CSE 21 Summer 2017 Homework 4

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1 CSE 21 Summer 201 Homework Key Concepts Minimum Spanning Trees, Directed Acyclic Graphs, Topological Sorting, Single source shortest paths, Counting, Basic probability principles, Independence, Linearity of expectation, Conditional probability, Expected value. Required Reading Rosen.1-.5,.1-., , DFS. Please use exactly the algorithm of the lecture. procedure DFS(G) t = 0 for u V if(color(u) = white), then DFS-Visit(u) procedure DFS-Visit(u) color(u) = grey t++ d[u] = t for all v V such that (u, v) E if(color(v) = white) then DFS-Visit(v) color(u) = black (a) Prove the correctness of DFS. (b) Perform a time analysis for DFS. 2. (Lecture Question) (a) Prove that there exists an eulerian circuit on a connected, undirected graph if and only if each node has even degree. (b) Prove there that there exists an Eulerian tour (circuit) on a connected, directed graph if and only of the total degree (in-degree - out-degree) of each node is 0.. Minimum Spanning Tree. procedureprim(g: undirected, weighted connected graph) T := a minimum weight edge for i := 1 to n 2 e := (u, v) an edge of minimum weight incident to a vertex in T and does not form a loop in T T := T {e} return T (a) Perform the algorithm of Prim on the following graph. Start node is node E. (b) Remember that n = V and m = E. Prim s algorithm needs running time O(n log n + m) if the priority queue (that is needed to keep track of the nodes that are not yet included in the minimum spanning tree) is implemented with e.g. a Fibonacci-Heap. Kruskal s algorithm needs running time O(m log m) Which algorithm is better, i.e. needs less time? Formulate your answer carefully and take into consideration different possible numbers m of edges in comparison to the number n of nodes. 1

2 A B C D E 9 F G H 9 I (c) Prove the correctness of Prim B (d) Prove the time complexity of Prim. 12 A D E G I C. Modelling problems with graphs: Topological Sorting. H Imagine you would have to perform in a project jobs A, B, C, D, E, F with the following F time dependencies: B must be finished before you can start with D, E, F. C must be finished before you can start with B, E. D must be finished before you can start with A, E. E must be finished before you can start with A. F must be finished before you can start with D. (a) Draw the corresponding graph to that problem (Why do you have to draw a directed graph?). (b) When you want to answer if all of the time dependencies can be fulfilled: what graph theoretic problem do you have to solve? Apply the corresponding algorithm to that problem. In what order will you have to perform the jobs? 5. Let G be a partially ordered set (DAG). (A minimal element is one with no incoming edges, u < v is there is a directed path from u to v.) procedure TopSort(G) k=1 while(g is not empty) if(g is non-empty and has no nodes with 0 in-degree) then return false {a k } = all nodes with 0 in-degree G = G - {a k } (remove the node and it s out-edges) k = k + {a k } return (a 1,..., a k ) (a) Prove the correctness of Topological Sort. (b) Prove the time complexity of Topological Sort.. Modelling problems with graphs: Directed Acyclic Graphs. A daily flight schedule is a list of all the flights taking place that day. In a daily flight schedule, each flight F i has an origin city OC i, a destination city DC i, a departure time d i and an arrival time a i > d i. This is an example of a daily flight schedule for August 21, 201, listing flights as F i = (OC i, DC i, d i, a i ): F 1 = (Portland, Los Angeles, :00am, 9:00am) 2

3 F 2 = (Portland, Seattle, 8:00am, 9:00am) F = (Los Angeles, San Francisco, 8:00am, 9:0am) F = (Seattle, Los Angeles, 9:0am, 11:0am) F 5 = (Los Angeles, San Francisco, 12:00pm, 1:00pm) F = (San Francisco, Portland, 1:0pm, :00pm) (a) Describe how to construct a Directed Acyclic Graph (DAG) so that paths in the DAG represent possible sequences of connecting flights a person could take. What are the vertices, and when are two vertices connected with an edge? (b) Why is your graph always a DAG? (c) Draw the DAG you described for the given example of August 21, 201. (d) Use your DAG to help you determine the maximum number of connecting flights a person could take on August 21, Counting. (a) How many different initials can a person have if they have a first name, a middle name, and a last name? (Hint: objects are A, B, C,..., Z). (b) A book publisher has 200 copies of Rosen. How many ways are there to store these books in their three warehouses? (Hint: objects are the 200 books). (c) How many different words can be made by rearranging the letters in the word RUMPELSTILT- SKIN (Hint: objects are the letters)? (d) In a twist of fate, the final becomes 50 True/False problems, 15 of which are True. How many different answer keys can there be? (Hint: objects are the location of either the Trues or the Falses). (e) Fourteen people show up to play for a community baseball game. How many different combinations of teams can be made? Baseball is played with 9 people. (Hint: the object (players) are numbered 1,2,..9). (f) How many bits does it take to store someone s initials in a database, assuming that each person has a first name, a middle name, and a last name? Simplify your answer. (g) How many 8-bit binary strings are there that have at least consecutive 1s (Hint: objects are 0, 1)? 8. Counting. For this problem, let G m,n represent the graph that looks like an m n grid. For example, this is the graph G, which looks like a grid with rows and columns. The questions that follow are about G m,n for general m and n, and all answers should be justified. (a) What is the length of the shortest path from the bottom left corner to the top right corner of G m,n? (b) How many paths of this length are there?

4 (c) How many of these paths start with a north step? (d) How many start with an east step? (e) The total number of paths p (part b) is clearly equal to the number of paths that start with a north step n (part c) and the number of paths that start with an east step e (part d). Give an algebraic proof that p = n + e. 9. (Lecture Question) For all comparison-based sorting algorithms, what is the lower bound complexity in the worst case? (i.e. what is the complexity for the best comparison-based sorting algorithm in the worst case?) 10. In a variant of the game Plinko from the gameshow The Price is Right, the player has one disk which they insert into the top of the board shown below. Each time the disk hits a peg (shown as a black circle), it has a 50% chance of falling to the left, and a 50% chance of falling to the right. Eventually, the disk lands in one of the bins at the bottom of the board. Each bin is marked with a dollar amount, and the player wins the amount of money shown on the bin in which the disk lands. $1000 $500 $100 $100 $500 $1000 (a) Suppose that bin 1 is the leftmost bin, and bins are numbered from 1 to reading from left to right. In terms of i, what is the probability that the disk falls into bin i? (b) How much money does the player expect to win at this game? (c) How much money does the player expect to win if you know that the disk falls to the left the first time it hits a peg? (d) How much money does the player expect to win if you know that the disk falls to the left the first and second time it hits a peg? 11. (a) A teacher randomly calls on students in the class to answer questions. If there are n students in the class, n questions are asked, and each question is asked to one student at random, find the expected number of students that never get called on. (b) Let G be a simple undirected graph with n vertices and m edges. We partition the vertices of G at random into two sets L and R, putting each vertex in L with probability 1/2, and in R with probability 1/2, (independently for each vertex). The cut is the set of edges with one endpoint in L and the other in R. Find the expected number of edges in the cut.

5 12. Suppose that A and B are independent events. Prove that P (A B) = P (A) and explain why this result makes sense. 1. Suppose you have a fair 8-sided die and a fair 15-sided die. How can you use the dice to pick a random integer between 1 and 50 (inclusive) so that each integer is equally likely? Show how you come to your answer. 1. (Lecture Question) What is the minimum number of people who need to be in a room so that the probability that at least two of them have the same birthday is greater than 10%? 0%? 0%? 5