Agenda

1. Agenda • Review: • Planar Graphs • Lecture Content: • Concepts of Trees • Spanning Trees • Binary Trees • Exercise

2. Review:Planar Graphs

3. 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

4. Example: Planar Graph

5. Example: Non-Planar Graph

6. 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

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

8. K3,3 dan K5 • Last week, we showed that K3,3 and K5 are not planar (using inequalities on previous slide) • If a graph contains K3,3 or K5 as a subgraph, then it cannot be planar • The converse of statement above is not true • Use concept of homeomorphic graph

9. Homeomorphic Graph Definition • If a graph G has a vertex v of degree 2 and edges (v,v1) and (v,v2) with v1≠ v2 ,  edges (v,v1) and (v,v2) are in series • A series reduction consists of deleting vertex v from G and replacing edges (v,v1) and (v,v2) by the edge (v1,v2). • The graph G’ is said to be obtained from G by a series of reduction or elementary subdivision • Graphs G1 and G2 are homeomorphicif G1 and G2 can be reduced to isomorphic graphs by performing a sequence of series reduction • Any graph is homeomorphic to itself • Graphs G1 and G2 are homeomorphic if G1 can be reduced to a graph isomorphic to G2 or if G2 can be reduced to a graph isomorphic to G1

10. Example

11. Kuratowski’s Theorem Theorem: A graph is nonplanar if and only if it contains a subgraph homeomorphic to K3,3 or K5 Example: Determine whether G is planar or nonplanar?

12. Trees

13. 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

14. 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

15. Which Ones are Trees?

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

17. Terminology & Characterization of Trees • Definition: • Let T be a tree with root v0. • Suppose that x, y, and z are vertices in T and that (v0, v1, ..., vn) is a simple path in T. • Then (a) vn-1 is the parent of vn (b) v0, ... , vn-1 are ancestors of vn (c) vn is a child of vn-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}.

18. 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.

19. Example

20. Spanning Trees

21. 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.

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

23. Removing edges to eliminate cycle/circuit A Spanning Tree

24. Other Spanning Trees of G

25. Algorithms for Creating Spanning Tree • Depth-First Search • Breadth-First Search

26. 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

27. Example Arbitrarily choose f as the root

28. 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

29. Example

30. Minimum Spanning Tree

31. 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

32. Prim’s Algorithm

33. Example

34. Binary Trees

35. 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.

36. Example

37. Binary Trees Theorem: • If T is a full binary tree with i internal vertices, then T has i+1terminal vertices and 2i+1total vertices. Theorem: • If a binary tree of height h has t terminal vertices, then lg2t≤ h Example

38. 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

39. Example • 01010111? • SOTO?

40. Algorithm to construct an optimal Huffman Code

41. 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 Construct tree as follows

42. Example

43. 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

44. Example