1 Greedy Graph Algorithms

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1 +//+ Algorithms & Data Structures (DL) Exam of 7 December 5 Instructions: This is a multiple-choice exam, to save ou the time of tiding up the presentation of our answers. There is exactl one correct answer per question. You can keep the question sheets and should hand in onl the answer sheet at the end: ou are not expected to explain our answers. Normall, a teacher will attend this exam from 5: to :. Also read the instructions on the answer sheet before starting. Greed Graph Algorithms Consider the following connected weighted graph, provided twice for our convenience: t s u t s u v w v w Recall that Prim s minimum spanning tree algorithm and Dijkstra s single-source shortest paths algorithm both take (V ) T Extract-Min + (E) T Decrease-Ke time on a graph with vertex set V and edge/arc set E. Answer the following questions after performing both algorithms (all edges being bidirectional when seen as arcs) starting from vertex s and, when dequeueing, breaking ties b choosing the vertex that has had the minimum priorit for the longest time: Question : What is the sum of the priorities of vertex v after Prim s nd and rd iterations? A B C D E 8 Question : What is the sum of the priorities of vertex v after Dijkstra s nd and rd iterations? A 9 B C D E Answer the following general questions about Prim s and Dijkstra s algorithms for a graph with vertex set V and edge/arc set E: Question : Under what condition on V and E should one implement the min-priorit queue of these algorithms with a binar heap or a binomial heap, rather than with an arra, indexed b the vertices, which is generall worse? C if E = V B if E < V / lg V D if E > V / lg V

2 +//59+ Question : Assuming distinct weights, under what condition on V and E does the edge with the lowest weight belong to a minimum spanning tree? B if E < V / lg V C if E > V D if E > V / lg V Question 5: Assuming distinct weights and at least three edges, under what condition on V and E does the edge with the highest weight belong to a minimum spanning tree? B if E < V / lg V C if E = V D if E > V / lg V Question : Under what condition do the shortest paths computed b Dijkstra s algorithm form a spanning tree? B if there is optimal substructure C if the graph is strongl connected D if all arc weights satisf the triangle inequalit Dnamic Programming Given n objects of n volumes v,...,v n and prices p,...,p n, as well as a bag of n volume V, we would like to determine the total price of the most expensive subset of the given objects that fits into the given bag. All numbers are positive integers. For example, given four objects of volumes,,, and prices,,, 5 respectivel, the most expensive subset fitting into a bag of volume V = 5 includes all except the third object; it has total volume + + = 5 apple V and total price = 7. Consider the following recurrence for a quantit T [i, v], parameterised b,,..., 5, and : 8 >< if T [i, v] = T [i, ] if and v< >: (T [i, ], T[i, ]+ 5 ) if and v Question 7: If T [n, V ] is returned b a correct algorithm for the specification above, then the integer T [i, v], with apple i apple n and apple v apple V, denotes the total price of... A... the most expensive subset of exactl i objects that fits into a bag of volume v. B... the most expensive subset of at least i objects that fits into a bag of volume v. C... the most expensive subset of at most i objects that fits into a bag of volume v. D... the most expensive subset of the first i objects that fits into a bag of volume v. E... the most expensive subset of i random objects that fits into a bag of volume v. Question 8: For the instance in the example above, what is the sum T [, ] + T [, ]? A B C 5 D 5 E other Question 9: What is the logical condition? A i = B v = C v = V D v = i E v + i =

3 +//58+ Question : What is the sum of the numeric expressions,..., 5? A v v i + p i C v + p i B v v i + p i D v + p i Question : What is the two-argument operator E v + v i + p i (written in infix form above)? A + B C D max E min Question : What is an ordering of the indices i and v under which the elements T [i, v] of the arra T can be filled without performing an redundant computations? A an order B b lexicographicall increasing hv, ii C b lexicographicall decreasing hi, vi D b lexicographicall increasing h E b lexicographicall increasing h i, vi v, ii We sa that ha, bi is lexicographicall smaller than hc, di if either a<cor a = c ^ b<d. Disjoint Sets Consider the forest on the right, representing disjoint sets for the integers to 8. Assume the ranks of the trees initiall are equal to their heights, measured as numbers of arcs. Assume the Find-Set operation performs path compression. Assume the Union operation follows the union-b-rank strateg. (Ignore the fact that this forest cannot be obtained using this Union operation when starting from singleton sets.) Answer the following questions, continuing at each question from the result of performing the operation of the previous question: Question : Which element is the parent of element after Union(, )? A is a root B C D E another element Question : What is the rank of the tree with element after Union(, )? A B C D E another value Question 5: Which element is the parent of element after Find-Set()? A is a root B C D 5 E another element Question : What is the rank of the tree with element 8 after Find-Set(8)? A B C D E another value

4 +//57+ P versus NP Complete the two sentences and answer the next question, under current ledge: Question 7: A decision problem is in NP if and onl if its answer takes... A... non-polnomial time to check. B... non-polnomial time to compute. C... polnomial time to check. D... polnomial time to compute. E... possibl forever to compute. Question 8: A decision problem is NP-hard if and onl if... A... it can be reduced in polnomial time to ever problem in NP. B... it can be reduced in polnomial time to ever NP-complete problem. C... ever problem in P can be reduced in polnomial time to it. D... ever problem in NP can be reduced in polnomial time to it. E... ever NP-complete problem can be reduced in polnomial time to it. Question 9: When can all instances of an NP-complete problem be solved in poltime? B if and onl if P ( NP C if and onl if P = NP D never Give the smallest complexit class, as currentl n, for each of the next problems: Question : The decision version of the problem of Section, when the bag volume V must be exactl reached and the notion of price is dropped. Question : The problem of determining the existence of a flow of at least a given value within a flow network with rational capacities. Question : The problem of determining the existence of an occurrence of a given string within a given longer string. Question : The problem of determining the existence of a plan for a robotic arm to drill holes at given points on a computer chip and then to return to its starting point, all this within a given amount of time.

5 Corrected Answer Sheet Exam DL of 7 December 5 Instructions: Do not alter the drawing above. Using a dark colour, fill in entirel at most one answer box (A to E) per question: we will use an optical character recognition (OCR) sstem that ignores crosses, circles, etc. Transfer our answers from the question sheets to this answer sheet just before handing in; if an answer becomes ambiguous to an OCR sstem, then please request another answer sheet. Ever correct answer gives points. Ever multiple answer or incorrect answer gives points. Partial credit of point ma be given in exceptional circumstances. If ou think a question is unclear or wrong, then mark its number with a? on this sheet, and explain on the backside of this sheet what our di cult with the question is and what additional assumption underlies the candidate answer that ou have chosen or the new answer that ou have indicated. Grading: Your grade is as follows, when our mark is e points: Greed Graph Algorithms Disjoint Sets Grade Condition 5 8 apple e apple apple e apple 7 apple e apple 9 U apple e apple Question : A B C D E Question : A B C D E Question : A B C D E Question : A B C D E Question 5: A B C D E Question : A B C D E Dnamic Programming Question 7: A B C D E Question 8: A B C D E Question 9: A B C D E Question : A B C D E Question : A B C D E Question : A B C D E Question : A B C D E Question : A B C D E Question 5: A B C D E Question : A B C D E P versus NP Question 7: A B C D E Question 8: A B C D E Question 9: A B C D E Question : A B C D E Question : A B C D E Question : A B C D E Question : A B C D E Again: Please use a dark colour to fill in our chosen boxes entirel! Your anonmous exam code: