Mathematics Masters Examination

Transcription

1 Mathematics Masters Examination OPTION 4 Fall 2011 COMPUTER SCIENCE??TIME?? NOTE: Any student whose answers require clarification may be required to submit to an oral examination. Each of the twelve numbered questions is worth 20 points. All questions will be graded, but your score for the examination will be the sum of your scores on your eight best questions. Please observe the following: DO NOT answer two or more questions on the same sheet (not even on both sides of the same sheet). DO NOT write your name on any of your answer sheets. You will be given separate instructions on the use of these answer sheets. When you have completed a question, place it in the large envelope provided. Fall 09 CS 1/4

2 Computer Algorithms 1. Let G = (V E) be a connected undirected graph and T be a depth-first tree of G. A vertex v is a cut vertex of G if the removal of v disconnects G. Let v be a vertex different from the root of T. Show that v is an articulation vertex iff v has a child u such that there is no back edge between a descendant of u and a proper ancestor of v. Solution: Use the fact that every edge is either a tree edge or a back edge of T. ( ) The removal of v disconnects u and the root of T. ( ) We show that if for every child u there is a back edge from a descendant of u to a proper ancestor of v then after removing v, for every vertex w there remains a path from w to the root (and thus v is not an articulation point). This is obvious if w is not a descendant of v. If w is a descendant of v then let u be the child of v that is an ancestor of w. The claim follows by considering the back edge from a descendant of u to a proper ancestor of v. 2. Give a O(n 2 ) dynamic programming algorithm to find the length of a longest monotonically increasing subsequence of a sequence of n numbers. Solution: Let the sequence be a 1... a n, and the length of the longest monotonically increasing subsequence ending with a i be l(i). Then l(1) = 1 and l(i) = max(l(j) : j < i a j < a i }+1. Combinatorics 3. Find and prove a formula for r,s,t A ( )( )( m1 m2 m3 where A is the set of all nonnegative integers r s t with sum r +s+t = n. Solution Using the identity a )( b ) ( i( i k i = a+b ) ( k twice we obtain m1 +m 2 +m 3 ) n. r 4. We are given 7 colors. What is the number of inequivalent ways to color the corners of a square? (here two colorings are equivalent if one can be obtained from another by a rotation or reflection) s Fall 09 CS 2/4 t )

3 Solution We plug in the values 7 in the cycle index of D 4, and obtain 3247/8 = 406 Graph Theory 5. Let G = (V E) be a simple undirected connected graph with n = V vertices and m = E -edges. 1. Define the adjacency matrix A and the incidence matrix M of G. 2. Let D := MM T A. Give the values of all the entries of the matrix D. 3. Assume that all cycles of G are even. Show that m n2. For which graphs equality 4 holds? (Justify.) Solution. a. Label the vertices of G as V := {1... n} and the edges of G as E := {e 1... e m }. Then A = [a ij ] where i j V. a ij = 0 if the edge (i j) is not in E. Otherwise a ij = 1. Note that a ii = 0 since G. M = [m ie ] where i V e E. m ie = 1 if i is an end point of e. Otherwise m ie = 0. b. D is a diagonal matrix where the element on the (i i) entry is d i, the degree of the vertex i. c. Since all cycles of G are even, G is bypartite. So V = V 1 V 2, where V 1 = p V 2 = q and p+q = n. Then number of maximal edges is pq, where each vertex in V 1 is connected to V 2. Since n = p+q caluclus implies that pq ( n 2 )2. Equality holds iff and only if p = q and G is the complete bipartite graph K p,p Define the graph Q d = (V d E d ), the d-dimensional cube. (Hint: The vertices of Q d are points in R d.) 2. How many vertices and edges Q d has? 3. Consider the vertices x = (0... 0) and y = (1... 1) of Q d. Describe a minimum vertex cut of Q d separating x and y, and a maximum collection of edge-disjoint xypaths. Fall 09 CS 3/4

4 Solution. a. VerticesofQ d areallpointsinr d whosecoordinatesareeither0or1. x = (x 1... x d ) y = (y 1... y d ) are joined by a an edge iff d i=1 x i y i = 1. b. Then number of vertices in Q d is 2 d. Each vertex has degree d. Hence the number of edges is d2 d 1 = d2d 2. c. Since the degree of each vertex is d the number of vertex disjoint paths is at most d. We claim that there exist exactly d vertex disjoint paths from x to y. Let e j = (δ j1... δ jd ). Then for each k let x 0,k = x and x j,k = x j 1,k +e k+j 1 for j = 1... d, where ed+i := e i for i 1. A vertex separating set can be all d neighbors of x. Error-Correcting Codes and Cryptography Suppose you discover that for n = we have mod n and mod n. Use this information to factorize n. 2. Show that if gcd(e 24) = 1, then e 2 1 mod 24. Solution State the Miller-Rabin Primality test. 2. Use this test with a=2 to show n = 561 is composite. Fall 09 CS 4/4

5 Solution Theory of Computation 9. Use the algorithm converting NFA to regular expressions to find a regular expression equivalenttothefollowingnfa:({a b} {q 0 q 1 } {q 0 } δ F),whereδ(q 0 a) = {q 0 q 1 } δ(q 0 b) = {q 1 } δ(q 1 a) = δ(q 1 b) = {q 0 } and F = {q 0 }. Note: no credit is given for finding the regular expression without using the algorithm. Solution: After ripping 1 and then 2 we get a (a b)(ba (a b)) ba. 10. Consider the language L = { M : M is a TM, which accepts some palindrome}. a) Is L Turing decidable? b) Is L Turing recognizable? Note: recall that a palindrome is a string which is equal to its reverse. Solution: a): not Turing decidable. This follows from Rice s theorem as the property defining L is not trivial and it depends on the language accepted by the Turing machine. b) Turing recognizable. The TM M recognizing L works as follows: given an input M, for every i = , it lists all strings w of length at most i, checks if w is a palindrome and if it is then it simulates M on w for i steps. If M accepts w then M accepts M, otherwise it continues. Numerical Analysis 11. Consider the points ( ), ( ), ( ), and ( ). 1. Compute the table of divided differences for Newton interpolation. 2. Give the Newton form of the polynomial interpolating through the given points. 12. Consider the quadrature formula b a f(x)dx wf(x 1 )+wf(x 2 ). Formulate the conditions on w, x 1, and x 2 so that the degree of the polynomials integrated exactly by this quadrature formula is as high as possible. Do not solve this system of conditions. Fall 09 CS 5/4