Opimal Saged Sel-Assemly o General Shapes Cameron Chalk, Eric Marinez, Roer Schweller 3, Luis Vega 4, Andrew Winslow 5, and im Wylie 6 Deparmen o Compuer Science, Universiy o eas Rio Grande Valley, Brownsville, USA cameron.chalk@urgv.edu Deparmen o Compuer Science, Universiy o eas Rio Grande Valley, Brownsville, USA eric.m.marinez@urgv.edu 3 Deparmen o Compuer Science, Universiy o eas Rio Grande Valley, Brownsville, USA roer.schweller@urgv.edu 4 Deparmen o Compuer Science, Universiy o eas Rio Grande Valley, Brownsville, USA luis.a.vega@urgv.edu 5 Déparemen d Inormaique, Universié Lire de Bruelles, Brussels, Belgium awinslow@ul.ac.e 6 Deparmen o Compuer Science, Universiy o eas Rio Grande Valley, Brownsville, USA imohy.wylie@urgv.edu Asrac We analyze he numer o sages, iles, and ins needed o consruc n n squares and scaled shapes in he saged ile assemly model. In paricular, we prove ha here eiss a saged sysem log n wih ins and ile ypes assemling an n n square using O( + ) sages and Ω( log n ) are necessary or almos all n. For a shape S, we prove O( K(S) K(S) + ) sages suice and Ω( ) are necessary or he assemly o a scaled version o S, where K(S) denoes he Kolmogorov compleiy o S. Similarly igh ounds are also oained when more powerul leile glue uncions are permied. hese are he irs saged resuls ha hold or all choices o and and generalize prior resuls. he upper ound consrucions use a new echnique or eicienly convering each oh sources o sysem compleiy, namely he ile ypes and miing graph, ino a i sring assemly. 998 ACM Sujec Classiicaion F Models o Compuaion Keywords and phrases ile sel-assemly, HAM, aam, DNA compuing, iocompuing Digial Ojec Ideniier.43/LIPIcs.ESA.6.6 Inroducion he saged sel-assemly model is a generalizaion o he wo-handed [, 4, 7, 8] or hierarchical [5, ] ile sel-assemly models. In ile sel-assemly, sysem monomers are uni squares wih edge laels ha collide randomly and aach permanenly i auing edge laels mach suicienly. his simple model is an asracion o a DNA-ased molecular implemenaion a Research suppored in par y Naional Science Foundaion Grans CCF-767, CCF-55566, and CCF-45. Cameron Chalk, Eric Marinez, Roer Schweller, Luis Vega, Andrew Winslow, and im Wylie; licensed under Creaive Commons License CC-BY 4rd Annual European Symposium on Algorihms (ESA 6). Ediors: Pior Sankowski and Chrisos Zaroliagis; Aricle No. 6; pp. 6: 6:7 Leiniz Inernaional Proceedings in Inormaics Schloss Dagsuhl Leiniz-Zenrum ür Inormaik, Dagsuhl Pulishing, Germany
6: Opimal Saged Sel-Assemly o General Shapes he nanoscale [3, ] and is compuaionally universal []. he saged varian is moivaed y eperimenal seings, where parallelism and miing can e achieved (e.g. es ues). Liquid-handling roos have een used o perorm comple miing insrucions in he la [5], similar o he miing algorihms o saged sel-assemly sysems. he saged model [8] eends he wo-handed model y carrying ou separae assemly processes in muliple ins. Assemly in each in egins wih inpu assemlies previously assemled in oher ins. hese ins are sraiied ino sages, and a mi graph speciies which ins in he previous sage supply each in wih inpu assemlies. he oupu o a saged sel-assemly sysem is he se o assemlies produced in he ins o he inal sage. A common goal in he design o sel-assemling sysems is he consrucion o a desired shape. Here we consider he design o eicien sysems wih minimal compleiy or a given shape. hree merics eis or saged sysems: he numer o disinc ile ypes used in he sysem (ile compleiy), he maimum numer o ins used in any sage (in compleiy), and he numer o sages (sage compleiy). Eicien consrucion or various classes o shapes [8, ] and paerns [9, ] have een considered, and urher eensions and varians o he saged sel-assemly model have also een sudied [, 3,, 6, 7, 8]. Our resuls. Here we sudy he wo classic enchmarks or he eiciency a ile sel-assemly model: he assemly o n n squares and arirary shapes (wih scaling permied). Previous works [9,, ] achieved maching upper and lower (univariae) ounds on he minimum compleiy o sysems ha assemle hese shape classes in he very irs ile assemly model []. Here we give nearly maching upper and lower (rivariae) ounds or assemling hese shapes in he saged model; our resuls are summarized in ale. For a given numer o ile ypes and ins, we prove ha any n n square is consruced y a sysem wih O( + ) sages and a scaled version o any shape S log n K(S) is assemled y a sysem wih O( + ) sages, where K(S) denoes he Kolmogorov compleiy o S wih respec o some ied universal uring machine. We pair hese resuls wih nearly maching lower ounds, proving ha or almos all naural numers n log n, Ω( ) sages are needed o assemle an n n square, and or all shapes S, K(S) Ω( ) sages are needed o assemle a scaled version o a given shape S. We urher eplore he sage compleiy o hese shapes wihin he leile glue model o ile aachmen [6] (where non-maching glue laels can have srengh), and prove ha n n squares and scaled shapes can e assemled using O( log n + ) and O( K(S) + ) sages, respecively. We pair his wih nearly maching lower ound sage compleiies o Ω( log n ) and Ω( K(S) ). Our upper ounds oh use a new echnique o eicienly assemle i sring pads: consan-widh assemlies wih an eposed sequence o glues encoding a given i sring. his echnique convers all hree orms o sysem compleiy (ile, in, and sage) ino is o he sring wih only a consan-acor loss o inormaion. In oher words, he numer o is in he i sring pad rises linearly wih he numer o is needed o speciy he ile ypes and mi graph o he consrucion. he scale acor is proporional o he produc o he ime and space used y he ied universal uring machine o encode S using K(S) is. he racion o values or which he saemen holds reaches in he limi as n.
C. Chalk e al. 6:3 ale he main resuls oained in his work: upper and lower ounds on he numer o sages o a saged sel-assemly sysem wih ins and ile ypes uniquely assemling n n squares and scaled shapes. K(S) denoes he Kolmogorov compleiy o a shape. Sandard Glue Sage Compleiy Resuls Shape Upper Bound heorem Lower Bound heorem n n Scaled shapes n n Scaled shapes log n log n O( + ) Ω( ) K(S) K(S) O( + ) 3 Ω( ) 4 Fleile Glue Sage Compleiy Resuls O( log n O( K(S) + + ) Ω( log n ) ) Ω( K(S) ) 3 Comparison wih prior work. In providing a class o nearly opimal saged sysems or any choice o in and ile coun, our resuls also generalize and improve on prior resuls. For insance, heorem implies consrucion o n n squares using O() ins, O( log n log log n ) ile ypes, and O() sages, maching a resul o [] (up o consan acors). For leile glues, his is improved o O( log n) ile ypes, a resul o [6]. he same heorem also yields consrucions using O() ins, O() ile ypes, and O(log n) sages (maching a resul o [8]) or O( log n) ins, O() ile ypes, and O(log log log n) sages, susanially improving over he O(log log n) sages used in [8]. For consrucing scaled shapes, heorem 3 implies sysems using O() ins, O( K(S) log K(S) ) ile ypes, and O() sages, a resul o []. he Saged Assemly Model iles. A ile is a non-roaale uni square wih each edge laeled wih a glue rom a se Σ. Each pair o glues g, g Σ has a non-negaive ineger srengh, denoed sr(g, g ). Every se Σ conains a special null glue whose srengh wih every oher glue is. I he glue srenghs do no oey sr(g, g ) = or all g g, hen he glues are leile. Unless oherwise saed, we assume ha glues are no leile. Coniguraions, assemlies, and shapes. A coniguraion is a parial uncion A : Z or some se o iles, i.e., an arrangemen o iles on a square grid. For a coniguraion A and vecor u = u, u y Z, A + u denoes he coniguraion A, where (, y) = ( + u, y + u y ). For wo coniguraions A and B, B is a ranslaion o A, wrien B A, provided ha B = A + u or some vecor u. For a coniguraion A, he assemly o A is he se à = {B : B A}. he shape o an assemly à is {dom(a) : A Ã} where dom() is he domain o a coniguraion. A shape S is a scaled version o shape S provided ha or some k N and D S, (,y) D (i,j) {,,,k } (k + i, ky + j) S. Bond graphs and sailiy. For a coniguraion A, deine he ond graph G A o e he weighed grid graph in which each elemen o dom(a) is a vere, and he weigh o he edge eween a pair o iles is equal o he srengh o he coinciden glue pair. A coniguraion is τ-sale or τ N i every edge cu o G A has srengh a leas τ, and is τ-unsale oherwise. Similarly, an assemly is τ-sale provided he coniguraions i conains are τ-sale. Assemlies à and B are τ-cominale ino an assemly C provided here eis A Ã, B B, and C C such ha A B = C and C is τ-sale. E S A 6
6:4 Opimal Saged Sel-Assemly o General Shapes wo-handed assemly and ins. We deine he assemly process via ins. A in is an ordered uple (S, τ) where S is a se o iniial assemlies and τ N is he emperaure. In his work, τ is always equal o. For a in (S, τ), he se o produced assemlies P (S,τ) is deined recursively as ollows:. S P (S,τ).. I A, B P (S,τ) are τ-cominale ino C, hen C P (S,τ). A produced assemly is erminal provided i is no τ-cominale wih any oher producile assemly, and he se o all erminal assemlies o a in (S, τ) is denoed P (S,τ). ha is, P (S,τ) represens he se o all possile superiles ha can assemle rom he iniial se S, whereas P (S,τ) represens only he se o superiles ha canno grow any urher. I all assemlies in P (S,τ) have inie size, hen he assemlies in P (S,τ) are uniquely produced y in (S, τ). Unique producion implies ha every producile assemly can e repeaedly comined wih ohers o orm an assemly in P (S,τ). Saged assemly sysems. An r-sage -in mi graph M is an acyclic r-parie digraph consising o r verices m i,j or i r and j, and edges o he orm (m i,j, m i+,j ) or some i, j, j. A saged assemly sysem is a 3-uple M r,, {,,..., }, τ where M r, is an r-sage -in mi graph, i is a se o ile ypes, and τ N is he emperaure. Given a saged assemly sysem, or each i r, j, a corresponding in (R i,j, τ) is deined as ollows:. R,j = j (his is a in in he irs sage); ( ). For i, R i,j =. k: (m i,k,m i,j) M r, P (R(i,k),τ i,k ) hus, ins in sage are ile ses j, and each in in any susequen sage receives an iniial se o assemlies consising o he erminally produced assemlies rom a suse o he ins in he previous sage as dicaed y he edges o he mi graph. 3 he oupu o a saged sysem is he union o he se o erminal assemlies o he ins in he inal sage. 4 he oupu o a saged sysem is uniquely produced provided each in in he saged sysem uniquely produces is erminal assemlies. 3 Key Lemmas Our resuls rely on wo key lemmas. he irs is an upper ound on he inormaion conen o a saged sysem ha implies he lower ounds on sysem compleiy. he second is a ormal saemen o he previously menioned i sring pad consrucion. Lemma. A saged sysem o ied emperaure τ wih ins, s sages, and ile ypes can e speciied using O( + s + ) is. Such a sysem wih leile glues can e speciied using O( + s + ) is. 3 he original saged model [8] only considered O() disinc ile ypes, and hus or simpliciy allowed iles o e added a any sage (since O() era ins could hold he individual ile ypes o mi a any sage). Because sysems here may have super-consan ile compleiy, we resric iles o only e added a he iniial sage. 4 his is a sligh modiicaion o he original saged model [8] in ha here is no requiremen o a inal sage wih a single oupu in. I may e easier in general o solve prolems in his varian o he model, so we consider i or lower ound purposes. However, all o our resuls apply o oh varians o he model.
C. Chalk e al. 6:5 (a) () Figure (a) he decomposiion o a i sring pad s is ino hose encoded y he hree seps o a saged sysem wih ile ypes and ins. () An eample i sring r = encoded as a widh-4 gap- -i sring pad where he op glues correspond o he is in r. Proo. A saged sysem can e speciied in our pars: he ile ypes, he glue uncion, he mi graph, and he assignmen o ile ypes o sage- ins. We separaely ound he numer o is required o speciy each. A se o ile ypes has up o 4 glue ypes, so speciying each ile requires O() is, and he enire ile se akes O( ) is. I he sysem does no have leile glues, hen he glue uncion can e speciied in O(4) = O() is, using O(log τ) = O() is per glue ype o speciy he glue s srengh. I he sysem has leile glues, hen he glue uncion can e speciied using O() is per pairwise glue ineracion and O((4) ) = O( ) is oal. he mi graph consiss o s nodes. Each pair o nodes in adjacen sages opionally share a direced edge poining upwards. hus speciying hese edges akes O( (s )) = O( s) is. he assignmen o ile ypes o sage- ins requires one i per each choice o ile ype and in, or O() is oal. hus a saged sysem wihou leile glues can e speciied in O( + + s + ) is, and oherwise in O( + + s + ) is. I immediaely ollows rom Lemma ha or mos i srings o lengh, any saged sysem wih ins and iles ha encodes he i sring mus have Ω( ) sages wih sandard glues and Ω( ) sages wih leile glues. he wo main posiive resuls o his work, eicien assemly o squares and general scaled shapes, oh rely mainly on eicien assemly o i sring pads: assemlies ha epose a sequence o norh glues ha encode a i sring. An eample is shown in Figure (). Squares and general scaled shapes are assemled y comining a universal se o compuaion iles wih eicienly assemled inpu i sring pads. Deiniion (i sring pad). A widh-k gap- r-i sring pad is a k ((r ) + ) recangular assemly wih r glues rom a se o wo glue ypes {, } eposed on he norh ace o he recangle a inervals o lengh, saring rom he lemos norh edge. Unless oherwise speciied, a i sring pad is gap-. All remaining eposed glues on he norh ile edges have some common lael. he remaining eposed souh, eas, and wes ile edges have glues g S, g E, and g W. A i sring pad represens a given sring o r is i he eposed and glues rom le o righ are equal o he given i sring. Bi sring pads are consruced y decomposing he pad ino hree supads and consrucing each in a separae sep using a dieren source o sysem compleiy (see Figure (a)): Sep : Θ() is rom assigning ile ypes o sage- ins (Secion 4.). Sep : Θ( ) is rom he ile ypes hemselves as in [, 6, 4, ] (Secion 4.3). Sep 3: Θ( ) is rom he mi graph using a varian o crazy miing [8] (Secion 4.4). hese supads are hen comined ino he complee pad. I leile glues are permied, Sep is modiied as in [6] o achieve O( + ) sages. E S A 6
6:6 Opimal Saged Sel-Assemly o General Shapes Lemma 3. here eis consans c, d N such ha, or any, N wih > c, > d and i sring S o lengh, here eiss a saged sysem wih ins, iles, and O( + sages ha assemles a widh-9 gap-θ(log ) l-i sring pad represening S. ) Proo. Le = 4 and = 4. Using an approach similar o ha in Secion 4., consruc a lengh log 5 9 + iller assemly using ile ypes and ins in O( ) sages such ha he assemly has glue e on is wes edge (maching ha o he eas side o he i sring pads) and glue w on is eas edge. Ne, use Lemma 5 wih ile ypes, ins, and O( ) sages o consruc a widh-9 gap- log 5 9 + Θ()-i sring pad. hen use Lemma 9 wih ile ypes, ins, and O() sages o consruc a widh-3, gap- log 5 9 +, Θ( )-i sring pad. So ar, Θ() + Θ( ) is have een encoded and so Θ( ) is remain. Invoke Lemma wih ile ypes, ins, and O( + ) sages o consruc a widh-9 gap- log 5 9 + Θ( )-i sring pad. In one inal sage, concaenae wo i sring pads using he iller assemly and in one more sage concaenae he hird. By concaenaing he lengh Θ()-i sring pad, he lengh Θ( )-i sring pad, and he Θ( )-i sring pad, each separaed y he log 5 9 + iller assemly, an -i sring pad wih O(log ) spacing is consruced; use addiional ile ypes (in ins) o ill in he porions o he assemly wih widh less han 9. he oal numer o ile ypes and ins used are 4 + = and 4 + =, respecively, wih 4 and 4 used or he hree i sring pads and one connecor assemly and he remainder or illing in he pad o widh 9. he oal numer o sages used is O( ) + O() + O( + ) = O( log + ). he addiive gap eween he upper and lower ounds implied y hese lemmas comes rom he O( ) addiional sages used o consruc some o he machinery needed o carry ou he hree seps o Lemma 3. 4 Bi Sring Pad Consrucion As menioned, i sring pads are assemled y comining hree supads consruced via separae and independen mehods ha uilize disinc sources o inormaion compleiy in a saged sel-assemly sysem. Each supad encodes a numer o is roughly proporional o he numer needed o descrie he corresponding porion o he saged sysem, i.e., an asympoiically opimal numer o is are encoded. 4. Wings he addiive gap in our upper and lower ounds come rom a helpul suconsrucion used in Seps and 3 descried here. his suconsrucion assemles all -gapped widh- i sring pads o a given lengh in separae ins: Lemma 4. here eis consans c, d N such ha, or any, N wih > c and > d, here is a saged sel-assemly sysem wih ins, ile ypes, and O( ) sages ha assemles all gap-, widh-, log()-i sring pads, each placed in a disinc in. Due o space consrains, he proo o his and some laer resuls are omied. We give proo skeches insead. Le γ = 6 and η = γ +. I γ log(), direcly uild all he i sring pads in O() sages. Oherwise, repeaedly apply a consan-sage round ha
C. Chalk e al. 6:7 r l l r r l r r r l l l (a) () Figure (a) he aachmen o era suassemlies ono i sring pads o creae le and righ wings. Each o he wo size 3 suassemlies use 3 iles o deerminisically assemle he respecive L shape in heir own ins. () he aachmen o wo i srips using maching wings. Noe ha he geomery aached o he sides o each wing preven misaligned, non-maching wings o aach. sars wih all inary gadges o a given lengh and yields all inary gadges o a acor o η longer, saring wih jus wo i sring pads encoding he wo i srings o lengh. Use O() addiional ile ypes, ins, and sages o augmen he he i sring pads assemled y Lemma 4 ino le and righ wings (seen in he le and righ porions o Figure (a)) ha aach when he underlying i srings are idenical. hese wings are used in Seps and 3 o achieve ordered assemly o i sring supads ino larger i sring pads. 4. Sep : encoding via iniial ile-o-in assignmen Recall ha in a saged sysem, each o he sysem s sage- ins is assigned a suse o oal ile ypes. Here we design an assignmen ha assemles a Θ()-i sring supad o he inal i sring pad using O(/ ) sages - enough o uilize he wings o Secion 4.. he assignmen yields ins ha conain assemlies encoding disinc equal-lengh susrings o he Θ() is. hese assemlies are hen comined using wings. Lemma 5. here eis c, d N such ha, or all, N wih > c and > d and i sring S o lengh Θ(), here is a saged sel-assemly sysem wih ins, iles, and O( ) sages ha assemles a gap-( log 5 9 + ) Θ()-i sring pad represening S. See Figure 3 or a skech o he idea. Le γ and β e consan racions o and, respecively. Use γ iles and β ins o consruc all le and righ log(β)-i wings according o Secion 4.. Also consruc γ consan-sized i srip suassemlies ha epose a or norh glue and have wings aached o heir righ and le sides such ha any γ -i sring pad can e assemled rom γ i srips aached sequenially. In each o β ins, assemle γ i srips ino a disinc γ -i sring supad o he desired pad. Comine hese β supads wih wings ha encode heir locaions in he pad, and hen comine hese wing-laeled supads o assemle he complee Θ()-i sring pad. he numer o sages used is O( ) (or he wings, see Lemma 4) plus O() (he supads o he desired pad). 4.3 Sep : encoding via ile ypes Here he goal is o design a collecion o ile ypes ha encodes Θ( ) is. he soluion is o uilize he ase conversion approach o [, 6, 4, ]. In his approach, ile E S A 6
6:8 Opimal Saged Sel-Assemly o General Shapes 3 3 3 3 Figure 3 he creaion o γβ-i sring pads. he squares laeled and represen i srips. he doed lines indicae ile o in assignmens eore he irs sage o he sysem; w r,i and w l,i represen he i h righ and le wings respecively. U = U = U = s s s s s s } = spacing } = spacing } = spacing Figure 4 Le: a widh- gap-log z decompression pad represening a i sring S = in ase z = 8. Righ: O(z) decompression iles inerac wih he norh glues o he decompression pad o comine ino a widh-3 i sring pad represening S in ase. ypes opimally encode ineger values in a high ase and hen decompressed ino a inary represenaion. In oal, ile ypes are used o encode (in a high ase) and decompress (ino a inary) Θ( ) is. Deiniion 6 (decompression pad). For k, r, N and u =, a widh-k, r-digi, ase-u decompression pad is a k r recangular assemly wih r glues rom a se o u glue ypes {,,, u } eposed on he norh ace o he recangle a inervals o lengh and saring rom he lemos norhern edge. All remaining glues on he norh surace have a common ype n. he remaining eposed souh, eas, and wes ile edges have glues g S, g E, and g W. A decompression pad represens a given sring o digis in ase u i he eposed glues rom le o righ, disregarding glues o ype n, are equal o he given digi sring in ase u. Consider he ollowing eample, also seen in Figure 4). Le S = (S = 4 in ase 8) e a i sring, wih he goal o consrucing a widh-3 9-i sring pad represening S. Firs, uild a decompression pad represening S in ase 8 y comining 3 dieren 3 log (8) locks. hen conver he decompression pad ino a i sring pad represening S using O(z) ile ypes. Lemma 7. Given inegers 3, d and z =, here eiss a -sage, -in saged sel-assemly sysem ha assemles a d-digi decompression pad o widh- and ase-z, using a mos 5d + log z ile ypes.
C. Chalk e al. 6:9 Figure 5 he creaion o β -i sring pads using β wings and O() sages. he recangles and represen i srips ha may aach wings on eiher side; w r,i and w l,i represen he i h righ and le wings respecively. Lemma 8. Given inegers d 3, 3, z =, and i sring S o lengh d log(z), here eiss a saged sel-assemly sysem wih in, 5d + z + log z 4 ile ypes, and sage ha assemles a widh-3 d log(z)-i sring pad represening S. Lemma 9. here eiss some consan c N such ha, or any c and i sring S o lengh Θ( ), here eiss a saged sel-assemly sysem wih in, ile ypes, and sage assemling a widh-3 Θ( )-i sring pad represening S. Omied addiional deails are needed o conver hese gap- pads o higher-gap pads consisen wih hose assemled in Secion 4.4. 4.4 Sep 3: encoding via mi graph his sep uses a mi graph o encode encodes a achieves he ollowing eicien assemly: Lemma. here eis c, d N such ha, or any > c and > d and i sring S o lengh, here is a saged sel-assemly sysem wih ins, ile ypes, and O( + ) sages ha assemles a widh-9 gap-( log 5 9 + ) -i sring pad represening S. An overview o he consrucion is shown in Figure 5. Le γ and β represen some consan racions o and respecively. Uilize γ iles and β ins o consruc all lengh-log (β) le and righ wings according o Secion 4. and denoe he i h le and wings y w l,i and w r,i, respecively. Also consruc wo consan-sized i srip suassemlies ha epose a or norh glue and allow wings o e aached o heir righ and le sides. In he irs sage and or all i β, mi w r,i and w l,i wih i srip ino a in denoed i. Similarly, mi w r,i and w l,i wih i srip ino a in denoed i or a oal o β ins. E S A 6
6: Opimal Saged Sel-Assemly o General Shapes In he second sage, selecively mi speciic or winged i srips o assemle speciic β-i sring pads across β ins. Speciically, mi eiher i or i or each i across β ins or a oal o β dieren β-i sring pads. In he hird sage, aach wings o each o he β-i sring pads. For each o he β ins, mi w r,i and w l,i ino he ins such ha w r, is mied wih he irs β is o he desired β -i sring pad, w r, and w l, are mied wih he second β is o he desired β -i sring pad, ec. In he inal sage, mi all β ins, each conaining a β-i sring pads, ino a common in o creae β -i sring pads. he wings ensure ha he i sring pads aach in he desired order. Repea his process β imes, each ime concaenaing he β -i sring pad ono each preceding i sring pad. In he end, a single -i sring pad resuls. In oal, O( ) sages are used o consruc he wings and O( ) sages are used o assemle β unique β -i sring pads. hus his sep has oal sage compleiy o O( log log β β + ) = O( + ). 5 Assemly o n n Squares Eicien assemly o n n squares is oained y comining i sring pads wih a echnique o Rohemund and Winree [9]. heir echnique uilizes a inary couning mechanism which consrucs a lengh Θ(n) recangle wih Θ(log n) widh. he mechanism uses O(log n) ile ypes o seed he couner a a cerain value, and hen O() ile ypes aach in a zig-zag paern, where zigs copy he value rom he row elow and zags incremen he he value y. Once he inary couner incremens o is maimum value (a sring o ), he assemly sops growing. wo recangles assemled his way can e comined o orm a ounding o ha is hen illed o orm a square. We uilize he i sring pad consrucion o Secion 4 o eicienly assemle he seed or he inary couning mechanism, requiring only an addiional O() ile ypes and sage o perorm he inary couning and square illing. heorem. here eis consans, N such ha or any,, n N wih,, here eiss a saged sel-assemly sysem wih ins, ile ypes, and O( ) sages ha uniquely produces an n n square. log n + Proo. Le c e he (consan) numer o ile ypes used o implemen he ied-widh zig-zag inary couning mechanism shown in [9]. Le = c, =, and n = log n. Le m = n (n )/ n ( log + ). Using Lemma 3, consruc wo Θ(log )-gap log m -i sring pads encoding m, where each consrucion each uses ins, ile ypes and log m O( + ) sages. Figure 6 shows he consrucion, including modiicaions o he echnique shown in [9]. On oh pads, a small modiicaion is made: he glues o he irs and las is are made unique and he irs i s glue srengh is se o. his modiicaion is necessary o implemen a ied-widh inary couning mechanism as in [9] and uses O() addiional ile ypes. Also, on he norh-acing (eas-acing) i sring pad, a unique srengh- glue C is placed on he souh (wes) ace o he pad s oommos righmos (opmos lemos) ile. his special glue is used o comine he wo pads wih a unique ile ype. Noe ha he i sring pads assemled in Secion 4 have susanial spacing eween he eposed inary glues, u he couner o [9] has spacing. his is resolved y adding generic iles which ranser inormaion horizonally. hese generic iles use cooperaive inding eween a souh-acing glue (which maches he glue ha spaces he is on he i sring pad) and wes/eas glues represening he inormaion o e passed horizonally across spacing o glues. he iles also epose a norh-acing glue o e used when he
C. Chalk e al. 6: * * C * p D D p * A * n n n n n n n n n n n n n n n n n n n n n n n n p * B * B p A * A n n n n n n n n n n n n n n n n n n * p * * * * p s s p p C C s C q C s p q q p q Figure 6 Consrucing a couner seed. he i sring pads are shown in gray. Glues wih a in he sring have srengh-, all oher glues have srengh. inormaion needs o e ranserred across he spacing in he row aove. Wihou loss o generaliy, roaed versions o hese iles are used in he eas-growing couner. log n he sage compleiy o he sysem is O( + ). Noe ha he lengh o he i sring pads assemled according o Lemma 3 is dependen on, he numer o ins used o consruc he i sring pad. I is so large ha he spacing eween is causes he widh o he i sring pad o eceed n (roughly log > n), we insead direcly consruc he appropriae i sring pad wih spacing using O( ) sages. he ollowing lower ound is derived rom Lemma y oserving ha or almos all n N, log n is are needed o represen n. heorem. For any, N and almos all n N, any saged sel-assemly sysem which uses a mos ins and ile ypes ha uniquely assemles an n n square mus use Ω( ) sages. log n 6 Assemly o Scaled Shapes Eicien assemly o arirary shapes (up o scaling) is achieved y comining i sring pads wih he shape-uilding scheme o Soloveichik and Winree []. heir consrucion uses wo suses o ile ypes: a varying se o encode he inary descripion o he arge shape and a ied se o decode he inary descripion and uild he shape. We replace he irs se wih a i sring pad encoding he same inormaion. E S A 6
6: Opimal Saged Sel-Assemly o General Shapes Figure 7 Consrucion o he modiied seed lock. Bi sring pads are colored in gray. We concaenae our K(S)-i sring pads represening S. heorem 3. here eis consans, N such ha or any shape S o Kolmogorov compleiy K(S) and, N such ha and, here eiss a saged sel-assemly K(S) sysem wih ins, ile ypes, and O( + ) sages ha uniquely produces S a some scale acor. Proo. Oserve ha he ile se descried in [] uniquely consrucs he same erminal assemly, namely a scaled version o S where each cell is replaced y a square lock o cells, when run a emperaure in he wo-handed miing model. I does so via a Kolmogorovcompleiy-opimal uring machine simulaion o a machine ha compues a spanning ree o he shape given a seed assemly or seed lock encoding he shape. he simulaion is hen run as i ills in he shape, eginning wih he seed lock. Here a similar seed lock is consruced and consiss o our i sring pads, a square core and addiional iller iles. Le = c 5 where c is he (consan) numer o ile ypes required y [] o carry ou he simulaion o a (ied) universal uring machine. Le = 5. Use he mehod o Lemma 3 o consruc he modiied seed lock y assemling our dieren K(S)-i sring pads represening a program ha oupus S, each using ins, ile ypes and O( K(S) + ) sages. hese our pads (each wih dimensions (K(S) log K(S) + ) O()) are aached o he our sides o a (K(S) log K(S) + ) (K(S) log K(S) + ) square consruced as in heorem using ile and ins in O( log(k(s) log K(S)+) + ) sages. An asrac igure o he compleed seed lock can e seen in Figure 7. he uring simulaion occurs in one sage y miing he our concaenaed i sring pads ino one in which conains he ied se o uring-machinesimulaion iles o []. he i sring pads conain spacing eween he eposed inary glues, while he simulaion ile ypes o [] epec adjacen glues. his is resolved y modiying he uring-machine-simulaion ile se o include generic iles or ranserring inormaion across spacing, similar o he iles o he same purpose discussed in he proo o heorem. We need a mos such ile or each ile in he (consan-sized) uring-machine-simulaion ile se, or a consan increase in ile compleiy. he sage compleiy is 4 O( K(S) ) + O( log(k(s) log K(S)+) + K(S) ) = O( + + ). he ollowing heorem ollows rom he inormaion-heoreic ound o Lemma.
C. Chalk e al. 6:3 ile ypes o conver modiied decompression pad o leile decompression pad: modiied decompression pad s' s ' s s ' s s ' sar s' s s s d- s ' ' ' d- (d-)' sar s' s ' s s ' s s ' s' s ' s s ' s s ' leile decompression pad Figure 8 he emplaes o conver a modiied decompression pad o a leile decompression pad using d + ile ypes, where ineger d 3, on he le. Using hese addiional ile ypes, a modiied decompression pad is convered ino a leile decompression pad. A modiied decompression pad has a wesmos norhmos glue o s and every non-s glue on he norh surace is a special prime version disinc rom oher similar glue ypes. On he le, a widh- modiied decompression pad represening he sring in ase-8 is convered o a widh-3 lengh-9 leile decompression pad. heorem 4. For any, N and shape S wih Kolmogorov compleiy K(S), any saged sel-assemly sysem which uses a mos ins and ile ypes ha uniquely assemles S mus use Ω( ) sages. K(S) 7 Fleile Glues Here, an alernae model permiing non-diagonal glue uncions, also called leile glues is considered. By modiying Sep o he i sring pad consrucion o Secion 4 o encode Θ( ) is raher han Θ( log()) is in ile ypes, similarly igh resuls are oained or he same prolems in his more powerul model. he echnique uses a modiied decompression ad, similar o he echnique inroduced in [6]. Deiniion 5 (leile decompression pad). A widh-k lengh-r leile decompression pad is a k r recangular assemly wih r norh glue ypes rom he se {sar,,,..., r} eposed on he norh ace o he recangle. he wesmos glue is sar, he ollowing r glues have ype, ollowed y r glues o ype, r glues o ype, and so on. he eposed souh, eas, and wes ile edges have glues g S, g E, and g W, respecively. In order o uild he leile decompression pad, a modiied decompression pad represening a numer C = c c... c d in ase d is needed. Lemma 6. Given an ineger d 3, here eiss a saged assemly sysem wih in, 8d ile ypes, and sage ha assemles a widh-3 lengh-d leile decompression pad. Proo. Sar wih he consrucion o Lemma 7 ha yields a a widh- lengh-d decompression pad encoding C. Modiy he ile ypes o his consrucion such ha he lemos norhmos glue is s and every non-s glue on he norh surace is a special prime version, o diereniae eween oher similar glue ypes. hen add d + iles ha modiy he norh surace decompression pad o yield widh-3 leile decompression pad, as seen in Figure 8. E S A 6
6:4 Opimal Saged Sel-Assemly o General Shapes leile decompression pad sar ile ypes o decompress leile decompression pad: a a c c d a d d a d i sring pad sar Figure 9 On he le, he emplaes or he decompression iles needed o decompress a leile decompression pad or any given d 3. In he op righ, an eample o a lengh-9 leile decompression pad. In he oom righ, he decompression iles inerac wih he leile decompression pad and glue uncion o assemle a i sring pad rom a leile decompression pad, represening he i sring. he leile glues orm a ond o srengh eween he glue pair ( sar, ), srengh eween glues pairs (, ), (, ), (, ), (, ), (, ), (, ), (, ), (, ), (, ), (, ), and (, ), and srengh eween all oher glue pairs. his sep requires 5d + log d ile ypes o uild a modiied decompression pad and d + iles o conver his modiied decompression pad ha ino a leile decompression pad. hus d + + 5d + log d 8d ile ypes are used in oal. Lemma 7. Given inegers d 3 and any lengh d i sring R, here eiss a sage, in, saged assemly sysem wih leile glues ha assemles a widh-4 gap- d -i sring pad represening R, using a mos d ile ypes. Proo. Consider a widh-3, lengh d leile decompression pad. he idea is o use d ile ypes and leile glues o uild a widh-4 gap- d -i sring pad rom he leile decompression pad (see Figure 9). Consider a sequence o d isrings D = D, D,..., D d wih each D i = s s s... s d such ha he in-order concaenaion o all isrings in D equals R. Le D i,j denoe he j h i o he i h i sring o D. he goal is o consruc a glue uncion such ha i speciies he iles ha can aached o he op o he leile decompression pad o e he concaenaion o he isrings in D. Noe ha he iles ha have a or glue as hose wih laels ha end in or, respecively. Le sr(g, g ) denoe he srengh eween glues g and g. Se he ile ha aaches o he sar glue o e one ha eposes or a y seing sr(sar, ) = or sr(sar, ) =, respecively. For all D i,j, we se sr(i, j) = i and only i D i,j = and sr(i, j) =, oherwise. In addiion, we se sr(a, a) =, sr(, ) =, and so on. Wih his, we uild a widh-4 gap- d -i sring pad rom he leile decompression pad. An eample o his can e seen in Figure 9. Also, 8d ile ypes are used o uild a widh-3, lengh d leile decompression pad. An addiional d ile ypes are needed o decompress, using leile glues, ino a widh-4 d -i sring pad. So he oal numer o ile ypes used is d.
C. Chalk e al. 6:5 Lemma 8. Given N, here eiss some consan c N, such ha or all cases where c, here eiss a saged sel-assemly sysem wih iles which assemles any widh-4 Θ( )-i sring pad using sage, in, and leile glues. Proo. Given ile ypes, consider how many is can e produced using Lemma 7. Le d = +. Invoke Lemma 7 o uild a widh-4 d -i sring pad wih leile glues using d iles. he numer o is produced is y = d = ( + ) = Θ( ). hen y Lemma 7, any widh-4 ( + ) -i sring pad can e uild in he leile glue model using a mos iles, sage, and in. he smalles choice o d requires d = + 3, implying 9. For all cases where c we have a consan, c = 9, where his lemma holds rue. he improvemens o Lemmas 7 and 8 allow or a larger i sring pad o e uil in Sep when compared o sandard glues, reducing sage compleiy o O( + ): Lemma 9. Given, N and a i sring r where = r. hen, here eis some consans c, d N, such ha or all cases where > c and > d, here is a saged selassemly sysem using leile glues wih ins and iles which assemles an -i sring pad represening r wih widh 9 and gap Θ(log ) using O( + ) sages. Nearly igh upper and lower ounds or square and general shape consrucion in he leile glue model are oained y replacing he i sring consrucion o Lemma 3 wih Lemma 9, and applying he leile glue lower ound o Lemma : heorem. For any,, n N and consans c, c such ha c and c, here eiss a saged sel-assemly sysem using leile glues wih ins and ile ypes ha uniquely produces an n n square using O( log n + ) sages. heorem. For any, N and almos all n N, any saged sel-assemly sysem wih leile glues which uses a mos ins and ile ypes ha uniquely assemles an n n square mus use Ω( log n ) sages. heorem. For any shape S and, N and consans c, c such ha c and c, here eiss a saged sel-assemly sysem using leile glues wih ins and ile ypes which uniquely produces S a some scale acor using O( K(S) + ) sages. heorem 3. For any, N and shape S wih Kolmogorov compleiy K(S), any saged sel-assemly sysem wih leile glues which uses a mos ins and ile ypes ha uniquely assemles S mus use Ω( K(S) ) sages. 8 Conclusion In his work, we achieved nearly opimal saged assemly o wo classic enchmark shape classes. hese consrucions generalize he known upper ounds o [, 6, 8, ] o arirary choices o ile ype and in couns, as well as o he leile glue model. he naural prolem le open is he eliminaion o he addiive O( ) gap eween he upper and lower ounds induced y he wings suconsrucion o Secion 4.. Alhough his suconsrucion is he cause o an addiive gap in an oherwise opimal resul, i is a useul approach or general assemly laeling and coordinaed aachmen and is likely useul in oher saged consrucions. he consan widh o our i sring pads can also poenially e eploied or eicien consrucion o shapes wih geomeric olenecks, e.g., hin recangles. E S A 6
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