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  1. Week 11 - Wednesday CS322

  2. Last time • What did we talk about last time? • Graphs • Euler paths and tours

  3. Questions?

  4. Logical warmup • One hundred ants are walking along a meter long stick • Each ant is traveling either to the left or the right with a constant speed of 1 meter per minute • When two ants meet, they bounce off each other and reverse direction • When an ant reaches an end of the stick, it falls off • Will all the ants fall off? • What is the longest amount of time that you would need to wait to guarantee that all ants have fallen off?

  5. Back to Graphs

  6. Hamiltonian circuits • An Euler circuit has to visit every edge of a graph exactly once • A Hamiltonian circuit must visit every vertex of a graph exactly once (except for the first and the last) • If a graph G has a Hamiltonian circuit, then G has a subgraphH with the following properties: • H contains every vertex of G • H is connected • H has the same number of edges as vertices • Every vertex of H has degree 2 • In some cases, you can use these properties to show that a graph does not have a Hamiltonian circuit • In general, showing that a graph has or does not have a Hamiltonian circuit is NP-complete (widely believed to take exponential time) • Does the following graph have a Hamiltonian circuit? a c b e d

  7. Matrix Representations of Graphs

  8. Matrices • As you presumably know, a matrix is a rectangular array of elements • An m x n matrix has m rows and n columns

  9. Graph representations • There are many, many different ways to represent a graph • If you get tired of drawing pictures or listing ordered pairs, a matrix is not a bad way • To represent a graph as an adjacency matrix, make an n x n matrix, where n is the number of vertices • Let the nonnegative integer at aij give the number of edges from vertex i to vertex j • A simple graph will always have either 1 or 0 for every location

  10. Graph to matrix examples • What is the adjacency matrix for the following graph? • What about for this one? v1 v3 v2 v1 v3 v2

  11. Matrix to graph example • Draw a graph corresponding to this matrix

  12. Another graph to matrix • What's the adjacency matrix of this graph? • Note that the matrix is symmetric • In a symmetric matrix, aij = aji for all 1 ≤ i ≤ n and 1 ≤ j ≤ n • All undirected graphs have a symmetric matrix representation v1 v2 v4 v3

  13. Matrix multiplications • To multiply matrices A and B, it must be the case that A is an m x k matrix and that B is a k x n matrix • Then, the ith row, jth column of the result is the dot product of the ith row of A with the jth column of B • In other words, we could compute element cij in the result matrix C as follows:

  14. Matrix multiplication practice • Multiply matrices A and B

  15. A few points about matrix multiplication • Matrix multiplication is associative • That is, A(BC) = (AB)C • Matrix multiplication is not commutative • AB is not always equal to BA (for one thing, BA might not even be legal if AB is) • There is an n x n identity matrix I such that, for any m x n matrix A, AI = A • I is all zeroes, except for the diagonal (where row = column) which is all ones • We can raise square matrices to powers using the following recursive definition • A0 = I, where I is the n x n identity matrix • Ak = AAk-1, for all integers k ≥ 1

  16. Finding powers of a matrix • Is A symmetric? • Compute A0 • Compute A1 • Compute A2 • Compute A3

  17. Matrix powers for graphs • We can find the number of walks of length k that connect two vertices in a graph by raising the adjacency matrix of the graph to the kth power • Raising a matrix to the zeroth power means you can only get from a vertex to itself (identity matrix) • Raising a matrix to the first power means that the number of paths of length one from one vertex to another is exactly the number of edges between them • The result holds for all k, but we aren't going to prove it

  18. Graph Isomorphism

  19. Isomorphism invariants • A property is called an isomorphism invariant if its truth or falsehood does not change when considering a different (but isomorphic) graph • 10 common isomorphism invariants: • Has n vertices • Has m edges • Has a vertex of degree k • Has m vertices of degree k • Has a circuit of length k • Has a simple circuit of length k • Has m simple circuits of length k • Is connected • Has an Euler circuit • Has a Hamiltonian circuit

  20. Using invariants to disprove isomorphism • If any of the invariants have different values for two different graphs, those graphs are not isomorphic • Use the 10 invariants given to show that the following pair of graphs is not isomorphic

  21. TreesWhat are they and how are they useful? Student Lecture

  22. Trees

  23. Trees • A tree is a graph that is circuit-free and connected • Examples: A graph made up of disconnected trees is called a forest

  24. Applications of trees • Trees have almost unlimited applications • You should all be familiar with the concept of a decision tree from programming > 10 Math 120 Score on Part II > 10 Math 110  10 Score on Part I = 8, 9, 10 Score on Part II > 6 Math 110 < 8 Math 100  6 Math 100

  25. Parse trees • A grammar for a formal language (such as we will discuss next week or the week after) is made up of rules that allow non-terminals to be turned into other non-terminals or terminals • For example: • <sentence>  <noun phrase><verb phrase> • <noun phrase>  <article><noun> | <article><adjective><noun> • <verb phrase>  <verb><noun phrase> • <article>  a | an | the • <adjective>  funky • <noun>  DJ | beat • <verb>  plays | spins • Make a parse tree corresponding to the sentence, "The DJ plays a funky beat"

  26. Describing trees • Any tree with more than one vertex has at least one vertex of degree 1 • If a vertex in a tree has degree 1 it is called a terminal vertex (or leaf) • All vertices of degree greater than 1 in a tree are called internal vertices (or branch vertices)

  27. A property of trees • For any positive integer n, a tree with n vertices must have n – 1 edges • Prove this by mathematical induction • Hint: Any tree with 2 or more nodes has a vertex of degree 1. What happens when you remove that vertex?

  28. Rooted trees • In a rooted tree, one vertex is distinguished and called the root • The level of a vertex is the number of edges along the unique path between it and the root • The height of a rooted tree is the maximum level of any vertex of the tree • The children of any vertex v in a rooted tree are all those nodes that are adjacent to v and one level further away from the root than v • Two distinct vertices that are children of the same parent are called siblings • If w is a child of v, then v is the parent of w • If v is on the unique path from w to the root, then v is an ancestor of w and w is a descendant of v

  29. Rooted tree example • Consider the following tree, rooted at 0 • What is the level of 5? • What is the level of 0? • What is the height of this tree? • What are the children of 3? • What is the parent of 2? • What are the siblings of 8? • What are the descendants of 3? 0 3 4 1 2 5 6 7 9 8

  30. Binary trees • A binary tree is a rooted tree in which every parent has at most two children • Each child is designated either the left child or the right child • In a full binary tree, each parent has exactly two children • Given a parent v in a binary tree, the left subtree of v is the binary tree whose root is the left child of v • Ditto for right subtree

  31. Binary tree applications • As we all know from data structures, a binary tree can be used to make a data structure that is efficient for insertions, deletions, and searches • But, it doesn't stop there! • We can represent binary arithmetic with a binary tree • Make a binary tree for the expression ((a – b)∙c) + (d/e) • The root of each subtree is an operator • Each subtree is either a single operand or another expression

  32. Full Binary Tree Theorem 1 • If k is a positive integer and T is a full binary tree with k internal vertices, then T has a total 2k + 1 vertices and has k + 1 terminal vertices • Prove it! • Hint: Induction isn't needed. We just need to relate the number of non-terminal nodes to the number of terminal nodes

  33. Full Binary Tree Theorem 2 • If T is a full binary tree with height h, then it has 2h+1 – 1 vertices • Prove it using induction!

  34. Generalizing that result • If T is a binary tree with t terminal vertices and height h, then t 2h • Prove it using strong induction on the height of the tree • Hint: Consider cases where the root of the tree has 1 child and 2 children separately

  35. Quiz

  36. Upcoming

  37. Next time… • Spanning trees • Graphing functions

  38. Reminders • Keep reading Chapter 10 • Start Chapter 11 • Start work on Assignment 9 • Due next Friday