Jonathan Turner Exam 2-12/17/03

Transcription

1 CS 54 Aotms n Poms Fn Pom Exm St Soutons 2 Soutons Jontn Tun Exm 2-2/7/3 /28/3. (5 onts) Sow tt t y mto fo t mnmum snnn t mntns t oo nvnt un n ton of t u u. Coo nvnt. T xsts mnmum snnn t T tt nus t u s n non of t ons. Lt {u,v} t st y n ton of t u u n t T t mnmum snnn t tt stsfs t nvnt fo t ton of t u u. If {u,v} s n T, tn y T st stsfs t nvnt ft t ton of t u u. Assum tn tt {u,v} s not n T. Lt (X,X ) t ut us y t u u to st {u,v}. Sn T s snnn t, t t fom u to v n T must nu t st on {x,y} tt osss t ut. Sn non of t s tt oss t ut fo t ton of t u u s u, {x,y} must unoo (t n t, sn T nus no s}. Sn {u,v} ws st to v mnmum ost mon unoo s ossn t ut, t ost of {u,v} n no tn t ost of {x,y}. If w mov {x,y} fom T, w st T nto two ts, on ontnn u n on ontnn v. If w tn t {u,v} w t nw snnn t tt osts no mo tn T n w stsfs t oo nvnt. - -

2 2. (5 onts) T t ow sows t unnn tms fo t tmn oton n t ny oton n Dst s otm usn -s wt vyn vus of. T unnn tms sown fo nom wt 5, vts n out, s. W uv s fo tmn n w s fo ny? Exn wy. 8 tm (s) mt ( ) T to uv must t on fo tmn, sn t unnn tm of tmn s O( o n) n ts nss s t mt nss, w t unnn tm of ny s O(o n) sn t y vus wys u, w os s t mt nss. If t num of s to ny s % of t wost-s num n t num of s to tmn qus t wost-s num wn =2, wt s t to twn t v unnn tm of on to tmn to t v unnn tm of on to ny. In t wost-s w n v, s to ny n 5, s to tmn. In ts s, t ony, s to ny n 5, s to tmn. If t unnn tms w qu, w wou xt t vu on t t fo ny to tw s s t vu fo tmn. In ft, t vu on t t fo ny s sty mo tn on fft s s t vu fo tmn. So t s tt to tmn ts ny tn tms s on s to ny wn =

3 3. (2 onts) T otnss of ny t stutu oton ns on ts mntnn tn ssnt ots of t t stutu. T t oton of t ss ton fo t C mmntton of t Fon t stutu n t st t stutu w t uss sown ow. Wt ots of t two t stutus must mntn y oms tt ot on tm? Lst s mny s you n tn of. You my stt ots nfomy (ut sy) n Ens. ss fs { nt n; // fs fn on tms,...,n stut no { yty yf; // y f nt nf; // n f t mf; // m f no f, f; //, nt onts } *v; st* ss; // oton of sn sts... }; ss st { nt N; // st fn on tms,...,n nt *nxt; // nxt[] s susso of n st nt *v; // v[] s sso of n st... }; n=n T nxt n v onts n st fom ouy n u sts tt tton t st of tms. Tt s, tm n,...,n s n xty on st (ossy snton st). Ts mns, n tu tt nxt[v[u]] = u fo u n f you foow nxt (o v) onts, you tun to t sm vtx. T n of no s qu to t num of n t s. Tt s, t num of tms n t st ontnn v[u].f s qu to v[u].nf fo nos u. A tms n t sm st wtn st v t sm vu of t nt f (so fo nos u v[u].f=v[nxt[u].f). If t nt f s not nu, tn tm snt y t nt f s f tt snts on of t tms n t vn st. Itms n ffnt sts v ffnt nt onts. T nt onts fn st of ts (t no ys fn y t nt onts). Fo tm x wt non-nu nt f, t y f of t nt s ss tn o qu to t y f of x. Tt s, v[v[x].f].yf v[x].yf. T vu of t n fs s t o n t n of n tm x s qu to t num of n tt x s ost sn t m t of som ot no

4 4. (2 onts) To f s ntv y, w n n ton sou vtx, wt zo ost to of t on vts n tn un t t-fst snnn otm. Ts ts O(mn) tm. W n otnty s u t otm fo s wt sm ntv ys (ys wt ust fw s) y n to s f t nt onts fom y t t n of ss. T suoutn sown ow s to s f t nt onts fom y. Wt s ts unnn tm? Wy? nt y(nt n, nt []) { // Rtun f t nt onts,, fom y. Es tun. } nt u, v; nt ount; nt *m = nw nt[n]; // m[u]= on w onfm tt u // s nu nsto, s m[u]= fo (u = ; u <= n; u) m[u] = ([u] == Nu? : ); fo (u = ; u <= n; u) { ount = ; fo (v = u; m[v] == ; v = [v]) f (ount == n) tun ; fo (v = u; m[v] == ; v = [v]) m[v] = ; } tun ; T unnn tm s O(n). Sn m[u] s ntz to fo vts wt nu nt ont, t nn oos unt to tmnt f u s not t of y. Lt x t num of ttons of of t nn oos wn t vu of t oo v s x. If x s t of y, tn t ount v nsus tmnton ft t most n ttons of t fst nn oo. If x s not t of y, tn x m ts st un t tton of t out oo wn u=x. On n m ts st no fut nn oo ttons n fom. Hn, t ov unnn tm s O(n). Suos w un ts suoutn ft ss of t t-fst snnn otm n tmnt y f y s foun. Wt s t unnn tm of ts mof otm f y s fom n t nt onts ft sss? Wy? O((mn)), sn ss ts O(m) tm n y ts O(n) tm. Suos w un t t-fst snnn otm on t sown ow (ft n t xt sou vtx n onntn s), n o to tt ntv ys. In t wost s, ow mny sss must xut y t otm fo t ntv y s ft y t nt onts? Wy? It w t 5 sss. In t wost-s, t quu w ontn vts,,, n tt o t t n of t fst ss, t stn zo wt nt s. At t n of t son ss,, n w v stn wt ()=, ()= n ()=. At t n of t t ss, n w v stn 2 wt no n n t nt onts. At t n of t fout ss, w v stn 3 wt no n n t nt onts. Fny, t t n of t fft ss, w v stn 3 n ()=, fomn y n t nt onts

5 5. (5 onts) In t two-s otm fo fnn mxmum fows n unt ntwos, w us t-fst s to fn umntn ts on (s) xs (n 2) /2. W v sown tt ts ows t otm to omt n O(mn /2 ) tm. If w o not swt to tfst s n t son s, t otm my not omt n O(mn /2 ) tm. Now, ons vson of t mss t otm n w xt stn s omut ft vy n ns n t unt umntn t s s stt t t sou vtx. Ts vson of t otm n fst tn t on otm n t, sn t n vo nums of n sts. Sow tt f ts otm s un on unt ntwo tt (s) xs (n 2) /2 ft O(mn /2 ) tm (you my us t ft tt t fst s of t two s otm omts n O(mn /2 ) tm). Sn t fst s of t two s otm uss t on mss t otm, t suffs to sow tt t tm qu fo omutn t xt stn s s O(mn /2 ). Sn non of t stn s xs (n 2) /2 un t fst s, t O(n 3/2 ) ns un t fst s n n O(n /2 ) omuttons of xt s. Sn omutton ts O(m) tm, t ton tm snt on omuttons s O(mn /2 ). Sow tt t mof mss t otm omts t omutton of mxmum fow n O(mn /2 ) tm, ft (s) xs (n 2) /2 (you my us fts ov n t on nyss of t two s otm, wtout -vn tm). As ws sown n t on nyss of t two s otm, t ffn twn t unt fow n t mxmum fow s O(n /2 ) ft (s)>(n 2) /2. Ts mns tt O(n /2 ) nw umntn ts mn to foun. Aft omutton of t vtx s, t nxt umntn t must foun fo t nxt nvu vtx n ous. Ts mns tt t O(n /2 ) omuttons n to omt t mx fow omutton, so t tm snt on ts omuttons s O(mn /2 ). Ts so ms tt t tm snt n nvu vts s O(n 3/2 )

6 6. (5 onts) T ffnt ost sntton us y t t sts t stutu qus tt mn n ost ut un otton oton. Sow tt t two qutons ow ot. 2, ost, mn x 4, z,, 2,4 ost, mn, z 4, 2,5 x 2,,2 ost'(z)= ost(z) mn(z) ost'(z)= ost'(z) mnost'(z) = ost(z) mnost(x) = ost(z) (mnost(z) mn(z)) = ost(z) mn(z) mn'()= mn() mn(z) mn'()= mnost'() mnost'(z) = mnost() mnost(x) = (mnost() mnost(z)) (mnost(z) mnost(x)) = mn() mn(z) - 6 -

7 7. (5 onts) T fu t t sows n ntmt stt n t xuton of t sn otm (wt ost tnsfomton) fo t mn-ost, mx fow om. In tu, t fu sows t stt of t otm t t n of t fst sn s. Sow t stt of t otm ust fo t fst umntn t s of t son sn s (tt s, ft s v n n fows ust to mnt votons of t n onton). 5, s s 2,, ty, ost, fow 5,2, 3,2, 5,2,5 6,,4 2,,,2 5,2 5,-2 4,- 5,, 6,3, 7,3,5 3,, 4,,4 5,2,4,3 5, 4,-2,2,4 t xss, su ty, ost 6,3 5,-3 t -5,5,2 4,- 2,,2 5,2, 3,2, 5,, 6,3, s 5,2,5 7,3,5 t 6,,6 2,, 3,,3 4,,4 2,2 5,2,4 2, 2,3 5,2 3,2 5, 6,3 3, s 5,-2 6, 2, 3,2 5,3 2,3 3, 4, t -5,5 2,2 4, ,4

8 ( onts) T fu ow sows n ntmt st n t xuton of t Emons- K otm fo fnn mxmum sz mtn. T mtn s not sown xty, ut nt onts sown n t tton t stutu s sown. Dw s oun t vtx sts w fom ossoms. M t s of t ossoms wt B. On t m, m vn vts wt us sn n o vts wt mnus sn. Int w s n t unt mtn, y mn tm wt n M. m q n m q n m q n B B B B