f02-hwex - CS 373: Combinatorial Algorithms, Fall 2002...

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Unformatted text preview: CS 373: Combinatorial Algorithms, Fall 2002 Homework 0, due September 5, 2002 at the beginning of class Name: Net ID: Alias: UG Neatly print your name (first name first, with no comma), your network ID, and an alias of your choice into the boxes above. Circle U if you are an undergraduate, and G if you are a graduate student. Do not sign your name. Do not write your Social Security number. Staple this sheet of paper to the top of your homework. Grades will be listed on the course web site by alias give us, so your alias should not resemble your name or your Net ID. If you don’t give yourself an alias, we’ll give you one that you won’t like. Before you do anything else, please read the Homework Instructions and FAQ on the CS 373 course web page (http://www-courses.cs.uiuc.edu/˜cs373/hwx/faq.html) and then check the box below. There are 300 students in CS 373 this semester; we are quite serious about giving zeros to homeworks that don’t follow the instructions. I have read the CS 373 Homework Instructions and FAQ. Every CS 373 homework has the same basic structure. There are six required problems, some with several subproblems. Each problem is worth 10 points. Only graduate students are required to answer problem 6; undergraduates can turn in a solution for extra credit. There are several practice problems at the end. Stars indicate problems we think are hard. This homework tests your familiarity with the prerequisite material from CS 173, CS 225, and CS 273, primarily to help you identify gaps in your knowledge. You are responsible for filling those gaps on your own. Rosen (the 173/273 textbook), CLRS (especially Chapters 1–7, 10, 12, and A–C), and the lecture notes on recurrences should be sufficient review, but you may want to consult other texts as well. CS 373 Homework 0 (due 9/5/02) Fall 2002 Required Problems 1. Sort the following functions from asymptotically smallest to asymptotically largest, indicating ties if there are any. Please don’t turn in proofs, but you should do them anyway to make sure you’re right (and for practice). 1 nlg n log1000 n n (lg n)n lg1000 n n2 (lg n)lg n lg(1000) n lg n nlg lg n lg(n1000 ) n lg n n1/ lg n 1+ n 1 1000 To simplify notation, write f (n) g (n) to mean f (n) = o(g (n)) and f (n) ≡ g (n) to mean f (n) = Θ(g (n)). For example, the functions n2 , n, n , n3 could be sorted either as n 2 n 2 n2 ≡ n n3 or as n n3 . 2 2 ≡n 2. Solve these recurrences. State tight asymptotic bounds for each function in the form Θ(f (n)) for some recognizable function f (n). Please don’t turn in proofs, but you should do them anyway just for practice. Assume reasonable but nontrivial base cases, and state them if they affect your solution. Extra credit will be given for more exact solutions. [Hint: Most of these are very easy.] A(n) = 2A(n/2) + n B (n) = 3B (n/2) + n C (n) = 2C (n/3) + n D(n) = 2D(n − 1) + 1 E (n) = max 1≤k ≤n/2 F (n) = 9F ( n/3 + 9) + n2 G(n) = 3G(n − 1)/5G(n − 2) √ H (n) = 2H ( n) + 1 I (n) = ∗ 1≤k ≤n/2 1≤k ≤n/2 min I (k ) + I (n − k ) + k J (k ) + J (n − k ) + k E (k ) + E (n − k ) + n J (n) = max 3. Recall that a binary tree is full if every node has either two children (an internal node) or no children (a leaf). Give at least four different proofs of the following fact: In any full binary tree, the number of leaves is exactly one more than the number of internal nodes. For full credit, each proof must be self-contained, the proof must be substantially different from each other, and at least one proof must not use induction. For each n, your nth correct proof is worth n points, so you need four proofs to get full credit. Each correct proof beyond the fourth earns you extra credit. [Hint: I know of at least six different proofs.] 2 CS 373 Homework 0 (due 9/5/02) Fall 2002 4. Most of you are probably familiar with the story behind the Tower of Hano¨ puzzle: 1 ı At the great temple of Benares, there is a brass plate on which three vertical diamond shafts are fixed. On the shafts are mounted n golden disks of decreasing size.2 At the time of creation, the god Brahma placed all of the disks on one pin, in order of size with the largest at the bottom. The Hindu priests unceasingly transfer the disks from peg to peg, one at a time, never placing a larger disk on a smaller one. When all of the disks have been transferred to the last pin, the universe will end. Recently the temple at Benares was relocated to southern California, where the monks are considerably more laid back about their job. At the “Towers of Hollywood”, the golden disks were replaced with painted plywood, and the diamond shafts were replaced with Plexiglas. More importantly, the restriction on the order of the disks was relaxed. While the disks are being moved, the bottom disk on any pin must be the largest disk on that pin, but disks further up in the stack can be in any order. However, after all the disks have been moved, they must be in sorted order again. The Towers of Hollywood. Describe an algorithm3 that moves a stack of n disks from one pin to the another using the smallest possible number of moves. For full credit, your algorithm should be non-recursive, but a recursive algorithm is worth significant partial credit. Exactly how many moves does your algorithm perform? [Hint: The Hollywood monks can bring about the end of the universe quite a bit faster than the original monks at Benares could.] The problem of computing the minimum number of moves was posed in the most recent issue of the American Mathematical Monthly (August/September 2002). No solution has been published yet. The puzzle and the accompanying story were both invented by the French mathematician Eduoard Lucas in 1883. See http://www.cs.wm.edu/˜pkstoc/toh.html 2 In the original legend, n = 64. In the 1883 wooden puzzle, n = 8. 3 Since you’ve read the Homework Instructions, you know exactly what this phrase means. 1 3 CS 373 Homework 0 (due 9/5/02) Fall 2002 5. On their long journey from Denmark to England, Rosencrantz and Guildenstern amuse themselves by playing the following game with a fair coin. First Rosencrantz flips the coin over and over until it comes up tails. Then Guildenstern flips the coin over and over until he gets as many heads in a row as Rosencrantz got on his turn. Here are three typical games: Rosencrantz: H H T Guildenstern: H T H H Rosencrantz: T Guildenstern: (no flips) Rosencrantz: H H H T Guildenstern: T H H T H H T H T H H H (a) What is the expected number of flips in one of Rosencrantz’s turns? (b) Suppose Rosencrantz flips k heads in a row on his turn. What is the expected number of flips in Guildenstern’s next turn? (c) What is the expected total number of flips (by both Rosencrantz and Guildenstern) in a single game? Prove your answers are correct. If you have to appeal to “intuition” or “common sense”, your answer is almost certainly wrong! You must give exact answers for full credit, but asymptotic bounds are worth significant partial credit. 6. [This problem is required only for graduate students (including I2CS students); undergrads can submit a solution for extra credit.] Tatami are rectangular mats used to tile floors in traditional Japanese houses. Exact dimensions of tatami mats vary from one region of Japan to the next, but they are always twice as long in one dimension than in the other. (In Tokyo, the standard size is 180cm×90cm.) (a) How many different ways are there to tile a 2 × n rectangular room with 1 × 2 tatami mats? Set up a recurrence and derive an exact closed-form solution. [Hint: The answer involves a familiar recursive sequence.] (b) According to tradition, tatami mats are always arranged so that four corners never meet. Thus, the first two arrangements below are traditional, but not the third. Two traditional tatami arrangements and one non-traditional arrangement. How many different traditional ways are there to tile a 3 × n rectangular room with 1 × 2 tatami mats? Set up a recurrence and derive an exact closed-form solution. (c) [5 points extra credit] How many different traditional ways are there to tile an n × n square with 1 × 2 tatami mats? Prove your answer is correct. 4 CS 373 Homework 0 (due 9/5/02) Fall 2002 Practice Problems These problems are only for your benefit; other problems can be found in previous semesters’ homeworks on the course web site. You are strongly encouraged to do some of these problems as additional practice. Think of them as potential exam questions (hint, hint). Feel free to ask about any of these questions on the course newsgroup, during office hours, or during review sessions. 1. Removing any edge from a binary tree with n nodes partitions it into two smaller binary trees. If both trees have at least (n − 1)/3 nodes, we say that the partition is balanced. (a) Prove that every binary tree with more than one vertex has a balanced partition. [Hint: I know of at least two different proofs.] (b) If each smaller tree has more than n/3 nodes, we say that the partition is strictly balanced. Show that for every n, there is an n-node binary tree with no strictly balanced partition. 2. Describe an algorithm CountToTenToThe(n) that prints the integers from 1 to 10n . Assume you have a subroutine PrintDigit(d) that prints any integer d between 0 and 9, and another subroutine PrintSpace that prints a space character. Both subroutines run in O(1) time. You may want to write (and analyze) a separate subroutine PrintInteger to print an arbitrary integer. Since integer variables cannot store arbitrarily large values in most programming languages, your algorithm must not store any value larger than max{10, n} in any single integer variable. Thus, the following algorithm is not correct: BogusCountToTenToThe(n): for i ← 1 to Power(10, n) PrintInteger(i) (So what exactly can you pass to PrintInteger?) What is the running time of your algorithm (as a function of n)? How many digits and spaces does it print? How much space does it use? 3. I’m sure you remember the following simple rules for taking derivatives: Simple cases: d dx α = 0 for any constant α, and d dx x =1 d Linearity: dx (f (x) + g (x)) = f (x) + g (x) d The product rule: dx (f (x) · g (x)) = f (x) · g (x) + d The chain rule: dx (f (g (x)) = f (g (x)) · g (x) f (x) · g (x) d Using only these rules and induction, prove that dx xc = cxc−1 for any integer c = −1. Do not use limits, integrals, or any other concepts from calculus, except for the simple identities listed above. [Hint: Don’t forget about negative values of c!] 5 CS 373 Homework 0 (due 9/5/02) Fall 2002 4. This problem asks you to calculate the total resistance between two points in a series-parallel resistor network. Don’t worry if you haven’t taken a circuits class; everything you need to know can be summed up in two sentences and a picture. The total resistance of two resistors in series is the sum of their individual resistances. The total resistance of two resistors in parallel is the reciprocal of the sum of the reciprocals of their individual resistances. x x y x+y y 1 1/x + 1/y Equivalence laws for series-parallel resistor networks. What is the exact total resistance4 of the following resistor networks as a function of n? Prove your answers are correct. [Hint: Use induction. Duh.] (a) A complete binary tree with depth n, with a 1Ω resistor at every node, and a common wire joining all the leaves. Resistance is measured between the root and the leaves. A balanced binary resistor tree with depth 3. (b) A totally unbalanced full binary tree with depth n (every internal node has two children, one of which is a leaf) with a 1Ω resistor at every node, and a common wire joining all the leaves. Resistance is measured between the root and the leaves. A totally unbalanced binary resistor tree with depth 4. (c) A ladder with n rungs, with a 1Ω resistor on every edge. Resistance is measured between the bottom of the legs. A resistor ladder with 5 rungs. 4 The ISO standard unit of resistance is the Ohm, written with the symbol Ω. 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Name: Net ID: Name: Net ID: Name: Net ID: Alias: U 3/4 1 Alias: U 3/4 1 Alias: U 3/4 1 Neatly print your name(s), NetID(s), and the alias(es) you used for Homework 0 in the boxes above. Please also tell us whether you are an undergraduate, 3/4-unit grad student, or 1-unit grad student by circling U, 3/4 , or 1, respectively. Staple this sheet to the top of your homework. Required Problems 1. (10 points) Prove that SAT is still a NP-complete problem even under the following constraints: each variable must show up once as a positive literal and once or twice as a negative literal in the whole expression. For ¯ ¯ ¯ ¯¯ ¯ ¯ ¯¯ instance, (A∨B)∧(A∨C∨ D) ∧ ( A∨B∨C ∨D) satisfies the constraints, while (A∨ B)∧(A∨C∨ D) ∧ (A∨ B∨C ∨D) does not, because positive literal A appears twice. 2. (10 points) A domino is 2 × 1 rectanble divided into two squares, with a certain number of pips(dots) in each square. In most domino games, the players lay down dominos at either end of a single chain. Adjacent dominos in the chain must have matching numbers. (See the figure below.) Describe and analyze an efficient algorithm, or prove that it is NP-complete, to determine wheter a given set of n dominos can be lined up in a single chain. For example, for the sets of dominos shown below, the correct output is TRUE. Top: A set of nine dominos Bottom:The entire set lined up in a single chain 3. (10 points) Prove that the following 2 problems are NP-complete. Given an undirected Graph G = (V, E), a subset of vertices V ⊆V , and a positive integer k: (a) determine whether there is a spanning tree T of G whose leaves are the same as V . (b) determine whether there is a spanning tree T of G whose degree of vertices are all less than k. 4. (10 points) An optimized version of Knapsack problem is defined as follows. Given a finite set of elements U where each element of the set u∈U has its own size s(u) > 0 and the value v(u) > 0, maximize A(U ) = v(u) under the condition prove it. G: Put all the elements u∈U into an array A[i] Sort A[i] by v(u)/ s(u) in a decreasing order S ←0 V ←0 for i ← 0 to NumOfElements if (S + s(u[i]) > B) break S ← S + s(u[i]) V ← V + v(u[i]) return V approximation algorithm. Determine the worst case approximation ratio R(U ) = max Opt(U )/Approx(U ) and U u∈U s(u)≤B and U ⊆U . This problem is NP-hard. Consider the following polynomial time u∈U AA: A1 ← Greedy() A2 ← S ingleElement() return max(A1, A2) SE: Put all the elements u∈U into an array A[i] V←0 for i ← 0 to NumOfElements if ( s(u[i]) ≤ B & V < v(u[i])) V ← v(u[i]) return V 5. (10 points) The recursion fairy’s distant cousin, the reduction genie, shows up one day with a magical gift for you: a box that determines in constant time whether or not a graph is 3-colorable.(A graph is 3-colorable if you can color each of the vertices red, green, or blue, so that every edge has do different colors.) The magic box does not tell you how to color the graph, just wheter or not it can be done. Devise and analyze an algorithm to 3-color any graph in polynomial time using the magic box. 6. (10 points) The following is an NP-hard version of PARTITION problem. PARTITION(NP-): Given a set of n positive integers S = {a i|i = 0 ... n − 1}, minimize max ai , ai ai ∈T ai ∈S −T where T is a subset of S . A polynomial time approximation algorithm is given in what follows. Determine the worst case approximation ratio min Approx(S )/Opt(S ) and prove it. S 2 AA: Sort S in an increasing order s1 ← 0 s2 ← 0 for i ← 0 to n if s1 ≤ s2 s1 ← s1 + ai else s2 ← s2 + a i result ← max( s1, s2) Practice Problems 1. Construct a linear time algorithm for 2 SAT problem. 2. Assume that P N P. Prove that there is no polynomial time approximation algorithm for an optimized version of Knapsack problem, which outputs A(I ) s.t. |Opt(I ) − A(I )| ≤ K for any instance I , where K is a constant. 3. Your friend Toidi is planning to hold a party for the coming Christmas. He wants to take a picture of all the participants including himself, but he is quite shy and thus cannot take a picture of a person whom he does not know very well. Since he has only shy friends, every participant coming to the party is also shy. After a long struggle of thought he came up with a seemingly good idea: • At the beginning, he has a camera. • A person, holding a camera, is able to take a picture of another participant whom the person knows very well, and pass a camera to that participant. • Since he does not want to waste films, everyone has to be taken a picture exactly once. Although there can be some people whom he does not know very well, he knows completely who knows whom well. Therefore, in theory, given a list of all the participants, he can determine if it is possible to take all the pictures using this idea. Since it takes only linear time to take all the pictures if he is brave enough (say “Say cheese!” N times, where N is the number of people), as a student taking CS373, you are highly expected to give him an advice: • show him an efficient algorithm to determine if it is possible to take pictures of all the participants using his idea, given a list of people coming to the party. • or prove that his idea is essentially facing a NP-complete problem, make him give up his idea, and give him an efficient algorithm to practice saying “Say cheese!”: for i ← 0 to N e.g., oops, it takes exponential time... Make him say “Say cheese!” 2i times 4. Show, given a set of numbers, that you can decide wheter it has a subset of size 3 that adds to zero in polynomial time. 3 5. Given a CNF-normalized form that has at most one negative literal in each clause, construct an efficient algorithm to solve the satisfiability problem for these clauses. For instance, ¯ ¯ (A ∨ B ∨ C ) ∧ (B ∨ A), ¯ ¯ (A ∨ B ∨ C) ∧ (B ∨ A ∨ D) ∧ (A ∨ D), ¯ ¯ (A ∨ B) ∧ (B ∨ A ∨ C) ∧ (C ∨ D) satisfy the condition, while ¯ ¯ ¯ (A ∨ B ∨ C ) ∧ (B ∨ A), ¯ ¯¯ (A ∨ B ∨ C) ∧ (B ∨ A ∨ D) ∧ (A ∨ D), ¯ ∨ B) ∧ (B ∨ A ∨ C) ∧ (C ∨ D) ¯ ¯¯ (A do not. 6. The ExactCoverByThrees problem is defined as follows: given a finite set X and a collection C of 3-element subsets of X , does C contain an exact cover for X , that is, a sub-collection C ⊆ C where every element of X occurs in exactly one member of C ? Given that ExactCoverByThrees is NP-complete, show that the similar problem ExactCoverByFours is also NP-complete. 7. The LongestS impleCycle problem is the problem of finding a simple cycle of maximum length in a graph. Convert this to a formal definition of a decision problem and show that it is NP-complete. 4 CS 373 Midterm 1 Questions (October 1, 2002) Fall 2002 Write your answers in the separate answer booklet. 1. Multiple Choice: Each question below has one of the following answers. A: Θ(1) B: Θ(log n) C: Θ(n) D: Θ(n log n) E: Θ(n2 ) X: I don’t know. For each question, write the letter that corresponds to your answer. You do not need to justify your answers. Each correct answer earns you 1 point. Each X earns you 1 point. 4 Each incorrect answer costs you 1 point. Your total score will be rounded down to an 2 integer. Negative scores will be rounded up to zero. i ? n i=1 nn (b) What is ? i=1 i (a) What is (c) How many bits do you need to write 10n in binary? (d) What is the solution of the recurrence T (n) = 9T (n/3) + n? 3 (e) What is the solution of the recurrence T (n) = T (n − 2) + n ? n−17 25 n √ (f) What is the solution of the recurrence T (n) = 5T − lg lg n + πn + 2 log∗ n − 6? (g) What is the worst-case running time of randomized quicksort? (h) The expected time for inserting one item into a randomized treap is O(log n). What is the worst-case time for a sequence of n insertions into an initially empty treap? (i) Suppose StupidAlgorithm produces the correct answer to some problem with probability 1/n. How many times do we have to run StupidAlgorithm to get the correct answer with high probability? (j) Suppose you correctly identify three of the possible answers to this question as obviously wrong. If you choose one of the three remaining answers at random, each with equal probability, what is your expected score for this question? 2. Consider the following algorithm for finding the smallest element in an unsorted array: RandomMin(A[1 .. n]): min ← ∞ for i ← 1 to n in random order if A[i] < min min ← A[i] ( ) return min (a) [1 point] In the worst case, how many times does RandomMin execute line ( )? (b) [3 points] What is the probability that line ( ) is executed during the nth iteration of the for loop? (c) [6 points] What is the exact expected number of executions of line ( )? (A correct Θ() bound is worth 4 points.) 1 CS 373 Midterm 1 Questions (October 1, 2002) Fall 2002 3. Algorithms and data structures were developed millions of years ago by the Martians, but not quite in the same way as the recent development here on Earth. Intelligent life evolved independently on Mars’ two moons, Phobos and Deimos.1 When the two races finally met on the surface of Mars, after thousands of Phobos-orbits2 of separate philosophical, cultural, religious, and scientific development, their disagreements over the proper structure of binary search trees led to a bloody (or more accurately, ichorous) war, ultimately leading to the destruction of all Martian life. A Phobian binary search tree is a full binary tree that stores a set X of search keys. The root of the tree stores the smallest element in X . If X has more than one element, then the left subtree stores all the elements less than some pivot value p, and the right subtree stores everything else. Both subtrees are nonempty Phobian binary search trees. The actual pivot value p is never stored in the tree. A A A A B C C E C H I I M N I N R S T I S T Y A Phobian binary search tree for the set {M, A, R, T, I, N, B, Y, S, C, H, E }. (a) [2 points] Describe and analyze an algorithm Find(x, T ) that returns True if x is stored in the Phobian binary search tree T , and False otherwise. (b) [2 points] Show how to perform a rotation in a Phobian binary search tree in O(1) time. (c) [6 points] A Deimoid binary search tree is almost exactly the same as its Phobian counterpart, except that the largest element is stored at the root, and both subtrees are Deimoid binary search trees. Describe and analyze an algorithm to transform an n-node Phobian binary search tree into a Deimoid binary search tree in O(n) time, using as little additional space as possible. 4. Suppose we are given an array A[1 .. n] with the special property that A[1] ≥ A[2] and A[n − 1] ≤ A[n]. We say that an element A[x] is a local minimum if it is less than or equal to both its neighbors, or more formally, if A[x − 1] ≥ A[x] and A[x] ≤ A[x + 1]. For example, there are five local minima in the following array: 9772137547334869 We can obviously find a local minimum in O(n) time by scanning through the array. Describe and analyze an algorithm that finds a local minimum in O(log n) time. [Hint: With the given boundary conditions, the array must have at least one local minimum. Why?] 1 2 Greek for “fear” and “panic”, respectively. Doesn’t that make you feel better? 1000 Phobos orbits ≈ 1 Earth year 2 CS 373 Midterm 1 Questions (October 1, 2002) Fall 2002 5. [Graduate students must answer this question.] A common supersequence of two strings A and B is a string of minimum total length that includes both the characters of A in order and the characters of B in order. Design and analyze and algorithm to compute the length of the shortest common supersequence of two strings A[1 .. m] and B [1 .. n]. For example, if the input strings are ANTHROHOPOBIOLOGICAL and PRETERDIPLOMATICALLY, your algorithm should output 31, since a shortest common supersequence of those two strings is PREANTHEROHODPOBIOPLOMATGICALLY. You do not need to compute an actual supersequence, just its length. For full credit, your algorithm must run in Θ(nm) time. 3 CS 373 Midterm 2 Questions (November 5, 2002) Fall 2002 Write your answers in the separate answer booklet. This is a 90-minute exam. The clock started when you got the questions. 1. Professor Quasimodo has built a device that automatically rings the bells in the tower of the Cath´drale de Notre Dame de Paris so he can finally visit his true love Esmerelda. Every e hour exactly on the hour (when the minute hand is pointing at the 12), the device rings at least one of the n bells in the tower. Specifically, the ith bell is rung once every i hours. For example, suppose n = 4. If Quasimodo starts his device just after midnight, then his device rings the bells according to the following twelve-hour schedule: 1:00 1 2:00 1 2 3 4 3:00 1 4:00 1 2 5:00 1 6:00 1 2 3 4 7:00 1 8:00 1 2 3 9:00 1 10:00 11:00 12:00 1 2 1 1 2 3 4 What is the amortized number of bells rung per hour, as a function of n? For full credit, give an exact closed-form solution; a correct Θ() bound is worth 5 points. 2. Let G be a directed graph, where every edge u → v has a weight w(u → v ). To compute the shortest paths from a start vertex s to every other node in the graph, the generic single-source shortest path algorithm calls InitSSSP once and then repeatedly calls Relax until there are no more tense edges. InitSSSP(s): dist(s) ← 0 pred(s) ← Null for all vertices v = s dist(v ) ← ∞ pred(v ) ← Null Relax(u → v ): if dist(v ) > dist(u) + w(u → v ) dist(v ) ← dist(u) + w(u → v ) pred(v ) ← u Suppose the input graph has no negative cycles. Let v be an arbitrary vertex in the input graph. Prove that after every call to Relax, if dist(v ) = ∞, then dist(v ) is the total weight of some path from s to v . 3. Suppose we want to maintain a dynamic set of values, subject to the following operations: • Insert(x): Add x to the set (if it isn’t already there). • Print&DeleteRange(a, b): Print and delete every element x in the range a ≤ x ≤ b. For example, if the current set is {1, 5, 3, 4, 8}, then Print&DeleteRange(4, 6) prints the numbers 4 and 5 and changes the set to {1, 3, 8}. Describe and analyze a data structure that supports these operations, each with amortized cost O(log n). 1 CS 373 Midterm 2 Questions (November 5, 2002) Fall 2002 4. (a) [4 pts] Describe and analyze an algorithm to compute the size of the largest connected component of black pixels in an n × n bitmap B [1 .. n, 1 .. n]. For example, given the bitmap below as input, your algorithm should return the number 9, because the largest conected black component (marked with white dots on the right) contains nine pixels. 9 (b) [4 pts] Design and analyze an algorithm Blacken(i, j ) that colors the pixel B [i, j ] black and returns the size of the largest black component in the bitmap. For full credit, the amortized running time of your algorithm (starting with an all-white bitmap) must be as small as possible. For example, at each step in the sequence below, we blacken the pixel marked with an X. The largest black component is marked with white dots; the number underneath shows the correct output of the Blacken algorithm. 9 14 14 16 17 (c) [2 pts] What is the worst-case running time of your Blacken algorithm? 5. [Graduate students must answer this question.] After a grueling 373 midterm, you decide to take the bus home. Since you planned ahead, you have a schedule that lists the times and locations of every stop of every bus in ChampaignUrbana. Unfortunately, there isn’t a single bus that visits both your exam building and your home; you must transfer between bus lines at least once. Describe and analyze an algorithm to determine the sequence of bus rides that will get you home as early as possible, assuming there are b different bus lines, and each bus stops n times per day. Your goal is to minimize your arrival time, not the time you actually spend travelling. Assume that the buses run exactly on schedule, that you have an accurate watch, and that you are too tired to walk between bus stops. 2 CS 373 Final Exam Questions (December 16, 2002) Fall 2002 Write your answers in the separate answer booklet. This is a 180-minute exam. The clock started when you got the questions. 1. The d-dimensional hypercube is the graph defined as follows. There are 2d vertices, each labeled with a different string of d bits. Two vertices are joined by an edge if their labels differ in exactly one bit. 110 111 101 1 01 11 001 010 011 100 0 00 10 000 The 1-dimensional, 2-dimensional, and 3-dimensional hypercubes. (b) [2 pts] Which hypercubes have an Eulerian circuit (a closed walk that visits every edge exactly once)? [Hint: This is very easy.] (a) [8 pts] Recall that a Hamiltonian cycle passes through every vertex in a graph exactly once. Prove that for all d ≥ 2, the d-dimensional hypercube has a Hamiltonian cycle. 2. A looped tree is a weighted, directed graph built from a binary tree by adding an edge from every leaf back to the root. Every edge has a non-negative weight. The number of nodes in the graph is n. 5 8 100 4 17 0 1 42 373 23 9 14 (a) How long would it take Dijkstra’s algorithm to compute the shortest path between two vertices u and v in a looped tree? (b) Describe and analyze a faster algorithm. 3. Prove that (x + y )p ≡ xp + y p (mod p) for any prime number p. 1 CS 373 Final Exam Questions (December 16, 2002) Fall 2002 4. A palindrome is a string that reads the same forwards and backwards, like X, 373, noon, redivider, or amanaplanacatahamayakayamahatacanalpanama. Any string can be written as a sequence of palindromes. For example, the string bubbaseesabanana (‘Bubba sees a banana.’) can be decomposed in several ways; for example: bub + baseesab + anana b + u + bb + a + sees + aba + nan + a b + u + bb + a + sees + a + b + anana b+u+b+b+a+s+e+e+s+a+b+a+n+a+n+a Describe an efficient algorithm to find the minimum number of palindromes that make up a given input string. For example, given the input string bubbaseesabanana, your algorithm would return the number 3. 5. Your boss wants you to find a perfect hash function for mapping a known set of n items into a table of size m. A hash function is perfect if there are no collisions; each of the n items is mapped to a different slot in the hash table. Of course, this requires that m ≥ n. After cursing your 373 instructor for not teaching you about perfect hashing, you decide to try something simple: repeatedly pick random hash functions until you find one that happens to be perfect. A random hash function h satisfies two properties: • Pr h(x) = h(y ) = • Pr h(x) = i = 1 m 1 m for any pair of items x = y . for any item x and any integer 1 ≤ i ≤ m. (a) [2 pts] Suppose you pick a random hash function h. What is the exact expected number of collisions, as a function of n (the number of items) and m (the size of the table)? Don’t worry about how to resolve collisions; just count them. (b) [2 pts] What is the exact probability that a random hash function is perfect? (c) [2 pts] What is the exact expected number of different random hash functions you have to test before you find a perfect hash function? (d) [2 pts] What is the exact probability that none of the first N random hash functions you try is perfect? (e) [2 pts] How many random hash functions do you have to test to find a perfect hash function with high probability ? To get full credit for parts (a)–(d), give exact closed-form solutions; correct Θ(·) bounds are worth significant partial credit. Part (e) requires only a Θ(·) bound; an exact answer is worth extra credit. 2 CS 373 Final Exam Questions (December 16, 2002) Fall 2002 6. Your friend Toidi is planning to hold a Christmas party. He wants to take a picture of all the participants, including himself, but he is quite shy and thus cannot take a picture of a person whom he does not know very well. Since he has only shy friends1 , everyone at the party is also shy. After thinking hard for a long time, he came up with a seemingly good idea: • Toidi brings a disposable camera to the party. • Anyone holding the camera can take a picture of someone they know very well, and then pass the camera to that person. • In order not to waste any film, every person must have their picture taken exactly once. Although there can be some people Toidi does not know very well, he knows completely who knows whom well. Thus, in principle, given a list of all the participants, he can determine whether it is possible to take all the pictures using this idea. But how quickly? Either describe an efficient algorithm to solve Toidi’s problem, or show that the problem is NP-complete. 7. The recursion fairy’s cousin, the reduction genie, shows up one day with a magical gift for you: a box that can solve the NP-complete Partition problem in constant time! Given a set of positive integers as input, the magic box can tell you in constant time it can be split into two subsets whose total weights are equal. For example, given the set {1, 4, 5, 7, 9} as input, the magic box cheerily yells “YES!”, because that set can be split into {1, 5, 7} and {4, 9}, which both add up to 13. Given the set {1, 4, 5, 7, 8}, however, the magic box mutters a sad “Sorry, no.” The magic box does not tell you how to partition the set, only whether or not it can be done. Describe an algorithm to actually split a set of numbers into two subsets whose sums are equal, in polynomial time, using this magic box.2 1 2 Except you, of course. Unfortunately, you can’t go to the party because you’re taking a final exam. Sorry! Your solution to problem 4 in homework 1 does not solve this problem in polynomial time. 3 ...
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