17 Basic Graph Properties

Transcription

1 Ltur 17: Bs Grp Proprts [Sp 10] O look t t sn y o. Tn t t twnty-svn 8 y 10 olor lossy pturs wt t rls n rrows n prrp on t k o on... n tn look t t sn y o. An tn t t twnty-svn 8 y 10 olor lossy pturs wt t rls n rrows n prrp on t k o on n n to ry. Bus O m to t rlzton tt t ws typl s o Amrn ln just, n tr wsn t notn oul o out t, n t ju wsn t onn look t t twntysvn 8 y 10 olor lossy pturs wt t rls n rrows n prrp on t k o on xplnn wt on ws, to us s vn nst us. An w ws n ty ollrs n to pk up t r. In t snow. But tt s not wt I m r to tll you out. Arlo Gutr, Al s Rsturnt (1966) I stuy my Bl s I tr ppls. Frst I sk t wol tr, tt t rpst mt ll. Tn I lm t tr n sk lm, n tn rn n tn tw, n tn I look unr l. Mrtn Lutr 17 Bs Grp Proprts 17.1 Dntons A rp G s pr o sts (V, E). V s st o rtrry ojts tt w ll vrts 1 or nos. E s st o vrtx prs, w w ll s or osonlly rs. In n unrt rp, t s r unorr prs, or just sts o two vrts; I wll usully wrt u v nst o {u.v} to not t unrt twn u n v. In rt rp, t s r orr prs o vrts; I wll usully wrt u v nst o (u.v) to not t rt rom u to v. W wll usully onrn only wt smpl rps, wr tr s no rom vrtx to tsl n tr s t most on rom ny vrtx to ny otr. Follown stnr (ut mttly onusn) prt, I ll lso us V to not t numr o vrts n rp, n E to not t numr o s. Tus, n n unrt rp, w v 0 E V 2, n n rt rp, 0 E V (V 1). I (u, v) s n n n unrt rp, tn u s nor or v n v vrs. T r o no s t numr o nors. In rt rps, w v two kns o nors. I u v s rt, tn u s prssor o v n v s sussor o u. T n-r o no s t numr o prssors, w s t sm s t numr o s on nto t no. T out-r s t numr o sussors, or t numr o s on out o t no. A rp G = (V, E ) s surp o G = (V, E) V V n E E. A pt s squn o s, wr sussv pr o s srs vrtx, n ll otr s r sjont. A rp s onnt tr s pt rom ny vrtx to ny otr vrtx. A sonnt rp onssts o svrl omponnts, w r ts mxml onnt surps. Two vrts r n t sm omponnt n only tr s pt twn tm. 1 T snulr o vrts s vrtx. T snulr o mtrs s mtrx. Unlss you r spkn Itln, tr s no su tn s vrt, mtr, n n, n ppn, l, n p, vort, r, smpl, o, rtr, omntr, Un, Kln, n Astr, n Ol, Domt, Gt, Coon, Vtlsttst, Grtr, or Jm Hnr! You wll los ponts or usn ny o ts so-ll wors. Copyrt 2010 J Erkson. Rls unr Crtv Commons Attruton-NonCommrl-SrAlk 3.0 Lns (ttp://rtvommons.or/lnss/y-n-s/3.0/). Fr struton s stronly nour; ommrl struton s xprssly orn. S ttp:// or t most rnt rvson. 1

2 Ltur 17: Bs Grp Proprts [Sp 10] A yl s pt tt strts n ns t t sm vrtx, n s t lst on. An unrt rp s yl no surp s yl; yl rps r lso ll orsts. Trs r spl rps tt n n n svrl rnt wys. You n sly prov y nuton (nt, nt, nt) tt t ollown ntons r quvlnt. A tr s onnt yl rp. A tr s onnt omponnt o orst. A tr s onnt rp wt t most V 1 s. A tr s mnml onnt rp; rmovn ny mks t rp sonnt. A tr s n yl rp wt t lst V 1 s. A tr s mxml yl rp; n n twn ny two vrts rts yl. A spnnn tr o rp G s surp tt s tr n ontns vry vrtx o G. O ours, rp n only v spnnn tr t s onnt. A spnnn orst o G s ollton o spnnn trs, on or onnt omponnt o G. Drt rps n ontn rt pts n rt yls. A rt rp s stronly onnt tr s rt pt rom ny vrtx to ny otr. A rt rp s yl t os not ontn rt yl; rt yl rps r otn ll s Astrt Rprsnttons n Exmpls T most ommon wy to vsully rprsnt rps s y lookn t n mn. An mn o rp mps vrtx to pont n t pln n to urv or strt ln smnt twn t two vrts. A rp s plnr t s n mn wr no two s ross. T sm rp n v mny rnt mns, so t s mportnt not to onus prtulr mn wt t rp tsl. In prtulr, plnr rps n v non-plnr mns! A non-plnr mn o plnr rp wt nn vrts, trtn s, n two onnt omponnts, n plnr mn o t sm rp. Howvr, mns r not t only usul rprsntton o rps. For xmpl, t ntrston rp o ollton o ojts s no or vry ojt n n or vry ntrstn pr. Wtr prtulr rp n rprsnt s n ntrston rp pns on wt kn o ojt you wnt to us or t vrts. Drnt typs o ojts ln smnts, rtnls, rls, t. n rnt lsss o rps. On prtulrly usul typ o ntrston rp s n ntrvl rp, wos vrts r ntrvls on t rl ln, wt n twn ny two ntrvls tt ovrlp. Anotr oo xmpl s t pnny rp o rursv lortm. Dpnny rps r rt yl rps. T vrts r ll t stnt rursv suprolms tt rs wn xutn t lortm on prtulr nput. Tr s n rom on suprolm to notr vlutn 2

3 Ltur 17: Bs Grp Proprts [Sp 10] () () () T xmpl rp s lso t ntrston rp o () st o ln smnts, () st o rls, or () st o ntrvls on t rl ln (stk or vslty). t son suprolm rqurs rursv vluton o t rst suprolm. For xmpl, or t Fon rurrn 0 n = 0, F n = 1 n = 1, F n 1 + F n 2 otrws, t vrts o t pnny rp r t ntrs 0, 1, 2,..., n, n or ntr rom 2 to n, n t s r ( 1) n ( 2) or vry ntr twn 2 n n. For t t stn rurrn j = 0 j = 0 Et(, j) = Et( 1, j) + 1, mn Et(, j 1) + 1, Et( 1, j 1) + A[] B[j] otrws t pnny rp s n m n r wt onls. Dynm prormmn works ntly or ny rurrn tt s smll pnny rp; propr vluton orr nsurs tt suprolm s vst tr ts prssors. A sltly mor rvolous xmpl o rp s t onurton rp o m, puzzl, or mnsm lk t-t-to, krs, t Ruk s Cu, t Towrs o Hno, or Turn mn. T vrts o t onurton rp r ll t vl onurtons o t puzzl; tr s n rom on onurton to notr t s possl to trnsorm on onurton nto t otr wt smpl mov. (Ovously, t prs nton pns on wt movs r llow.) Evn or rsonly smpl mnsms, t onurton rp n xtrmly omplx, n w typlly only v ss to lol normton out t rp. T onurton rp o t 4-sk Towr o Hno Fnlly, t nt-stt utomt us n orml lnu tory r just ll rt rps. A trmnst nt-stt utomton s usully ormlly n s 5-tupl M = (Q, Σ, δ, q 0, A), wr Q 3

4 Ltur 17: Bs Grp Proprts [Sp 10] s nt st o stts, Σ s nt st ll t lpt, δ: Q Σ Q s trnston unton, q 0 Q s t ntl stt, n F Q s t st o ptn stts. But t s otn usul to tnk o M s rt rp G M wos vrts r t stts Q, n wos s v t orm q δ(q, x) or vry stt q Q n rtr x Σ. Tn s qustons out t lnu pt y M n prs s qustons out t rp G M. For xmpl, t lnu pt y M s mpty n only tr s no pt n G M rom q 0 to n ptn stt. It s mportnt not to onus ts xmpls/rprsnttons o rps wt t tul nton: A rp s pr o sts (V, E), wr V s n rtrry nt st, n E s st o prs (tr orr or unorr) o lmnts o V Grp Dt Struturs Tr r two ommon t struturs us to xpltly rprsnt rps: jny mtrs 2 n jny lsts. T jny mtrx o rp G s V V mtrx, n w ntry nts wtr prtulr s or s not n t rp: A[, j] := (, j) E. For unrt rps, t jny mtrx s lwys symmtr: A[, j] = A[ j, ]. Sn w on t llow s rom vrtx to tsl, t onl lmnts A[, ] r ll zros. Gvn n jny mtrx, w n n Θ(1) tm wtr two vrts r onnt y n just y lookn n t pproprt slot n t mtrx. W n lso lst ll t nors o vrtx n Θ(V ) tm y snnn t orrsponn row (or olumn). Ts s optml n t worst s, sn vrtx n v up to V 1 nors; owvr, vrtx s w nors, w my stll v to xmn vry ntry n t row to s tm ll. Smlrly, jny mtrs rqur Θ(V 2 ) sp, rrlss o ow mny s t rp tully s, so t s only sp-nt or vry ns rps Ajny mtrx n jny lst rprsnttons or t xmpl rp. For sprs rps rps wt rltvly w s jny lsts r usully ttr o. An jny lst s n rry o lnk lsts, on lst pr vrtx. E lnk lst stors t nors o t orrsponn vrtx. For unrt rps, (u, v) s stor tw, on n u s nor lst n on n v s nor lst; or rt rps, s stors only on. Etr wy, t ovrll sp rqur or n jny lst s O(V + E). Lstn t nors o no v tks O(1+(v)) tm; just sn t nor lst. Smlrly, w n trmn wtr (u, v) s n n O(1 + (u)) tm y snnn t nor lst o u. For unrt rps, w n sp up t sr y smultnously snnn 2 S ootnot 1. 4

5 Ltur 17: Bs Grp Proprts [Sp 10] t nor lsts o ot u n v, stoppn tr w lot t or wn w ll o t n o lst. Ts tks O(1 + mn{(u), (v)}) tm. T jny lst strutur soul mmtly rmn you o s tls wt nn. Just s wt s tls, w n mk jny lst strutur mor nt y usn somtn ss lnk lst to stor t nors. For xmpl, w us s tl wt onstnt lo tor, wn w n tt s n O(1) xpt tm, just s wt n jny lst. In prt, ts s only usul or vrts wt vry lr r, us t onstnt ovr n ot t sp n sr tm s lrr or s tls tn or smpl lnk lsts. At ts pont, you mt rsonly sk wy nyon woul vr us n jny mtrx. Atr ll, you us s tls to stor t nors o vrtx, you n o vrytn s st or str wt n jny lst s wt n jny mtrx, only usn lss sp. On omplln nswr s tt mny rps r mpltly rprsnt y jny mtrs. For xmpl, ntrston rps r usully rprsnt s n rry or lst o t unrlyn omtr ojts. As lon s w n tst wtr two ojts ovrlp n onstnt tm, w n pply ny rp lortm to n ntrston rp y prtnn tt t s stor xpltly s n jny mtrx. On t otr n, ny t strutur ul rom rors wt pontrs twn tm n sn s rt rp. Alortms or srn rps n ppl to ts t struturs y prtnn tt t rp s rprsnt xpltly usn n jny lst. Smlrly, w n pply ny rp lortm to onurton rp s tou t wr vn to us s n jny lst, prov w n numrt ll possl movs rom vn onurton n onstnt tm. To kp tns smpl, w ll onsr only unrt rps or t rst o ts ltur, ltou t lortms I ll sr lso work or rt rps Trvrsn onnt rps Suppos w wnt to vst vry no n onnt rp (rprsnt tr xpltly or mpltly). T smplst mto to o ts s n lortm ll pt-rst sr, w n wrttn tr rursvly or trtvly. It s xtly t sm lortm tr wy; t only rn s tt w n tully s t rurson stk n t non-rursv vrson. Bot vrsons r ntlly pss sour vrtx s. RECURSIVEDFS(v): v s unmrk mrk v or vw RECURSIVEDFS(w) ITERATIVEDFS(s): PUSH(s) wl t stk s not mpty v POP v s unmrk mrk v or vw PUSH(w) Dpt-rst sr s on (prps t most ommon) nstn o nrl mly o rp trvrsl lortms. T nr rp trvrsl lortm stors st o nt s n som t strutur tt I ll ll. T only mportnt proprts o r tt w n put stu nto t n tn ltr tk stu k out. (In C ++ trms, tnk o t s tmplt or rl t strutur.) A stk s prtulr typ o, ut rtnly not t only on. Hr s t nr trvrsl lortm: 5

6 Ltur 17: Bs Grp Proprts [Sp 10] TRAVERSE(s): put s n wl t s not mpty tk v rom t v s unmrk mrk v or vw put w nto t Ts trvrsl lortm lrly mrks vrtx n t rp t most on. In orr to sow tt t vsts vry no n t rp t lst on, w moy t lortm sltly; t motons r lt n r. Inst o kpn vrts n t, t mo lortm stors prs o vrts. Ts llows us to rmmr, wnvr w vst vrtx v or t rst tm, w prvously-vst vrtx p put v nto t. Ts vrtx p s ll t prnt o v. TRAVERSE(s): put (, s) n wl t s not mpty tk (p, v) rom t ( ) v s unmrk mrk v prnt(v) p or vw ( ) put (v, w) nto t ( ) Lmm 1. TRAVERSE(s) mrks vry vrtx n ny onnt rp xtly on, n t st o prs (v, prnt(v)) wt prnt(v) ns spnnn tr o t rp. Proo: T lortm ovously mrks s. Lt v ny vrtx otr tn s, n lt s u v t rt pt rom s to v wt t mnmum numr o s. Sn t rp s onnt, su pt lwys xsts. (I s n v r nors, tn u = s, n t pt s just on.) I t lortm mrks u, tn t must put (u, v) nto t, so t must ltr tk (u, v) out o t, t w pont v must mrk. Tus, y nuton on t sortst-pt stn rom s, t lortm mrks vry vrtx n t rp. Ts mpls tt prnt(v) s wll-n or vry vrtx v. T lortm lrly mrks vry vrtx t most on, so t must mrk vry vrtx xtly on. Cll ny pr (v, prnt(v)) wt prnt(v) prnt. For ny no v, t rt pt o prnt s v prnt(v) prnt(prnt(v)) vntully ls k to s, so t st o prnt s orm onnt rp. Clrly, ot nponts o vry prnt r mrk, n t numr o prnt s s xtly on lss tn t numr o vrts. Tus, t prnt s orm spnnn tr. T xt runnn tm o t trvrsl lortm pns on ow t rp s rprsnt n wt t strutur s us s t, ut w n mk w nrl osrvtons. Bus vrtx s mrk t most on, t or loop ( ) s xut t most V tms. E uv s put nto t xtly tw; on s t pr (u, v) n on s t pr (v, u), so ln ( ) s xut t most 2E tms. Fnlly, w n t tk mor tns out o t tn w put n, so ln ( ) s xut t most 2E + 1 tms Exmpls Lt s rst ssum tt t rp s rprsnt y n jny lst, so tt t ovr o t or loop ( ) s only onstnt tm pr. 6

7 Ltur 17: Bs Grp Proprts [Sp 10] I w mplmnt t usn stk, w rovr our ornl pt-rst sr lortm. E xuton o ( ) or ( ) tks onstnt tm, so t ovrll runnn tm s O(V + E). I t rp s onnt, w v V E + 1, n so w n smply t runnn tm to O(E). T spnnn tr orm y t prnt s s ll pt-rst spnnn tr. T xt sp o t tr pns on t strt vrtx n on t orr tt nors r vst n t or loop ( ), ut n nrl, pt-rst spnnn trs r lon n sknny. I w us quu nst o stk, w t rt-rst sr. An, xuton o ( ) or ( ) tks onstnt tm, so t ovrll runnn tm or onnt rps s stll O(E). In ts s, t rt-rst spnnn tr orm y t prnt s ontns sortst pts rom t strt vrtx s to vry otr vrtx n ts onnt omponnt. W ll s sortst pts n n utur ltur. An, xt sp o rt-rst spnnn tr pns on t strt vrtx n on t orr tt nors r vst n t or loop ( ), ut n nrl, sortst pt trs r sort n usy. A pt-rst spnnn tr n rt-rst spnnn tr o on omponnt o t xmpl rp, wt strt vrtx. Now suppos t s o t rp r wt. I w mplmnt t usn prorty quu, lwys xtrtn t mnmum-wt n ln ( ), t rsultn lortm s rsonly ll sortst-rst sr. In ts s, xuton o ( ) or ( ) tks O(lo E) tm, so t ovrll runnn tm s O(V + E lo E), w smpls to O(E lo E) t rp s onnt. For ts lortm, t st o prnt s orm t mnmum spnnn tr o t onnt omponnt o s. Surprsnly, s lon s ll t wts r stnt, t rsultn tr os not pn on t strt vrtx or t orr tt nors r vst; n ts s, tr s only on mnmum spnnn tr. W ll s mnmum spnnn trs n n t nxt ltur. I t rp s rprsnt usn n jny mtrx nst o n jny lst, nn ll t nors o vrtx n ln ( ) tks O(V ) tm. Tus, pt- n rt-rst sr run n O(V 2 ) tm, n sortst-rst sr runs n O(V 2 + E lo E) = O(V 2 lo V ) tm Srn sonnt rps I t rp s sonnt, tn TRAVERSE(s) only vsts t nos n t onnt omponnt o t strt vrtx s. I w wnt to vst ll t nos n vry omponnt, w n us t ollown wrppr roun our nr trvrsl lortm. Sn TRAVERSE omputs spnnn tr o on omponnt, TRAVERSEALL omputs spnnn orst o t ntr rp. TRAVERSEALL(s): or ll vrts v v s unmrk TRAVERSE(v) Som txtooks lm tt ts wrppr n only us wt pt-rst sr; ty r wron. 7

8 Ltur 17: Bs Grp Proprts [Sp 10] Exrss 1. Prov tt t ollown ntons r ll quvlnt. A tr s onnt yl rp. A tr s onnt omponnt o orst. A tr s onnt rp wt t most V 1 s. A tr s mnml onnt rp; rmovn ny mks t rp sonnt. A tr s n yl rp wt t lst V 1 s. A tr s mxml yl rp; n n twn ny two vrts rts yl. 2. Prov tt ny onnt yl rp wt n 2 vrts s t lst two vrts wt r 1. Do not us t wors tr o l, or ny wll-known proprts o trs; your proo soul ollow ntrly rom t ntons. 3. Lt G onnt rp, n lt T pt-rst spnnn tr o G root t som no v. Prov tt T s lso rt-rst spnnn tr o G root t v, tn G = T. 4. Wnvr roups o pons tr, ty nstntvly stls pkn orr. For ny pr o pons, on pon lwys pks t otr, rvn t wy rom oo or potntl mts. T sm pr o pons lwys ooss t sm pkn orr, vn tr yrs o sprton, no mttr wt otr pons r roun. Surprsnly, t ovrll pkn orr n ontn yls or xmpl, pon A pks pon B, w pks pon C, w pks pon A. () Prov tt ny nt st o pons n rrn n row rom lt to rt so tt vry pon pks t pon mmtly to ts lt. Prtty pls. () Suppos you r vn rt rp rprsntn t pkn rltonsps mon st o n pons. T rp ontns on vrtx pr pon, n t ontns n j n only pon pks pon j. Dsr n nlyz n lortm to omput pkn orr or t pons, s urnt y prt (). 5. You r lpn roup o tnorprs nlyz som orl story t ty v ollt y ntrvwn mmrs o vll to lrn out t lvs o popl lv tr ovr t lst two unr yrs. From t ntrvws, you v lrn out st o popl, ll now s, wom w wll not P 1, P 2,..., P n. T tnorprs v ollt svrl ts out t lspns o ts popl. Splly, or som prs (P, P j ), t tnorprs v lrn on o t ollown ts: () P or P j ws orn. () P n P j wr ot lv t som momnt. Nturlly, t tnorprs r not sur tt tr ts r orrt; mmors r not so oo, n ll ts normton ws pss own y wor o mout. So ty lk you to trmn wtr t t ty v ollt s t lst ntrnlly onsstnt, n t sns tt tr oul v xst st o popl or w ll t ts ty v lrn smultnously ol. 8

9 Ltur 17: Bs Grp Proprts [Sp 10] Dsr n nlyz n lortm to nswr t tnorprs prolm. Your lortm soul tr output possl ts o rt n t tt r onsstnt wt ll t stt ts, or t soul rport orrtly tt no su ts xst. 6. Lt G = (V, E) vn rt rp. () T trnstv losur G T s rt rp wt t sm vrts s G, tt ontns ny u v n only tr s rt pt rom u to v n G. Dsr n nt lortm to omput t trnstv losur o G. () T trnstv ruton G TR s t smllst rp (mnn wst s) wos trnstv losur s G T. Dsr n nt lortm to omput t trnstv ruton o G. 7. A rp (V, E) s prtt t vrts V n prtton nto two susts L n R, su tt vry s on vrtx n L n t otr n R. () Prov tt vry tr s prtt rp. () Dsr n nlyz n nt lortm tt trmns wtr vn unrt rp s prtt. 8. An Eulr tour o rp G s los wlk trou G tt trvrss vry o G xtly on. () Prov tt onnt rp G s n Eulr tour n only vry vrtx s vn r. () Dsr n nlyz n lortm to omput n Eulr tour n vn rp, or orrtly rport tt no su rp xsts. 9. T -mnsonl ypru s t rp n s ollows. Tr r 2 vrts, ll wt rnt strn o ts. Two vrts r jon y n tr lls r n xtly on t. () A Hmltonn yl n rp G s yl o s n G tt vsts vry vrtx o G xtly on. Prov tt or ll 2, t -mnsonl ypru s Hmltonn yl. () W yprus v n Eulr tour ( los wlk tt trvrss vry xtly on)? [Hnt: Ts s vry sy.] 10. Rtrk (lso known s Grp Rrs n Vtor Rlly) s two-plyr ppr-n-pnl rn m tt J ply on t us n 5t r. 3 T m s ply wt trk rwn on st o rp ppr. T plyrs ltrntly oos squn o r ponts tt rprsnt t moton o r roun t trk, sujt to rtn onstrnts xpln low. E r s poston n vloty, ot wt ntr x- n y-oornts. T ntl poston s pont on t strtn ln, osn y t plyr; t ntl vloty s lwys (0, 0). At stp, t plyr optonlly nrmnts or rmnts tr or ot oornts o t r s 3 T tul m s t mor omplt tn t vrson sr r. In prtulr, n t tul m, t ounrs o t trk r r-orm urv, n (t lst y ult) t ntr ln smnt twn ny two onsutv postons must l ns t trk. In t vrson J ply, r os run o t trk, t r strts ts nxt turn wt zro vloty, t t ll r pont losst to wr t r lt t trk. 9

10 Ltur 17: Bs Grp Proprts [Sp 10] vloty; n otr wors, omponnt o t vloty n n y t most 1 n snl stp. T r s nw poston s tn trmn y n t nw vloty to t r s prvous poston. T nw poston must ns t trk; otrws, t r rss n tt plyr loss t r. T r ns wn t rst r rs poston on t ns ln. Suppos t rtrk s rprsnt y n n n rry o ts, wr 0 t rprsnts r pont ns t trk, 1 t rprsnts r pont outs t trk, t strtn ln s t rst olumn, n t ns ln s t lst olumn. Dsr n nlyz n lortm to n t mnmum numr o stps rqur to mov r rom t strtn ln to t ns ln o vn rtrk. [Hnt: Bul rp. Wt r t vrts? Wt r t s? Wt prolm s ts?] vloty poston (0, 0) (1, 5) (1, 0) (2, 5) (2, 1) (4, 4) (3, 0) (7, 4) (2, 1) (9, 5) (1, 2) (10, 7) (0, 3) (10, 10) ( 1, 4) (9, 14) (0, 3) (9, 17) (1, 2) (10, 19) (2, 2) (12, 21) (2, 1) (14, 22) (2, 0) (16, 22) (1, 1) (17, 21) (2, 1) (19, 20) (3, 0) (22, 20) (3, 1) (25, 21) START FINISH A 16-stp Rtrk run, on trk. Ts s not t sortst run on ts trk. 11. Druts/krs s m ply on n m m r o squrs, ltrntly olor lt n rk. (T m s usully ply on n 8 8 or or, ut t ruls sly nrlz to ny or sz.) E rk squr s oup y t most on m p (usully ll kr n t U.S.), w s tr lk or wt; lt squrs r lwys mpty. On plyr ( Wt ) movs t wt ps; t otr ( Blk ) movs t lk ps. Consr t ollown smpl vrson o t m, ssntlly Amrn krs or Brts ruts, ut wr vry p s kn. 4 Ps n mov n ny o t our onl rtons, tr on or two stps t tm. On turn, plyr tr movs on o r ps on stp onlly nto n mpty squr, or mks srs o jumps wt on o r krs. In snl jump, p movs to n mpty squr two stps wy n ny onl rton, ut only t ntrmt squr s oup y p o t oppost olor; ts nmy p s ptur n mmtly rmov rom t or. Multpl jumps r llow n snl turn s lon s ty r m y t sm p. A plyr wns r opponnt s no ps lt on t or. Dsr n lortm tt orrtly trmns wtr Wt n ptur vry lk p, try wnnn t m, n snl turn. T nput onssts o t wt o t or (m), lst o postons o wt ps, n lst o postons o lk ps. For ull rt, your lortm 4 Most otr vrnts o ruts v lyn kns, w v vry rntly tn wt s sr r. 10

11 Ltur 17: Bs Grp Proprts [Sp 10] soul run n O(n) tm, wr n s t totl numr o ps. [Hnt: T ry strty mk rtrry jumps untl you t stuk os not lwys n wnnn squn o jumps vn wn on xsts. S prolm 8. Prty, prty, prty.] Wt wns n on turn. Wt nnot wn n on turn rom tr o ts postons. Copyrt 2010 J Erkson. Rls unr Crtv Commons Attruton-NonCommrl-SrAlk 3.0 Lns (ttp://rtvommons.or/lnss/y-n-s/3.0/). Fr struton s stronly nour; ommrl struton s xprssly orn. S ttp:// or t most rnt rvson. 11