CS322

1. Week 11 - Friday CS322

2. Last time • What did we talk about last time? • Graph isomorphism • Trees

3. Questions?

4. Logical warmup • You are the despotic ruler of an ancient empire • Tomorrow is the 25th anniversary of your reign • You have 1,000 bottles of wine you were planning to open for the celebration • Your Grand Vizier uncovered a plot to murder you: • 10 years ago, a rebel worked in the vineyard that makes your wine • He poisoned one of the 1,000 bottles you are going to serve tonight and has been waiting for revenge • The rebel died an accidental death, and his papers revealed his plan, but not which bottle was poisoned • The poison exhibits no symptoms until death • Death occurs within ten to twenty hours after consuming even the tiniest amount of poison • You have over a thousand slaves at your disposal and just under 24 hours to determine which single bottle is poisoned • You have a few hundred prisoners, sentenced to death • Can you use just the prisoners and risk no slaves? • If so, what's the smallest number of prisoners you can risk and still be sure to find the bottle?

5. Finishing Trees

6. 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"

7. 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)

8. 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?

9. 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

10. 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

11. 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

12. 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

13. 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 of 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

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

15. 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

16. Graphing Functions

17. Graphing functions • I am assuming that you have been graphing functions since about 6th or 7th grade • Still, the formal definition of a graph of a function is: • Let f be a real-valued function of a real variable • The graph of f is the set of all points (x, y) in the Cartesian coordinate plane such that x is in the domain of f and y = f(x)

18. Power functions • In computer science, we are often interested in power functions of x written pa(x) where pa(x) = xa where x and a are nonnegative • Power functions are the building blocks of polynomial functions • Graph the following on the same coordinate axes • p0 • p1/2 • p1 • p2

19. Discontinuous functions • Recall the definition of the floor of x: • x = the largest integer that is less than or equal to x • Graph f(x) = x • Defining functions on integers instead of real values affects their graphs a great deal • Graph p1(x) = x, x  R • Graph f(n) = n, n  N

20. Multiples of functions • There is a strong visual (and of course mathematical) correlation a function that is the multiple of another function • Examples: • g(x) = x + 2 • 2g(x) = 2x + 4 • Given f graphed below, sketch 2f 2 1 -6 -5 -4 -3 -2 -1 -1 -2 1 2 3 4 5 6

21. Absolute value • Consider the absolute value function • f(x) = |x| • Left of the origin it is constantly decreasing • Right of the origin it is constantly increasing 2 1 -6 -5 -4 -3 -2 -1 -1 -2 1 2 3 4 5 6

22. Increasing and decreasing functions • We say that f is decreasing on the set Siff for all real numbers x1 and x2 in S, if x1 < x2, then f(x1) > f(x2) • We say that f is increasing on the set Siff for all real numbers x1 and x2 in S, if x1 < x2, then f(x1) < f(x2) • We say that f is an increasing (or decreasing) function ifff is increasing (or decreasing) on its entire domain • Clearly, a positive multiple of an increasing function is increasing • Virtually all running time functions are increasing functions

23. Asymptotic Notation (Big Oh) Student Lecture

24. Asymptotic Notations

25. Growth of functions • Mathematicians worry about the growth of various functions • They usually express such things in terms of limits, maybe with derivatives • We are focused primarily on functions that bound running time spent and memory consumed • We just need a rough guide • We want to know the order of the growth

26. Definitions • Let f and g be real-valued functions defined on the same set of nonnegative real numbers • f is of order at least g, written f(x) is(g(x)), iff there is a positive A  R and a nonnegative a  R such that • A|g(x)| ≤ |f(x)| for all x > a • f is of order at most g, written f(x) isO(g(x)), iff there is a positive B  R and a nonnegative b  R such that • |f(x)| ≤ B|g(x)| for all x > b • f is of order g, written f(x) is(g(x)), iff there are positive A, B  R and a nonnegative k  R such that • A|g(x)| ≤ |f(x)| ≤ B|g(x)| for all x > k

27. Using the notation • Express the following statements using appropriate notation: • 10|x6| ≤ |17x6 – 45x3 + 2x + 8| ≤ 30|x6|, for x > 2 • Justify the following: • is (x)

28. Properties of -, O-, and - notation • Let f and g be real-valued functions defined on the same set of nonnegative real numbers • f(x) is (g(x)) and f(x) is O(g(x)) ifff(x) is (g(x)) • f(x) is (g(x)) iffg(x) is O(f(x)) • f(x) is (f(x)), f(x) is O(f(x)), and f(x) is (f(x)) • If f(x) is O(g(x)) and g(x) is O(h(x)) then f(x) is O(h(x)) • If f(x) is O(g(x)) and c is a positive real, then cf(x) is O(g(x)) • If f(x) is O(h(x)) and g(x) is O(k(x)) then f(x) + g(x) is O(G(x)) where G(x) = max(|h(x)|,|k(x)|) for all x • If f(x) is O(h(x)) and g(x) is O(k(x)) then f(x)g(x) is O(h(x)k(x))

29. Orders of functions • If 1 < x, then • x < x2 • x2 < x3 • … • So, for r, sR, where r < s and x > 1, • xr < xs • By extension, xr is O(xs)

30. Proving bounds • Prove a  bound for g(x) = (1/4)(x – 1)(x + 1) for x R • Prove that x2 is not O(x) • Hint: Proof by contradiction

31. Polynomials • Let f(x) be a polynomial with degree n • f(x) = anxn + an-1xn-1 + an-2xn-2 … + a1x + a0 • By extension from the previous results, if an is a positive real, then • f(x) is O(xs) for all integers s n • f(x) is (xr) for all integers r≤n • f(x) is (xn) • Furthermore, let g(x) be a polynomial with degree m • g(x) = bmxm + bm-1xm-1 + bm-2xm-2 … + b1x + b0 • If an and bm are positive reals, then • f(x)/g(x) is O(xc) for real numbers c> n - m • f(x)/g(x) is not O(xc) for all integers c > n -m • f(x)/g(x) is (xn- m)

32. Upcoming

33. Next time… • More on asymptotic notation

34. Reminders • Keep reading Chapter 11 • Keep working on Assignment 9 • Due next Friday