Tiling Layouts with Dominoes

Transcription

1 +, 16th Canadian Confeence on Computational Geomety, 2004 Tiling Layouts with Dominoes Mak Watson Chis Woman c 1 We exploe the complexity of tiling finite subsets of the plane, which we call layouts, with a finite set of tiles The tiles ae inspied by Wang tiles and the domino game piece Each tile is composed of a pai of faces Each face is coloed with one of possible colos We want to know if a given layout is tileable by a given set of dominoes In a tiling, dominoes that touch must do so at like-coloed domino faces We povide an time algoithm fo tiling layouts that ae paths o cycles We also show that if the layout is patially tiled at the outset of the poblem, then the tiling decision poblem is N-complete We also show that the poblem emains N-complete even if the layout is a tee (a) (b) c 1! "# $% In a geometic tiling poblem we wish to fill all o some of the plane with non-ovelapping polygons called tiles The tiling poblems studied heein ae motivated by ecent esults concening Wang tiles Wang tiles ae non-otatable unit squaes that have coloed edges [5] In a tiling that uses Wang tiles, neighboing tiles must have the same colo on adjacent edges In a typical Wang tiling poblem, we ae given a finite numbe of types of tiles and an infinite numbe of each type, and we ae asked to tile some subset of the plane Bege showed that deciding if the entie plane can be tiled by a given set of Wang tiles is undecidable [2] Motivated by a connection between Wang tilings and self assembly in DNA computing, eseaches have begun to study tiling pope infinite subsets of the plane [1, 3] In [1], the authos show that the poblem of tiling a ibbon, which is an infinite path in the plane, is undecidable This esult is extended in [3] to show that the poblem of tiling a ibbon that is a cycle is undecidable We study a vaiation of Wang tiles, which we call dominoes, that ae &('*) ectangles that ae patitioned into 2 coloed faces Thus unlike Wang tiles, the faces ae coloed athe than the edges Also unlike Wang tiles, we allow otation of the tiles and we conside finite sets of dominoes Thus although ou tiles have a connection to Wang tiles, they ae essentially a genealization of the commonly used domino game piece Depatment of Compute Science, Univesity of Saskatchewan, mdw655@mailusaskca Depatment of Compute Science, Univesity of Saskatchewan, cmw133@mailusaskca (c) Figue 1: (a) A set of dominoes - (d) ealized as othogonal ectangles with two faces (b) A layout (c) The gaph /10 (d) The ealization of a tiling 2 on the set $ ;:% <;= Let >@?BADCDEGF8CIHFIJKJKJLFMCINO be a finite set of colos A domino is a pai QCSE#F8CITHD of colos fom > We will efe to the cuent set of dominoes that is unde consideation as -, and we will efe to the numbe of dominoes in - as We will only conside finite sets of dominoes Now we discuss how we can embed a domino in the intege plane When embedding a domino in the plane, we intepet a domino?uqc E F8C TH as a &V'W) ectangle that is patitioned into exactly 2 squaes We call these squaes faces and efe to them as X E and X H The face X Y will be coloed colo C Y (Figue 1(a)) We say that a function 2[Z\-^]_a` H 'b` H is a domino tiling function if 2 has cetain popeties that we now explain Fistly, 2 must map a domino to a pai of hoizontally o vetically aligned points that ae 1 unit apat Futhemoe, 2 must map dominoes such that adjacent domino faces have the same colo Finally, 2 must map dominoes to non-ovelapping locations These popeties ae fomalized below: 1 2c d?efgfih kjml f!hln?o) 2 2c Y W?pQf6F8hqFi2c # [?stufiv!f and lfwtxly?p)bj C TzQE?{C M ME 3 2c Y }?eqf6f8hqfi2c # }?oqt(fiv F and l hvgl~?o) jc TzH? CI H 4 2c Y d?ofgfih and 2c d?@qf6f8v!dj Y? 86

2 CCCG 2004, Monteal, Quebec, August 9 11, c Y k?efgfih and 2c k?et(f8hqdj whee l f!hl is the Euclidean distance between f and h, and F Y F # - We use 2c to detemine whee a domino is located If 2c?ƒfgFih then X E will be centeed at the point f, and X H centeed at the point h We note hee that 2 may be a patial function, in which case not all dominoes ae positioned on the plane A layout is a subset of ` H 'u` H such that fo all fgfih, lfwh\l[? ), and fo all f Y F8h Y FSQf F8h we have f Yu? f F8h and h Y?ˆf Fih In othe wods, consists of pais of unique points fom ` H that ae eithe hoizontally o vetically aligned (Figue 1(b)) We will also stipulate that?š, ie the numbe of dominoes is exactly the ight numbe to cove the layout Let f and h be two points that ae components of some elements of Then we say f and h ae Œ CI S\Ž in iff l f!hl?@) If two membes Y FM have adjacent components, then Y and ae also deemed Œ qc D\Ž If 2 maps two domino faces to positions that ae adjacent in, then those faces ae deemed adjacent If thee exists a domino tiling function 2 that is sujective on fo a set of dominoes - then we say that is tileable by - and we efe to 2 as a tiling of using - Since we ae inteested in the computation of tilings, we define the following poblems: Definition 1 In the DOMINO TILING poblem we ae given a set - of dominoes and a layout and we ae asked to compute a tiling 2 of if one exists We also study a domino tiling poblem whee cetain dominoes have aleady been positioned on the layout Thus the layout is patially tiled with dominoes, and we wish to complete the tiling Definition 2 In the ATIAL DOMINO TILING poblem, we ae given a patial tiling function 2, a set - of dominoes, a layout and we ae asked to compute a tiling function 2 such that fo all - such that 2 is defined, we have the following: 2 d?efgfih j 2c Y? d?efgfih In the both the DOMINO TILING poblem and the A- TIAL DOMINO TILING poblem, 2 may o may not exist In Section 3 we study the algoithmic aspects of DOMINO TILING with espect to the topology of layouts We classify the topology of a layout by using a gaph theoetic chaacteization of We define a gaph / 0?e df8 x whose nodes and edges ae defined using the components of a layout in the following way (see Figue 1(c)): o?@ash h!fif\ o fgfih, fo some f6o o?@aqf6f8hq f and h ae adjacent in no We will use the gaph popeties of / 0 to chaacteize If / 0 is a path o a cycle, fo example, then we also say that is a path o a cycle, espectively We will also use a gaph chaacteization of a set of dominoes Using a set - of dominoes, we build an undiected pseudogaph 1 whose nodes coespond to colos and whose edges coespond to dominoes Given a set of dominoes - Gšœ that ž qÿ use the colos >?aagc E FMC H FSJLJKJKF8C N O, we define the D /g {?@ kfm u : o? > o?eiqf6f8hq fgfih -9 Thus each domino fom - edge in, while each colo fom > single node in 4 $ : $ < c ( «ªk= : 5 coesponds to exactly one is epesented by a In this Section, we focus on the DOMINO TILING poblem whee the povided layout is a path o a cycle The following Lemma povides the motivation fo ou algoithm fo computing tilings of layout that ae paths o cycles Lemma 1 A layout, which is a path, is tileable by - /g contains an Eule tail o cicuit oof ecall that an Eule tail (esp cicuit) of a gaph is a path (esp cycle) that uses evey edge of exactly once (j ) We define a c F8 ~ d using 2 : w?badh h is a face of a domino fom -±O ~?@AQf6F8hq faces u and v ae adjacent unde 2O The gaph will be isomophic to the layout gaph /g0 since each face lies a point fom a component of, and faces ae only adjacent if they ae one unit apat Since is a path, / 0 is also a path, and hence is a path ecall that has exactly one edge fo each domino in - If the faces that ae at the endpoints of the path iff ae diffeent colos then descibes a path in /d If they ae the same colos then they descibe a cycle in /ž Moeove, this path o cycle will be Euleian since it uses evey edge (ie domino) exactly once (² ) Let ³?µ EGFM DH FIJKJLJKF8 G be an Eule tail o cicuit fom / that is given by its edges We will use ³ to position dominoes fom - onto, thus computing 2 Now we descibe the pocedue fo positioning the dominoes of - onto by using ³ We begin by specifying a membe of that is at an endpoint of the path /~0 to be NEXT Then we pefom the following pocedue fo?o) FT& FIJKJKJLF8 : 1 lace the domino that coesponds to the edge Y at location NEXT accoding to Y Now location NEXT is deemed occupied 1 A ¹8º8» ¼½M¾ Ài Á is a gaph that allows both loops and multiple edges between nodes 87

3 Ç 16th Canadian Confeence on Computational Geomety, NEXT Â Ã the next unoccupied membe of that is adjacent to NEXT We ae assued that adjacent dominoes positioned using this pocedue have like-coloed faces since consecutive edges in ³ have a common endpoint 2 We ae also guaanteed that all dominoes fom - ae used since ³ is Euleian and edges in /g have a one-to-one coespondence with the dominoes in - Ä We can easily extend this poof to deal with layouts that ae cycles The only diffeence is that NEXT is initially any membe of, and we update NEXT cyclically aound the cycle Thus we have the following: Lemma 2 A layout, which is a cycle, is tileable by - /g contains an Eule cicuit GÆB It is well-known that Eule cicuits and paths can be computed on a gaph?å df8 u in time ecall that nodes in / coespond to colos, while edges coespond to dominoes Futhemoe, since evey domino is coloed with at most 2 colos we have that Thus we have the following: Theoem 3 DOMINO TILING can be solved in Q time if is a path o a cycle 4 $ : $ < c È $% : : = 4 $ : 5 Éq#=#w M In this Section we show that the decision vesion of A- TIAL DOMINO TILING is N-complete We use the following poblem in ou poof, which was shown to be Ncomplete in [4]: Definition 3 In the 3,4-SAT poblem we ae given a boolean expession Ê in CNF with exactly 3 vaiables pe clause and each vaiable appeas at most 4 times in Ê, and we ae asked to decide whethe Ê is satisfiable Theoem 4 ATIAL DOMINO TILING is Ncomplete oof ATIAL DOMINO TILING is in N since we can guess a tiling, and in polynomial time we can veify that the tiling is valid We will educe 3,4-SAT to ATIAL DOMINO TILING We will constuct a layout, a set of dominoes -, and a patial tiling function 2 accoding to the boolean expession Ê such that a total tiling function 2 will exist iff Ê is satisfiable The colos we use fo the faces of dominoes come fom diffeent aspects of Ê Fo each vaiable h Y we intoduce a unique colo CGË z Fo each clause t Y we intoduce a unique colo CSÌ z We also intoduce a unique colo C z fo each not Ï 2 ecall that these endpoints, ie the vetices of Í Î, ae the colos fom iff Figue 2: Left, a vaiable layout fo an unnegated vaiable ight, a vaiable layout with one negated output opeato that appeas in Ê Finally, we have one colo called C Ð that epesents tue, and anothe colo called C"Ñ that epesents false Now we descibe how is constucted fom a collection of othe layouts that coespond to diffeent aspects of Ê We will epesent vaiables that ae neve negated in Ê by constucting a layout and adding some dominoes to - The layout that epesents an unnegated vaiable h Y is shown in Figue 2 Fou points fom this layout have been emphasized We will efe to these points ae teminals since layouts that epesent clauses will be positioned at these points A domino QCDË z FMCË z, which we will efe to as a vaiable domino, has been positioned on this layout The position of each vaiable domino will be eflected in ou patial tiling function 2 We efe to the potion of a vaiable layout that is above the vaiable domino as the value zone, while the potion below is called the esevoi Fo each unnegated vaiable in Ê we will also add a collection of five QC Ð FMC Ð dominoes and five C#Ñ;F8CIÑ dominoes to - These dominoes will be used to tansmit the tuth value of a vaiable to a clause We will also add two impotant dominoes of the fom QC Ð FMCË z and QC Ñ FMCË z to - These dominoes will be used to set the tuth value of a vaiable Negated vaiables will be handled in a vey simila fashion to unnegated vaiables We simply inset negation dominoes nea the teminal on the layout fo an unnegated vaiable (Figue 2) These dominoes will be of the fom QC z F8C z The fact that such negation dominoes exist will be eflected in 2 Fo each negation domino we add two dominoes QCI z F8C Ð and QCS z F8CSÑ to - We epesent each clause t Y by exactly thee layouts that will be adjacent to the layouts fo the vaiables that ae in the clause t Y The thee layouts fo clause t Y ae shown in Figue 3 The emphasized points in Figue 3 ae called teminals Thee ae thee dominoes of the fom C Ìwz FMC Ìwz that have been positioned on these clause layouts We will efe to these dominoes as clause dominoes The positions of clause dominoes will be eflected in 2 The potion of the layout that is to the left of the clause domino will be efeed to as the esevoi of the clause layout Notice that 88

4 CCCG 2004, Monteal, Quebec, August 9 11, 2004 Figue 3: The thee layouts that epesent a clause one of the clause layouts does not have a esevoi Each of these layouts coesponds to exactly one of the vaiables that is in clause t Y Each of them will be positioned so that thei teminal is adjacent to one of the fou teminals in the layout fo the vaiable they coespond with Fo this to be accomplished, clause layouts may need to be otated Fo each clause t Y, we add thee dominoes of the fom QCÌ z FMC Ð and two dominoes of the fom C Ì z F8CIÑ to - Figue 4 shows a vaiable layout that has two clause layouts attached to it c wj c wj Figue 4: Vaiable h Y appeas negated in clause t Y and unnegated in clause t Now suppose that thee exists some tiling function 2 that is a solution to ATIAL DOMINO TILING fo, -, and 2 as descibed above We will show that if 2 exists then Ê is satisfiable Specifically, we will use a tuth assignment of Ê that is implied by 2 Fist let us examine the dominoes that have been positioned next to vaiable dominoes Vaiable dominoes ae only adjacent to two othe dominoes fom - ecall that the vaiable domino fo h Y is coloed CË z In ou constuction of -, we placed exactly two dominoes that have faces with colo C Ë z These two dominoes ae of the fom QCDÐTF8C Ë z and QCDÑ"FMC Ë z Thus we ae guaanteed that these two dominoes ae adjacent to the vaiable domino fo h Y We will assign a tuth value to vaiable h Y in Ê accoding to which of these two dominoes gets placed in the value zone of the vaiable layout fo h Y : if QCÐ F8C Ë z is positioned in the value zone then h Y? tue, and h Y? false othewise Now conside negation dominoes The ÐQÒ negation domino is only adjacent to two othe dominoes, and due ou constuction, these two dominoes will be C z FMC Ð and QC z FMC Ñ Now conside clause dominoes Due to ou constuction, all dominoes with faces coloed C Ì!z must be positioned next to a coesponding clause domino This ensues that all clause layout esevois get filled by eithe C Ì z F8C Ð o CIÌ z FMCIÑ dominoes Also, since thee ae only two dominoes of the fom QCSÌ z F8CSÑ, we ae guaanteed that at least one layout component fo each clause will have a face of colo C Ð on its teminal This coesponds to each clause fom Ê being satisfied We still must show that if a clause layout has a C Ð face on its teminal then the vaiable associated with that teminal has eithe been set of tue, o it has been set to false and it appeas negated in the clause Fist conside the case whee a vaiable h Y appeas unnegated in clause t Y and suppose a clause layout fo t Y has a CÐ face on the teminal associated with h Y Since this teminal is adjacent to a teminal in the h Y s vaiable layout, all the positions in h Y s value zone will be occupied by dominoes with C Ð faces The same agument woks fo clause teminals that ae coveed by C Ñ faces Thus fo unnegated vaiables, tuth values ae popely popagated to clauses We can easily extend this agument fo negated vaiables by noticing that negation dominoes simply flip the tuth value Thus we have that evey clause is satisfied Now suppose that Ê is satisfiable We will compute a tiling function 2 fom a satisfying tuth assignment of Ê If a vaiable h Y is set to tue then we position domino QC Ë z FMC Ð in h Y s vaiable layout value zone along with five QCÐMFMC Ð dominoes We also place QC Ë z F8CSÑ along with five QCSÑ"FMCIÑ dominoes in h Y s esevoi lacing the five QCGÐ FMC Ð dominoes in h Y s value zone will ensue that all unnegated teminals coespond to clauses that have a QC Ì z FMC Ð domino at thei teminal If a vaiable has a negated teminal, then this foces the QCD z F8CÐ to be adjacent to the vaiable layout s teminal, which in tun foces the C z FMCIÑ domino to be adjacent with the teminal fo the coesponding clause This causes the clause to have a QC Ì!z F8C Ñ domino at its teminal Since Ê is satisfiable, this can only occu at at most two vaiable layouts ecall that we ve only added two dominoes of the fom QC Ì!z F8C Ñ fo each clause This implies that vaiable layouts that coespond to a vaiable that has been set to tue can be tiled If the vaiable h Y is false, we do the exact opposite: we fill the value zone with QC Ë z FMCIÑ and five QCDÑwF8CSÑ dominoes, and the esevoi with C Ë z F8CÐ and five CIÐ F8C Ð dominoes We can ague that such layouts can be tiled by using the evese of the agument given above The C Ñ"FMCIÑ dominoes in the value zone will foce unnegated teminals to be associated with clauses that have a QC Ì!z FMC Ñ at thei teminal Since Ê is satisfiable, at most two vaiables will have this popety, which coesponds to the fact that thei ae only two QC Ìwz FMC Ñ dominoes pe clause Ä The layout constucted in the poof of Theoem 4 is disconnected in the sense that / 0 is disconnected We can easily augment so that it is connected We can achieve this by daisy chaining the vaiable layouts with fixed position dominoes: the vaiable layout fo h Y is connected to the vaiable layout fo h YKÓ E by a path that contains fixed position dominoes It is easily seen that this new connected layout is also a tee in the sense that /k0 is a tee Thus we have the following: 89

5 Õ Ü Æ 16th Canadian Confeence on Computational Geomety, 2004 Coollay Ô 5 ATIAL DOMINO TILING is Ncomplete even if is a tee ªd :Öq I$ Ø"q I ; 59Ù 8Ú We ve shown that domino tilings of layouts that ae eithe paths o cycles can be computed in " time, whee is the numbe of dominoes and is the numbe of colos on the domino faces We ve also shown that if a layout is patially tiled at the outset of the poblem, then the poblem is N-complete, even if the layout is a tee Many questions egading domino tilings emain open We ae inteested in the time complexity of DOMINO TILING whee is a tee o a gaph Also, we have only consideed domino tiling poblems Many inteesting poblems aise when we conside so-called impefect tilings whee Û In this situation, we wish to find subsets of - that can tile It would also be inteesting to exploe the similaities and diffeences between domino tilings and Wang tilings ~ Ú Ý1: 5 "< T We would like to thank D Mak Daley, D Gant Cheston, and D Mak Keil fo poviding useful insights into the poblems discussed heein Þ 5ßÈ5 5 5 [1] L Adleman, J Kai, L Kai, and D eishus On the decidability of self-assembly of infinite ibbons oceedings of FOCS 2002, IEEE Symposium on Foundations of Compute Science, pages , 2002 [2] Bege The undecidability of the domino poblem Memois of the Ameican Mathematical Society, (66):1 72, 1966 [3] J Kai Infinite snake tiling poblems In M Ito and M Toyama, editos, Developments in Language Theoy, 6th Intenational Confeence, DLT 2002, Kyoto, Japan, Septembe 18-21, 2002, evised apes, volume 2450 of Lectue Notes in Compute Science, pages Spinge, 2003 [4] C Tovey A simplified N-complete satisfiability poblem Discete Applied Mathematics, (8):85 89, 1984 [5] H Wang oving theoems by patten ecognition ii Bell Systems Technical Jounal, (40):1 41,