Midterm 2 March 10, 2014 Name: NetID: # Total Score

Transcription

1 CS 3 : Algorithm and Model of Computation, Spring 0 Midterm March 0, 0 Name: NetID: # 3 Total Score Max Grader Don t panic! Pleae print your name and your NetID in the boxe above. Thi i a cloed-book, cloed-note, cloed-electronic exam. If you brought anything except your writing implement and your double-ided ½" " cheat heet, pleae put it away for the duration of the exam. In particular, you may not ue any electronic device. Pleae read the entire exam before writing anything. Pleae ak for clarification if any quetion i unclear. You have 0 minute. If you run out of pace for an anwer, continue on the back of the page, or on the blank page at the end of thi booklet, but pleae tell u where to look. Alternatively, feel free to tear out the blank page and ue them a cratch paper. Pleae return your cheat heet and all cratch paper with your anwer booklet. We will return everything with your graded exam. If you ue a greedy algorithm, you mut prove that it i correct to receive credit. Otherwie, proof are required only if we pecifically ak for them. A uual, anwering any (ub)problem with I don t know (and nothing ele) i worth % partial credit. Correct, complete, but uboptimal olution are alway worth more than %. A blank anwer i not the ame a I don t know.

2 . [0 point] Clearly indicate the four indicated panning tree of the ame undirected edgeweighted graph (a) A breadth-firt earch tree rooted at. (b) A hortet-path tree rooted at (c) The minimum panning tree. (d) The maximum panning tree (extra copie for cratch work)

3 . [ point] Let L be the et of tring {0 n n n 0}. For example, the tring 00 and 0000 are in L, but the tring 00 and 00 are not. (a) Decribe a context-free grammar for L. (b) Decribe a context-free grammar for the complement {0, } \ L. In both grammar, give a brief decription of the language generated by each non-terminal.

4 3. [0 point] Recall that a palindrome i any tring that i the ame a it reveral. For example, I, DAD, HANNAH, AIBOHPHOBIA (fear of palindrome), and the empty tring are all palindrome. (a) Decribe and analyze an algorithm to find the length of the longet ubtring (not ubequence!) of a given input tring that i a palindrome. For example, BASEESAB i the longet palindrome ubtring of BUBBASEESABANANA ( Bubba ee a banana. ). Thu, given the input tring BUBBASEESABANANA, your algorithm hould return the integer. (b) Any tring can be decompoed into a equence of palindrome ubtring. For example, the tring BUBBASEESABANANA can be broken into palindrome in the following way (and many other): 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 Decribe and analyze an algorithm to find the mallet number of palindrome that make up a given input tring. For example, given the input tring BUBBASEESABANANA, your algorithm hould return the integer 3. 3

5 (additional pace for problem 3)

6 . [ point] Binaria ue coin whoe value are,,,..., k, the firt k power of two, for ome integer k. A in mot countrie, Binarian hopkeeper alway make change uing the following greedy algorithm: MAKECHANGE(N): if N = 0 ay Thank you, come again! ele c larget coin value uch that c N give the cutomer one c coin MAKECHANGE(N c) For example, to make 3 in change, the hopkeeper would give the cutomer a 3 coin, a coin, and a coin, and then ay Thank you, come again! (For purpoe of thi problem, aume that every hopkeeper ha an unlimited upply of each type of coin.) Prove that thi greedy algorithm alway ue the mallet poible number of coin.

7 . [0 point] You jut dicovered your bet friend from elementary chool on Twitbook. You both want to meet a oon a poible, but you live in two different cite that are far apart. To minimize travel time, you agree to meet at an intermediate city, and then you imultaneouly hop in your car and tart driving toward each other. But where exactly hould you meet? You are given a weighted graph G = (V, E), where the vertice V repreent citie and the edge E repreent road that directly connect citie. Each edge e ha a weight w(e) equal to the time required to travel between the two citie. You are alo given a vertex p, repreenting your tarting location, and a vertex q, repreenting your friend tarting location. Decribe and analyze an algorithm to find the target vertex t that allow you and your friend to meet a quickly a poible.

8 (cratch paper)

9 (cratch paper)