This course covers the essential information that every serious programmer needs to know about algorithms and data structures, with emphasis on applications and scientific performance analysis of Java implementations. Part I covers elementary data structures, sorting, and searching algorithms. Part II focuses on graph- and string-processing algorithms. All the features of this course are available for free. It does not offer a certificate upon completion.
Graph API
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Algorithms, Part II
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Undirected Graphs
We define an undirected graph API and consider the adjacency-matrix and adjacency-lists representations. We introduce two classic algorithms for searching a graph—depth-first search and breadth-first search. We also consider the problem of computing connected components and conclude with related problems and applications.
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Robert SedgewickWilliam O. Baker *39 Professor of Computer Science
Computer Science
Kevin WaynePhillip Y. Goldman '86 Senior Lecturer
Computer Science
. As we'll see, in little bit later in the
lecture the way that we're going to set up our graph processing algorithms, is to
develop an API to cover our representation of the graph and provide simple set of
methods for clients to call to process the graph.
So, let's take a look at that in detail. So, the idea is we have to represent, a,
graph within the computer. One of the first things, to remember is
that, you can, draw a graph, that maybe, that provides some kind of intuition,
about the structure, an, but you could have two drawings that represent,
represent the same graph that look pretty different.
And so one of the things to remember in any graph representation is, it can give
you some intuition. But that intuition may be misleading.
And we'll just remember that as we look at different representations.
So first thing is how to represent the vertices.
So what we're going to do in this lecture is just use integers between zero and V
minus one. The reason we do that is it allows us, to
use vertex indexed arrays, within the graph representation.
And we understand, from earlier algorithms, lectures, that we can use a
symbol table to convert names to integers, with a symbol table.
And so, we'll, leave that part as a symbol table application.
And, just work with graphs with vertex names between zero and V minus one where v
is the number of vertices. Now we have to remember when we're doing
representation we can have various anomalies in the graph.
So we can we draw edges, but actually, in real data we might have multiple edges or
we might have a self loop. And we'll take that into account when we
look at the representation, graph representation.
So here is the API that we're going to use.
So again our graph processing algorithms are going to be clients of this API.
The idea is that will use this to build graphs.
And then our processing programs will be client at this program,
In this, of this API. And the idea is most of them have a very
simple operations that they need to do, and those are the one's that we put in the
API. So we have two constructors.
One that creates an empty graph would be vertices.
Another one that creates a graph from an input stream.
Then the basic operation for building a graph is just add edge, so that adds an
edge connecting two vertices, V and W. The basic operations for processing our
graph, well there's V and E that gives a number of edges and vertices.
But then there's an iterator that takes the vertex's argument,
Then iterates through the vertices adjacent to that vertex.
All of our graph processing can be cast in terms of this iterator.
So down here is an example of a client program that prints out every edge in the
graph. So we created an input stream, maybe with
a given final name, create a graph from that input stream,
And we'll look at the code for that. And then the client program for every
vertex, So remember the vertices are numbered
between zero and G dot the number of vertices minus one.
So for every vertex, we iterate through all the vertices adjacent to that vertex
and print out an edge, if, if there's an edge connecting v and w we print out v and
then a little dash to indicate an edge and then w.
This actually prints out every edge twice and in an undirected graph because if the
v and w are connected by an edge then w appears in v's adjacency list and v
appears in w's adjacency list. So heres an example of a running that
client, if, if we have a file tinyG.txt, our standard is we have a number of
vertices as an integer in the first is the first integer in the file.
Number of edges is the second integer in the file and then pairs of vertex names.
And so. The constructor will read those two things
and then call that edge for all of these pairs of things and that enables this
client, to, if we run it for that graph, to print out all the edges.
So everybody adjacent to zero, 61 and five, everybody adjacent to one is just
zero, So you notice the edge 0-1 appears twice in the list.
So that's the sample client of our basic graph in API.
And so here's some typical and simpicle, simple graph processing code that uses the
API. So you can write a static method that
takes a graph and a vertex as argument. And returns the degree, the number of
edges that are connect, number of vertices that are connected by an edge to V.
So all it does is set a local variable degree to zero, And then iterate through
all the vertices adjacent to V and increment that and return it.
Similarly, you can compute the maximum degree of a vertex in a graph.
And that's for every vertex, compute the degree and find the biggest one or the
average degree. Well, the average degree, if you think
about it, it's just twice the number of edges divided by the vertex or you could
go through all the way through every vertex and edge and compute the total and
divide, but this is probably a much more efficient way to do it or say number of
self loops. and so that involves going through the whole graph of every vertex
for every edge adjacent to that vertex you check whether it's v and we've got a self
loop. And if it does then your return the number
of self loops that divided by two because every edge is counted twice.
So those are examples of static methods that a client might use.
In just example of the use of the ATI. So now how we going to implement that?
That's our usual standard of lets look at some clients and now let's talk about a
representation that we can use to implement the graph API.
So one possible representation is, set of edges representation, where, for every
edge, we just maintain a list, maybe an array of edges or a linked list, of edges.
So for every edge in the graph, there's, a representation of it.
And that one is a possible representation, but it leads to, inefficient,
implementation, much less efficient, that would make it unusable for, the huge
graphs that we see in practice. Another one is called the adjacency matrix
representation, Here we maintain a 2-dimensional v x v
array, It's a boolean array, 0-1 or true or
false. And for every edge v-w in the graph you
put true for row v in column w and for row w in column v.
So it's actually two representations of each edge in an adjacency matrix graph
representation. S, you can immediately, give a v and w
test whether there's an edge connecting v and w.
But that's one of the few operations that's efficient with this representation
and it's not very widely used, because for a huge graph, say with billions of, of,
vertices, You would have to have billion squared
number of entries in this array, which is going to be too big for your computer,
most likely. So this one actually isn't that widely
used. A, The one that is most widely used in
practice, and the one that we'll stick with, is called the adjacency list
representation. And that's where we keep a vertex index
array where, for every vertex, we maintain a list of the vertices that are adjacent
to that. So, for example, vertex four, Has, five,
is connected to five, six, and three, so its list has five, six, and three on it.
Now, on lower level representations we've talk about using linked width or array for
these, But actually in modern lingo with the
background that we'd built with algorisms what we're going to use is an abstract
data type. Our bag representation for this, which is
implemented with a linked list, but we don't have to think about it when we're
talking about graphs. we'd keep the vertices,
Names of, numbers of the vertices that are adjacent to each given vertex in a bag.
And we know that we can implement it, such that we can iterate through and time
proportional to the number of entries and the space taken is also proportional to
the number of entries,, And that's going to enable us to process
huge graphs. So here's the full implementation of the
Jason C, List graph representation.
So, the private instance variables that we're gonna use area the number of
vertices in the graph and then a array of nags of integers. so data the types and
set of values, set operations on those values, so those are the sets of values
for a graph. So here's the constructor of an empty
graph with V vertices. We keep the value v in the instance
variable as usual. Then we create.
An array of size V. And, of bags of integers.
And, store that array in, [inaudible] as the adjacency array of the graph.
Adjacency lists array. And then, as usual.
When we create, a, a, an array of objects. We go through.
And for every entry in the array, we initialize with, a, empty object.
So, after this code, we have the empty bags.
And so that's the constructor and then the other main engine and building graphs is
at edge and so, to add an edge between V and W, we add W to V's bag, and we add V
to W's bag. That's it.
And to iterate through all the vertices adjacent to a given vertex we simply
return the bag which is iterable. This is a nice example, illustrating the
power of abstraction. Because we did the low level processing,
for, that, that's involved, with our bag implementation in one of the early,
lectures. And now we, we get to use that to give a
very compact implementation, and efficient implementation, of the, graph data
structure. So it's really important to understand
this code. And you should make sure, that you study
it. So, as I mentioned in practice.
We're gonna be using this adjacency list representation.
Because all the algorithms are based on iterating over the vertices adjacent to V.
And this gets that done in time proportional to the number of such
vertices. So that's number one and number two, in
the real world, the graphs have lots of num, lots of vertices,
But a pretty small vertex degre. so number one, we can afford to represent, represent
the graph when we use the adjacency list representation because basically our space
is proportional to the number of edges. And number two, we can afford to process
it because our time taken is proportional to the number of edges that, that we
examined, with the ray of edges representation in the adjacency matrix
representation it gets very slow for some very simple task,
But, mostly, it's very slow for iterating over the vertices given to, adjacent to a
given vertex. Which is the key operation.
A list of edges, you have to go through all the edges to find the ones adjacent to
a given vertex. Adjacency matrix, you have to go through,
all possible, vertices adjacent and that's just going to be much too slow in
practice, Because adjacency list gets it done, in
time proportional degree of v, which is much smaller, for the huge graph that we
see in the real world. So that's our basic API.
Next, we'll look at some algorithms that are clients of this API.
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