Fall 2014 David Wagner MT2 Soln

Transcription

1 CS 70 Algorihm Fall 04 Daid Wagner MT Soln Problem. [True or fale] (9 poin) Circle TRUE or FALSE. Do no juify your 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 capaciy of he edge (u,) necearily increae he maximum flo of G. Explanaion: In he folloing graph, S = {, u} form a minimum (,)-cu, bu increaing he capaciy of he edge (u,) doen increae he maximum flo of G. u (b) TRUE or FALSE: All inance of linear programming hae exacly one opimum. (c) (d) Commen: We decided no o grade hi queion (all aner ill be acceped a correc). The inended aner a Fale : a linear program can hae muliple oluion ha all aain he opimum alue (e.g., maximize x + y, ubjec o he requiremen ha x + y 5). Hoeer, ome uden poined ou ha i a no enirely clear heher opimum refer o he alue of he oluion or o he oluion ielf. Therefore, e decided no o grade hi queion, o aoid deducing poin from omeone ho did underand he maerial bu ere hron off by he ambiguiy in he queion. Some linear program hae no feaible oluion and hu arguably hae no opimum (e.g., maximize x, ubjec o he requiremen ha x 5 and x 6), bu one could alo conider ha cae o be one here he maximum alue i ; in any cae, e judged ha hi i oo ricky o make a good bai for a queion. TRUE or FALSE: If all of he edge capaciie in a graph are an ineger muliple of 7, hen he alue of he maximum flo ill be a muliple of 7. Explanaion: One proof i o noice ha each ieraion of he Ford-Fulkeron algorihm ill increae he alue of he flo by an ineger muliple of 7. Therefore, by inducion, Ford- Fulkeron ill oupu a flo hoe alue i a muliple of 7. Since Ford-Fulkeron oupu a maximum flo, he alue of he maximum 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). CS 70, Fall 04, MT Soln

2 Explanaion: For inance, SAT P SAT (ince you 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 eery graph G and eery maximum flo on G, here alay exi an edge uch ha increaing he capaciy on ha edge ill increae he maximum flo ha poible in he graph. Explanaion: See he example graph hon in par (a). (g) TRUE or FALSE : Suppoe he maximum (,)-flo of ome graph ha alue f. No e increae he capaciy of eery edge by. Then he maximum (,)-flo in hi modified graph ill hae alue a mo f +. Explanaion: In he folloing graph, he maximum flo ha alue f = 4. Increaing he capaciy of eery edge by caue he maximum flo in he modified graph o hae alue 6. u (h) TRUE or FALSE : There i no knon polynomial-ime algorihm o ole maximum 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 any earch problem, hen X can be reduced o INDEPENDENT SET. Explanaion: INDEPENDENT SET i NP-complee. (k) TRUE or FALSE : If e can find a ingle problem in NP ha ha a polynomial-ime algorihm, hen here i a polynomial-ime algorihm for SAT. (l) (m) (n) Explanaion: SAT i in NP and ha a polynomial-ime algorihm, bu ha doen necearily mean ha SAT ha a polynomial-ime algorihm. TRUE or FALSE: If here i a polynomial-ime algorihm for SAT, hen eery problem in NP ha a polynomial-ime algorihm. Explanaion: 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). Explanaion: We a in cla ho e can expre he maximum 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). CS 70, Fall 04, MT Soln

3 (o) Explanaion: You can reduce anyhing in P o anyhing in NP, and LINEAR PROGRAMMING i in P and INTEGER LINEAR PROGRAMMING i in NP. Or: You can reduce anyhing in NP o anyhing NP-complee, and LINEAR PROGRAMMING i in NP and INTEGER LINEAR PROGRAMMING i NP-complee. TRUE or FALSE: Eery problem in P can be reduced o SAT. Explanaion: Thi follo from he Cook-Lein heorem. Eery problem in P i in NP. The Cook-Lein heorem ay ha eeryhing in NP can be reduced o CircuiSAT, hich can in urn be reduced o SAT. Or: Eery problem in P i in NP, and eery problem in NP can be reduced o any NP-complee problem. SAT i NP-complee. (p) TRUE or FALSE : Suppoe e hae a daa rucure here he amorized running ime of Iner and Delee i O(lgn). Then in any equence of n call o Iner and Delee, he or-cae running ime for he nh call i O(lgn). Explanaion: The fir n call migh ake Θ() 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 by rank and pah compreion. Then he la call o Union ake O(lg m) ime. Explanaion: The or cae ime for a ingle call o Union migh be much larger han i amorized running ime. Problem. [Shor aner] ( poin) Aner he folloing queion, giing a hor juificaion (a enence or o). (a) If P NP, could here be a polynomial-ime algorihm for SAT? Aner: No. SAT i NP-complee, o a polynomial-ime algorihm for SAT ould imply a polynomial-ime algorihm for eery problem in NP i ould imply ha P = NP. (b) If P NP, could GRAPH -COLORING be NP-complee? Aner: No. There a polynomial-ime algorihm for GRAPH -COLORING, o if i ere NP-complee, e ould hae a polynomial-ime algorihm for eery problem in NP, hich ould mean ha P = NP. (c) If e hae a dynamic programming algorihm ih n ubproblem, i i poible ha he running ime could be aympoically ricly more han Θ(n )? Aner: Ye, for inance, if he running ime per ubproblem i Θ(n) (or anyhing larger han Θ()). (d) If e hae a dynamic programming algorihm ih n ubproblem, i i poible ha he pace uage could be O(n)? CS 70, Fall 04, MT Soln

4 Aner: Ye, if e don need o keep he oluion o all maller ubproblem bu only he la O(n) of hem or o. For example, Fibonacci i an example here e only need o ore he oluion o he la fe ubproblem (i ha n ubproblem and only need o ore he la, o i pace uage i O()), a doe he oluion o Problem 5. Commen: We didn accep here migh be n ubproblem bu ome of hem migh be oled repeaedly, o memoizaion ould ore only he diinc one. In dynamic programming, hen e coun he number of ubproblem, e coun he number of diinc ubproblem, ihou regard o ho many ime heir oluion i ued hen oling ome oher ubproblem. We didn accep ome of he n ubproblem migh neer need o be oled. Implicily, hen e re couning he number of ubproblem, e only coun he one e need o ole. We didn accep if you ue a recurie algorihm bu don memoize and ju re-calculae he oluion o ome ubproblem many ime, you don need much pace. A ha poin he algorihm no longer coun a dynamic programming. (e) Suppoe ha e implemen an append-only log daa rucure, here he oal running ime o perform any equence of n Append operaion i a mo n. Wha i he amorized running ime of Append? Aner: n/n = O(), a he amorized running ime i he oal ime for many operaion diided by he number of operaion. (Incidenally, hi i Q from HW7.) (f) Suppoe e an o find an opimal binary earch ree, gien he frequency of bunch of ord. In oher ord, he ak i: Inpu: n ord (in ored order); frequencie of hee ord: p, p,..., p n. Oupu: The binary earch ree of loe co (defined a he expeced number of comparion in looking up a ord). Prof. Cerie ugge defining C(i) = he co of he opimal binary earch ree for ord...i, riing a recurie formula for C(i), and hen uing dynamic programming o find all he C(i). Prof. Ru ugge defining R(i, j) = he co of he opimal binary earch ree for ord i... j, riing a recurie formula for R(i, j), and hen uing dynamic 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 you chooe Cerie approach) or R alue (if you chooe Ru approach)? Aner: Ru (here no eay ay o compue C(n) from C(),...,C(n ), bu a e a in he homeork, Ru approach doe ork). By increaing alue of j i. (Incidenally, hi i Q4 from HW.) CS 70, Fall 04, MT Soln 4

5 Problem. [Max flo] (0 poin) Conider he folloing graph G. The number on he edge repreen he capaciie of he edge. 7 x 9 y (a) Ho much flo can e end along he pah x y? Aner: (b) Dra he reuling reidual graph afer ending a much flo a poible along he pah x y, by filling in he keleon belo ih he edge of he reidual graph. Label each edge of he reidual graph ih i capaciy. Aner: 4 5 x 6 y (c) Find a maximum (,)-flo for G. Label each edge belo ih he amoun of flo en along ha edge, in your flo. (You can ue he blank pace on he nex page for crach pace if you like.) Aner: There are muliple alid aner. Here i one maximum flo: CS 70, Fall 04, MT Soln 5

6 x y 0 6 Anoher maximum flo: x y 7 6 Ye anoher alid maximum flo: x y (d) Dra a minimum (,)-cu for he graph G (hon belo again). Aner: S = {,,x,y}. CS 70, Fall 04, MT Soln 6

7 7 x 9 y Problem 4. [Max flo] ( poin) We ould like an efficien algorihm for he folloing ak: Inpu: A direced graph G = (V,E), here each edge ha capaciy ; erice, V ; a number k N Goal: find k edge ha, hen deleed, reduce he maximum flo in he graph by a much a poible Conider he folloing approach:. Compue a maximum (,)-flo f for G.. Le G f be he reidual graph for G ih flo f.. Define a e S of erice by (omehing). 4. Define a e T of edge by (omehing). 5. Reurn any 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).) Explanaion: (S,V S) i a minimum (,)-cu, and by he min-cu max-flo heorem, i capaciy i alue( f ). If e remoe any k edge from T, hen he capaciy of he cu (S,V S) ill drop o alue( f ) k, and i ll be a minimum (,)-cu in he modified graph. Therefore, by he min-cu max-flo heorem, he alue of he maximum in he modified graph ill be alue( f ) k. (Deleing k edge can reduce he alue of he maximum flo more han ha, o hi i opimal.) Commen : T = edge ha are auraed (i.e., here he flo on ha edge i equal o he capaciy) i no correc. There can be edge ha are auraed bu hoe remoal doe no reduce he ize of he maximum flo, e.g., becaue here i an alernae roue flo could ake. Conider, e.g., he folloing graph; e migh hae one uni of flo going ia u, bu deleing he auraed edge doe no reduce he alue of he maximum flo. CS 70, Fall 04, MT Soln 7

8 u Commen : We expec definiion of he e S and T ha are mahemaically correc. We graded he aner o Problem 4(a) ricly. I a no enough o hae an inuiie underanding; you needed o be able o expre your definiion in erm ha are mahemaically accurae. We did no gie any credi if your definiion of S and/or T a imprecie, ague, or conain miake. (For inance, noe ha S hould be a e of erice and T hould be a e of edge; if you defined T o be a e of erice, you did no receie any credi for T.) Alo, e do no gie any parial credi for loppy definiion like: S = erice of a minimum cu T = edge of a minimum cu An (,)-cu coni of o dijoin e of erice. Thee e hould be appropriaely defined if ued in he definiion of S and/or T. Alo, he direcion of he edge ha are choen o be added in T hould be explicily pecified, and mu be pecified correcly o receie any credi. Noe ha edge (,) E uch ha / S and S hould no be included in T. A a pecial cae, e did gie parial credi for definiion of S like he folloing, if hey had no error: S = one of he o dijoin e of erice of a minimum cu Thi definiion doe no explain ho o generae he e S and hu i doe no gie direcly an algorihm, o i did no receie full credi. We deduced one poin from uch definiion. Commen : Noe ha he definiion: S = { V : i reachable from in G f } i no correc. For example, conider he folloing graph: Such a definiion of S ould imply S = {,,,} hich i no one of he o dijoin e of erice of a minimum (,)-cu. (b) I here an algorihm o implemen ep in O( V E ) ime or le? If ye, ha algorihm hould e ue? If no, hy no? Eiher ay, juify your aner in a enence or o. Aner: Ye. Ford-Fulkeron: he alue of he max flo i a mo V (ince here are a mo V edge ou of, each of hich can carry a mo uni of flo), o Ford-Fulkeron doe a mo V ieraion, and each ieraion ake O( E ) ime. CS 70, Fall 04, MT Soln

9 Commen: Karp-Edmond running ime i O( V E ) in general, oo lo. (There are apparenly algorihm ha can compue he max flo in an arbirary graph in O( V E ) ime, bu hey are uper-complex mehod: e.g., Orlin algorihm plu he King-Rao-Tarjan algorihm.) Problem 5. [Dynamic programming] ( poin) Le A[..n] be a li of ineger, poibly negaie. I play a game, here in each urn, I can chooe beeen o poible moe: (a) delee he fir ineger from he li, leaing my core unchanged, or (b) add he um of he fir o ineger o my core and hen delee he fir hree ineger from he li. (If I reach a poin here only one or o ineger remain, I m forced o chooe moe (a).) I an o maximize my core. Deign a dynamic programming algorihm for hi ak. Formally: Inpu: A[..n] Oupu: he maximum core aainable, by ome equence of legal moe For inance, if he li i A = [,5,7,,0,0,], he be oluion i =. You do no need o explain or juify your aner on any of he par of hi queion. (a) Define f ( j) = he maximum core aainable by 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 ) = 0 f (n ) = max(0,a[n ] + A[n ]) (b) Wrie a recurie formula for f ( j). You can aume j n. Aner: f ( j) = max( f ( j + ), A[ j] + A[ j + ] + f ( j + )) (c) Suppoe e ue he formula from par (a) and (b) o ole hi problem ih dynamic programming. Wha ill he aympoic running ime of he reuling algorihm be? Aner: Θ(n). Explanaion: We hae o ole n ubproblem, and each one ake Θ() ime o ole (ince ha i he ime o ealuae he formula in par (b) for a ingle alue of j). (d) Suppoe e an o minimize he amoun of pace (memory) ued by he algorihm o ore inermediae alue. Aympoically, 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(). Explanaion: A e ole he ubproblem in he order f (n), f (n ),..., f (), e don need o ore all of he oluion; e only need o ore he hree la alue of f ( ). CS 70, Fall 04, MT Soln 9

10 Problem 6. [Greedy card] ( poin) Ning and Ean are playing a game, here here are n card in a line. The card are all face-up (o hey 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 player i o maximize he um of he card hey e colleced. (a) Ning decide o ue a greedy raegy: on my urn, I ill ake he larger of he o card aailable o me. Sho a mall counerexample (n 5) here Ning ill loe if he play hi greedy raegy, auming Ning goe fir and Ean play opimally, bu he could hae on if he had played opimally. Aner: [,,9,]. Explanaion: Ning fir greedily ake he from he righ end, and hen Ean nache he 9, o Ean ge and Ning ge a mierly 5. If Ning had ared by crafily aking he from he lef end, he d guaranee ha he ould ge and poor Ean ould be uck ih 5. There are many oher counerexample. They re all of lengh a lea 4. (b) Ean decide o ue dynamic programming o find an algorihm o maximize hi core, auming he i playing again Ning and Ning i uing he greedy raegy from par (a). Le A[..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 you implify your expreion by expreing (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 expreion for all hree ha Ean could ue in hi dynamic programming algorihm. You can aume i < j n and j i. Don orry abou bae cae. Aner: (i, j) = max(l(i, j),r(i, j)) { A[i] + (i +, j ) if A[ j] > A[i + ] here l(i, j) = A[i] + (i +, j) oherie. { A[ j] + (i +, j ) if A[i] A[ j ] r(i, j) = A[ j] + (i, j ) oherie. Commen: We didn pecify ho Ning reole ie, if he card on he lef end ha he ame alue a he card on he righ end. Therefore, your recurie formula can reole ie in any manner you an: any correc formula ill be acceped a correc regardle of ho you 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 dynamic programming algorihm be, if e ue your formula from par (b)? You don need o juify your aner. CS 70, Fall 04, MT Soln 0

11 Aner: Θ(n ). Aner: There are n(n + )/ ubproblem and each one can be oled in Θ() ime (ha he ime o ealuae he recurie formula in par (b) for a ingle alue of i, j). Problem 7. [Tile Daid alkay] (4 poin) Daid i going o lay ile for a long alkay leading up o hi houe, and he an an algorihm o figure ou hich paern of ile are achieable. The alkay i a long rip, n meer long and meer ide. Each meer quare can be colored eiher hie or black. The inpu o he algorihm i a paern P[..n] ha pecifie he equence of color ha hould appear on he alkay. There are hree kind of ile aailable from he local ile ore: a hie ile (W), a all-black ile (BB), and a black-hie-black ile (BWB). Unforunaely, here i a limied upply of each: he ile ore only 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[..n], ineger p, q, r N Queion: I here a ay o ile he alkay ih paern P, uing a mo p W, q BB, and r BWB? For example, if he paern P i WBWBWBBWWBWBW and p = 5, q =, and r =, he aner i ye: i can be iled a follo: W BWB W BB W W BWB W. For hi problem, e ugge you ue dynamic programming. Define { True if here a ay 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 juify or explain your aner o any of he folloing par. You can aume omeone ele ha aken care of he bae cae ( j = n,n,n), e.g., f (n, p,q,r) = (P[n] = W p ) and o on; you don need o orry abou hem. (a) Wrie a recurie formula for f. You can aume j n. Aner: f ( j, p,q,r) =(P[ j] = W p > 0 f ( j +, p,q,r)) (P[ j.. j + ] = BB q > 0 f ( j +, p,q,r)) (P[ j.. j + ] = BWB r > 0 f ( j +, p,q,r )). (b) If e ue your formula from par (a) o creae a dynamic programming algorihm for hi problem, ha ill i aympoic running ime be? Aner: Θ(npqr) or Θ(n(p + )(q + )(r + )). (There are n(p + )(q + )(r + ) ubproblem. Each ubproblem can be oled in Θ() ime uing he formula in par (a).) CS 70, Fall 04, MT Soln

12 Θ(n 4 ) i alo OK (ince ihou lo of generaliy e can aume p < n, q < n, and r < n; here no ay o ue more han n ile). Θ(pqr) i alo a reaonable aner: ih he recurie formula in par (a), j i deermined by he p,q,r argumen o f. There alo a ay o make hi run in Θ(n) ime, bu no oling all of he ubproblem: you look a heher P[ j..n] ar ih W, BB, or BWB, hen ealuae f a he correponding alue. Thi lead o a linear can hrough he P array. We decided o accep all of Θ(n), Θ(n ), Θ(n ), Θ(n 4 ), ince one can make an argumen for each of hem, a ell a Θ(npqr) and Θ(pqr). Problem. [Walkay, ih more ile] ( poin) No le conider SUPERTILE, a generalizaion of Problem 7. The SUPERTILE problem i Inpu: A paern P[..n], ile,,..., m Oupu: A ay o ile he alkay uing a ube of he proided ile in any order, or NO if here no ay 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 any order. For example, if he paern P i WBBBWWB and he ile are = BBW, = B, = WB, 4 = WB, hen he aner i ye, 4 (hi correpond o WB BBW WB ). (a) I SUPERTILE in NP? Juify your aner in a enence or o. Aner: Ye. There i a polynomial-ime oluion-checking algorihm: check ha he ile ued are a ube of,..., m and no ile i ued more ime han i appear in,..., m. (b) Prof. Maue beliee he ha found a ay o reduce SUPERTILE o SAT. In oher ord, he beliee he ha proen ha SUPERTILE P SAT. If he i correc, doe hi imply ha SUPERTILE i NP-hard? Juify your aner in a enence or o. Aner: No. SUPERTILE could be much eaier han SAT. SAT i NP-complee, o eeryhing in NP reduce o SAT. (c) Prof. Argyle beliee he ha found a ay o reduce SAT o SUPERTILE. In oher ord, he beliee he ha proen ha SAT P SUPERTILE. If he i correc, doe hi imply ha SUPERTILE i NP-hard? Juify your 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[..n], ile,,..., m Oupu: A ay o ile he alkay uing a ube of he ile, in he proided order, or NO if here no ay o do i CS 70, Fall 04, MT Soln

13 In hi arian, you canno re-order he ile. For inance, if P i WBBBWWB and he ile are = BBW, = B, = WB, 4 = WB, hen he oupu i NO ; hoeer, if he ile are = WB, = B, = BBW, 4 = WB, hen he aner i ye, 4. Rohi adior aked him o proe ha ORDEREDTILE i NP-complee. Rohi ha pen he pa fe day rying o proe i, bu ihou any ucce. He ondering heher he ju need o ry harder or if i hopele. Baed on ha you e learned from hi cla, hould you encourage him o keep rying, or hould you adie him o gie up? Explain hy, in enence. Aner: Gie up. There a polynomial-ime algorihm for ORDEREDTILE uing dynamic programming, o any proof ha ORDEREDTILE i NP-complee ould imply ha P = NP. Gien ho many oher reearcher hae ried, i no reaonable o ak Rohi o proe P = NP. Explanaion: The dynamic programming algorihm: le f (i, j) be rue if here i a ay o ile P[i..n] uing ile j,..., m. Then f (i, j) = f (i, j + ) (P[i..i + len( j ) ] = j f (i + len( j ), j + )), o e obain a Θ(nm) ime algorihm for ORDEREDTILE. CS 70, Fall 04, MT Soln