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CS231 Algorithms Handout # Prof Lyn Turbak February 8, 2001 Wellesley College
PROBLEM SET 2 Due: Thursday, February 15
Reading: CLR Sections 6.1, 6.2, 6.3, & 6.6; Start reading Chapters 7 & 8.
Suggested Problems: 6.1-1, 6.1-2, 6.1-3, 6.1-4; 6.2-2, 6.2-3, 6.2-4, 6.2-11; 6.3-1, 6.3-2, 6.3-3 ; 6.6-1; 6-2;
Required Problems. Write up and turn in the solutions to the problems listed below. Notes:
- The last problem (Problem 5) does not depend on combinatorics or probability, so you may want to do it first. It is last only because its problem statement is longer than that of the other problems.
- Problems 4a and 5d require you to design algorithms rather than just analyze them. I recommend that you look at these problems early on so that you have sufficient time to think about them. Just having them in the back of your mind can sometimes lead to unexpected progress. The kind of creativity involved in algorithm design is difficult to summon at the last minute!
Problem 1: Let Me Count the Ways ... [20]
In the following parts,. please show how you derive your answers:
a [5] How many ways can 12 students be assigned to the 15 computers in E101? (Assume each student is assigned to one computer.)
b [5] Before we started running out of telephone numbers and changes were made to the system, it used to be that a long distance phone number was an area code followed by a 7- digit local number. An area code was a 3-digit number whose middle digit had to be a 0 or 1, whose first digit could not be 0 or 1, and whose last two digits could not be 11. The first 3 digits of the 7-digit local number could not be an area code. Under this scheme, how many long-distance phone numbers were there?
c [5] How many terms are there in the expansion of (w + x + y + z)^7? Each term has the
form cwe1xe2ye3ze4 , where e1 + e2 + e3 + e4 = 7.
d [5] How many ways can 100 students be scheduled into 5 different writing classes so that each class has exactly 20 students and each student takes one class?
Problem 2: Randomizing Arrays [15]
Many array manipulation algorithms assume that the elements of an input array are randomly permuted. An array of n distinct items is randomly permuted if each of the n! permutations is equally likely. In many cases, the array input to such an algorithm is the output of another process that is decidedly non-random. So it is desirable to find an efficient way to randomly permute a given array.
Suppose that:
- A is a length n array containing all the integers from 1 to n.
- swap(A, i, j) swaps the values in A[i] and A[j].
- random(x, y) is a routine that returns a random integer between x and y, inclusive.
For each of the following algorithms, indicate whether the algorithm randomly permutes the input array A. Justify your answer in each case. ( Hint: try the algorithms for n = 3, and generalize.)
a [5] for i <- 1 to n do A[i] <- random(1, n)
b [5] for i <- 1 to n do swap(A, i, random(1, n))
c [5] for i <- 1 to n do swap(A, i, random(i, n))
Problem 3: The Infamous Monty Hall Problem [10]
Do CLR Exercise 6.2-10 (p. 110). Justify your answer with a probability tree. It is important to note that the emcee knows where the prize is hidden, and so will always be able to reveal an empty stage behind one of the curtains you have not picked. (This is a classic problem that got national attention a few years back when it appeared in a popular column by Marilyn vos Savant. Many readers of her column, including those with math PhDs, made fools of themselves by adamantly arguing the wrong answer.)
Problem 4: Coin Flipping [25]
We all know that a flip of a fair coin can be used to choose one of two possibilities with
equal probability. It is not hard to see that a n flips of a fair coin can choose one of 2 n possibilities with equal probability. But did you know that a fair coin can be used to choose among m possibilities, where m is not a power of 2? In this problem, you will explore the case of m = 3, but the strategy generalizes to any m.
Suppose you are given a black-box procedure Flip() that returns one of the symbols H or T, each with a probability of 1/2.
a [15] Using Flip(), write pseudocode for a procedure One-Third() that returns one of the symbols A, B, or C, each with a probability of 1/3. ( Hints: (1) think of the paper- tearing exercise from class and (2) make One-Third() recursive!)
b [5] Demonstrate that your One-Third() procedure from part a is correct by showing that the probability of returning A is 1/3. (Presumably, the argument for B and C will be similar, so you need not show it.)
c [5] Determine the expected number of times that your One-Third() procedure will call Flip() before returning a result.
c. [5] Another way to search for k in a 2d-sorted n x n array is to use divide and conquer to generalize Binary-Search to work on a 2d-sorted array A. Here’s the idea: suppose that p is the value in the array at A[n/2, n/2]. (See the following picture. For simplicity, assume that n is a power of 2 throughout this part.)
X
Y
W
Z
p
n/
n/
Then if k = p , the search terminates with true ; if k < p , the search continues in the three square quadrants W, X, and Y; and if k > p , the search continues in the three square quadrants X, Y, and Z. Give the worst-case running time of this algorithm as a function of n. Use the iteration method on a recurrence equation to justify your answer.
d. [10] Develop an a Θ(n) worst-case running time algorithm that implements 2D-Search on a 2d-sorted n x n array. Give pseudocode for your algorithm, and justify why it is
Θ(n). Hint: Any number k can be used to divide a 2d-sorted array into two parts: one consiting of the elements <= k , and the other consisting of the elements > k. For example, the following figure shows a thick line separating the two parts for the sample array B and k = 20.
2 4 7 11 12 13 19 23
3
6
9
11
14
e. [5] Rank the algorithms from parts a through d by running time. An algorithm with
running time f should appear above an algorithm with running time g if f = ω( g ), and
should appear on the same line if f = Θ( g ).
Extra Credit Problem 1: Envelopes [20] (The best probability problem I know of! The following wording comes from Uri Wilensky’s MIT Phd thesis)
There are two envelopes in front of you. One contains x dollars and the other contains 2x dollars. You choose one, open it, and get $100. You are then offered a switch if you want it. You can take the other envelope instead.
Here are two arguments:
- There were 2 envelopes, one contained x dollars and the other 2x. You got either x or 2x. Each was equally likely, so switching doesn’t help.
- The other envelope contains either $200 or $50. These are equally likely. So the expected value of the other envelope is ($200 + $50)/2 = $125. You should switch.
Which if any of these arguments do you believe? What is the bug in the other argument?
Extra Credit Problem 2: Poker Time! [30]
There are many variants of poker; here we describe the form most commonly played on video poker machines.
A poker hand is a collection of five cards. We will write cards as a value (2 through 10, J (jack), Q (queen), K (king), or A (ace)) followed by a suit (C (clubs), D (diamonds), H (hearts), S (spades)). We will write a poker hand as comma delimited cards within square brackets, e.g. [2H, 5C, 6H, JS, AD]. The order of cards in a hand does not matter, but by convention we will write them in sorted order from low to high. The order of a card is first determined by its value (2 is lowest, ace is highest); two cards with the same value are order by suit, in the sequence (from low to high) clubs, diamonds, hearts, spades.
A poker hand can be classified into the following categories; in cases where more than one category applies, the one appearing earliest in the list is chosen:
Royal Flush: a ten, jack, queen, king, and ace of the same suit. E.g.: [10H, JH, QH, KH, AH]
Straight Flush: a sequence of five consecutive values of the same suit. E.g.: [AC, 2C, 3C, 4C, 5C], [8S, 9S, 10S, JS, QS]. Note that for the purposes of a straight, an ace can count as the lowest card as well as the highest.
Four of a Kind: a hand that contains four cards with the same value: E.g.: [6C, 6D, 6H, 6S, 9D], [2S, 3C, 3D, 3H, 3S]
Full House : a hand that contains three of one value of card and two of another. E.g.: [7C, 7H, JD, JH, JS], [2C, 2H, 2S, 8D, 8H]
Flush: a hand whose cards are all of the same suit. E.g.: [3C, 5C, 8C, 9C, QC], [8S, 9S, 10S, JS, KS]
Straight: a sequence of five consecutive values. Note that for the purposes of a straight, an ace can count as the lowest card as well as the highest. E.g.: [AC, 2H, 3H, 4D, 5S] , [4D, 5H, 6D, 7C, 8S], [10H, JH, QH, KC, AH]
advertising to attract others.
There are a few video poker machines with a different payoff schedule that has greater than 100% return for perfect play. These machines are coveted by professional gamblers, who will often play at them for stretches of 24 hours or longer. The key to winning on such a machine is to play until hitting a royal flush, which has a high relative payoff. Why would casinos have such machines? Because they attract customers, and few people can play them optimally. Any deviance from optimal play usually leads to an advantage for the casino.
Problem Set Header Page Please make this the first page of your hardcopy submission.
CS231 Problem Set 2
Due Thursday, February 15, 2001
Name:
Date & Time Submitted (only if late) :
Collaborators ( anyone you collaborated with in the process of doing the problem set) :
In the Time column, please estimate the time you spent on the parts of this problem set. Please try to be as accurate as possible; this information will help me to design future problem sets. I will fill out the Score column when grading your problem set.
Part Time Score
General Reading
Problem 1 [20]
Problem 2 [15]
Problem 3 [10]
Problem 4 [25]
Problem 5 [30]
Extra Credit 1 [20]
Extra Credit 2 [30]
Total
