Week 3 Day 4 Part 1, Graphs¶
Question 0: Run this cell¶
RUN THE FOLLOWING CELL
In [1]:
%config InteractiveShell.ast_node_interactivity="none"
%pip install networkx
%pip install matplotlib
%pip install scipy
%matplotlib inline
import matplotlib.pyplot as plt
import networkx as nx
import numpy as np
import warnings
warnings.simplefilter(action='ignore', category=FutureWarning)
from collections import deque
def f(globals, locals):
import base64
CODE = ("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", "def make_pretty_assert():
    import sys
    import IPython
    import ast
    import inspect
    import io
    import itertools
    
    global make_pretty_assert
    del make_pretty_assert

    if IPython.__version__ == "5.5.0":
        # colab TODO
        def get_assert_line(frame):
            return frame.f_back.f_back.f_locals["node"]
        def at_top_level(frame):
            code = frame.f_back.f_back.f_code
            return code.co_filename.endswith("IPython/core/interactiveshell.py")
        def can_annotate(frame):
            return False

    elif IPython.__version__ == "8.4.0":
        # lab computers
        def get_assert_line(frame):
            return frame.f_back.f_back.f_locals["node"]
        def at_top_level(frame):
            code = frame.f_back.f_back.f_code
            return code.co_filename.endswith("IPython/core/interactiveshell.py") and code.co_name == "run_ast_nodes"
        def can_annotate(frame):
            return at_top_level(frame) and False

    def is_literal(a):
        try:
            ast.literal_eval(a)
            return True
        except ValueError:
            return False
        return False


    def annotate_call(frame):
        if not can_annotate(frame):
            return
        
        expr: ast.Expr = get_assert_line(frame)
        for kwarg in expr.value.keywords:
            if kwarg.arg == "got":
                got = kwarg.value

        if isinstance(got, ast.Call):
            if not isinstance(got.func, ast.Name):
                return
            name = got.func.id
            func = frame.f_locals[name]
            if not hasattr(func, DEBUGGABLE_ARGS):
                return
            args = getattr(func, DEBUGGABLE_ARGS)
            kwargs = getattr(func, DEBUGGABLE_KWARGS)
            boundargs = inspect.signature(func).bind(*args, **kwargs)

            result = io.StringIO()
            pos = result.write(name)
            pos += result.write("(")

            sep = ', '
            arg_tuples = []  # (ast_arg_str, evaluated_arg, start, limit)

            # Use an iterator here. If a boundarg is not consumed here,
            # it may be consumed later by a pos+kw arg.
            boundargs_iter = iter(boundargs.args)
            for ast_arg, arg in zip(got.args, boundargs_iter):
                ast_arg_str = ast.unparse(ast_arg)
                arg_tuples.append((ast_arg_str, arg, pos, pos + len(ast_arg_str)))
                pos += len(ast_arg_str) + len(sep)

            for ast_kwarg, kwarg in itertools.zip_longest(got.keywords, boundargs_iter):
                ast_arg_str = ast.unparse(ast_kwarg)
                arg_tuples.append((ast_arg_str, kwarg, pos + len(ast_kwarg.arg) + len('='), pos + len(ast_arg_str)))
                pos += len(ast_arg_str) + len(sep)

            pos -= len(sep)
            result.write(sep.join(t[0] for t in arg_tuples))
            result.write(')')
            result_str = result.getvalue()

            yield result_str

            nonliterals = []  # (evaluated_arg, result_str, start, limit)
            for _, evaluated_arg, start, limit in arg_tuples:
                if not is_literal(result_str[start:limit]):
                    nonliterals.append((evaluated_arg, result_str[start:limit], start, limit))

            underlines = io.StringIO()
            pos = 0
            for _, _, start, limit in nonliterals:
                if (limit - start) < 3:
                    pos += underlines.write(' ' * (start - pos))
                    pos += underlines.write('↑')
                    pos += underlines.write('↑' * (limit - pos))
                    continue
                idx = start + (limit - start) // 2
                pos += underlines.write(' ' * (start - pos))
                #pos += underlines.write('╙')
                pos += underlines.write('↑' * (idx - pos))
                pos += underlines.write('↑')
                pos += underlines.write('↑' * (limit - pos))
                #pos += underlines.write('╜')
            yield underlines.getvalue()

            def make_bars_buf(nonliterals, end, evaled):
                buf = io.StringIO()
                pos = 0
                for i in range(len(nonliterals) - 1):
                    _, _, start, limit = nonliterals[i]
                    if (limit - start) < 3:
                        idx = start
                    else:
                        idx = start + (limit - start) // 2
                    pos += buf.write(' ' * (idx - pos))
                    pos += buf.write('│')
                    pos += buf.write(' ' * (limit - pos))

                _, result_str, start, limit = nonliterals[-1]
                if (limit - start) < 3:
                    idx = start
                else:
                    idx = start + (limit - start) // 2                
                pos += buf.write(' ' * (idx - pos))
                pos += buf.write('└')
                pos += buf.write('─' * (end - 1 - pos))
                pos += buf.write('╴')
                pos += buf.write(result_str)
                pos += buf.write(' ≔ ')
                pos += buf.write(str(evaled))

                return buf, pos

            while nonliterals:
                buf, pos = make_bars_buf(nonliterals, len(result_str) + 4, nonliterals[-1][0])
                yield buf.getvalue()
                last = nonliterals.pop()

    def assert_equal(*, want, got, out=sys.stdout):
        if want == got:
            print("Test case passed.")
            return

        frame = inspect.currentframe().f_back

        box_padding = 2
        header = " Test case failed. "
        want_line = f"{box_padding * ' '}Want: {repr(want)} (type: {type(want).__name__})"
        got_line = f"{box_padding * ' '}Got:  {repr(got)} (type: {type(got).__name__})"
        
        if can_annotate(frame):
            assert_line = f"{box_padding * ' '}>>> {ast.unparse(get_assert_line(frame))}"
        else:
            assert_line = ""
            
        debug_lines = list(annotate_call(frame))
        if debug_lines:
            got_line += " ← "
            padding = len(got_line)
            got_line += debug_lines[0]

        padded_debug_lines = []
        for i, line in enumerate(debug_lines):
                if i == 0:
                    continue
                padded_debug_lines.append(' ' * padding + line)
                
        line_max_len = max(len(l) + box_padding for l in (assert_line, want_line, got_line, *padded_debug_lines))
        line_max_len = max(32, line_max_len)
        header_dashes = line_max_len - len(header)


        print(file=out)
        print('-' * (header_dashes // 2), end="", file=out)
        print(header, end="", file=out)
        print('-' * ((header_dashes + 1) // 2), end="", file=out)
        print(file=out)
        print(file=out)

        
        if assert_line:
            print(assert_line, file=out)
            print(file=out)

        print(want_line, file=out)
        print(got_line, file=out)
        for line in padded_debug_lines:
            print(line, file=out)
        
        print(file=out)
        print('-' * line_max_len, file=out)
        print(file=out)
        
    DEBUGGABLE_ARGS = "_debuggable_args"
    DEBUGGABLE_KWARGS = "_debuggable_kwargs"

    def debuggable(f):
        import functools
        @functools.wraps(f)
        def g(*args, **kwargs):
            result = f(*args, **kwargs)
            setattr(g, DEBUGGABLE_ARGS, args)
            setattr(g, DEBUGGABLE_KWARGS, kwargs)
            return result
        return g
    
    return assert_equal, debuggable

assert_equal, debuggable = make_pretty_assert()
")
for code in CODE:
exec(base64.b64decode(code), globals, locals)
f(globals(), locals())
del f
Question 1: Definitions¶
A graph consists of two main things:
- nodes or vertices: points in the graph
- edges: lines that connect two nodes, usually indicating some relationship between different nodes. Edge
ecan be represeted by the two nodes it connects:e = (a, b)means that edgeeconnects nodeato nodeb.
Note: Here are some additional exercises we recommend that go over the basics of graphs.
1.1
Run the following to generate a graph
In [2]:
# Run this cell
G = nx.Graph()
G.add_nodes_from('ABCDE')
G.add_edges_from([('A', 'B'), ('A', 'C'), ('C', 'D'), ('D', 'E'), ('C', 'E'), ('A', 'E')])
nx.draw(G, with_labels=True)
1.1
Name a node in this graph
In [3]:
# Answer here
1.2
Name an edge in this graph
In [4]:
# Answer here
1.3
How many edges are there in the graph?
In [5]:
# Answer here
1.4
How many nodes are there in the graph?
In [6]:
# Answer here
1.5
Run the following to verify your answer to 1.3 and 1.4
In [7]:
# The solution to 1.4 and 1.5
print(f'There are {G.number_of_edges()} edges and {G.number_of_nodes()} nodes in graph G.')
Question 2: More definitions¶
Here are some more definitions:
- The neighbors of a node are all the nodes that directly connected to that node.
- The degree of a node is the number of neighbors
- A cycle is a
list(or path) of edges that begin and end at the same vertex.
Note: Sometimes a node will have an edge that connects directly to itself. This means the node is its own neighbor! We will ignore such cases.
2.1
Run the following to generate a graph
In [8]:
# Run this cell
G = nx.Graph()
G.add_nodes_from('ABCDE')
G.add_edges_from([ ('A', 'D'), ('C', 'D'), ('D', 'E'), ('C', 'E'), ('A', 'E')])
pos = nx.circular_layout(G)
nx.draw(G, with_labels=True, pos=pos)
2.2
Name one neighbor of node A.
In [9]:
# Answer here
2.3
What is the degree of node E?
In [10]:
# Answer here
2.4
What is the degree of node B?
In [11]:
# Answer here
2.5
Can you name one cycle in the graph?
In [12]:
# Give a path of nodes that form a cycle
2.6
Run the next cell to check your answers to 2.2, 2.3, 2.4. 2.5
In [13]:
print(f'Node A has degree {len(G["A"])} and neighbors: {list(G["A"])}')
print(f'Node E has degree {len(G["E"])} and neighbors: {list(G["E"])}')
print(f'Node B has degree {len(G["B"])} and neighbors: {list(G["B"])}')
print(f'Example Cycle: {nx.find_cycle(G)}')
Question 3: More More Definitions¶
Here are many different properties of graphs:
- undirected vs directed:
- undirected: this means edges are symmetric.
e = (A, B)means you can useeto go from nodeAto nodeBand you can useeto go from nodeBto nodeA. - directed: this means that each edge has a direction.
e = (A, B)means you can useeonly to go fromAtoB. This means that each node will have an indegree (the number of edges that lead into the node) and an outdegree (the number of edges that lead out of the node).
- undirected: this means edges are symmetric.
- unweighted vs weighted:
- unweighted: edges are either there or not.
- weighted: edges have a weight that captures the relationship between two nodes.
- cyclic vs acyclic:
- cyclic: means the graph has a cycle
- acyclic: means the graph does not have a cycle
3.1
Run the following code to generate a graph
In [14]:
# Run this cell
G = nx.Graph()
G.add_nodes_from('ABC')
G.add_edges_from([('A', 'B'), ('B', 'C'), ('A', 'C')])
pos = nx.circular_layout(G)
nx.draw(G, with_labels=True, pos=pos)
3.2
Is the graph directed? Is the graph weighted? Is the graph cyclic?
In [15]:
# Your answers here.
3.3
Run this to check your answers
In [16]:
print(f'Is G directed? {nx.is_directed(G)}')
print(f'Is G weighted? {nx.is_weighted(G)}')
# Will throw error if acyclic
print(f'Is G cyclic? Cycle: {nx.find_cycle(G)}')
3.4
Run the following code to generate a graph
In [17]:
DG = nx.DiGraph()
DG.add_weighted_edges_from([(1, 2, 0.5), (3, 1, 0.75), (3, 2, 0.6), (1, 4, 0.25)])
pos = nx.circular_layout(DG)
nx.draw_networkx_edge_labels(DG, pos, nx.get_edge_attributes(DG, "weight"))
nx.draw(DG, with_labels=True, pos=pos)
3.5
Is the graph directed? Is the graph weighted? Is the graph cyclic?
In [18]:
# Your answers here.
3.6
Run this to check your answers
In [19]:
print(f'Is G directed? {nx.is_directed(DG)}')
print(f'Is G weighted? {nx.is_weighted(DG)}')
# Will throw error if acyclic
print(f'Is G cyclic? Cycles: {list(nx.simple_cycles(DG))}')
3.7
Give a short real life example of when you might need a directed graph instead of an undirected graph. (For instance, one example could be Instagram where node/person A might follow node/person B, but node/person B might not follow node/person A)
In [20]:
# A few word here, doesn't need to be full sentences
3.8
Give an example where you might need a weighted graph instead of an unweighted one. (For instance, one example could be if the graph represents roads, you might need to keep track of road lengths).
In [21]:
# A few words here, doesn't need to be full sentences
Question 4: Representation¶
Recall that you can represent a graph in two ways:
- An adjaceny list that records a list of neighbors for each node
- and an adjacency matrix that records whether or not there is an edge in a matrix. If
matrix[row][col] == 1, then there is an edge from node represented byrowto the node represented bycol
4.1
Run the following cell to generate a graph
In [22]:
G = nx.DiGraph()
G.add_edges_from([('A', 'B'), ('A', 'C'), ('C', 'D'), ('B', 'C'), ('D', 'B')])
pos = nx.circular_layout(G)
nx.draw(G, with_labels=True, pos=pos)
4.2
Generate the 4x4 adjency matrix representation of the graph by hand
In [23]:
# Your answer here
# A = 0, B = 1, C = 2, D = 3.
# Then adj[row][col] == 1 means there is an edge from node row to node col
adj = [[0, 0, 0, 0],
[0, 0, 0, 0],
[0, 0, 0, 0],
[0, 0, 0, 0]]
4.3
Check your answer to 4.2 in the cell below
In [24]:
# A = 0, B = 1, C = 2, D = 3.
# Then adj[row][col] == 1 means there is an edge from node row to node col
print(nx.adjacency_matrix(G).toarray())
4.4
Generate the adjacency list representation of the graph by hand
In [25]:
#
4.5
Check your answer to 4.4 in the cell below
In [26]:
# There are a number of cool things we can do:
# (1) nx.to_dict_of_dicts(G) OR nx.to_dict_of_lists
print(f'Adjaceny list: {nx.to_dict_of_lists(G)}')
print()
# (2) nx.generate_adjlist(G)
lines = [x[0] + '->' + x[1:] for x in nx.generate_adjlist(G)]
for line in lines:
print(line)
print()
# (3) G.adj
print(f'Adjacency list: {G.adj}')
# If you have anything like any of these answers, you are correct!
Question 5: get_neighbors()¶
5.1¶
Write a function that takes in a graph G and a node n, and returns a list of all the neighbors of n. If n has no neighbors, return the empty list [].
Hint: The format of G is confusing, so try printing G out! G is a dictionary where each node is a key. The value for a key (which is a node) is the list of neighbors reachable from that key
In [27]:
def get_neighbors(G, n):
""" returns the list of neighbors of node n in graph G
Inputs:
G: The graph
type: dict{list[string or int]}
n: The node
type: string or int
Returns: The list of neighbors
type: list[string or int]
"""
return []
In [28]:
# Test cases:
G = nx.Graph()
G.add_nodes_from('ABCDE')
G.add_edges_from([('A', 'B'), ('B', 'C'), ('A', 'C'), ('C', 'D')])
pos = nx.circular_layout(G)
nx.draw(G, with_labels=True, pos=pos)
graph = nx.to_dict_of_lists(G)
print(f'The graph G is: {graph}')
print()
assert_equal(want=set(['B', 'C']), got=set(get_neighbors(graph, 'A')))
assert_equal(want=set(['A', 'B', 'D']), got=set(get_neighbors(graph, 'C')))
assert_equal(want=set([]) , got=set(get_neighbors(graph, 'E')))
5.2¶
You are given a graph and two nodes n1 and n2. Check if there is a edge from n1 to n2.
In [29]:
def is_neighboring(G, n1, n2):
""" returns True if there is an edge from n1 to n2 in G
Inputs:
G: The graph
type: dict{list[string or int]}
n1: A node
type: string or int
n2: Another node
type: string or int
Returns:
type: bool
"""
return False
In [30]:
# Test cases:
G = nx.Graph()
G.add_nodes_from('ABCDE')
G.add_edges_from([('A', 'B'), ('B', 'C'), ('A', 'C'), ('C', 'D')])
pos = nx.circular_layout(G)
nx.draw(G, with_labels=True, pos=pos)
graph = nx.to_dict_of_lists(G)
print(f'The graph G is: {graph}')
print()
assert_equal(want=False, got=is_neighboring(graph, 'A', 'D'))
assert_equal(want=True , got=is_neighboring(graph, 'C', 'D'))
assert_equal(want=False,got=is_neighboring(graph, 'E', 'B'))
assert_equal(want=True, got=is_neighboring(graph, 'C', 'B'))
5.3¶
You are given a graph and two nodes n1 and n2. Check if there is an path of length 2 or less that starts from n1 and ends at `n2.
For example in the graph above, D -> C -> A is a path of length 2, so path_2(G, 'D', 'A') should return True. D -> C is a path with one edge, so path_2(G, 'D', 'C') should also return True. There is no path to E, so path_2(G, 'C', 'E') should return False.
Hint: It may save some time if you use is_neighboring() in your code!
In [31]:
def path_2(G, n1, n2):
""" returns True if there is a path of length 2 or smaller from n1 to n2 in G
Inputs:
G: The graph
type: dict{list[string or int]}
n1: A node
type: string or int
n2: Another node
type: string or int
Returns:
type: bool
"""
return False
In [32]:
# Test cases:
G = nx.DiGraph()
n = 10
G.add_nodes_from(range(n))
edges = []
for i in range(n - 1):
for j in range(i + 1, n):
if j <= i ** 2 and j % 2 == i % 2:
edges.append((i, j))
G.add_edges_from(edges)
pos = nx.circular_layout(G)
nx.draw(G, with_labels=True, pos=pos)
graph = nx.to_dict_of_lists(G)
print(f'The graph G is: {graph}')
print()
assert_equal(want=False, got=path_2(graph, 1, 4))
assert_equal(want=False, got=path_2(graph, 7, 3))
assert_equal(want=True, got=path_2(graph, 3, 7))
assert_equal(want=False, got=path_2(graph, 2, 8))
assert_equal(want=True, got=path_2(graph, 4, 6))
assert_equal(want=True, got=path_2(graph, 2, 6))
assert_equal(want=False, got=path_2(graph, 8, 1))
Question 6: is_connected() - challenge¶
Write a function to check if there is a path from node n1 to node n2 in G.
Hint: Either this lecture or next lecture we will show an O(n) algorithm to do this called Depth First Search. If we have not covered this yet, feel free to read up on it here.
Hint-2: If we haven't learned DFS, we can do inefficient algorithms as well! For example, we can try all paths of length |V| where |V| is the number of nodes in the graph. Alternatively, we've coded path_2(). Can you code path_4()? Can you code path_8()? Can you code path_n()?
In [33]:
def is_connected(G, n1, n2):
""" returns True if there is a path from n1 to n2 in G
Inputs:
G: The graph
type: dict{list[string or int]}
n1: A node
type: string or int
n2: Another node
type: string or int
Returns:
type: bool
"""
return False
In [34]:
# Run below to test your code
# Staff solution (BFS, something we have not learned)
from collections import deque
def staff_solution(G, n1, n2):
visited = set()
q = deque()
q.append(n1)
while len(q) > 0:
node = q.popleft()
for neighbor in G[node]:
if neighbor not in visited:
q.append(neighbor)
visited.add(node)
if node == n2:
return True
return False
# Graph:
G = nx.DiGraph()
n = 10
G.add_nodes_from(range(n))
edges = []
for i in range(n - 1):
for j in range(i + 1, n):
if j <= i ** 2 and j % 2 == i % 2:
edges.append((i, j))
G.add_edges_from(edges)
pos = nx.circular_layout(G)
nx.draw(G, with_labels=True, pos=pos)
graph = nx.to_dict_of_lists(G)
# Test cases:
print(f'The graph G is: {graph}')
print()
assert_equal(want=is_connected(graph, 1, 4), got=staff_solution(graph, 1, 4))
assert_equal(want=is_connected(graph, 7, 3), got=staff_solution(graph, 7, 3))
assert_equal(want=is_connected(graph, 3, 7), got=staff_solution(graph, 3, 7))
assert_equal(want=is_connected(graph, 2, 8), got=staff_solution(graph, 2, 8))
assert_equal(want=is_connected(graph, 4, 6), got=staff_solution(graph, 4, 6))
assert_equal(want=is_connected(graph, 2, 6), got=staff_solution(graph, 2, 6))
assert_equal(want=is_connected(graph, 8, 1), got=staff_solution(graph, 8, 1))
Question 7: Word Questions¶
How many edges are there in a fully connected graph with n nodes? That is, if you have n nodes, what is the maximum number of unique edges between two distinct nodes you can have?
In [35]:
# Answer here
What if the graph is directed? Then what is the maximum number of edges?
In [36]:
# Answer here
Why might you be able to think of an undirected graph as a special case of a directed graph?
In [37]:
# Answer here