Data structures: Introduction to graphs

Hello everyone. So far in the series on
data structures, we have talked about some of the linear
data structures like array, linked list, stack and queue. In all these
structure, data is arranged in a linear or sequential manner, so we can call
them linear data structures and we’ve also talked about tree which
is a non-linear data structure. Tree is a hierarchical structure. Now as
we understand data structures are ways to store and organize data, and for different kinds of data we use
different kinds of data structures. In this lesson, we’re going to introduce
you to another non linear data structure that has got its application in a wide
number of scenarios in computer science. It is used to model and represent a
variety of systems and this data structure is graph. When we study data structures, we often
first study them as mathematical or logical models. Here
also, we will first study graph as a
mathematical or logical model and we will go into implementation
details later. Okay so let’s get started. A graph just
like a tree is a collection of objects or entities that
we call nodes or vertices, connected to each
other through a set of edges. But in a tree connections are bound to be in a certain
way. In a tree that our rules dictating the connection among the nodes.
In a tree with N Nodes, we must have exactly N – 1 edges. One edge for each parent child relationship. As
we know an edge in a tree is for a parent child
relationship and all nodes in a tree except the root node would have apparent
would have exactly 1 parent and that’s why if they are N nodes, it
must be exactly N – 1 edges. In a tree, all nodes must be reachable
from the root and there must be exactly one possible
path from root to a node. Now in a graph there are no
rules dictating the connection among the nodes.
A graph contains a set of nodes and a set of edges and edges can be connecting nodes in any
possible way. Tree is only a special kind of graph.
Now graph as a concept has been studied
extensively in mathematics. If you have taken a course on discrete
mathematics then you must be knowing about graphs
already. In computer science, we basically study and implement the same concept
of graph from mathematics. The study of graph is often referred to
as graph theory. In pure mathematical terms we can define graph something like this. A graph G is in order pair of a set
V of vertices and a set E of edges. Now I’m using
some mathematical jargon here. An ordered pair is just a pair of
mathematical objects in which the order of objects in the pair matters. This is how we write and
represent an ordered pair, objects separated by comma put
within parenthesis. Now because the order here matters. We
can say that V is the first object in the pair and E
is the second object. An ordered pair A, B is not equal to B, A unless A and B are equal. In our definition of graph here, first
object in the pair must always be a set of vertices and
the second object must be a set of edges that’s why we are calling the pair
an ordered pair. We also have concept of unordered pair.
An unordered pair is simply a set of two elements. Order is
not important here. We write an unordered pair using curly
brackets or braces. Because the order is not important here,
unordered pair A, B is equal to B, A. It doesn’t matter
which object is first and which object is second. Okay coming back,
so a graph of is an ordered pair of a set of
vertices and a set of edges and G=(V,E) is a formal
mathematical notation that we use to define a graph. Now I
have a graph drawn here in the right. This graph is 8 vertices
and 10 edges. What I want to do is I want give some
names to these vertices because each node in a graph must
have some identification. It can be a name or it can be an index. I’m naming these vertices as V1, V2 V3, V4, V5 and so on, and this
naming is not indicative of any order. There is no 1st, 2nd and 3rd Node here. I could
give any name to any node. So my set of
vertices here is this. We have 8 elements in the
set V1, V2, V3, V4, V5, V6, V7 and V8. So this is my set of
vertices for this graph. Now what’s my set of edges. To answer
this we first need to know how to represent
an edge. An edge is uniquely identified by it’s
2 endpoints, so we can just write the names
of the two endpoint of an edge as a pair and it can be
a representation for the edge. But edges can be of two types. We can
have a directed edge in which connection is
one-way or we can have an undirected edge in
which connection is two way. In this example graph that I’m showing
here, edges are undirected but if you remember the tree that I had
shown earlier then we had directed edges in that tree.
With this directed edge that I’m showing you here, we are saying that there is link or path from vertex U to V but we cannot
assume a path from V to U. This connection is one way. For
a directed edge, one of the endpoints would be the
origin and other end point would be the destination and we draw the edge with an arrow head pointing towards the
destination. For our edge here, origin is U and
destination is V. A directed edge can be to represented as
an ordered pair, first element in the pair can be the
origin and second element can be the destination. So with this directed edge represented
as ordered pair (U,V), we have a path from U to V. If we want a path from V to U, when need
to draw another directed edge here with V as
origin and U as destination and this edge can be
the represented as ordered pair (V,U), the upper one here is (U,V) and the
below one is (V,U) and they are not same. Now if the edge is undirected, the
connection is 2 way and undirected edge can be
to represented as an unordered pair here because the edge is
bi directional origin and destination are not fixed. We
only need to know what two end points have been connected by the edge. So now that we know how to present edges,
we can write the set of edges for this example graph here. We have an undirected edge between V1
and V2 then we have 1 between V1 and V3 and
then be have V1 V4. This is really simple and just go ahead
and write all of them. So this is my set of edges. Typically in
a graph, all edges would either be directed or
undirected. It’s possible for a graph to have both
directed and undirected edges but we are not going to study such
graphs, we are only going to study graphs in which all edges would either be
directed or undirected. A graph with all directed
edges is called a directed graph or digraph and a graph with all undirected ages is
called an undirected graph. There is no special name for an
undirected graph. Usually, if the graph directed, we
explicitly say that it’s directed graph or digraph. So these are
two types of graphs. Directed graph or digraph in which edges are uni-directional or ordered pairs and undirected graph in which edges are
bi-directional or unordered pairs. Now many real-world
systems and problems can be modeled using a graph.
Graphs can be used to represent any collection of objects having some
kind of pairwise relationship. Let’s have a look
at some of the interesting examples. A social network like Facebook can be
represented as an undirected graph. A user would be a node in
the graph and if 2 user are friends, there
would be an edge connecting them. A real social network would have millions
and billions of nodes. I can show only few in my diagram here
because I’m short of space. Now social network is an undirected
graphs because friendship is a mutual relationship. If I’m your
friend, you are my friend too. So connections have to be 2 way. Now once
a system is modeled as a graph a lot of problems
can easily be solved by applying standard algorithms
in graph theory. Like here in this social network, let’s
say we want to do something like suggest friends to a user. Let’s say we
want to suggest some connections to Rama. One possible approach to do so can be
suggesting friends of friends who are not connected already. Rama has
3 friends, Ella, Bob and Katie and friends of 3 that are not connected to Rama
already can be suggested. There is no friend of
Ella which is not connected to Rama already. Bob however, has 3 friends Tom, Sam, and Lee that are not friends with Rama so
they can be suggested and katie has two friends Lee and Swati that are not connected to Rama.We have
counted Lee already, so in all we can suggest these for users
to Rama. Now even though we described this
problem in context of a social network. This is a standard graph problem. The
problem here in pure graph terms is finding all nodes
having lenght of shortest path from a given
node equal to 2. Standard algorithms can be applied to
solve this problem. We’ll talk about concepts like path in
a graph in some time. For now just know that the problem that
we just described in context of a social network is a standard graph
problem. Okay so a social network like Facebook
is an undirected graph Now let’s have a look at another example.
Interlinked web pages on the internet or the World
Wide Web can be represented as a directed graph.
A web page that would have a unique address or URL
would be a node in the graph and we can have a directed edge if a
page contains link to another page. Now once again, there are billions of pages
on the web but I can show only few here. The edges
in this graph are directed because that relationship is not mutual this
time. If page A has a link to page B then
it’s not necessary that page B will also have a link to page A. Let’s say one of the pages on has a tutorial on graph and on this page I have put a link to Wikipedia article on
graph. Let’s assume that in this example
graph that I am showing you here. Page P is my mycodeschool tutorial
on graph with this address or URL and lets say, page Q is the
Wikipedia article on graph with this URL Wikipedia/org/wiki/graph. Now on my page that is page P, I have put a link to the Wikipedia page on
graph. If you are on page P, you can click on
this link and go to page Q but Wikipedia has not reciprocated
to my favor by putting a link back to my page. So if you are on
page Q you cannot click on the link can
come to page P. Connection here is one way and that’s why we have
drawn a directed egde here. Okay now once again if we are able to
present web as a directed graph, we can apply
standard graph theory algorithms to solve problems and to perform tasks. One of the tasks that search engines
like Google perform very regularly is web crawling. Search engines use a
program called web crawler that systematically browsers the
World Wide Web to collect and store data about web
pages. Search engines can then use this data to provide quick and accurate results
against search queries. Now even though in this context, we are
using a nice and heavy term like web crawling. Web crawling is basically graph
traversal or in simpler words, act of visiting all
nodes in a graph and no prizes for guessing that there
are standard algorithms for graph traversal. We will be studying graph traversal
algorithms in a later lessons. Okay now the next thing that I want to
talk about is concept of a weighted graph. Sometimes in a graph, all
connections cannot be treated as equal. Some connections can be preferable to
others like for example we can represent intercity road network that is the
network of highways and free ways between cities as an
undirected graph. I’m assuming that all highways would be
bi-directional. Intra-city road network that is road
network within a city would definitely have one-way roads and so Intra-city network must be
represented as a directed graph but intercity road network in my opinion
can be represented as an undirected graph. Now clearly we cannot
treat all connections as equal here. Roads would be of different lengths and
to perform a lot of tasks to solve a lot of problems, we need to
take length of roads into account. In such cases, we associate some weight or cost with every edge. We label the edges with
their weights. In this case weight can be lenght of the roads,
so what to do here is I’ll just label this edges with some values for
the lenghts. Let’s say these values are in kilometers and now edges in this graph are weighted and
this graph can be called weighted graph. Let’s say in this graph, we want to
pick the best route from city A to city D. Have a look at these four possible
routes, I am showing them in different colors. Now if I would treat all edges as equal then I would say that the green route
through B and C and a red route through E and F are equally
good. Both these paths have to three edges and
this yellow route through E is the best because we have only two
edges in this path. But with different weights assigned to
the connections, I need to add up weights of edges in a path to calculate
total cost. When I’m taking weight into account shortest
route is through B and C. Connections have
different weights and this is really important here in this
graph. Actually, we can look at all the graphs as weighted graphs An unweighted graph can basically be seen
as a weighted graph in which weight of all
the edges is same and typically we assume to weight
as one. Okay so we have represented inter-city cities
road network as weighted undirected graph. Social network
was an unweighted undirected graph and World Wide Web was an unweighted
directed graph and this one is a weighted undirected graph. Now this was inter-city road network. I
think intra-city road network that is road network within a city can be
modeled as a weighted directed graph because in a
city that would be some one ways. Intersections in interest city’s road
network would be Nodes and road segments would be our edges, and by the way we can also draw an
undirected graph as directed. It’s just that for each undirected edge
we will have 2 directed edges. We may not be able to redraw a directed
graph has undirected but we can always redraw an undirected
graph as directed. Okay I’ll stop here now. This much is
good for an introductory lesson. In next lesson, we will talk about some more
properties of graph. This is it for this lesson. Thanks for
watching !

100 thoughts on “Data structures: Introduction to graphs

  1. I had to prepare for an interview and needed to brush up on some old forgotten information. Your videos fantastic and I found that my knowledge came flowing back again. So thanks very much for taking so much time to put together such a great set of information!

  2. I would be wary when representing an undirected graph as a directed graph with double directed edges, as this messes up certain algorithms. Other than that, spectacular lesson!

  3. This is the most cogent and easily understandable explanation of basic computer science I have seen. I look forward to watching the rest of your videos.

  4. So if Facebook can be represented by undirected graph then instagram could be represented by directed grapth?

  5. +mycodeschool please sir can you insert some videos to explain the ideas of topological sort ,backtracking and dynamic programming because I found your videos so helpful for me I'll be pleased if you answerd my request and thank you so much

  6. Hello. Is one vertex also a graph? is that graph connected if it only has one vertex? is then minimum graph degree 0? Pls help, tnx

  7. Directed graph is a graph with all edges unidirectional. Then why in the given example at 7:46 has one unidrectional edge??

  8. F***ing genius! God, thanks for being born, dude! I read a whole 28 page chapter and I was totally clueless… I watch our 16 minute video and I totally get it now…

  9. E -> D is the shortest route, based on weights. Otherwise, E -> A -> B -> C -> D has a total weight of 680. If E -> is congested, then E –> A … -> D is the shortest route.

  10. superb work!!!!! thankyou so much for making me understand the concept of data structures so beautifully !!!!

  11. Can anyone help me to clarify some questions in data structures ..please help me anyone help me. I can post my questions if u help me.. plsssss

  12. One of the best tutorials on Datastructures I have come accross, Thank you so much for sharing this knowledge this way.. !! kudos to your teaching skills

  13. Came for a refresher on these things and will probably end up watching your videos all day. Your videos really hit all points and are delivered so well, Thank you:)

  14. Great Video I really enjoyed your dtata structures playlist.
    Sir i am trying to open mycode website for problem solving it gives bad gateway error.

  15. Wikipedia has not returned my favor by giving a link to my video… ha ha awesomely scripted.. beautiful way to teach.. kudos… keep up the good effort.

  16. So a graph in c++ is like a non-sequential linked list where all the nodes are pretty much pointing to another node randomly? And the the nodes that are being pointed to are the "neighbors" of that node that is doing the pointing?

  17. You are a great man and more better than proffessors in my college, I have learned a lot from you . I felt like I was waste fellow , I couldn't understand data structures, algorithms and pointers
    even if I had studied many times. but after watching your videos ,those concepts became easy .only because of you dude. you are a life saver,thanks a lot.But wanna see you back in you tube, why don't you upload more videos to save people like us.

  18. Go back by clicking, left pointing arrow icon on the top-left corner of your page, just below the tab button…
    and what if I am following a certain someone on Facebook.

  19. fascinating, I had never considered that the entire graph could be a data structure. I was introduced to trees in my studies, but for some reason, I never considered it a subset of the set of all graphs.

  20. You are like a great teacher who goes deep into making the concepts as lucid as possible. I am a working professional, if you need anything that I can be helpful to this channel, please do email me. I strongly feel this channel has the potential to change billion minds across globe.

  21. I was struggling with my data structures assignment. I watched the 50 minute lecture over and over and still didn't get it. This video is a life saver and I can finally get on with the assignment.

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