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Algorithm Design and Analysis

Final Assessment (Weightage: 50%)

Due Wednesday 16 December 2020

Submission Instructions

Create a single .zip archive containing the following components:

• For all Software development tasks, please include .java files in the correct

folder format. Please ensure code files do not contain any non-English

characters (such as comments in Chinese) because my AUT computer cannot

display these characters. Worse, these code files do not compile for

me.

• For demonstration videos, please include video files in .mp4 format, of no

more than 30 MB in size (for each video).

Please submit the .zip file on Blackboard→Assessment→Final Assessment.

Part 1

A persistent dynamic set is a set which maintains past versions of itself

as it gets updated. One simple implementation of a persistent dynamic set

would be to copy the entire data structure whenever it gets modified, but this

approach becomes Θ(n) and may require substantial memory, so instead the

data structure just keeps track of changes made.

The purpose of this assignment is to develop an efficient data structure for

a persistent dynamic set of Comparable elements.

The assignment should include the following components, each with some

suitable test examples:

Binary Search Tree which is a binary search tree with add, contains, and

remove methods. Each node contains a Comparable element, and links to

its left and right children (or null if it doesn’t have a child). Note that

each node will NOT have a link to its parent (to simplify effort later in the

assignment). It is suggested that the binary search tree class be arranged

to be extensible, and hook methods included that get called whenever a

node is visited during add or remove. Please include a suitable way to

visualise the tree. (10 marks)

1

Persistent Dynamic Set which is an implementation of a persistent dynamic

set for Comparable elements which utilises the binary search tree. When

an element is added or removed only the nodes that are affected from the

root down to that element are copied and links made to unaffected nodes

from the previous version of the binary search tree.

For example if “dog” is added to the tree on the left a second tree with four

new nodes is created on the right, including a new root node to represent

the updated version.

Please include some suitable way to visualise the different version of the

tree. (20 marks)

Balanced Persistent Dynamic Set which is a red-black tree subclass of the

binary search tree that keeps the tree balanced so the persistent dynamic

set is guaranteed to have O (log n) operations. It should utilise the tree’s

hook methods to avoid traversing the tree multiple times during rebalancing,

and care should be used during left and right rotations to correctly

update the version. The eight cases to be considered in the remove method

will be limited to a maximum four marks. (15 marks)

Demonstration video ( 3 minutes) showing the test cases and features of the

programs. (5 marks)

2

Part 2

A currency exchange graph is a graph whose n nodes represent different curriencies,

and whose directed edges represent exchange rates ruv between those

currencies (for example re$

≈ 1.78 for exchanging Euro to New Zealand dollars).

For convenience the edges can be weighted using wuv = log 1

ruv

(for example

we$

≈ −0.58), allowing the weight of adjacent edges to be added together to

find combined exchange rates, and shorter paths result in higher exchange rates.

The purpose of this assignment is to develop useful graph tools for currency

trading.

The assignment should include the following components, each with some

suitable test examples:

Best Conversion Finder which accepts an n × n connectivity table of exchange

rates (where 0 denotes no direct exchange rate between two currencies),

and can calculate the best conversion rate between any two specified

currencies (both ways), and how that rate can be obtained both ways as

sequences of exchanges. (10 marks)

Arbitrage Finder which accepts an n×n table of exchange rates and efficiently

finds whether there is arbitrage in the system, a closed loop of exchanges

that results in a profit. If so then all the occurrences of arbitrage, currencies

v0, v1, v2, . . . , vk−1 for which rv0v1

· rv1v2

· . . . · rvk−1v0 > 1 should be

found (allowing a currency trader to exploit a price differential to make a

profit). (20 marks)

Bridge Exchange Finder which accepts an n×n table of exchange rates and

forms an undirected (and unweighted) graph which has edges between

currencies if those two currencies can be directly exchanged. It then finds

any exchanges (called bridge edges in the graph) which if were unavailable

(its edge removed from graph) would mean some currencies could no longer

be traded with each other via a sequence of exchanges (the graph would

be disconnected).

The bridges can be found by first performing a depth first search of the

undirected graph, labelling each vertex u with a counter d(u) as the vertex

is discovered, and with a value m(u) as the search is finished with the

3

vertex. The value m(u) is taken to be the smallest value between d(u), the

values of m(v) found while visiting each adjacent currently black (child)

vertex v, and the value of d(v) for any adjacent currently grey (already

discovered) vertex v other than its parent u.

For example, if in the illustrated diagram a depth-first search starting at

a might visit the vertices in the following order:

v a b c d e f g h i j k l m

d(v) 0 1 2 3 4 5 6 7 8 9 10 11 12

and when finished with each vertex the following values of m(v) would be

calculated:

v a b c d e f g h i j k l m

m(v) 0 1 1 1 4 4 4 7 8 9 9 9 9

An edge between u and v is a bridge if and only if it is included in the

depth first tree from u to v and m(v) > d(u). (15 marks)

Demonstration video of the programs with sufficient test cases. (5 marks)

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