Information Science 2

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1 Information Science 2 - Applica(ons of Basic ata Structures- Week 03 College of Information Science and Engineering Ritsumeikan University

2 Agenda l Week 02 review l Introduction to Graph Theory - Basic definitions - Traversable and non-traversable graphs l Graphs in computer memory l Binary tree: concept and applications l Test 2

3 Recall concepts from Week 02 l Array l Node l Logical and Physical structures l Singly-linked list l oubly-linked list l Stack, Push, Pop l Queue, Enqueue, equeue 3

4 Class objectives l Understand basics of Graph Theory and binary trees l After this week s lecture and study, you must be able to: raw and describe (undirected) graphs and multigraphs using correct terms Show how to represent an (undirected) graph in computer memory Show how to represent a binary tree 4

5 Introduction to Graph Theory l We have been learning about data structures with interconnected nodes of different types l Structures formed by connected nodes are widely used in information science to represent and analyze various networks (e.g. computer and social networks), electronic circuits, natural and programming languages, complex real-world objects (e.g. NA, transportation systems), etc. l A structure of linked nodes can be called a graph l Graph Theory is an extremely useful field of mathematics for describing and studying all these structures and more 5

pairs in E is not important, the graph is called undirected.

6 Graphs: Terminology l A graph G comprises a set V of vertices and a set E of edges l Each edge in E is a pair (a,b) of vertices in V l If the ordering of the pairs in E is not important, the graph is called undirected. Otherwise, the graph is directed l An ordered pair (a,b) of vertices in a directed graph is called arc 6

7 Graph drawing l If (a,b) is an edge in E, we connect points a and b with a line in the graph drawing of G A B e 4 C e 1 e 5 e 3 e 2 A, B, C, are nodes e 1, e 2, e 5 are edges l If (a,b) is an arc in E, we connect a and b with an arrow from a to b in the graph drawing of G A, B, C, are nodes e 1, e 2, e 5 are arcs A B e 4 C e 1 e 5 e 3 e 2 7

8 Multiple edges and loops l In this course, we will mainly deal with undirected graphs l Consider the following graph: A e 1 e 2 e 4 C e 3 e6 B and C are connected by more than one edge: e 2 and e 5 B e 5 edge e 6 connects vertex C to itself l e 2 and e 5 are called multiple edges l Edge e 6 is called a loop 8

9 Multigraphs and simple graphs l If a graph contains one or more loops or multiple edges then it is a multigraph. For example: l A (simple) graph contains no loops or multiple edges. For example: 9

10 A B Graph (formal) notation e 4 C e 1 e 5 e 3 e 2 The size of G = G(V, E) is the number of edges in E The order of G = G(V, E) is the number of vertices in V l The graph G = G(V, E) depicted above consists of a set of vertices V = {A,B,C,} and a set of edges E = {e 1, e 2, e 3, e 4, e 5 } = {(A,B),(B,),(,C),(,A),(C,B)} l In G, vertices A and B are adjacent because there exists at least one edge between them, e 1 l A and C are not adjacent because there is no edge between them 10

11 ensity l l For a graph G = G(V, E), its density is the ratio of its size (number of edges) to the maximum possible number of edges Formally, density(g) = 2 size(g)/(order(g) (order(g)-1)) = 2xE / (V x (V-1)) For example: A C density = 2*5/(4*3) = 5/6 B 11

(edge e is also incident on vertex A) The vertex degree is the number of edges incident on the vertex

12 Incidence and degrees l l For an edge e = (A,B), there are three ways to describe its vertices A and B 1) A and B are the endpoints of edge e 2) e connects A and B 3) Edge e is incident on vertex B (edge e is also incident on vertex A) The vertex degree is the number of edges incident on the vertex For example: A B C The degree of vertex A is 2; i.e. formally deg(a) = 2 deg(b) = 3 deg() = 4 deg(c) = 3 12

13 Graphs: Traversable and non-traversable l A graph is called traversable if there exists at least one path through all of the edges and vertices without crossing the same edge twice For example: l Not all graphs are traversable. For example: 13

The traversable problem l For a multigraph, to be traversable:

All of the other vertices must have an even degree one pair of edges

if and only if it has two or fewer vertices with odd degree numbers

14 The traversable problem l For a multigraph, to be traversable: Vertices with an odd degree must be the start point or the end point All of the other vertices must have an even degree one pair of edges for each entrance and exit l Therefore, a multigraph is traversable if and only if it has two or fewer vertices with odd degree numbers For example: traversable 3 not 1 2 traversable not 3 14

15 Some other graphs l The trivial graph is one vertex with no edges. For example: l A complete graph is one in which each of the vertices are linked with each other vertex. For example: l In a connected graph, each vertex has a path to every other l A regular graph is connected and all vertices have the same degree. For example: Connected, not regular Not connected Regular and connected 15

16 Representing graphs in computer memory l Two common ways of representing graphs in a computer are by linked lists or by a matrix (a set of array data structures) l Linked lists are more common and convenient for graphs with few vertices l Matrices are more common and convenient for graphs with many vertices l The graph is stored as vertex adjacencies or as vertices and their incident edges 16

17 Adjacency matrix l To store a graph, an n n matrix can be used, where n is the number of vertices in the graph l In this matrix (called adjacency matrix), for each vertex pair (i, j) the value of the matrix element at i, j is set to 1 if adjacent, and to 0 if not A B C A B C A B C " % $ ' $ ' $ ' $ # ' & l The adjacency matrix is symmetrical: each pair is represented twice 17

18 Multigraph by matrix l Since a multigraph has multiple connections and/or loops, its adjacency matrix stores number of connecting edges for each pair: A B C A B C A B C " % $ ' $ ' $ ' $ # ' & 18

isadvantages of representation by matrix l Listing each pair twice may be inefficient use of memory resources l Some graphs contain relatively few connections and the matrix will have

19 isadvantages of representation by matrix l Listing each pair twice may be inefficient use of memory resources l Some graphs contain relatively few connections and the matrix will have many zeros, possibly wasting memory l The same graph or multigraph may have different matrices depending on vertex order l Adding or removing vertices requires rebuilding the whole matrix 19

20 Representation by linked lists l An array or linked list is used to store the vertex structures in a vertex file. Each vertex structure in the file holds a name of the vertex, and is associated (connected) with a linked list that stores references (pointers) to the adjacent vertices A B C E A B C E B A B B C C 20

21 Tree graphs l A (Hamiltonian) circuit is a path that touches every vertex only once and returns to the start l A tree graph is any connected graph for which there is no circuit. For example: 21

l A binary tree is a tree graph that (if not empty or null) starts at a root node, usually shown on top l Each node, in addition to having a path up to the root node, may also have up to two nodes

22 l A binary tree is a tree graph that (if not empty or null) starts at a root node, usually shown on top l Each node, in addition to having a path up to the root node, may also have up to two nodes below it l Nodes below are left or right l Nodes below, if they have one or more nodes linked below them, may be called left and right subtrees l Terminal nodes have no connections below Binary trees root two of the five terminal nodes in this binary tree right subtree of root 22

Uses of binary trees l Binary trees have many uses

language can be represented by a binary tree, then parsed with a

+ precedence are closer to the root - l Terminal nodes in these

23 Uses of binary trees l Binary trees have many uses (abs(w)+(x-y))/(z*2) l For example, expressions of a programming language can be represented by a binary tree, then parsed with a stack as shown in the previous class / l Operators with lower * + precedence are closer to the root - l Terminal nodes in these abs(w) 2 z parsing trees may be constants, variables, or function calls x y 23

24 Representing binary trees l A natural way to store a binary tree in memory is as nodes with two pointers each, similar to a doubly-linked list root B E G A C J F I K E B C A F J I K H G H 24

(arrays), as linked lists, or as a combination l Binary trees are a special type of graph

25 Summary l Graphs and multigraphs are extremely useful mathematical constructs consisting of nodes (vertices) linked by edges l Graphs and multigraphs can be stored as matrices (arrays), as linked lists, or as a combination l Binary trees are a special type of graph that can be represented with nodes that have two pointers each, a left node and a right node 25

26 Homework l Read the slides l Learn the English terms new for you l o the self-preparation assignments 26

27 Next class l Basic search algorithms - Linear - Binary l Search on tree structures - epth-first - Breadth-first 27

28 1. Test 01 28

(5.2) 151 Math Exercises. Graph Terminology and Special Types of Graphs. Malek Zein AL-Abidin

(5.2) 151 Math Exercises. Graph Terminology and Special Types of Graphs. Malek Zein AL-Abidin King Saud University College of Science Department of Mathematics 151 Math Exercises (5.2) Graph Terminology and Special Types of Graphs Malek Zein AL-Abidin ه Basic Terminology First, we give some terminology