CS 170: Midterm Exam II University of California at Berkeley Department of Electrical Engineering and Computer Sciences Computer Science Division

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Darrell Young
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1 1 1 April 000 Demmel / Shewchuk CS 170: Midterm Exam II Univerity of California at Berkeley Department of Electrical Engineering and Computer Science Computer Science Diviion hi i a cloed book, cloed calculator, cloed computer, cloed network, open brain exam, but you are permitted a one-page, double-ided et of note Write all your anwer on thi exam If you need cratch paper, ak for it, write your name on each heet, and attach it when you turn it in (we have a tapler) Do not open your exam until you are told to do o! (1 point) Name: (1 point) Login: (1 point) Lab A: (1 point) Neighbor to your left: (1 point) Neighbor to your right: Do not write in thee boxe Problem # Poible Score Name, etc 5 1 Network Flow 1 rue/fale 16 NP-Completene 1 4 Dynamic Programming 15 otal 60

2 Name: Login: Problem 1 (1 point) Network Flow In the flow network illutrated below, each directed edge i labeled with it capacity We are uing the Ford-Fulkeron algorithm to find the maximum flow he firt augmenting path i S-A-C-D-, and the econd augmenting path i S-A-B-C-D- S 4 A C B 1 1 D 4 a (6 point) Draw the reidual network after we have updated the flow uing thee two augmenting path (in the order given) S 4 A B C D b (4 point) Lit all of the augmenting path that could be choen for the third augmentation tep S-C-A-B-, S-C-B-, S-C-D-B-, and S-C-D- c ( point) What i the numerical value of the maximum flow? Draw a dotted line through the original graph to repreent the minimum cut 5 See top figure above for the minimum cut

3 Name: Login: Problem (16 point) rue or Fale? Each true/fale quetion i worth two point You will loe two point for a wrong anwer, o only anwer if you think you are likely to be right a for all rue Oberve that! " $# %'&( )# b rue Oberve that # *+ # -, (hink about the definition of # to ee why) c A grandchild in a foret-baed dijoint-et data tructure i any node that ha a grandparent (In other word, neither the node nor it parent i a root) If we ue union-by-rank, a ingle union operation cannot increae the total number of grandchildren by more than /10, where i the total number of node Fale he number of grandchildren can increae by up to, Conider the following example, with both root having rank 1 A union operation might make the right root a child of the left d Let 4 be a non-empty file that ue -bit fixed-length codeword for the character a h he file cannot ue any character other than thee eight rue or fale: uffman coding can alway compre any uch file (ie, produce an encoding with fewer bit) (Don t count the bit needed to expre the mapping from bit to alphabet) Fale For example, if all eight character occur with the ame frequency, the uffman encoding will aign each character a -bit codeword e If A i an NP-complete problem, and problem B reduce (polynomially) to A, then B i an NPcomplete problem Fale B might be much eaier For example, the problem of determining whether a tring of bit include a 1 can be eaily reduced to SA, but it certainly not an NP-hard problem f he problem of finding the length of the hortet path between two node in a directed graph (which we would normally olve uing Dijktra algorithm) can be expreed a a linear program (An ordinary, continuou linear program; we re not talking about integer linear program) Aume that there i only one hortet path rue For example, we can formulate thi a a min-cot flow problem (Note that I didn t ay maxflow) Aign every edge of the graph a capacity of 1, and a cot equal to it weight We treat thi

4 Name: Login: 4 flow problem a a linear program with the uual capacity contraint (the flow through any edge i not le than zero and not more than one), the uual conervation contraint (total flow into a node equal total flow out of the node, except for the ource and ink), plu the contraint that the amount of flow out of the ource i one he objective function i the uual for min-cot flow: the um over all edge of the flow through an edge time the cot of the edge Since there i only one hortet path, the entire one unit of flow will move through the hortet path, and there will be no fractional flow g Conider a linear program with three variable (dimenion) If 5 i an optimal point of the linear program, then 5 mut be a vertex of the polyhedron formed by the contraint (inequalitie) Fale If the objective vector i orthogonal to an edge or a face of the polyhedron, 5 might lie anywhere within that edge or face h Imagine we have a computer with an infinite amount of RAM, in which any memory addre can be acceed in contant time here exit a computer program of infinite length that olve SA in linear time rue he implet approach i to have an infinite table that map every poible input to the correct anwer for that input Our program merely convert the input into a memory addre in the table, and look up the olution at that addre If you think it cheating to have an infinite table of data, we can olve SA with program code alone We walk through the input one bit at a time For each bit, ue if tatement to branch to code for one of three cae: a zero bit, a one bit, or the end of the input tring he branch for the end of a tring return/print the correct olution for the input tring he other two branche continue by reading the next bit, and by having three-way branche baed on that bit

5 Name: Login: 5 Problem (1 point) NP-Completene We know that max-flow problem can be olved in polynomial time owever, conider a modified max-flow problem in which every edge mut either have zero flow or be completely aturated In other word, if there i any flow through an edge at all, the flow through the edge i the capacity of the edge Let call a flow a aturated flow if it atifie thi additional contraint, a well a all the uual contraint of flow problem he modified problem ak u to find the maximum aturated flow a (1 point) Formulate thi problem a a deciion (ye/no) problem (We ll call it MAX-SAURAED- FLOW) Write your anwer in the form of a quetion I there a aturated flow of value 6 or greater? (Alternative: i there a aturated flow of value 6?) ere, 6 i an arbitrary number b ( point) Show that MAX-SAURAED-FLOW i in NP For a certificate, we ue a lit of all the aturated edge Given uch a lit, we can check in linear time the flow out of the ource node, and whether the conervation condition for each node (except the ource and ink) i atified c (8 point) Show that MAX-SAURAED-FLOW i NP-hard int: the eaiet reduction i not from one of the NP-complete graph problem you know What other NP-complete problem do you know? It relatively eay to reduce SUBSE-SUM to MAX-SAURAED-FLOW Conider the NP-hard problem of finding a ubet of 798 :<;=89>1;=8@?A;<B<B<B@;=8@CED that um to F hi problem i olved if we find a aturated flow of value F in the following graph (or alternatively, the maximum aturated flow of thi graph) 1 1 t n n

6 f m m 7 0 d d ; 0 ; ; ; ; ; J, J,, 0 d ; ; 7 Name: Login: 6 Problem 4 (15 point) Dynamic Programming + Conider the Maximum Alternating Sum Subequence (MASS) problem Given a equence M G 5:;I5>1; B<B<B;I5CKJ of poitive integer, find the ubequence L 5ONQP<;I5ON R9;<B<B<B;I5ON SJ, where I:VUWX>UZY<Y<Y[U\X], that maximize the alternating um 5ON P 5ON R 5ON^ 5Ǹ _ Y<Y<Y ab5on S B c d h,b0idv, For example, if G ;=e$; ;gf@j, the MASS i L e$; ;gf@j which evaluate to j c h h e f If G f);=k$;=l$; J, the MASS i L f@j Clearly, the length of the MASS depend on the actual value in G Aume that the length of G i at leat one, and all it element are integer greater than zero he following quetion ak you to develop a dynamic programming algorithm to find the value of the MASS (but not the actual ubequence) for a given equence We would like a olution with optimal running time, but you will only loe a few point if you have a correct, uboptimal algorithm a ( point) Decribe the ubproblem you will need to olve ow many table (of ubproblem olution) do you need? What i the dimenion of the table()? What do the table entrie repreent? (int: it help to treat ubequence of even and odd length eparately) hi quetion can be anwered in everal way; here one Let m and be two one-dimenional table with indice from zero to nj will tore the value of the maximum odd-length alternating um ubequence of the firt element of G, and m oj will tore the value of the maximum even-length alternating um ubequence of the firt element of G b (7 point) Write a recurion for the value your algorithm will fill into the table() Alo write the bae cae(); ie the table value() that boottrap() the recurion Recurion: p rqpat Bae cae: K!y,{z 7@m ; m,@ K 5Nu;,@ D ; m!qpat p,@[, 5Nv;wm,x@ D ( K i acceptable, but note that it wouldn t work if we allowed G to contain negative integer) c (4 point) Write iterative, non-recurive peudocode for your algorithm,~z J} J for to p qpat nj qpat nj return J 7@m p,x,x p 5ON ; 5ON ;wm,, JnD JnD If G could contain negative integer, the lat line would have to return qat 7 Jo;wm JnD d (1 point) What i the running time of your algorithm? Our run in &( Q time

Problem Set 8 Solutions

Problem Set 8 Solutions Deign and Analyi of Algorithm April 29, 2015 Maachuett Intitute of Technology 6.046J/18.410J Prof. Erik Demaine, Srini Devada, and Nancy Lynch Problem Set 8 Solution Problem Set 8 Solution Thi problem