross-Kanten vorhanden? Wenn ja, heben

Sie diese hervor.

d) Sind im Graph bzgl. dieser Breitensuche irgendwelche forward-Kanten vorhanden? Wenn ja, heben

Sie diese hervor.

Musterlösung:

Die Knoten werden von einer Breitensuche in der Reihenfolge A, B, C, E, D, G, F besucht.

Im folgenden Graphen sind die tree- bzw. Baumkanten als normale Linien, die cross- bzw. Querkanten

als gestrichelte Linien und die einzige backward- bzw. Rückwärtskante als gepunktete Linie eingezeichnet.

Bei einer Breitensuche können keine forward- bzw. Vorwärtskanten auftreten.

1

i. List all vertices in G in the order they are visited when running a BFS search on G that starts at

A and picks nodes in alphabetical ordering when several choices are possible.

(1 mark)

ii. Considering the order with which the BFS search in (a) has visited the nodes of G, classify each arc

in G as a tree, forward, back, or cross arc.

(2 marks)

2. AVL trees

(6 marks)

CompSci 220 (2020 S2) Assignment 2 Page 2 of 4

For a given adjacency matrix of a rooted tree you need to decide whether this tree can be labeled to become

an avl tree. Vertices of the tree do not have keys but numbered from 0 to n − 1 in order to write the adjacency

matrix.

INPUT format: The input consists of blocks. Blocks are written one after another without empty lines

between them. Every block starts with a new line and consists of several lines:

• the first line contains a positive integer number n which corresponds to the number of nodes in a tree and

the root index;

• the next n lines correspond to the adjacency matrix of this tree.

OUTPUT format: The answer for every block should consist of the height of the tree and True/False (True

if the tree is an avl tree). Answers for different blocks should be located in different lines.

3. Page ranking

(7 marks)

You are asked to create a simplified version of the Google page ranking algorithm. You are given a map of links

between n websites which is represented by a digraph. Websites are nodes marked from 0 to n − 1. An arc from

i to j means that website i has a link to website j. Suppose you are searching for something and the result of

your search is these n websites. Usually the search engine outputs these websites in order of popularity. In this

problem, let the popularity index of a website be the number of websites which have a link to this website. So

the first website in the list is the one with the largest popularity index. The one after it is the the one with the

second largest popularity index, etc. If there are several websites with identical popularity index, then they are

ordered according to their numbers in the ascending order. You need to output the number of the kth website

in this list. Note that k is between 1 and n inclusive.

INPUT format: The input consists of blocks. Blocks are written one after another without empty lines

between them. Every block starts from a new line and consists of several lines:

• the first line contains a positive integer number n which corresponds to the number of websites in this

block and number k, 1 ≤ k ≤ n;

• the next n lines correspond to the adjacency list of the digraph.

For example, the block:

3 1

1 2

0

0 1

corresponds to the linking map graph

You need to find the first most popular website, which is 0.

OUTPUT format: For every block you need to output the kth most popular website. Answers for different

blocks should be located in different lines.

Questions involving programming

• Supported languages are Java, C,C++, PyPy, Python, Go, C#, Rust, F#, Javascript.

• Your answer to each question should be a single file (containing all nonstandard classes you use).

CompSci 220 (2020 S2) Assignment 2 Page 3 of 4

• A sample input and output file for each question will be available from Canvas. The markers may check

the output with a text comparison program, so it must be in EXACTLY the right format.

• The input is taken from stdin.

• You have an unlimited number of submissions.

• After the 10th submission there is a penalty of 25%.

• After the 20th submission there is a penalty of 50%. That is if you got the result correct only on your

21st try, then you get 50% of the mark. This adjustment will be made by instructors after the assignment

due date.

• If you were not able to pass the automarker’s test you still have a chance to get up to 25% for detailed

explanation of your algorithm. Markers will check your code for comments only if you have not passed

the automarker.

CompSci 220 (2020 S2) Assignment 2 Page 4 of 4