3.4. Lab 3: Graphs¶
Due Date: 11:59pm February 6, 2024 Labs should be submitted as a single Jupyter notebook to CourseLink Dropbox This lab counts for 4% of the final grade |
To do in advance of this lab:
Make sure you have run the code in Allen Downey’s Chapter 2 notebook (we will review this in lecture)
3.4.1. Exercises¶
Complete the exercises corresponding to Exercise 2.1 – 2.4 in Think Complexity (version 2):
There are a few short exercises embedded in Chapter 2 notebook. Complete these in-place.
In Section 2.8 of the text, we analyzed the performance of
reachable_nodes
and classified it in \(O(n+m)\), where \(n\) is the number of nodes and \(m\) is the number of edges. Continuing the analysis, what is the order of growth foris_connected
?def is_connected(G): start = next(G.nodes_iter()) reachable = reachable_nodes(G, start) return len(reachable) == len(G)
You can indicate your answer directly in the notebook, using a Markdown cell with LaTeX.
In Allen Downey’s implementation of
reachable_nodes
, you might be bothered by the apparent inefficiency of adding all neighbours to the stack without checking whether they are already inseen
. Write a version of this function that checks the neighbours before adding them to the stack. Does this “optimization” change the order of growth? Does it make the functions faster?There are actually two kinds of ER graphs. The one we generated in Chapter 2, \(G(n,p)\), is characterized by two parameters, the number of nodes and the probability of an edge between the nodes.
An alternative definition, denoted \(G(n,m)\), is also characterized by two parameters: the number of nodes, \(n\), and the number of edges, \(m\). Under this definition, the number of edges is fixed, but their location is random.
Repeat the experiments we did in this chapter using this alternative definition. Here are a few suggestions for how to proceed:
Write a function called
m_pairs
that takes a list of nodes and the number of edges, \(m\), and returns a random selection of \(m\) edges. A simple way to do that is to generate a list of all possible edges and userandom.sample
.Write a function called
make_m_graph
that takes \(n\) and \(m\) and returns a random graph with \(n\) nodes and \(m\) edges.Make a version of
prob_connected
that usesmake_m_graph
instead of make_random_graph.Compute the probability of connectivity for a range of values of \(m\).
How do the results of this experiment compare to the results using the first type of ER graph?
3.4.2. Submission¶
Please submit a single Jupyter notebook which completes the above exercises, filling in Allen Downey’s Chapter 2 notebook. There are already placeholders in the notebook for your work.