## Description

## Objectives

The assignment aims to give you more independent, self-directed practice with

advanced data structures, especially graphs

graph algorithms

asymptotic runtime analysis

Admin

## Background

A partially ordered set (“poset”) is a set S together with a partial order ≼ on the elements from S.

A partial order graph for a finite poset (S,≼) is a directed graph (“digraph”) with

the elements in S as vertices

a directional edge from s to t if, and only if, s ≼ t and s ≠ t

Example:

where

S = {1, 11, 13, 143}

s ≼ t iff s is a divisor of t

A monotonically increasing sequence of length k over a poset (S,≼) is a sequence of elements from S,

s1 ≺ s2 ≺ … sk-1 ≺ sk

such that si ≼ si+1 and si ≠ si+1, for all i=1…k-1. Examples:

1 ≺ 11 ≺ 143 and 1 ≺ 13 ≺ 143 are monotonically increasing sequences of length 3 over the poset from

above.

1 ≺ 143 121 is a monotonically increasing sequence of length 2 over this poset.

## Aim

Your task is to write a program poG.c for computing a partial order graph from a given specification and then

find and output all longest monotonically increasing sequences that can be constructed over this poset.

Your program should:

accept a single positive number p on the command line;

compute the set Sp of all (positive) divisors of p;

## Task A:

build and output the partial order graph over Sp corresponding to a specific partial order (see

below);

## Task B:

output all longest monotonically increasing sequences over this partial order.

Your program should include a time complexity analysis, in Big-Oh notation, for

1. your implementation for Task A, depending on the number n of divisors of p and the length m of the

decimal p;

2. your implementation for Task B, depending on the number n of divisors of p.

## Hints

You may assume that

the command line argument is correct (a number p ≥1);

p is at most 2,147,483,647 (the maximum 4-byte int);

p will have no more than 1000 divisors.

If you find any of the following ADTs from the lectures useful, then you can, and indeed are encouraged to,

use them with your program:

stack ADT : stack.h, stack.c

queue ADT : queue.h, queue.c

graph ADT : Graph.h, Graph.c

prompt$ ./poG 121

Partial order:

1: 11 121

11: 121

121:

Longest monotonically increasing

sequences:

1 < 11 < 121

prompt$ ./poG 9481

Partial order:

1: 19 9481

19: 9481

499: 9481

9481:

Longest monotonically increasing

sequences:

1 < 19 < 9481

weighted graph ADT : WGraph.h, WGraph.c

You are free to modify any of the four ADTs for the purpose of the assignment (but without changing the file

names). If your program is using one or more of these ADTs, you should submit both the header and

implementation file, even if you have not changed them.

Your main program file should start with a comment: /* … */ that contains the time complexity of your

solutions for Task A and Task B, together with an explanation.

## Stage 1 (2 marks)

For stage 1, you should demonstrate that you can build the underlying graph correctly from all divisors of

input p and the following partial order:

x ≼ y iff x is a divisor of y

All you need to do for Task B at this stage is to output all nodes of the graph in ascending order.

Here is an example to show the desired behaviour of your program for a stage 1 test:

Hint: The only tests for this stage will be with numbers p for which the above order is identical to the stricter partial order for stages 2–4, and such

that all of the divisors of p together form a monotonically increasing sequence.

## Stage 2 (3 marks)

For stage 2, you should extend your program for stage 1 such that it puts an additional constraint on the

partial order of the divisors of p:

x ≼ y iff

x is a divisor of y, and

all digits in x also occur in y

For example, 11 ≺ 143 since 1 is also contained in 143, but 16 ⊀ 128 since 6 is not a digit in 128.

All you need to do for Task B at this stage is to find and oputput the path that starts in 1 and always selects

the next neighbour in ascending order until you reach a node without outgoing edge.

Here is an example to show the desired behaviour of your program for a stage 2 test:

prompt$ ./poG 125

Partial order:

1: 125

5: 25 125

25: 125

125:

Longest monotonically increasing

sequences:

5 < 25 < 125

prompt$ ./poG 143

Partial order:

1: 11 13 143

11: 143

13: 143

143:

Longest monotonically increasing

sequences:

1 < 11 < 143

1 < 13 < 143

Hint: All tests for this stage will be such that the only longest monotonically increasing sequence is the unique path from 1 to p obtained by always

moving to the next neighbour in ascending order.

## Stage 3 (3 marks)

For stage 3, you should demonstrate that you can find a single longest monotonically increasing sequence.

All tests for this stage will be such that there is a unique longest sequence.

Here is an example to show the desired behaviour of your program for a stage 3 test:

## Stage 4 (4 marks)

For stage 4, you should extend your program for stage 3 such that it outputs, in ascending order, all

monotonically increasing sequences of maximal length.

Here is an example to show the desired behaviour of your program for a stage 4 test:

Note:

It is required that the sequences be printed in ascending order.

## Testing

We have created a script that can automatically test your program. To run this test you can execute the

dryrun program for the corresponding assignment, i.e. assn2. It expects to find, in the current directory,

the program poG.c and any of the admissible ADTs (Graph,WGraph,stack,queue) that your program

is using, even if you use them unchanged. You can use dryrun as follows:

prompt$ ~cs9024/bin/dryrun assn2

Please note: Passing the dryrun tests does not guarantee that your program is correct. You should

thoroughly test your program with your own test cases.

## Submit

For this project you will need to submit a file named poG.c and, optionally, any of the ADTs named

Graph,WGraph,stack,queue that your program is using, even if you have not changed them. You can

either submit through WebCMS3 or use a command line. For example, if your program uses the Graph ADT

and the queue ADT, then you should submit:

prompt$ give cs9024 assn2 poG.c Graph.h Graph.c queue.h queue.c

Do not forget to add the time complexity to your main source code file poG.c.

You can submit as many times as you like — later submissions will overwrite earlier ones. You can check

that your submission has been received on WebCMS3 or by using the following command:

prompt$ 9024 classrun -check assn2

## Marking

This project will be marked on functionality in the first instance, so it is very important that the output of your

program be exactly correct as shown in the examples above. Submissions which score very low on the

automarking will be looked at by a human and may receive a few marks, provided the code is well-structured

and commented.

Programs that generate compilation errors will receive a very low mark, no matter what other virtues they

may have. In general, a program that attempts a substantial part of the job and does that part correctly will

receive more marks than one attempting to do the entire job but with many errors.

Style considerations include: readability, structured programming and good commenting.

## Plagiarism

Group submissions will not be allowed. Your program must be entirely your own work. Plagiarism detection

software will be used to compare all submissions pairwise (including submissions for similar projects in

previous years, if applicable) and serious penalties will be applied, particularly in the case of repeat offences.

Do not copy ideas or code from others

Do not use a publicly accessible repository or allow anyone to see your code, not even after the

deadline

Please refer to the on-line sources to help you understand what plagiarism is and how it is dealt with at

UNSW:

Plagiarism and Academic Integrity

UNSW Plagiarism Policy Statement

UNSW Plagiarism Procedure

Help

See FAQ for some additional hints.

Finally …

Best of luck and have fun! Michael