Qualifying Exam Theory

Transcription

1 Theory This exam contains 8 problems, some with multiple parts. (including 2 blank pages). There are 11 pages to the exam Write your solutions in the space provided. If you need more space, write on the back of the sheet containing the problem or use the last two blank pages. Do not put part of the answer to one problem on the back of the sheet for another problem. Show your work, as partial credit will be given. You will be graded not only on the correctness of your answer, but also on the clarity with which you express it. Be neat! Good Luck! Problem 1. 2.a 2.b 3. 4.a 4.b 4.c 5.a 5.b Final Result Grade 1

2 1. You are placed inside of a forest (a forest of actual living trees, not the graph theoretic kind). Assume that you are given a list containing the (x, y) coordinates of each of the trees. You are also given your own (x, y) coordinate at which you are standing. You have a camera with a fixed field of view, of which you know the angle. Devise an efficient algorithm to determine by which angle you should rotate to take a picture such that the picture will contain the maximum number of trees. Assume that occlusion is not a problem (i.e., each tree trunk is infinitely thin). 2

3 2. The Approximate searching problem concerns the design of a data structure for representing a set of numbers S upon which the two operations Insert(x) and Approx-Search(x) can be performed. Approx-Search(x) operation returns a value z in S satisfying the condition that number of values w in S that lie between x and z does not exceed n/ lg n, where n is current size of S. No other requirement is imposed upon z, and any such z is considered an appropriate response to this operation. a. Give an order magnitude lower bound for the worst case cost for the Approx-Search(x) operation, assuming a comparison based data structure. b. Sketch a solution for the approximate searching problem with good amortized costs. 3

4 3. Given a directed graph G = (V, E) with two distinct vertices s, t V along with a non-negative number k(v) on each vertex v V, you are required to determine whether or not you can disconnect all paths from s to t by deleting, for each vertex v V, at most k(v) arcs incident from v. State an efficient algorithm for this problem and specify its worst-case running time as a function of m = E and/or n = V. 4

5 Theory Spring A list L = {a 1,..., a n } of n distinct elements is said to be k-sorted if i Rank(a i ) k, i {1,..., n}, where Rank(a i ) counts the number of elements less than a i (no item is more than k places from its position in the sorted list). We work in a decision tree model. a. Give an efficient algorithm to k-sort a list and state the complexity of your algorithm as a function of n and k. For what values of k does your algorithm run in time o(n lg n). b. If list L is k-sorted, give an efficient algorithm to select the item of rank j. What is its complexity? c. If list L is k-sorted, give an efficient algorithm to sort L completely. What is its complexity? 5

6 5. This problem has multiple parts. a. Let Σ = {0, 1, +, =} and Add = {x = y + z x, y, z are binary integers, and x is the sum of y and z}. Show that Add is not a regular language. b. Let D = {w w contains an equal number of the substrings 01 and 10}. Thus 101 D, but 1010 D. Show that D is a regular language. 6

7 6. Prove the following stronger form of the pumping lemma, wherein we require both pieces v and y to be nonempty when the string s is broken up. If A is a context-free language, then there is a number k where, if s is any string in A of length at least k, then s may b divided into five pieces s = uvxyz, satisfying the conditions: I. For each i 0, uv i xy i z A. II. v ɛ and y ɛ, and III. vxy k. 7

8 7. A Turing Machine with Stay Put Instead of Left (TMSPIL) is similar to an ordinary Turing Machine (TM) except that the transition funcion has the form δ : Q Γ Q Γ {R, S}. At each point the TMSPIL can move its head right or let it stay in the same position. Show that TMSPIL is not equivalent to the usual version of TM. What class of languages do these machines recognize? 8

9 Theory Spring A coin collector has an opportunity to purchase n coin collections S 1, S 2,..., S n. Some collections may contain coins in common with the others. No collection contains more than n 2 coins. The collector wants to buy at least k collections out of the n, but must avoid duplicates. The Coin Collector s Problem (CCP) is whether there is a selection of at least k mutually disjoint collection out of n. Show that CCP is NP-complete or devise a polynomial time algorithm for it. 9

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