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
Marks 2 marks for stage 1 (correctness)
3 marks for stage 2 (correctness)
3 marks for stage 3 (correctness)
4 marks for stage 4 (correctness)
2 marks for complexity analysis
1 mark for style
———————
Total: 15 marks
Due 23:59 on Monday 8 October (week 11)
Late 2.25 marks (15%) off the ceiling per day late
(e.g. if you are 25 hours late, your maximum possible mark is 10.5)
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.
