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...

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 ≺ … s_k-1 ≺ s_k such that s_i ≼ s_i+1 and s_i ≠ s_i+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 ≺ 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 S_p of all (positive) divisors of p;
• Task A:
  • build and output the partial order graph over S_p 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
• 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:

./poG 121 Partial order: 1: XXXXXXXXXX: XXXXXXXXXX:  Longest monotonically increasing sequences: 1 

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:

./poG 9481 Partial order: 1: XXXXXXXXXX: XXXXXXXXXX: XXXXXXXXXX:  Longest monotonically increasing sequences: 1 

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:

./poG 125 Partial order: 1: 125 5: XXXXXXXXXX: XXXXXXXXXX:  Longest monotonically increasing sequences: 5 

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:

./poG 143 Partial order: 1: XXXXXXXXXX: 143 13: XXXXXXXXXX:  Longest monotonically increasing sequences: 1 

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:

~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:

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:

9024 classrun -check assn2