CS 2210 Discrete Math Graphs Fall 2019 Sukumar Ghosh Seven Bridges of Knigsberg Is it possible to walk along a route that crosses each bridge exactly once?

Seven Bridges of Knigsberg A Graph What is a Graph A graph G = (V, E) consists of V, a nonempty set of vertices (or nodes) and E, a set of edges. Each edge connects a pair of nodes that are called its endpoints. Graphs are widely used to model various systems in the

real world Back to the Bridges of Knigsberg Eulers solution Euler path Simple graph

Types of graph At most one edge between a pair of nodes Multiple edges between some pair of nodes More examples of graphs

Each node denotes an actor or an actress, and each edge between P and Q denotes that P, Q worked together in some movie. It is an undirected graph Hollywood graph Each node denotes a web page, and each edge from page P to Q Q denotes a link on page P pointing to page Q.

It is a directed graph Web graph Vertex degree The max degree of any node in a simple graph is |V|-1 Degree sequence

Handshaking theorem Proof. Each edge contributes 2 to the sum on the RHS A theorem THEOREM. An undirected graph has even number of vertices of odd degree. Proof. Should follow from the handshaking theorem

Given a degree sequence such that the sum of the degrees is even, does there always exist a graph with that degree sequence? No, for simple graphs. Consider (3, 3, 3, 1). Can you draw a graph With the degree sequence (3, 3, 3, 1)? No (try this out) What is the maximum possible degree of a node in a simple graph? It is (n-1) if there are n vertices. Review of basic definitions

Types of graphs A cycle of a graph is a subset of its edge set that forms a path such that the first node of the path corresponds to the last. A connected graph with no cycle is a tree A set of one or more trees not necessarily

connected No directed cycle in a DAG Types of graphs The n-cube graph n=3

Complete graph or Clique: All vertices are adjacent to one another. A clique with n vertices is called a Kn Wheel graph Types of graphs Bipartite graph A simple graph is called bipartite if its vertex set V can be

partitioned into two disjoint subsets V1 and V2, such that every edge in the graph connects one vertex in V1 to another vertex in V2. V1 V2 Can always be colored using two colors. Subgraphs

Computer representation of graphs ADJACENCY MATRIX (taken from Wolfram Mathworld) Computer representation of graphs ADJACENCY LIST 1 2

4 Vertex Adjacent to 1 3, 4

2 3, 4 3 1, 2 4

1, 2 3 Can be represented as a linked list Graph isomorphism Taken from MIT 6.042J/18.062J

Graph isomorphism Taken from MIT 6.042J/18.062J Graph isomorphism Taken from MIT 6.042J/18.062J Connectivity An undirected graph is connected if there is a path between

every pair of distinct vertices of the graph. A connected component is the maximal connected subgraph of the given graph. Connectivity issues Erds number in academic collaboration graph Erds number = n means the person collaborated with someone whose Erds number is (n-1) Kevin Bacon number in Hollywood graph

Actor Kevin Bacon once remarked that he worked with everybody in Hollywood, or someone who worked with them. Cut vertex, cut set, cut edge A cut vertex (or articulation point ) is a vertex, by removing which one can partition the graph. A cut edge is an edge by removing which one can partition the graph. If multiple edges need to be remove to partition the graph, then the minimal set of such edges a cut set.

Examples taken from Wikipedia Connectivity in directed graphs A directed graph is strongly connected if there is a path from any vertex a to any other vertex b of the graph. A directed graph is weakly connected if there is a path between any two vertices of the underlying undirected graph. Strongly or weakly connected ?

Euler path vs. Hamiltonian path Hamiltonian path = A path that passes through every vertex exactly once. A closed path is a Hamiltonian circuit or cycle. Euler path = A path that includes every edge exactly once. A closed path is a Euler circuit or cycle. We have reviewed Euler path in the 7-bridges of Konigsberg Problem. It did not have an Euler path. Does it have a Hamiltonian path?

Hamiltonian path 1 2 5 4 3

Hamiltonian circuit/cycle colored red Does the above graph have a Hamiltonian cycle? No! Shortest path In the above weighted graph, compute the shortest path from a to z

Shortest path: Dijkstras algorithm Read the algorithm from page 747-748 of your text book Shortest path: Dijkstras algorithm Shortest path: Dijkstras algorithm Shortest path: Dijkstras algorithm Computes the shortest path from a source node to each target node

L (source) = 0, and for all other node u, L(u) := infinity, S:= null while z is not in S u := a vertex not in S with L(u) minimal; S := S {u}; {u}; for all vertices v not in S if L(u) + w(u, v) < L(v) then L(v) := L(u) + w(u, v) {known as relaxation: this adds a vertex with minimal label to S and updates the labels of vertices not in S} return L(z)

Traveling Salesman Problem (TSP) A traveling salesman wants to visit each of n cities exactly once, and then return to the starting point. In which order should he visit the cities to travel the minimum total distance? TSP = Computing the minimum cost Hamiltonian circuit. TSP is an extremely hard problem to solve (NP-complete)

An optimal TSP tour through Germanys largest cities (Source: Wikipedia) Planar Graph A planar graph is one that can be embedded in the plane, i.e., it can be drawn on the plane in such a way that its edges do not intersect except only at their endpoints.

planar K4 Butterfly planar Non-planar Non-planar

K5 K3,3 Planar Graph How to verify that a graph is a planar graph? It should not depend upon how you draw the graph. Any graph that contains a K5 or K3,3 as its sub-graph is not planar

Graph Coloring Let G be a graph, and C be a set of colors. Graph coloring finds an assignment of colors to the different nodes of G, so that no two adjacent nodes have the same color. The problem becomes challenging when the |C| is small. Chromatic number. The smallest number of colors needs to color a graph is called its chromatic number. The chromatic number of a tree is 2. Why?

Graph Coloring What are the chromatic numbers of these two graphs? Graph Coloring What are the chromatic numbers of these two graphs? Four color theorem Theorem. Any planar graph can be colored using

at most four colors. It all started with map coloring bordering states or counties must be colored with different colors. In 1852, an ex-student of De Morgan, Francis Guthrie, noticed that the counties in England could be colored using four colors so that no adjacent counties were assigned the same color. On this evidence, he conjectured the four-color theorem. It took nearly 124 years to find a proof. It was presented by Andrew Appel and Wolfgang Haken.

Trees Rooted tree: recursive definition Base case. A single node is a tree Recursive step. A graph obtained by connecting the roots of a set of trees to a new root is a tree leaf

Rooted tree: recursive definition Base case. A single node is a tree leaf leaf leaf Recursive step. A graph obtained by

connecting the roots of a set of trees to a new root is a tree leaf A Subree of T leaf leaf

T leaf Rooted tree terminology Level 0 v has a depth of 2 r3 has a height of 3 The tree has a height of 4

Level 1 Level 2 Level 3 Level 4 Important properties of trees Theorem. Every tree is a bipartite graph. Theorem. Every tree is a planar graph.

Binary and m-ary tree Binary tree. Each non-leaf node has up to 2 children. If every non-leaf node has exactly two nodes, then it becomes a full binary tree. Complete binary tree: all levels are completely filled except possibly the last level. Perfect binary tree: a full binary tree where all leaves are at the same depth (a.k.a balanced binary tree) m-ary tree. Each non-leaf node has up to m children. If every non-leaf node has exactly m nodes, then it becomes a full m-ary tree

Examples of various binary trees More examples of balanced trees Here is a balanced rooted ternary tree all all leaves are at levels h = 3 or h 1 = 2. There are at most mh leaves in an m-ary tree of height h. If an m-ary tree of height h has l leaves, then A rooted ternary tree

Binary search tree Ordered binary tree. For any non-leaf node The left subtree contains the lower keys. The right subtree contains the higher keys. How can you search an item? How many steps A binary search tree of size 9 and depth 3, with root 8 and leaves 1, 4, 7 and 13

does each search take? Binary search tree Decision tree Decision trees generate solutions via a sequence of decisions. Example 1. There are seven coins, all of which are of equal weight, and one counterfeit coin that is lighter than the rest. Given a weighing scale, in how many times do you need to

weigh (each weighing determines the relative weights of the objects on the the two pans) to identify the counterfeit coin? {We will solve it in the class}. Spanning tree Consider a connected graph G. A spanning tree is a tree that contains every vertex of G Many other spanning trees of this graph exist

Computing a spanning tree Given a connected graph G, remove the edges (in some order) without disrupting the connectivity, i.e. not causing a partition of the graph. When no further edges can be removed, a spanning tree is generated. Graph G Computing a spanning tree

Spanning tree of G Enumeration of spanning trees How many spanning trees do the following graphs have? B C C

B E D A A D

G1 B G2 C C B

E D A G1 A D

G2 Minimum spanning tree A minimum spanning tree (MST) of a connected weighted graph is a spanning tree for which the sum of the edge weights is the minimum. B 4 B

G1 C 3 1 A 5

C 2 4 1 E 6

D A 3 C 3 2

G2 7 D 4 B 1 D

A B 4 E 1 A

5 6 C 3 D

7 Depth First Search procedure DFS (G: connected graph with vertices v1vn) T := tree consisting only of the vertex v1 visit(v1) {Recursive procedure} procedure visit (v: vertex of G) for each vertex w adjacent to v and not yet in T add vertex w and edge {v, w} to T

visit (w) The visited nodes and the edges connecting them form a spanning tree. DFS can also be used as a search or traversal algorithm Depth First Search: example Breadth First Search ALGORITHM. Breadth-First Search. procedure BFS (G: connected graph with vertices v1, v2, vn)

T := tree consisting only of vertex v1 L := empty list put v1 in the list L of unprocessed vertices while L is not empty remove the first vertex v from L for each neighbor w of v if w is not in L and not in T then add w to the end of the list L add w and edge {v, w} to T

Breadth First Search A different way of generating a spanning tree Given graph G Spanning tree Huffman coding Consider the problem of coding the letters of the English

alphabet using bit-strings. One easy solution is to use 5 bits for each letter (25 > 26). Another such example is The ASCII code. These are static codes, and do not make use of the frequency of usage of the letters to reduce the size of the bit string. One method of reducing the size of the bit pattern is to use prefix codes. Prefix codes 0

e In typical English texts, e is most frequent, followed by, l, n, s, t The prefix tree assigns to each letter of the alphabet a code whose length depends on the frequency: 1

0 1 a 0 1

l 0 1 n 0 s

e = 0, a = 10, l= 110, n = 1110 etc 1 t Such techniques are popular for data compression purposes. The resulting code is a variable-length code.

Huffman codes Another data compression technique first developed By David Huffman when he was a graduate student at MIT in 1951. (see pp. 763-764 of the textbook) Huffman coding is a fundamental algorithm in data compression, the subject devoted to reducing the number of bits required to represent information. Huffman codes

Example. Use Huffman coding to encode the following symbols with the frequencies listed: A: 0.08, B: 0.10, C: 0.12, D: 0.15, E: 0.20, F: 0.35. What is the average number of bits used to encode a character? Huffman coding example Huffman coding example

Huffman coding example Huffman coding example So, in this example, what is the average number of bits needed to encode each letter? 3x 0.08 + 3 x 0.10 + 3 x 0.12 + 3 x 0.15 + 2 x 0.20 + 2 x 0.35 = 2.45 instead of 3-bits per letter.