Agenda Review: –Planar Graphs Lecture Content:  Concepts of Trees  Spanning Trees  Binary Trees Exercise

Review: Planar Graphs

Planar Graphs Definition: A graph is called planar if it can be drawn in the plane without any edges crossing (where a crossing of edges is the intersection of the lines or arcs representing them at a point other than their common endpoint). Such a drawing is called a planar representation of the graph

Example: Planar Graph

Example: Non-Planar Graph

Euler’s Formula A planar representation of a graph splits the plane into regions or faces, including an unbounded region Euler’s Formula: Let G be a connected planar simple graph with e edges and v vertices. Let r be the number of regions in a planar representation of G, then r = e – v + 2 Boundary is formed by cycle

Some inequalities for planar graphs Euler’s formula can be used to establish some inequalities that must be satisfied by planar graphs, i.e:

K 3,3 dan K 5  Last week, we showed that K 3,3 and K 5 are not planar (using inequalities on previous slide)  If a graph contains K 3,3 or K 5 as a subgraph, then it cannot be planar  The converse of statement above is not true  Use concept of homeomorphic graph

Homeomorphic Graph Definition  If a graph G has a vertex v of degree 2 and edges (v,v 1 ) and (v,v 2 ) with v 1 ≠ v 2,  edges (v,v 1 ) and (v,v 2 ) are in series  A series reduction consists of deleting vertex v from G and replacing edges (v,v 1 ) and (v,v 2 ) by the edge (v 1,v 2 ).  The graph G’ is said to be obtained from G by a series of reduction or elementary subdivision  Graphs G 1 and G 2 are homeomorphic if G 1 and G 2 can be reduced to isomorphic graphs by performing a sequence of series reduction Any graph is homeomorphic to itself Graphs G 1 and G 2 are homeomorphic if G 1 can be reduced to a graph isomorphic to G 2 or if G 2 can be reduced to a graph isomorphic to G 1

Example

Kuratowski’s Theorem Theorem: A graph is nonplanar if and only if it contains a subgraph homeomorphic to K 3,3 or K 5 Example: Determine whether G is planar or nonplanar?

Trees

 Very useful in computer science applications: –File Computer file system –IP Multicast routing Multimedia transmission to a group of receivers –To construct efficient algorithm E.g.: searching: Depth-first search, breadth-first search –Construct efficient codes for data transmission and storage –Games –Decision making –Object-oriented programming: parent child relation

Trees Definition:  A (Free) Tree T is a simple graph satisfying the following: –If v and w are vertices in T, there is a unique simple path from v to w.  A Rooted Tree is a tree in which a particular vertex is designated the root, and every edge is directed away from the root  Note: The level of a vertex v is the length of the simple path from the root to v. The height of a rooted tree is the maximum level number that occurs

Which Ones are Trees?

Example Rooted Tree with root vertex = a Rooted Tree with root vertex = c

Terminology & Characterization of Trees Definition:  Let T be a tree with root v 0.  Suppose that x, y, and z are vertices in T and that (v 0, v 1,..., v n ) is a simple path in T.  Then (a) v n-1 is the parent of v n (b) v 0,..., v n-1 are ancestors of v n (c) v n is a child of v n-1 (d) If x is an ancestor of y, y is a descendant of x. (e) If x and y are children of z, x and y are siblings. (f) If x has no children, x is a terminal vertex (or a leaf). (g) If x is not a terminal vertex, x is an internal vertex (or branch) (h) The subtree of T rooted at x is the graph with vertex set V and edge set E, where V is x together with the descendants of x and E = {e| e is an edge on a simple path from x to some vertex in V}.

Terminology & Characterization of Trees Theorem:  Let T be a graph with n vertices. The following are equivalent. (a) T is a tree. (b) T is connected and acyclic (graph with no cycles) (c) T is connected and has n–1 edges. (d) T is acyclic and has n–1 edges.

Example

Spanning Trees

Definition  A tree T is a spanning tree of a graph G if T is a subgraph of G that contains all of the vertices of G. Theorem  A graph G has a spanning tree if and only if G is connected.

Find Spanning Tree of G Graph G Graph G is connected, but it is not a tree because it contains simple cycles/circuits

Removing edges to eliminate cycle/circuit A Spanning Tree

Other Spanning Trees of G

Algorithms for Creating Spanning Tree  Depth-First Search  Breadth-First Search

Depth-First Search Arbitrarily choose a vertex of the graph as the root. Form a path starting at this vertex by successively adding vertices and edges, where each new edge is incident with the last vertex in the path and a vertex not already in the path. Continue adding vertices and edges to this path as long as possible. Move back to the last vertex in the path, and, if possible, form a new path starting at this vertex passing through vertices that were not already visited. Move back another vertex in the path (two vertices in the path) and try again. Repeat this procedure  Backtracking

Example Arbitrarily choose f as the root

Breadth-First Search Arbitrarily choose a vertex of the graph as the root. Add all edges incident to this vertex. The new vertices added at this stage become the vertices at level 1 in the spanning tree. For each level in level 1, add all edges incident to this vertex to the tree as long as it does not produce a simple circuit. This produce the vertices at level 2 in the tree. Repeat this procedure

Example

Minimum Spanning Tree

Definition:  Let G be a weighted graph. A minimum spanning tree of G is a spanning tree of G with minimum weight. Algorithms:  Prim’s algorithm  Kruskal’s algorithm

Prim’s Algorithm

Example

Binary Trees

Definition:  A binary tree is a rooted tree in which each vertex has either no children, one child, or two children.  If a vertex has a child, that child is designated as either a left child or a right child (but not both).  If a vertex has two children, one child is designated a left child and the other child is designated a right child. Note:  A full binary tree is a binary tree in which each vertex has either two children or zero children.

Example

Binary Trees Theorem:  If T is a full binary tree with i internal vertices, then T has i+1 terminal vertices and 2i+1 total vertices. Theorem:  If a binary tree of height h has t terminal vertices, then lg 2 t ≤ h Example

Application: Huffman Coding  Huffman code represents characters by variable- length bit strings  alternatives to ASCII or other fixed-length codes  Objective to use fewer bits: to save minimize storage and transmission time  The idea is to use short bit strings to represent most frequently used characters and longer bit strings to represent less frequently used characters  Easily defined by binary tree

Example ? SOTO?

Algorithm to construct an optimal Huffman Code

Example  Construct Huffman code for the following characters, given the frequency of occurrence as follows:  Algorithm begins by repeatedly replacing the smallest two frequencies with the sum until two-element sequence is obtained 2,3,7,8,12  2+3,7,8,12 5,7,8,12  5+7, 8,12 8,12,12  8+12, 12 12, 20 CharacterFrequency #7 $8 %12 Construct tree as follows

Example

Binary Search Tree How should items in a list be stored so that an item can be easily located? Definition  A binary search tree is a binary tree T in which data are associated with the vertices. The data are arranged so that, for each vertex v in T, each data item in the left subtree of v is less than the data item in v, and each data item in the right subtree of v is greater than the data item in v Example  Construct a binary search tree from the following words: OLD PROGRAMMERS NEVER DIE THEY JUST LOSE THEIR MEMORIES

Example