Fall 2014 David Wagner MT2 Soln

Transcription

1 CS 170 Algorihm Fall 014 Daid Wagner MT Soln Problem 1. [True or fale] (9 poin) Circle TRUE or FALSE. Do no juif our aner on hi problem. (a) TRUE or FALSE : Le (S,V S) be a minimum (,)-cu in he neork flo graph G. Le (u,) be an edge ha croe he cu in he forard direcion, i.e., u S and V S. Then increaing he capaci of he edge (u,) necearil increae he maimum flo of G. Eplanaion: In he folloing graph, S = {, u} form a minimum (,)-cu, bu increaing he capaci of he edge (u,) doen increae he maimum flo of G. u (b) TRUE or FALSE : All inance of linear programming hae eacl one opimum. (c) (d) TRUE or FALSE: If all of he edge capaciie in a graph are an ineger muliple of 7, hen he alue of he maimum flo ill be a muliple of 7. Eplanaion: One proof i o noice ha each ieraion of he Ford-Fulkeron algorihm ill increae he alue of he flo b an ineger muliple of 7. Therefore, b inducion, Ford- Fulkeron ill oupu a flo hoe alue i a muliple of 7. Since Ford-Fulkeron oupu a maimum flo, he alue of he maimum flo mu be a muliple of 7. TRUE or FALSE: If e an o proe ha a earch problem X i NP-complee, i enough o reduce SAT o X (in oher ord, i enough o proe SAT P X). (e) TRUE or FALSE : If e an o proe ha a earch problem X i NP-complee, i enough o reduce X o SAT (in oher ord, i enough o proe X P SAT). Eplanaion: For inance, SAT P SAT (ince ou can ole SAT uing an algorihm for SAT), bu SAT i in P and hence no NP-complee (unle P = NP). (f) TRUE or FALSE : For eer graph G and eer maimum flo on G, here ala ei an edge uch ha increaing he capaci on ha edge ill increae he maimum flo ha poible in he graph. Eplanaion: See he eample graph hon in par (a). (g) TRUE or FALSE : Suppoe he maimum (,)-flo of ome graph ha alue f. No e increae he capaci of eer edge b 1. Then he maimum (,)-flo in hi modified graph ill hae alue a mo f + 1. CS 170, Fall 014, MT Soln 1

2 Eplanaion: In he folloing graph, he maimum flo ha alue f = 4. Increaing he capaci of eer edge b 1 caue he maimum flo in he modified graph o hae alue 6. u (h) TRUE or FALSE : There i no knon polnomial-ime algorihm o ole maimum flo. (i) TRUE or FALSE: If problem A can be reduced o problem B, and B can be reduced o C, hen A can alo be reduced o C. (j) TRUE or FALSE: If X i an earch problem, hen X can be reduced o INDEPENDENT SET. Eplanaion: INDEPENDENT SET i NP-complee. (k) TRUE or FALSE : If e can find a ingle problem in NP ha ha a polnomial-ime algorihm, hen here i a polnomial-ime algorihm for SAT. (l) (m) (n) (o) Eplanaion: SAT i in NP and ha a polnomial-ime algorihm, bu ha doen necearil mean ha SAT ha a polnomial-ime algorihm. TRUE or FALSE: If here i a polnomial-ime algorihm for SAT, hen eer problem in NP ha a polnomial-ime algorihm. Eplanaion: SAT i NP-complee. TRUE or FALSE: We can reduce he earch problem MAXIMUM FLOW o he earch problem LINEAR PROGRAMMING (in oher ord, MAXIMUM FLOW P LINEAR PROGRAMMING). Eplanaion: We a in cla ho e can epre he maimum flo problem a a linear program. TRUE or FALSE: We can reduce he earch problem LINEAR PROGRAMMING o he earch problem INTEGER LINEAR PROGRAMMING (in oher ord, LINEAR PROGRAMMING P INTEGER LINEAR PROGRAMMING). Eplanaion: You can reduce anhing in P o anhing in NP, and LINEAR PROGRAMMING i in P and INTEGER LINEAR PROGRAMMING i in NP. Or: You can reduce anhing in NP o anhing NP-complee, and LINEAR PROGRAMMING i in NP and INTEGER LINEAR PROGRAMMING i NP-complee. TRUE or FALSE: Eer problem in P can be reduced o SAT. Eplanaion: Thi follo from he Cook-Lein heorem. Eer problem in P i in NP. The Cook-Lein heorem a ha eerhing in NP can be reduced o CircuiSAT, hich can in urn be reduced o SAT. Or: Eer problem in P i in NP, and eer problem in NP can be reduced o an NP-complee problem. SAT i NP-complee. CS 170, Fall 014, MT Soln

3 (p) TRUE or FALSE : Suppoe e hae a daa rucure here he amorized running ime of Iner and Delee i O(lgn). Then in an equence of n call o Iner and Delee, he or-cae running ime for he nh call i O(lgn). Eplanaion: The fir n 1 call migh ake Θ(1) ime and he nh call migh ake Θ(nlgn) ime. Thi ould be conien ih he premie, a i ould mean ha he oal ime for he fir n call i Θ(nlgn). (q) TRUE or FALSE : Suppoe e do a equence of m call o Find and m call o Union, in ome order, uing he union-find daa rucure ih union b rank and pah compreion. Then he la call o Union ake O(lg m) ime. Eplanaion: The or cae ime for a ingle call o Union migh be much larger han i amorized running ime. Problem. [Shor aner] (1 poin) Aner he folloing queion, giing a hor juificaion (a enence or o). (a) If P NP, could here be a polnomial-ime algorihm for SAT? Aner: No. SAT i NP-complee, o a polnomial-ime algorihm for SAT ould impl a polnomial-ime algorihm for eer problem in NP i ould impl ha P = NP. (b) If P NP, could GRAPH -COLORING be NP-complee? Aner: No. There a polnomial-ime algorihm for GRAPH -COLORING, o if i ere NP-complee, e ould hae a polnomial-ime algorihm for eer problem in NP, hich ould mean ha P = NP. (c) If e hae a dnamic programming algorihm ih n ubproblem, i i poible ha he running ime could be ampoicall ricl more han Θ(n )? Aner: Ye, for inance, if he running ime per ubproblem i Θ(n) (or anhing larger han Θ(1)). (d) If e hae a dnamic programming algorihm ih n ubproblem, i i poible ha he pace uage could be O(n)? Aner: Ye, if e don need o keep he oluion o all maller ubproblem bu onl he la O(n) of hem or o. (e) Suppoe ha e implemen an append-onl log daa rucure, here he oal running ime o perform an equence of n Append operaion i a mo n. Wha i he amorized running ime of Append? Aner: n/n = O(1), a he amorized running ime i he oal ime for man operaion diided b he number of operaion. (Incidenall, hi i Q1 from HW7.) (f) Suppoe e an o find an opimal binar earch ree, gien he frequenc of bunch of ord. In oher ord, he ak i: CS 170, Fall 014, MT Soln

4 Inpu: n ord (in ored order); frequencie of hee ord: p 1, p,..., p n. Oupu: The binar earch ree of loe co (defined a he epeced number of comparion in looking up a ord). Prof. Cerie ugge defining C(i) = he co of he opimal binar earch ree for ord 1...i, riing a recurie formula for C(i), and hen uing dnamic programming o find all he C(i). Prof. Ru ugge defining R(i, j) = he co of he opimal binar earch ree for ord i... j, riing a recurie formula for R(i, j), and hen uing dnamic programming o find all he R(i, j). One of he profeor ha an approach ha ork, and one ha an approach ha doen ork. Which profeor approach can be made o ork? In ha order hould e compue he C alue (if ou chooe Cerie approach) or R alue (if ou chooe Ru approach)? Aner: Ru (here no ea a o compue C(n) from C(1),...,C(n 1), bu a e a in he homeork, Ru approach doe ork). B increaing alue of j i. (Incidenall, hi i Q4 from HW.) Problem. [Ma flo] (10 poin) Conider he folloing graph G. The number on he edge repreen he capaciie of he edge. 7 9 (a) Ho much flo can e end along he pah? Aner: (b) Dra he reuling reidual graph afer ending a much flo a poible along he pah, b filling in he keleon belo ih he edge of he reidual graph. Label each edge of he reidual graph ih i capaci. Aner: CS 170, Fall 014, MT Soln 4

5 5 6 4 (c) Find a maimum (,)-flo for G. Label each edge belo ih he amoun of flo en along ha edge, in our flo. (You can ue he blank pace on he ne page for crach pace if ou like.) Aner: There are muliple alid aner. Here i one maimum flo: Anoher maimum flo: Ye anoher alid maimum flo: CS 170, Fall 014, MT Soln 5

6 (d) Dra a minimum (,)-cu for he graph G (hon belo again). Aner: S = {,,,}. 7 9 Problem 4. [Ma flo] (1 poin) We ould like an efficien algorihm for he folloing ak: Inpu: A direced graph G = (V,E), here each edge ha capaci 1; erice, V ; a number k N Goal: find k edge ha, hen deleed, reduce he maimum flo in he graph b a much a poible Conider he folloing approach: 1. Compue a maimum (,)-flo f for G.. Le G f be he reidual graph for G ih flo f.. Define a e S of erice b (omehing). 4. Define a e T of edge b (omehing). 5. Reurn an k edge in T. (a) Ho could e define he e S and T in ep 4, o ge an efficien algorihm for hi problem? Aner: S = { V : i reachable from in G f }. T = {(,) E : S, / S}. (or: T = he edge ha cro he cu (S,V S).) CS 170, Fall 014, MT Soln 6

7 Eplanaion: (S,V S) i a minimum (,)-cu, and b he min-cu ma-flo heorem, i capaci i alue( f ). If e remoe an k edge from T, hen he capaci of he cu (S,V S) ill drop o alue( f ) k, and i ll be a minimum (,)-cu in he modified graph. Therefore, b he min-cu ma-flo heorem, he alue of he maimum in he modified graph ill be alue( f ) k. (Deleing k edge can reduce he alue of he maimum flo more han ha, o hi i opimal.) (b) I here an algorihm o implemen ep 1 in O( V E ) ime or le? If e, ha algorihm hould e ue? If no, h no? Eiher a, juif our aner in a enence or o. Aner: Ye. Ford-Fulkeron: he alue of he ma flo i a mo V 1 (ince here are a mo V 1 edge ou of, each of hich can carr a mo 1 uni of flo), o Ford-Fulkeron doe a mo V 1 ieraion, and each ieraion ake O( E ) ime. Commen: Karp-Edmond running ime i O( V E ) in general, oo lo. (There are apparenl algorihm ha can compue he ma flo in an arbirar graph in O( V E ) ime, bu he are uper-comple mehod: e.g., Orlin algorihm plu he King-Rao-Tarjan algorihm.) Problem 5. [Dnamic programming] (1 poin) Le A[1..n] be a li of ineger, poibl negaie. I pla a game, here in each urn, I can chooe beeen o poible moe: (a) delee he fir ineger from he li, leaing m core unchanged, or (b) add he um of he fir o ineger o m core and hen delee he fir hree ineger from he li. (If I reach a poin here onl one or o ineger remain, I m forced o chooe moe (a).) I an o maimize m core. Deign a dnamic programming algorihm for hi ak. Formall: Inpu: A[1..n] Oupu: he maimum core aainable, b ome equence of legal moe For inance, if he li i A = [,5,7,,10,10,1], he be oluion i =. You do no need o eplain or juif our aner on an of he par of hi queion. (a) Define f ( j) = he maimum core aainable b ome equence of legal moe, if e ar ih he li A[ j..n]. Fill in he folloing bae cae: Aner: f (n) = 0 f (n 1) = 0 f (n ) = ma(0,a[n ] + A[n 1]) (b) Wrie a recurie formula for f ( j). You can aume 1 j n. Aner: f ( j) = ma( f ( j + 1), A[ j] + A[ j + 1] + f ( j + )) (c) Suppoe e ue he formula from par (a) and (b) o ole hi problem ih dnamic programming. Wha ill he ampoic running ime of he reuling algorihm be? Aner: Θ(n). Eplanaion: We hae o ole n ubproblem, and each one ake Θ(1) ime o ole (ince ha i he ime o ealuae he formula in par (b) for a ingle alue of j). CS 170, Fall 014, MT Soln 7

8 (d) Suppoe e an o minimize he amoun of pace (memor) ued b he algorihm o ore inermediae alue. Ampoicall, ho much pace ill be needed, a a funcion of n? (Don coun he amoun of pace o ore he inpu.) Ue Θ( ) noaion. Aner: Θ(1). Eplanaion: A e ole he ubproblem in he order f (n), f (n 1),..., f (1), e don need o ore all of he oluion; e onl need o ore he hree la alue of f ( ). Problem 6. [Greed card] (1 poin) Ning and Ean are plaing a game, here here are n card in a line. The card are all face-up (o he can boh ee all card in he line) and numbered 9. Ning and Ean ake urn. Whoeer urn i i can ake one card from eiher he righ end or he lef end of he line. The goal for each plaer i o maimize he um of he card he e colleced. (a) Ning decide o ue a greed raeg: on m urn, I ill ake he larger of he o card aailable o me. Sho a mall counereample (n 5) here Ning ill loe if he pla hi greed raeg, auming Ning goe fir and Ean pla opimall, bu he could hae on if he had plaed opimall. Aner: [1,1,9,]. Eplanaion: Ning fir greedil ake he from he righ end, and hen Ean nache he 9, o Ean ge 10 and Ning ge a mierl. If Ning had ared b crafil aking he 1 from he lef end, he d guaranee ha he ould ge 10 and poor Ean ould be uck ih. There are man oher counereample. The re all of lengh a lea 4. (b) Ean decide o ue dnamic programming o find an algorihm o maimize hi core, auming he i plaing again Ning and Ning i uing he greed raeg from par (a). Le A[1..n] denoe he n card in he line. Ean define (i, j) o be he highe core he can achiee if i hi urn and he line conain card A[i.. j]. Ean need a recurie formula for (i, j). Fill in a formula he could ue, belo. Ean ugge ou implif our epreion b epreing (i, j) a a funcion of l(i, j) and r(i, j), here l(i, j) i defined a he highe core Ean can achiee if i hi urn and he line conain card A[i.. j], if he ake A[i]; alo, r(i, j) i defined o be he highe core Ean can achiee if i hi urn and he line conain card A[i.. j], if he ake A[ j]. Wrie an epreion for all hree ha Ean could ue in hi dnamic programming algorihm. You can aume 1 i < j n and j i. Don orr abou bae cae. Aner: (i, j) = ma(l(i, j),r(i, j)) { A[i] + (i + 1, j 1) if A[ j] > A[i + 1] here l(i, j) = A[i] + (i +, j) oherie. { A[ j] + (i + 1, j 1) if A[i] A[ j 1] r(i, j) = A[ j] + (i, j ) oherie. CS 170, Fall 014, MT Soln

9 Commen: We didn pecif ho Ning reole ie, if he card on he lef end ha he ame alue a he card on he righ end. Therefore, our recurie formula can reole ie in an manner ou an: an correc formula ill be acceped a correc regardle of ho ou decided o reole ie. (For inance, he formula aboe aume ha if here i a ie, Ning ake he card on he lef end.) (c) Wha ill he running ime of he dnamic programming algorihm be, if e ue our formula from par (b)? You don need o juif our aner. Aner: Θ(n ). Aner: There are n(n + 1)/ ubproblem and each one can be oled in Θ(1) ime (ha he ime o ealuae he recurie formula in par (b) for a ingle alue of i, j). Problem 7. [Tile Daid alka] (14 poin) Daid i going o la ile for a long alka leading up o hi houe, and he an an algorihm o figure ou hich paern of ile are achieable. The alka i a long rip, n meer long and 1 meer ide. Each 1 1 meer quare can be colored eiher hie or black. The inpu o he algorihm i a paern P[1..n] ha pecifie he equence of color ha hould appear on he alka. There are hree kind of ile aailable from he local ile ore: a 1 1 hie ile (W), a 1 all-black ile (BB), and a 1 black-hie-black ile (BWB). Unforunael, here i a limied uppl of each: he ile ore onl ha p W ile, q BB, and r BWB in ock. Deie an efficien algorihm o deermine heher a gien paern can be iled, uing he ile in ock. In oher ord, e an an efficien algorihm for he folloing ak: Inpu: A paern P[1..n], ineger p, q, r N Queion: I here a a o ile he alka ih paern P, uing a mo p W, q BB, and r BWB? For eample, if he paern P i WBWBWBBWWBWBW and p = 5, q =, and r =, he aner i e: i can be iled a follo: W BWB W BB W W BWB W. For hi problem, e ugge ou ue dnamic programming. Define { True if here a a o ile P[ j..n] uing a mo p W, q BB, and r BWB f ( j, p,q,r) = Fale oherie. You don need o juif or eplain our aner o an of he folloing par. You can aume omeone ele ha aken care of he bae cae ( j = n,n 1,n), e.g., f (n, p,q,r) = (P[n] = W p 1) and o on; ou don need o orr abou hem. (a) Wrie a recurie formula for f. You can aume 1 j n. Aner: f ( j, p,q,r) =(P[ j] = W p > 0 f ( j + 1, p 1,q,r)) (P[ j.. j + 1] = BB q > 0 f ( j +, p,q 1,r)) (P[ j.. j + ] = BWB r > 0 f ( j +, p,q,r 1)). CS 170, Fall 014, MT Soln 9

10 (b) If e ue our formula from par (a) o creae a dnamic programming algorihm for hi problem, ha ill i ampoic running ime be? Aner: Θ(n(p + 1)(q + 1)(r + 1)). We ll alo accep Θ(n 4 ) (ince ihou lo of generali e can aume p < n, q < n, and r < n; here no a o ue more han n ile). Θ(npqr) i echnicall no quie righ if p, q, or r i zero, bu if e had a promie ha he ere non-zero i ould alo be correc. Eplanaion: There are n(p + 1)(q + 1)(r + 1) ubproblem. Each ubproblem can be oled in Θ(1) ime uing he formula in par (a). Problem. [Walka, ih more ile] (1 poin) No le conider SUPERTILE, a generalizaion of Problem 7. The SUPERTILE problem i Inpu: A paern P[1..n], ile 1,,..., m Oupu: A a o ile he alka uing a ube of he proided ile in an order, or NO if here no a o do i Each ile i i proided a a equence of color (W or B). Each paricular ile i can be ued a mo once, bu he ame ile-equence can appear muliple ime in he inpu (e.g., e can hae i = j ). The ile can be ued in an order. For eample, if he paern P i WBBBWWB and he ile are 1 = BBW, = B, = WB, 4 = WB, hen he aner i e, 1 4 (hi correpond o WB BBW WB ). (a) I SUPERTILE in NP? Juif our aner in a enence or o. Aner: Ye. There i a polnomial-ime oluion-checking algorihm: check ha he ile ued are a ube of 1,..., m and no ile i ued more ime han i appear in 1,..., m. (b) Prof. Maue beliee he ha found a a o reduce SUPERTILE o SAT. In oher ord, he beliee he ha proen ha SUPERTILE P SAT. If he i correc, doe hi impl ha SUPERTILE i NP-hard? Juif our aner in a enence or o. Aner: No. SUPERTILE could be much eaier han SAT. SAT i NP-complee, o eerhing in NP reduce o SAT. (c) Prof. Argle beliee he ha found a a o reduce SAT o SUPERTILE. In oher ord, he beliee he ha proen ha SAT P SUPERTILE. If he i correc, doe hi impl ha SUPERTILE i NP-hard? Juif our aner in a enence or o. Aner: Ye. Thi reducion ho ha SUPERTILE i a lea a hard a SAT, hich i NP-complee. (d) Conider he folloing arian problem, ORDEREDTILE: Inpu: A paern P[1..n], ile 1,,..., m CS 170, Fall 014, MT Soln 10

11 Oupu: A a o ile he alka uing a ube of he ile, in he proided order, or NO if here no a o do i In hi arian, ou canno re-order he ile. For inance, if P i WBBBWWB and he ile are 1 = BBW, = B, = WB, 4 = WB, hen he oupu i NO ; hoeer, if he ile are 1 = WB, = B, = BBW, 4 = WB, hen he aner i e, 1 4. Rohi adior aked him o proe ha ORDEREDTILE i NP-complee. Rohi ha pen he pa fe da ring o proe i, bu ihou an ucce. He ondering heher he ju need o r harder or if i hopele. Baed on ha ou e learned from hi cla, hould ou encourage him o keep ring, or hould ou adie him o gie up? Eplain h, in enence. Aner: Gie up. There a polnomial-ime algorihm for ORDEREDTILE uing dnamic programming, o an proof ha ORDEREDTILE i NP-complee ould impl ha P = NP. Gien ho man oher reearcher hae ried, i no reaonable o ak Rohi o proe P = NP. Eplanaion: The dnamic programming algorihm: le f (i, j) be rue if here i a a o ile P[i..n] uing ile j,..., m. Then f (i, j) = f (i, j + 1) (P[i..i + len( j ) 1] = j f (i + len( j ), j + 1)), o e obain a Θ(nm) ime algorihm for ORDEREDTILE. CS 170, Fall 014, MT Soln 11