Graphs, Trees, Paths and Their Representations

CS 6213 –
Advanced
Data
Structures
Lecture 2
GRAPHS AND THEIR
REPRESENTATION
TREES, PATHS, CYCLES
TOPOLOGICAL SORTING

 Instructor
Prof. Amrinder Arora
amrinder@gwu.edu
Please copy TA on emails
Please feel free to call as well
 TA
Iswarya Parupudi
iswarya2291@gwmail.gwu.edu
CS 6213 – Arora – L2 Advanced Data Structures - Graphs 2
LOGISTICS

L4 - BTrees CS 6213 - Advanced Data Structures - Arora 3
CS 6213
Basics
Record /
Struct
Arrays / Linked
Lists / Stacks
/ Queues
Graphs / Trees
/ BSTs
Advanced
Trie, B-Tree
Splay Trees
R-Trees
Heaps and PQs
Union Find

 Graphs – Basics
 Degrees, Number of Edges, Min/Max Degree
 Kinds of Graphs
 How to Represent in Data Structures
 Trees, Paths, Cycles
 Journeys
 Topological Sorting, DAGs
AGENDA

 A graph G=(V,E) consists of a finite set V, which is the
set of vertices, and set E, which is the set of edges.
Each edge in E connects two vertices v1 and v2,
which are in V.
 Can be directed or undirected
Not to be confused with a bar graph!!!! 
GRAPH

A core data structure that shows up in many
circumstances:
 Transportation and Logistics (paths, etc.)
 Circuit Design
 Social Networking – Model connections between people
 Sociology – Model influence
 Zoology and Wildlife
 Project Task Management
 Job Scheduling and Resource Assignment (matching)
 Time table scheduling
 Task parallelization (graph coloring)
GRAPH – APPLICATIONS

 (Undirected) Degree of a node
 (Directed) Indegree / Outdegree
 Min degree in a graph: 
 Max degree in a graph: 
 Basic observations
 (Undirected) Sum of degrees = 2 x number of edges
 (Directed) Sum of indegree = Sum of outdegree = number of
edges
DEGREE

 If (x,y) is an edge, then x is said to be adjacent to y, and y is adjacent
from x.
 In the case of undirected graphs, if (x,y) is an edge, we just say that x
and y are adjacent (or x is adjacent to y, or y is adjacent to x). Also, we
say that x is the neighbor of y.
 The indegree of a node x is the number of nodes adjacent to x
 The outdegree of a node x is the number of nodes adjacent from x
 The degree of a node x in an undirected graph is the number of
neighbors of x
 A path from a node x to a node y in a graph is a sequence of node x,
x1,x2,...,xn,y, such that x is adjacent to x1, x1 is adjacent to x2, ..., and xn
is adjacent to y.
 The length of a path is the number of its edges.
 A cycle is a path that begins and ends at the same node
 The distance from node x to node y is the length of the shortest path
from x to y.
GRAPH DEFINITIONS

 Using a matrix A[1..n,1..n] where A[i,j] = 1 if (i,j) is an
edge, and is 0 otherwise. This representation is called
the adjacency matrix representation. If the graph is
undirected, then the adjacency matrix is symmetric about
the main diagonal.
 Using an array Adj[1..n] of pointers, which Adj[i] is a
linked list of nodes which are adjacent to i.
 The matrix representation requires more memory, since it
has a matrix cell for each possible edge, whether that
edge exists or not. In adjacency list representation, the
space used is directly proportional to the number of
edges.
 If the graph is sparse (very few edges), then adjacency
list may be a more efficient choice.
GRAPH REPRESENTATIONS

 A very practical choice is to use graphing libraries,
such as:
 JGraphT (Java)
 Boost (C++)
 GraphStream (Java)
 JUNG (Java)
GRAPH REPRESENTATION (CONT.)

 Graphs can be characterized in many ways. Two
important ones being:
 Directed or Undirected
 Weighted or Unweighted
 Both Adjacency Matrix (AM) and Adjacency List (AL)
representations can be used for graphs – weighted
or unweighted, directed or undirected.
 A[i,j] = A[j,i] if graph is undirected. So, we could decide to use
just the upper triangle.
 If graph is weighted, in adjacency list, we can also store the
weight. AL[i] = [(j,w(i,j)), (k,w(i,k)), …]
GRAPH CHARACTERIZATIONS

 A tree is a connected acyclic graph (i.e., it has no
cycles)
 Rooted tree: A tree in which one node is designated
as a root (the top node)
TREE
Example:
Node A is root node
F and D are child nodes of A.
P and Q are child nodes of J.
Etc.

 Definitions
 Leaf is a node that has no children
 Ancestors of a node x are all the nodes on the path from x to
the root, including x and the root
 Subtree rooted at x is the tree consisting of x, its children and
their children, and so on and so forth all the way down
 Height of a tree is the maximum distance from the root to any
node
TREE (CONT.)

Option 1
•Trees are
graphs, so we
can use
standard graph
representation
– Adjacency
Matrix or
Adjacency List
Option 2
•Use parent
node and list
for child nodes
Option 3
•Use parent
node, and two
pointers – one
for first child
and the other
for nextSibling
REPRESENTING TREES
3 Basic Options

 We can use standard graph representation –
Adjacency Matrix or Adjacency List
 These options are overkill for trees, but in some
instances, the graph is not known to be a tree
beforehand, so this option works.
TREE REPRESENTATION – OPTION 1

 Rather than using Adjacency Matrix
or Adjacency List representation,
we can use a simpler representation
 Each node has a pointer to parent,
and a linked list of child nodes
 The root node’s parent pointer is null
 This representation works for “rooted” trees. If the
tree is not rooted, we can designate one node as
root.
Node {
Node parent,
List<Node> childNodes
}

 Another representation for Trees: Left
Child, Right Sibling representation for a
tree. Each node has 3 pointers:
 Parent – Points to parent (null if this is the root
node)
 Left pointer – Points to first child (null if this is
a leaf node)
 Right pointer – Points to right sibling (null if no
more siblings)
 For example:
 is represented by
Node {
Node parent,
Node firstChild,
Node nextSibling
}

 When updating values in a tree, such as the size of the
subtree rooted at a node, the weight of the subtree
rooted at a node, etc, there are two main methods:
 Recompute whenever there is a modify operation (addition of a
node, deletion of a node, changing the weight of a node, etc).
Recomputations usually only need to propagate from the change
node upwards to the root. [Advantage: Values always up to date,
Disadvantages: Lot of time spent in Recompute, Method cannot
be run concurrently.]
 Use a dirty flag and set to true when there is a change. Set dirty
flag to true for all ancestors (navigate to parent, until the root).
Recompute when there is a need, or as per a schedule.
[Advantages: Efficient, Methods (other than the “recomputed”
operation can be run concurrently. Disadvantages: Values are out
of date at times.]
UPDATING VALUES

 Able to hold the entire graph in memory?
 If your graph is large and changing fast, like the
Facebook interconnection graph, you simply cannot
hold it in memory using traditional methods. You
need to replicate it across multiple servers and use
reliable services to get partial data out from the
graph. Your graph may simply be backed by a
database with which you interact directly (without
ever loading a complete “graph” object.)
LARGE GRAPHS

 Given each of the graph representations, how do we
find a path?
 Shortest path algorithms
 Dijkstra
 All Pairs Shortest Paths
 [Refer to CS 6212 Notes for details]
PATHS

 Path, spread over time
 Useful concept if the graph changes over time
 Specifically, consider this scenario:
 Edge e1 existed from x to y at time t1
 Edge e2 existed from y to z at time t2
 t1 < t2
 Then, we say that there exists a journey from x to z
JOURNEY

 How can we detect cycles in a graph?
CYCLES

 Assuming a graph is a Directed Acyclic Graph,
topological ordering can be produced in linear time.
O(n + m)
TOPOLOGICAL SORTING

 Graphs are important data structures with numerous
applications
 Graphs can be of different kinds
 Many convenient ways to model graphs
 Adjacency Matrix
 Adjacency List
 Special structures for trees
 Many commercial and open source libraries exist, such as
JGraphT
SUMMARY

Graphs, Trees, Paths and Their Representations

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