Week 11 - Wednesday CS322
Last time • What did we talk about last time? • Graphs • Euler paths and tours
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?
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
Matrices • As you presumably know, a matrix is a rectangular array of elements • An m x n matrix has m rows and n columns
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
Graph to matrix examples • What is the adjacency matrix for the following graph? • What about for this one? v1 v3 v2 v1 v3 v2
Matrix to graph example • Draw a graph corresponding to this matrix
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
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:
Matrix multiplication practice • Multiply matrices A and B
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
Finding powers of a matrix • Is A symmetric? • Compute A0 • Compute A1 • Compute A2 • Compute A3
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
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
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
TreesWhat are they and how are they useful? Student Lecture
Trees • A tree is a graph that is circuit-free and connected • Examples: A graph made up of disconnected trees is called a forest
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
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"
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)
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?
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
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
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
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
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
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!
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
Next time… • Spanning trees • Graphing functions
Reminders • Keep reading Chapter 10 • Start Chapter 11 • Start work on Assignment 9 • Due next Friday