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There's considerable literature on graph algorithms, which are an important part of discrete mathematics. Graphs also have much practical use in computer algorithms. Obvious examples can be found in the management of networks, but examples abound in many other areas. For instance, caller-callee relationships in a computer program can be seen as a graph (where cycles indicate recursion, and unreachable nodes represent dead code).
Few programming languages provide direct support for graphs as a data type, and Python is no exception. However, graphs are easily built out of lists and dictionaries. For instance, here's a simple graph (I can't use drawings in these columns, so I write down the graph's arcs):
A -> B
A -> C
B -> C
B -> D
C -> D
D -> C
E -> F
F -> C
This graph has six nodes (A-F) and eight arcs. It can be represented by
the following Python data structure:
graph = {'A': ['B', 'C'],
'B': ['C', 'D'],
'C': ['D'],
'D': ['C'],
'E': ['F'],
'F': ['C']}
This is a dictionary whose keys are the nodes of the graph. For each key,
the corresponding value is a list containing the nodes that are connected
by a direct arc from this node. This is about as simple as it gets (even
simpler, the nodes could be represented by numbers instead of names, but
names are more convenient and can easily be made to carry more
information, such as city names).Let's write a simple function to determine a path between two nodes. It takes a graph and the start and end nodes as arguments. It will return a list of nodes (including the start and end nodes) comprising the path. When no path can be found, it returns None. The same node will not occur more than once on the path returned (i.e. it won't contain cycles). The algorithm uses an important technique called backtracking: it tries each possibility in turn until it finds a solution.
def find_path(graph, start, end, path=[]):
path = path + [start]
if start == end:
return path
if not graph.has_key(start):
return None
for node in graph[start]:
if node not in path:
newpath = find_path(graph, node, end, path)
if newpath: return newpath
return None
A sample run (using the graph above):
>>> find_path(graph, 'A', 'D')
['A', 'B', 'C', 'D']
>>>
The second 'if' statement is necessary only in case there are nodes that are
listed as end points for arcs but that don't have outgoing arcs
themselves, and aren't listed in the graph at all. Such nodes could also
be contained in the graph, with an empty list of outgoing arcs, but
sometimes it is more convenient not to require this.Note that while the user calls find_graph() with three arguments, it calls itself with a fourth argument: the path that has already been traversed. The default value for this argument is the empty list, '[]', meaning no nodes have been traversed yet. This argument is used to avoid cycles (the first 'if' inside the 'for' loop). The 'path' argument is not modified: the assignment "path = path + [start]" creates a new list. If we had written "path.append(start)" instead, we would have modified the variable 'path' in the caller, with disastrous results. (Using tuples, we could have been sure this would not happen, at the cost of having to write "path = path + (start,)" since "(start)" isn't a singleton tuple -- it is just a parenthesized expression.)
It is simple to change this function to return a list of all paths (without cycles) instead of the first path it finds:
def find_all_paths(graph, start, end, path=[]):
path = path + [start]
if start == end:
return [path]
if not graph.has_key(start):
return []
paths = []
for node in graph[start]:
if node not in path:
newpaths = find_all_paths(graph, node, end, path)
for newpath in newpaths:
paths.append(newpath)
return paths
A sample run:
>>> find_all_paths(graph, 'A', 'D')
[['A', 'B', 'C', 'D'], ['A', 'B', 'D'], ['A', 'C', 'D']]
>>>
Another variant finds the shortest path:
def find_shortest_path(graph, start, end, path=[]):
path = path + [start]
if start == end:
return path
if not graph.has_key(start):
return None
shortest = None
for node in graph[start]:
if node not in path:
newpath = find_shortest_path(graph, node, end, path)
if newpath:
if not shortest or len(newpath) < len(shortest):
shortest = newpath
return shortest
Sample run:
>>> find_shortest_path(graph, 'A', 'D')
['A', 'C', 'D']
>>>
These functions are about as simple as they get. Yet, they are nearly
optimal (for code written in Python). In another Python Patterns
column, I will try to analyze their running speed and improve their
performance, at the cost of more code.Another variation would be to add more data abstraction: create a class to represent graphs, whose methods implement the various algorithms. While this appeals to the desire for structured programming, it doesn't make the code any more efficient (to the contrary). It does make it easier to add various labels to the nodes or arcs and to add algorithms that take those labels into account (e.g. to find the shortest route between two cities on a map). This, too, will be the subject of another column.