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MTH5001 Introduction to Computer Programming
Task:
The project
We now outline your specifific tasks for the project. These are broken up into four parts, each containing several questions. Remember that you should attempt each part of each question.
Read the questions carefully and make sure to answer everything that is asked.
2.0.1 Part I: Randomly generating and drawing self-avoiding paths [35 marks]
1. [10 marks] Explain in detail how the function enumerate_paths and the related code defifined above works. More specififically:
– Explain the purpose of the variable edges.
– Explain the purpose of the function make_step.
– Explain how the function enumerate_paths works. In particular:
– Describe the input parameters and the output of this function, in particular the meaning of Path=[[0,0]].
– Explain the purpose of the variable next_points.
– Explain the purpose of the variable allowed_points.
– Most importantly, explain the meaning of count=sum([enumerate_paths (n-1, Path+[point]) for point in allowed_points]).
– Run the command [enumerate_paths(n, [[0, 0], [1, 0]]) for n in range(10)] and explain the output.
Your explanation(s) should make it clear how the function enumerate_paths actually does its job, i.e. enumerating all self-avoiding lattice paths with a given number of steps. For example, it is not enough to say “the variable edges is a list with four elements”. That is obvious. You should instead explain what those four elements represent in the context of the problem, and how they are used in the function enumerate_paths.
2. [10 marks] By modifying the function enumerate_paths, or otherwise, write a function generate_path which returns a random self-avoiding lattice path of length n (on Z2 , starting at the origin). By “random” we mean that the direction of each step should be randomly chosen; you may wish to use list indexing and an appropriate function from the package random for this purpose. Your function generate_path should satisfy the following properties: * It should have a single required input, an integer n. You may also fifind it useful to have an optional input Path, as in the function enumerate_paths. * It should return a list whose elements are lists of length 2, representing points in Z2 visited by the lattice path. * It may happen that a partially completed path cannot be extended to one with n steps without intersecting itself; in this case, the function should simply return the ‘trapped’ path of length
3. [5 marks] Use your function generate_path from Question 2 to produce two fifigures:
(i) Create two self-avoiding lattice paths using generate_path(100) and draw these together in one fifigure.
(ii) Create 10 self-avoiding lattice paths using generate_path(10000) and draw these together in a second fifigure.
Ensure for each fifigure that the horizontal and vertical axes are drawn to the same scale.
MTH5001 Introduction to Computer Programming
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