On Self-Avoiding Walks across n-dimensional Dice and Combinatorial Optimization: An Introduction

Transcription

1 On Sf-Avoiing Waks acoss n-dimnsiona Dic an ombinatoia Optimization: An Intoction Fanc Bgz ompt Scinc Raigh, N, USA Vsion: Sn Sp 5:5: EDT 3 ontnts Th fab -- Abot Gt an Hans saching fo ks (an abot Jok hiing th ks) Notation an finitions -- ombinatoia pobm: fin b fnction an (concatnat) cooinat tp(s) Th foing pobm amps: n pan A, pan B, an pan -- cooinat nighbohoo(s) fin b cooinat tp(s) -- a sf-avoiing wak (SAW): a sqnc of stps that chain a niq st of pivot cooinats Goba sach n SAW an pimnta sts -- potin foing on ctanga attic in D Smma an concsions Appni -- Epimnts with SAW in pogss: gaph {V, iq, IS, A}, abs, Goomb, masat, npp, jobs,...

2 Th Fab (abot Gt, Hans, an Jok who kps hiing th ks) -- Jok isgiss th fist st of ks as ab tickts in two ns: Gt an Hans sach fo ks b tiving thm fom th spctiv ns nti ach fins a tickt whos ab psnt th combination fo th ock on th oo to thi apatmnt. -- Onc Gt an Hans opn oos to thi apatmnts, ach stps onto a patfom (o a fac) of a hg patonic soi (pohon) -- so th think at fist In fact, Jok pac ach n not with a pohon bt with a hphon (sam nmb of facs, iffnt fac ajacnc stct) an hi anoth st of ks, this tim n th ab tickts, attach to fac cnts of ach hphon. -- Jok asks Gt an Hans to sach fo ks b waking thi spctiv hphons, fom fac-to-fac, nti th fin th a k to thi apatmnt. 3 Gt, Hans, an Jok (invisib bt omnipsnt) Majoing in ompt Scinc Majoing in an Sving Both a tning to thi ajacnt apatmnts (aft a pat) 4

3 Jok pacs ocks on oos an posts two ns with cs Gt Gt taks th tickt fom th n an if sh os not scc in opning th oo, sh pts th tickt into h hanbag an tivs anoth tickt. W sa that Gt is samping contnts of th n withot pacmnt: th pobabiit of Gt fining th coct tickt on tia k foows nifom istibtion, givn k maining tickts: pobabiit is /k, man va is (k + )/, an vaianc is (k )/. Gt an Hans iscov not on that ocks hav bn chang on both apatmnt oos with pnch-k ocks bt aso that mats that hi th ks w pac with two ns, ach containing a st 36 tickts. Each tickt has a pint ab with fiv igits in th fomat.:z (. is a bina/tna cooinat, z is a va) (^)*(3^) = 36 niq cooinats! On on ab opns Gt s oo, an on on opns Hans s oo. Th two sts a intica. Who gts in fist? Hans Hans, who ha a fw inks at th pat, taks th tickt an if h os not scc in opning th oo, tns th tickt to th n. W sa that Hans is samping contnts of th n with pacmnt: th pobabiit of Hans fining th coct tickt on tia k foows gomtic istibtion: pobabiit is (/n)( (/n)) k, man va is n, an vaianc is n ( (/n)). Th point of th fab so fa: w an that in a sach scnaios sch as scib h, on can impov th chanc of fist sccss b namica cing th sach spac aft ach tia. In th avag cas, Gt s sach, vn with hanbag of imit capacit, awas qis fw tias than Hans s. 5 On nting spctiv apatmnts, ach stps onto what appas as a patonic soi, bt on Jok knows what it is....:9.:.:9.:.:.:.:6.:5.:6.:5 Gt s Ent Hans s Ent () Upon nt, both a staning on a fat patfom (a fac) of an objct, with fo ajacnt facs soping ownwas -- an a th can s is th ab of th fac on which th a staning an th fo abs on th ajacnt facs. Each objct has 36 facs with 36 abs takn fom ach n, spctiv. () Jok asks thm to wak fom fac-to-fac an sach fo th ab that his th k to thi a oo (th k is now n th ab). H givs thm on, an on on, c abot th ab: th ab with va of o ss than has th k. 6

4 On Jok has a goba viw of this objct: a hphon Jok cats this viw as foows: Hass ank istanc fom th initia cooinat (th bottom vt) :9.:9.:9.:6.:5.:9.:9.:9.:9.:.:4.:.:.:9.:9.:9.:.:.:3.:3.:.:.:.:9.:9.:9.:.:4.:.:.:.:9.:6.:5.:.: fnction = BT.in. V = 36, E = vtics an abs a o -> R b fnction vas (fo cootp=bt, vt istibtion at ach ank ma pn on cooinit) -- H movs insi th hphon, fins th cnt of th fac, an attachs on n of a sting to th cnt an attachs th oth sting to th cnt of th ajacnt fac. -- H pats th pocss fo a facs an ths cats a gaph; in this cas an nict gaph with 36 facs an 4 gs. Not that th pohon with 36 facs, with ach fac having on 4 nighbos, has gs tota! -- H aso assigns fnction vas to ach cooinat: his choic of vas is pct to confon Gt an Hans in thi sach. -- To psnt this gaph in th pan, h fins a istanc btwn th cooinats assign to ach ab an maks a pojction of th gaph as a a gaph. -- This gaph (now a Hass gaph) is not visib to Gt an Hans, howv, th gaph nabs Jok to tac ach stp th wo mak ing thi sach. Tacing th sf-avoiing wak b Gt :9.:9.:9.:6.:5.:9.:9.:9.:9.:.:4.:.:.:9.:9.:9.:.:.:3.:3.:.:.:.:9.:9.:9.:.:4.:.:.:.:9.:6.:5.:.: Sia nmb fo Gt's k:.: Wak hains Un Hamitonian chain = soi wak, b Gt taks chain 5 stps = ott to fin Hans's chain k = soi an gn 3 stps chain h own? = tc... k. 4 6 vtics an abs a o -> R b fnction vas (fo cootp=bt, vt istibtion at ach ank ma pn on cooinit) Gt is a compt scinc majo an mmbs a ct abot Hamitonian waks in gaphs. Sh knows that sh is staning on on of 36 facs an that if sh associats ach fac with a vt in a gaph an th gs btwn ajacnt facs with gs in this gaph, sh can compt an mmb th path that visits ach fac on onc. In th wost cas, sh wi tak 35 stps to fin th k. Th poc sh ss to compt th cooinats fo ach stp in th Hamitonian wak is not as simp to pain as th poc s b Hans an pain nt. Sffic it to sa that fnction vas associat with ach cooinat hav no o in tmining th Hamitonian path in th gaph. An amp of Gt s wak, as tac b Jok, is shown on ft. Sh taks 5 stps to fin Hans s k an ns to contin fo a tota of 3 stps bfo fining h k. Th fist stp, fom.: to.:6, is a ibat stp in this Hamitonian wak a stp that Hans wo nv hav takn fom this stating position fo asons w pain nt.

5 Tacing th sf-avoiing wak b Hans B-nighbos T-nighbos.:9.:9.:9.:6.:5 Sia nmb fo Hans's k:.:.:9.:9.:9.:9.:.:4.:.:.:9.:9.:9.:.:.:3.:3.:.:.:.:9.:9.:9.:.:4.:.:.:.:9.:6.:5.:.: fnction = BT.in. V = 36, E = vtics an abs a o -> R b fnction vas (fo cootp=bt, vt istibtion at ach ank ma pn on cooinit) Hans s majo is an sving an taks th hint abot fnction vas sios. H viss a fw s bfo stating th wak: () mak th fac fom wh th wak stats with an as-to-spot tokn; at on in th pap, w ca this fac th initia pivot. () pob ach ajacnt fac that has not t bn mak an wit its va on a ist. (3) sct th ajacnt fac with th smast va, stp on this fac, ca it a cnt pivot, an mak it with a nw tokn. If th a sva facs with th sam va, mak a anom sction. (4) pat stp () fom th cnt pivot nti aching th tagt va. Th pocss of making th pivots ing th wak with tokns maks this wak sf-avoiing. Hans can n into a pobm with ths s in two cass: () h ns ot of tokns, () th wak is tapp. In ith cas, Hans stats th wak fom a nw fac. Hans can fin th tickt that his his pass-k in 3 stps o ss fom man initia positions. What w show is ik his ongst wak: stps. 9 Notation an finitions -- ombinatoia pobm: fin b fnction an (concatnat) cooinat tp(s) th foing pobm: amps n pan A, pan B, an pan -- cooinat nighbohoo(s) fin b cooinat tp(s) -- a sf-avoiing wak (SAW): a sqnc of stps that chain a niq st of pivot cooinats

6 A combinatoia pobm amp (th foing pobm, D) -- fin b fnction va(s) an (concatnat) cooinat tp(s) cooinat: concatnation of tp B(ina) an tp T(na). bbwwbwbwwb.ccc Two-coo configation (=, wight=5) onfomation in D (ctanga attic) c c b b w w b w b w w b c Fnction va: ngativ sm of bb bons = -4 A combinatoia pobm amp (th foing pobm) -- n pan A, pan B, an pan pan A (th taitiona foing pobm fomation) Inpt: a fi configation -- a bina cooinat of ngth an wight W Sach fo minimm ng confimation in E (fnction va): tn a bst-va confomation as a tna cooinat of ngth - pan B (aka as th invs foing pobm fomation) Inpt: a fi confomation -- a tna cooinat of ngth -, wight Wma Sach fo minimm ng configation in E (fnction va): tn a bst-va configation as a bina cooinat of ngth an wight <= Wma pan (os ma w b th fist sts n this fomation) Inpt: a anom configation AND a anom confomation, i.. a bina cooinat of ngth an wight Wma an a tna cooinat of ngth - Sach fo minimm ng E (fnction va): tn a bst-va configation AND confomation as a bst-va bina cooinat of ngth an wight <= Wma AND a bst-va tna cooinat of ngth -

7 ooinat nighbohoo(s) fin b cooinat tp(s) Pob th nighbohoo bfo making th nt stp! (caw bfo o wak).:9.:9.:9.:6.:5.:9.:9.:9.:9.:.:4.:.:.:9.:9.:9.:.:.:3.:3.:.:.:.:9.:9.:9.:.:4.:.:.:.:9.:6.:5.:.: B-nighbos T-nighbos 4 6 vtics an abs a o -> R b fnction vas (fo cootp=bt, vt istibtion at ach ank ma pn on cooinit) Th Hass gaph is th on intoc b Jok to tac waks b Gt an Hans. abs in this gaph a a concatnation of bina cooinats an tna cooinats, a tota of (^) * (3^) = 36 Each bina cooinat has awas a nighbohoo of (ott gs). Each tna cooinat has awas a nighbohoo that vais fom to 4 (ott gs). Fo amp, th cooinat. has bina an tna nighbos; th cooinat. has bina an 4 tna nighbos. 3 ooinat nighbohoo(s) fin b cooinat tp(s) (pobing n pan... th most gna cas) % fnc.bt.nighb.saw fohp. ib it coob.coot Bst n p coot NA NA NA NA NA NA NA NA NA NA NA NA NA. - + NA NA NA * NA NA NA NA NA <-- th nt stp to tak In this amp, = an th a (^)*(3^9) =,55,39 cooinats -- w cannot show thm in a Hass gaph -- w can on obsv th pobing of nighbohoo fom a tab sch as shown h. Inics ib an it that ass vas in bina an tna cooinats a awas anom pmt in o to pvnt biasing th o of choics fo bst fnction va. Fnction vas > psnt not on nfasib confomations bt aso th ativ v of nfasibiit. Th conts n an p pot th siz of th nighbohoo an th nmb pobs to fin ach va of Bst. Th pobing of 9 nighbos of an initia pivot cooinat. as, in this amp -- an in a sing stp, to an optima confomation with a tagt ng of

8 ---> A sf-avoiing wak (SAW): Hamitonian of ngth 4 -- a sqnc of stps that chain a niq st of pivot cooinats (a) - - : :4 : :6 :9 :3 : :5 : : : :4 : : :5 :3 ---> - - Hass ank istanc fom th bottom fac of th ic 3 4 : :5 : :4 : : : : :6 :3 : :4 : :9 :5 :3 Wak hains chain = soi b chain = ott chain = soi gn chain? = tc vtics an abs a o > R b fnction vas (fo cootp=b, vt istibtion at ach ank is binomia) Two sis of th sam coin: an instanc in -D in a nit c, sbsts of points on a gi in a attic, vss an instanc of a Hamitonian wak of th sam ngth in a Hass gaph fin b imnsion 4 (wt to bas ). Th wak in th nit c is contigos on with spct to cooinats fin in th attic. Simia, th wak in Hass gaphs is contigos on with spct to cooinats fin in th Hass gaph. 5 A sf-avoiing wak (SAW): Hamitonian of ngth 4 -- a sqnc of stps that chain a niq st of pivot cooinats (b) U = ft R = ight U = p D = own U R U R R U R R U D D Hass ank istanc fom th bottom fac of th ic : 3:9 3:6 3: :5 3: 3:4 3: :4 : :3 :3 : :5 : : Wak hains chain = soi b chain = ott chain = soi gn chain? = tc... R vtics an abs a o > R b fnction vas (fo cootp=q, vt istibtion at ach ank pns on cooinit) Two sis of th sam coin: an instanc in 3-D in a nit c, sbsts of points on a gi in a attic, vss an instanc of a Hamitonian wak of th sam ngth in a Hass gaph fin b imnsion (wt to bas 4). Th wak in th nit c is contigos on with spct to cooinats fin in th attic. Simia, th wak in Hass gaphs is contigos on with spct to cooinats fin in th Hass gaph. 6

9 A sf-avoiing wak (SAW): Hamitonian of ngth a sqnc of stps that chain a niq st of pivot cooinats (c) U = contin = ft U = p U Hass ank istanc fom th bottom fac of th ic :6 :5 :3 : :5 :4 : : :6 :4 :4 : : :9 :5 :3 : :3 : :9 : : : :6 : : : Wak hains chain = soi b chain = ott chain = soi gn chain? = tc vtics an abs a o > R b fnction vas (fo cootp=t, vt istibtion at ach ank pns on cooinit) Two sis of th sam coin: an instanc in 3-D in a nit c, sbsts of points on a gi in a attic, vss an instanc of a Hamitonian wak of th sam ngth in a Hass gaph fin b imnsion 3 (wt to bas 3). Th wak in th nit c is contigos on with spct to cooinats fin in th attic. Simia, th wak in Hass gaphs is contigos on with spct to cooinats fin in th Hass gaph. A sf-avoiing wak (SAW): amp of Hans s wak -- a sqnc of stps that chain a niq st of pivot cooinats Hass ank istanc fom th initia cooinat (th bottom vt) B-nighbos T-nighbos.:9.:9.:9.:6.:5 Sia nmb fo Hans's k:.:.:9.:9.:9.:9.:.:4.:.:.:9.:9.:9.:.:.:3.:3.:.:.:.:9.:9.:9.:.:4.:.:.:.:9.:6.:5.:.: fnction = BT.in. V = 36, E = vtics an abs a o -> R b fnction vas (fo cootp=bt, vt istibtion at ach ank ma pn on cooinit) Pobing fom th initia pivot.:9 th bst-va nighbo is a T- nighbo.: foow b a B-nighbo.: which tns p th Gt s k, so Hans ns to contin th wak. H, Hans wo ach his tagt in two stps if h choos.: as th nt pivot. Howv, b anom choic, his nt pivot is a T- nighbo.: fom wh h ns to contin th wak fo mo stps nti aching th pivot with tagt va that aso tns p his k:.:

10 Goba sach n SAW an Smma of pimnts Goba sach n SAW (an fnction, an cooinats) -- compt pso co -- tai: poc cooupat.saw Smma of pimnts (th potin foing pobm, D) -- Fo =, wight =, 3, 4, 5, -- Fo =, wight = ** -- Fo = 4, wight = ** -- Fo = 5, wight = 9** ** Aso fating significant impov sotions vss th sotions pot in [] 9 SAW in contt of goba sach (): tais in th pap A concpts in this pso co hav bn infoma intoc b Gt, Hans, an Jok ai.

11 SAW in contt of goba sach (): tais in th pap A concpts in this pso co hav bn infoma intoc b Gt, Hans, an Jok ai. =, w =,3,4 n pan ; initia cooinats wakngth mian man stv ma pobspstp mian man stv..3. bon_tagt niq_sotions 3 wakngth mian man stv ma pobspstp mian man stv bon_tagt niq_sotions 5 wakngth mian man stv ma pobspstp mian man stv bon_tagt niq_sotions 95 4 f q n c Fqnc Histogam of wight = ng = wakngth f q n c Fqnc 4 6 Histogam of wight = 3 ng = wakngth f q n c Fqnc 4 6 Histogam of wight = 4 ng = wakngth wight = : ng = -: wight = 3: ng = -: wight = 4: ng = -3:

12 =, w = 4,5, n pan ; initia cooinats wakngth mian man stv ma pobspstp mian man stv bon_tagt niq_sotions wakngth mian man stv ma pobspstp mian man stv bon_tagt niq_sotions 5 wakngth mian man stv ma pobspstp mian man stv bon_tagt niq_sotions 9 f q n c Fqnc Histogam of wight = 4 ng = wakngth f q n c Fqnc 4 6 Histogam of wight = 5 ng = wakngth f q n c Fqnc 4 6 Histogam of wight = ng = wakngth wight = 4: ng = -4: wight = 5: ng = -4: wight = : ng = -4: 3 =, w = 4 n pan ; initia cooinats wakngth mian man stv ma pobspstp mian man stv bon_tagt niq_sotions f q n c Fqnc Histogam of wight = 4 ng = wakngth Impotant obsvations fo this cas: -- on two niq sotions (fom initia cooinats) -- wakngth has amost ctain gomtic istibtion -- th avag wakngth to ach on of th two sotions n Hamitonian wak =.5 ( 3 9 ) = 5,3,4 wight = 4: ng = -4: -- th avag wakngth to ach on of th two sotions n th Hans s wak, aso a SAW, is,4.3 4

13 =, fom fnc, to qivant, to btt sotions statistics a bas on anom initia cooinats Pan A Pan Pan vabst = -9 bbsiz = vabst = -9 bbsiz = vabst = - bbsiz = <wakngth> =.5+4 (wost-cas = (9/)*(^)*(3^9) = = 4, fom fnc, to qivant, to btt sotions statistics a bas on anom initia cooinats Pan A vabst = -9 bbsiz = Pan vabst = -9 bbsiz = Pan 9 vabst = - bbsiz = <wakngth> = (wost-cas = (5/)*(^4)*(3^3) =

14 = 5, fom fnc, to qivant, to btt sotions statistics a bas on anom initia cooinats Tt... bas on th samp siz of 6 ath than Pan A vabst = - bbsiz = Pan 4 vabst = - bbsiz = 3 <wakngth> =.43+6 (wost-cas = (/6)*(^5)*(3^4) =.636+) Pan vabst = - bbsiz = oncsions an Ft Wok -- O pimnts with a SAW sov ais th pctation that th sotion of th potin foing pobm, wh th chain configation an its confimation a optimiz simtanos, ma b fasib at an accptab cost. -- Epimnts on potin foing pobm a bing pann aso fo tianga an hagona gis in - an 3-imnsions. -- W a fining SAW sovs to sca to v ag pobms in a nmb of iffnt omains n iffnt cooinat tps: -- ow atocoation bina sqnc (abs, aka in phsics as th apioic on-imnsiona Ising spin sstm with ong ang 4-spin intactions) -- gaph {vt cov, ciq, inpnnt st, ina aangmnt,...} -- optima Goomb (og) -- masat -- pasimon -- nmb patitioning pobm (npp) -- job sching pobm (jobs) -- cptogaph -- tc