Scheduling Sensors by Tiling Lattices

Sc hedulin g Sensors b y Tiling L attices ∗ Andreas Klapp enec k er, Hyuny oung Lee, and Je nnifer L. W elc h Abstract Supp ose that wirelessl y communicating sensors are placed in a regular fashion on th e p oints of a lattice. Common comm unication proto cols allo w the sensors to broadcast messages at arbitrary times, which can lead to problems should tw o sensors broadcast at the same time. It is sho wn that one can ex p loit a tiling of the lattice to d erive a deterministic p erio dic sc hedule for th e broadcast communication of sensors th at is guaran teed to b e collision-free. The proposed schedule is shown to be optimal in the num b er of time slots. Keyw ords: distributed computing, sc heduling sensors, lattice tiling, wir eless c o mmun icatio n. 1 In tro duction Sensors ar e sometimes distributed in a regular fa s hion to monitor an a rea. W e assume that the sensors use wireless communication. Mo s t wireless co mmun icatio n pro to cols allow the se ns ors to send at a rbitrary times. How ever, this can ca use the following c ol lision pr oblems : If t wo distinct sens ors A and B se nd a t the same time and B is within the in terference range of A , then frequen tly hardware limitations preven t B from receiving the message o f A cor rectly . In addition, if tw o distinct s e nsors A and B s end at the s ame time a nd a senso r C is within interference r ange of bo th A and B , then C will not b e a ble to correc tly receive either message. In these cases, the sensors A and B need to resend their messages , which is evidently a waste of energy . Let us assume that the senso rs hav e a c cess to the curre n t time, represented by a n integer t . One can assign each sensor no de an integer k and set up a p erio dic s chedule such that a no de with integer k is allowed to broadcas t message s at time t if a nd o nly if t ≡ k (mod m ). The goa l of this pap er is to g ive a conv enient combinatorial formu lation using lattice tilings that allows one to assign optimal schedules with minimal n umber of time slots m s uch that no t wo sensors that are scheduled to bro adcast simultaneously hav e int ers e cting int erfer e nce r anges; w e call such sc hedules c ol lision-fr e e . Related W ork. Since most communication proto cols for wir eless s ensor netw or k s a r e pr obabilistic in na ture, there e x ist few pr ior works that ar e dir ectly r elated to o ur approach. How ever, there exist a few no table exceptions that w e w ant to discuss here. ∗ The research b y A.K. was supp orted b y NSF CAREER a ward CCF 0347310 and NSF grant CCF 0622201. The r esearc h b y H.L. w as supp orted in part by NSF grant CNS- 0614929. The research by J.W. was supp orted in part by NSF grant 0500265 and T exas Higher Education Coor dinating Board grant ARP-00512-0007-2006. 1 Suppo se fo r the moment that we are given a finite set of k sensors that share the sa me freq uency band for communication. The simplest wa y to ensure that the communication will be collisio n- free, is to use a time division multiple access (TDMA) scheme. Here each of the k senso rs is as s igned a differ e n t time slot and scheduling is done in a r ound ro bin fas hion. Becaus e of its simplicit y , this sc heme is used in ma ny s ystems, see e.g. [8, Chapter 3.4]. The obvious disadv antage of TDMA is that it do es no t scale: If the num ber k of sensor s is large, then the sensor s cannot comm unicate frequen tly enough. The bas ic TDMA s cheme doe s not ta ke adv an tage o f the fact that ea ch sensor typically a ffects only small nu mber of neighbo ring sensors by its r adio communication. This pr o mpts the q uestion whether o ne can mo dify the TDMA scheme and find a s chedule with m time slots that is collision-fr ee. T o answer this question, conside r a directed g r aph that has a no de for each sensor and an edge from vertex v to vertex u if and o nly if u is a ffected by the ra dio communication of v . A v alid schedule with m time slo ts corr esp onds to a distance-2 coloring with m colors , that is, a ll vertices of distance ≤ 2 must be as signed a different colo r (= time slot) to avoid collision problems. Therefore, the n umber of time s lots m of an optimal co llis ion-free schedule coincides with the chromatic n umber of a distance- 2 coloring. The distance - 2 color ing problem is also known as the broadcast scheduling problem in the netw ork ing communit y . McCormick has shown that the decision pro blem whether a given gr aph ha s a distance-2 coloring with m colors is NP-complete [6]. Llo yd and Ramanatha n showed that the broadca st schedule pro blem even remains NP-complete when restricted to planar graphs and m = 7 time slo ts [5 ]. Due to these intractability results, muc h of the subseq uent resear ch fo cus ed on heur istics for finding o pti- mal schedules; for insta nce, W ang and Ansar i used simulated annea ling [12], and Shi and W ang use d neural net works [9] to find optimal s chedules. Another po pular dir ection of resear ch are approximation alg orithms for broadcas t scheduling alg o rithms, see e.g. [7]. Con tributions. The main contributions of this paper can b e briefly summarized as follows (the terminology is explained in the subsequent sections ): 1) W e develop a metho d that a llows one to derive an o ptimal collision-free s chedule from the tiling of a lattice. 2) Our scheme sca les to a n arbitra ry num ber of sens ors; in fact, we formulate our schedules fo r an infinite nu mber of sensors. Schedules for a finite n umber of sensors ar e obtained by restriction, and these schedules remain optimal under very mild co nditio ns (g iven in the c onclusions). 3) Our assumption o n the set of pr ototiles ensur e s that a n optimal schedule is obtained r egardles s of the chosen tiling. In Section 4 , w e show that if our assumption on the set o f prototiles is remov ed, then in g e ne r al o ne will not obtain an optimal schedule. W e formulate our results for ar bitrary la ttices in arbitrary dimensions, since the pro o fs ar e not more complicated than in the familia r case of the tw o-dimensional square lattice. F or the squa re lattice, ther e are p olyno mial-time algorithms av ailable to chec k whether a g iven prototile can tile the la ttice; th us, despite the fact that finding optimal schedules is NP-hard in general, one can use our method to easily construct optimal sc hedules in the case 2 of a single pr ototile. This method of creating s imple instances of an NP-hard problem mig ht b e of independent int eres t. 2 Lattice Tilings and Optimal Sc h edules A Euclidean lattice L is a discr ete subg roup of R d that spans the Euclidean space R d as a r eal vector space. In o ther words, there exis t d v ectors { v 1 , . . . , v d } in L that a re linear ly indep endent ov er the r eal num ber s such that L = ( d X k =1 a k v k      a k ∈ Z for 1 ≤ k ≤ d ) , and for each vector v in L there exis ts an o p en set c ontaining v bu t no other ele men t of L . In particular , the g roup L is is omorphic to the additive a b elian g roup Z d . Two examples of lattices in tw o dimensions are illustrated in Fig ure 1. 0 v 1 v 2 v 1 + v 2 0 u 1 u 2 u 1 + u 2 Figure 1: The figure on the left sho ws part of t he sq uare lattice L S = Z 2 that is generated by the vector s v 1 = (1 , 0) and v 2 = (0 , 1 ). The figure on the right shows the hexagonal lattice L H that is generated by the vectors u 1 = (1 , 0 ) and u 2 = ( 1 2 , 1 2 √ 3). Our goal is to find a deterministic collision-free p erio dic s chedule for sensors lo cated at the po int s o f a lattice L that is optimal in the num ber of time s lots, i.e. , no p er io dic schedule with a shorter p erio d ca n be found that is collision-free. W e call a finite subset N o f L a pr ototile or a neighb orho o d of the po int 0 if and only if it con tains 0 itself. The par ticular nature o f N will b e determined for instance by the type o f antenna and by the sig nal strength used by the senso r. The elements in N are the sensor s affected by wireless communication of the sensor lo ca ted at the p oint 0 (that is, only the elements in N a re within interference range of the sensor lo ca ted at the po int 0). W e will firs t assume a homogeneous situation, namely the neighborho o d affected by co mm unication of the sensor lo cated at a po int t in L is of the form t + N = { t + n | n ∈ N } , wher e the a ddition denotes the usual addition of vectors in R d . The set t + N co ntains t , since 0 is contained in N . So me exa mples of neighbo rho o ds N are given in Figure 2. Our schedule will be a deterministic p erio dic schedule, that is, each sensor is assigned a cer tain time slot and it is only allow ed to send during that time slot. Since our schedule is re quired to be free of collision proble ms , it follo ws that the sensor s lo ca ted at distinct p oints s and t in L cannot broadcas t at the s a me time unless ( s + N ) ∩ ( t + N ) = ∅ . 3 0 0 0 Figure 2: The three figures illustrate some p ossible shap es of the n eigh b orho o d N of 0. The elemen ts in N are marked by small crosses. The left fi gure is a b all of radius 1 in the Chebyc heff (or ℓ ∞ ) metric. The figure in the middle is a ball of radius 1 in the Euclidean (or ℓ 2 ) metric. The figure on the right pro vides an examp le of a neighborho od where th e sensor at 0 uses a directional antenna. Let T denote a subset o f L . W e say tha t T provides a tiling o f L with neighborho o ds (or tiles) of the form N if and only if the following tw o conditions hold: T1. S t ∈ T ( t + N ) = T + N = L , T2. ( s + N ) ∩ ( t + N ) = ∅ for all distinct s, t in T . The set T con tains all the vectors that tra ns late the prototile N . Condition T1 s ays that the whole lattice L is cov ered by the translates t + N of the prototile N , when t ranges ov er the elemen ts of T . Co ndition T2 simply says that the translates of the tile N do not overlap. The tiling s pr ovide us with an elegant means to constr uct an optimal deterministic schedule. Theorem 1. Le t T b e a tiling of a Euclide an lattic e L in R d with neighb orho o ds of the form N . Then ther e exists a deterministic p erio dic sche dule that avoids c ol lision pr oblems using m = | N | time slots. The sche dule is optimal in the sense t hat one c annot achieve this pr op erty with fewer than m time slots. Pr o of. Supp ose that N = { n 1 , . . . , n m } is the neighborho o d of 0. F o r k in the r ange 1 ≤ k ≤ m , we schedule the sensors lo cated at the p oints n k + T at time t ≡ k (mod m ). W e first no tice that each senso r lo cated at a po int in L is scheduled at some po int in time, since N + T = L by pro pe rty T1 of a tiling. Seeking a contradiction, w e ass ume that the s chedule is not collisio n-free. This means that at s ome time k in the range 1 ≤ k ≤ m there e x ist s ensors lo cated at the positions n k + s and n k + t with distinct s and t in T such that ( n k + s + N ) ∩ ( n k + t + N ) 6 = ∅ . How ever, this w ould imply that ( s + N ) ∩ ( t + N ) 6 = ∅ for distinct s and t in T , contradicting prop erty T2 of a tiling. It follows that our sc hedule is co llision-free. It remains to prove the optimality of the s chedule. Seek ing a contradiction, we assume tha t there ex is ts a schedule with m 0 < m time s lots that is collisio n-free. This mea ns that for some time slot k in the range 1 ≤ k ≤ m 0 t wo ele men ts n ′ and n ′′ of N must be scheduled. How ever, this would imply that the element n ′ + n ′′ is contained in b oth s e ts ( n ′ + N ) and ( n ′′ + N ), contradicting the assumption that the schedule with m 0 time slots is collision- fr ee. W e illustrate some a sp ects of the pro of of the pre vious theorem in Figure 3. 4 1 2 3 4 5 6 7 8 1 2 3 4 5 6 7 8 1 2 3 4 5 6 7 8 1 2 3 4 5 6 7 8 1 2 3 4 5 6 7 8 1 2 3 4 5 6 7 8 1 2 3 4 5 6 7 8 1 2 3 4 5 6 7 8 · · · · · · 1 2 3 4 5 6 7 8 1 2 3 4 5 6 7 8 1 2 3 4 5 6 7 8 1 2 3 4 5 6 7 8 1 2 3 4 5 6 7 8 1 2 3 4 5 6 7 8 1 2 3 4 5 6 7 8 1 2 3 4 5 6 7 8 · · · · · · Figure 3: The fi gure on the left illustrates the prev ious theorem using a tiling with t h e n eigh b orho o d N given by the righ tmost example of Figure 2. Each of the eight elements of N is assigned a time slot from 1 to 8, the tran slated versi ons t + N with t in T have the time slots at t h e corresp onding translated p ositions. As a result, t h e broadcast during time slot 1 affects only th e n eigh b orho o ds shown in t he tili ng dep icted b y the figure. The figure on the righ t sho ws one dashed neighborho od of a sensor broadcasting during time slot 2. Considering the neighborho od s of all sensors broadcasting during t ime slot 2 one obt ains once again a tiling, namely the tiling of L S obtained by right shifting the (solid gray) neighborho ods of the sensors broadcasting during time step 1. 3 Existence of Tilings Our concept o f tiling a lattice with translates o f a pr ototile N turned o ut to b e conv enient for our purposes . In this sectio n, we relate the tilings of a lattice to tilings of the Euclidea n space R d , so that we can b enefit from the large num b er of results that ar e av ailable in the literature. An y tiling of a lattice L can b e co nv e rted in to a tiling of R d as follows. Let K denote the unio n of the closed V oro no i reg ions ab o ut the p oints in N . Then the translates t + K with t in T yield a tiling of R d . Co nversely , any tiling of R d with translates o f a tile consisting of the union of V orono i regions of p oints in L evidently yields a tiling o f the la ttice L in our sense. Figure 4 shows s ome t wo-dimensional examples of V oronoi regions. Figure 4: (a) The figure on the left show s that th e V oronoi region ab out a p oint in the square lattice L S is given by a square of unit length; a tile K in the plane obtained by a un ion of unit sq uares ab out points in L S is called a quasi-p olyomino. (b ) A V oronoi region ab out a p oin t in the hexagonal lattice L H is a hexagon; a tile K in the p lane obtained from a un ion of these V oronoi-hex agons ab out points in L H is called a quasi-p olyhex. The union of V or onoi regions ab out p oints in a lattice a re also known a s qua si-p olyforms. A q uasi-p olyfor m that is ho meomorphic to the unit ball in R d is known as a po lyform. The b o o ks by Gr¨ un baum a nd Shepherd [4 ] and by Stein a nd Sza b´ o [1 0] c o ntain n umerous examples of tilings obtained by translating quasi-p oly forms (and esp ecially poly fo rms). The p olyforms in the s quare grid L S = Z 2 are called p o lyominoes, the most w ell-known t yp e o f p o lyforms; see Golo m b’s b o ok [3]. By a buse of languag e , we will a lso refer to a pro to tile N in L S as a po lyomino if the union of the V oro noi reg ions of N fo rm a p o lyomino. 5 A pro totile N in a lattice L that admits a tiling is called exact . It is natural to ask the following question: Q1. When is a given prototile N exa ct, i.e. , when do es there ex ist a subset T of L such that the co nditions T1 and T2 are satisfied? Beauquier and Niv at gave a simple criterio n that allows one to answer Q1 for po lynominos in the s q uare lattice L S . Roughly sp eak ing , their criter ion says that if N can b e surro unded by translates of itself such that there are no gaps or ho les, then N is exa ct; see [1] for deta ils. In particular, it immediately follows that each prototile shown in Figure 2 is exact. Algorithmic criter ia for deciding the questio n Q1 are particularly interesting. F or p olyomino es in the square lattice L S , one can dec ide this questio n in time p olyno mial in the length of the b oundary of the p olyomino (describ ed by a word ov er the alphab et { u, d, l , r } , which is sho r t for up, down, left, and r ight), as Wijshoff and v an Leeuw en hav e shown [1 3]. The characterization of exa ctness of a po lyomino by Beauquier and Niv at [1] mentioned ab ov e leads to an O ( n 4 ) algorithm, wher e n is the leng th o f the word desc r ibing the b o undary . Recently , Ga mb ini and V uillon [2] der ived an improv e d O ( n 2 ) algorithm for this problem. Less is known for arbitr ary (not necessa rily connected) proto tiles in a g eneral lattice. Szegedy [11] der ived an algor ithm to decide whether a pr ototile N in a lattice L is exact assuming that the car dinality o f N is a prime or is equal to 4. 4 Generalization to Sev eral Prototiles W e ha ve seen that the conditions for tiling a lattice with a single pro totile ar e so mew ha t restrictive. F or example, we might wan t to allow different ro tated versions of the tile if the ra diation pattern of the a ntenna used by a sensor is asymmetrica l. W e mig ht wan t to cons ider different tiles co rresp onding to v a rious different signal strength settings. F urthermore, w e might wan t to allow senso rs with v a rious differen t st yles of ant enna. W e can acco mmo da te all these different situa tio ns by allowing trans lates o f several prototiles ins tead of just a single one. In this s ection, we show that o ne can still o btain an optimal p erio dic schedule which guarantees that the sch edule is collision-free, as long a s sensors of the same type and setting are deplo yed within each tile and a c o nstraint on the tiles is satisfied. Let L b e a lattice in R d . Let N 1 , . . . , N n be prototiles in the lattice L , that is, N k is a subset of L that contains 0 for 1 ≤ k ≤ n . Let T 1 , . . . , T n be pairwise disjoint nonempty subsets of L . W e say that T 1 , . . . , T n provide a tiling o f L with prototiles N 1 , . . . , N n if and only if the following tw o conditions are satisfied: GT1. n [ k =1 [ t k ∈ T k ( t k + N k ) = n [ k =1 ( T k + N k ) = L. GT2. F or all k, ℓ ∈ { 1 , . . . , n } , we have ( s k + N k ) ∩ ( t ℓ + N ℓ ) = ∅ for all s k in T k and t ℓ in T ℓ such that s k 6 = t ℓ . Condition GT1 ensures tha t the la ttice L is c ov ered by translates of the prototiles N 1 , . . . , N n . Condition GT2 ensures that tw o distinct tiles will no t overlap. The set T k contains all vectors tha t are used to translate the tile N k , that is, the set { t k + N k | t k ∈ T k } contains all shifted versions of N k that o ccur in the tiling of L . Since 6 the sets T 1 , . . . , T n are pair wise disjoint, it is c le ar that ( s k + N k ) ∩ ( t ℓ + N ℓ ) = ∅ whenever k 6 = ℓ . Condition GT2 requires further that the translates of the prototile N k with elemen ts in T k do not overlap. W e will call a tiling of L r esp e ctable if and only if the prototile N 1 contains all other prototiles N k , that is, N 1 ⊇ N k for 2 ≤ k ≤ n . If this is the case, then we call N 1 the r esp e ctable pr ototile . Suppo se that we are g iven a tiling T 1 , . . . , T n of L r esp ectively with neighborho o ds of the for m N 1 , . . . , N n . W e will assume that the sensor s ar e deploy ed in the following fashio n: D1. A sens or a t lo cation s k in the neighborho o d t k + N k of a n ele men t t k in T k affects precis ely the neigh b ors s k + N k by interference, wher e k is in the r a nge 1 ≤ k ≤ n . Lo osely s p ea king, c o ndition D1 s ays that all element s in the neighborho o d t k + N k hav e neighborho o d t yp e N k . Theorem 2. L et T 1 , . . . , T n b e a r esp e ctable tiling of a Euclide an lattic e L with neighb orho o ds of the typ e N 1 , . . . , N n . Supp ose that the sensors ar e deploy e d ac c or ding to the scheme D1 . Then ther e exists a deterministic p erio dic sche dule that avoids c ol lision pr oblems using m = | N 1 | time slots. The sche dule is optimal in the sense that one c ann ot achieve t his pr op erty with fewer than m time slots. Pr o of. The pe r io dic s chedule is sp ecified as follows. Let N = S n k =1 N k = { n 1 , . . . , n m } . F or a ll ℓ in the ra nge 1 ≤ ℓ ≤ n , we schedule the elements n k + T ℓ at time t ≡ k ( mo d m ) if and only if n k is c o ntained in the neighborho o d N ℓ . Notice that all elements in L will b e scheduled at some p oint in time by pro per ty GT1 . F ur thermore, condition GT2 ensures that an elemen t in L is not scheduled mor e than o nce within m consecutive time steps. W e claim that this schedule is collis io n-free. Seeking a contradiction, w e assume that tw o distinct elements in L a re s cheduled at the same time, but yield a collision problem. In other words, there must exist in teger s k and ℓ in the range 1 ≤ k , ℓ ≤ n , a n element n ∈ N such that n is co ntained in b oth N k and N ℓ , and elements s k ∈ T k and t ℓ ∈ T ℓ with s k 6 = t ℓ such that ( n + s k + N k ) ∩ ( n + t ℓ + N ℓ ) 6 = ∅ . This implies tha t ( s k + N k ) ∩ ( t ℓ + N ℓ ) 6 = ∅ for s k 6 = t ℓ , con tra dic ting prop erty GT2 . Therefor e, our schedule is collision- fr ee. Without los s of g enerality , we may ass ume that the p oint 0 in L has a resp ectable neigh b orho o d N 1 (other- wise, simply shift the tiling such that this condition is satisfied). Seeking a contradiction, we ass ume that there exists a deterministic p erio dic sc hedule with m 0 < m time slots that is collision-free. It follows that ther e must exist tw o distinct elements n ′ and n ′′ in N 1 that ar e scheduled at the sa me time. How ever, this w ould imply that the element n ′ + n ′′ is contained in bo th n ′ + N 1 and n ′′ + N 1 ; th us, ( n ′ + N 1 ) ∩ ( n ′′ + N 1 ) 6 = ∅ , co n tra dicting the fact that the schedule with m 0 time slots is collision- fr ee. The prev io us theo rem is a natura l genera lization of Theor em 1. A salie n t feature of The o rems 1 a nd 2 is that the o ptimal sc hedule is indep endent of the nature of the tiling of L . Notice that one ca n obtain a collis ion-free p erio dic schedule even when ther e do es not e x ist a resp ectable prototile. In fact, the resp ecta ble prototile w as o nly used in the last part of the pro o f o f Theorem 2 to establish the optimalit y of the schedule. Therefor e, o ne migh t wonder what will happ en in the non-resp ectable ca s e. 7 Let us ag ree o n so me gro und rules. W e would like to maintain the fact that for each translated version of a prototile the schedule is the s ame, as this s implifies co nfig uring the sensor net work. How ever, in the non-resp ectable case we might hav e different proto tiles of the same size, s o we allow that the schedules in the different prototiles can b e indep endently chosen, as lo ng a s this doe s not lead to collisio n pro blems. Figure 5 shows that the num b er of time steps in an optimal schedule depends on the chosen tiling when the tiling is non-resp ectable. 1 2 5 6 1 2 5 6 1 2 5 6 1 2 5 6 1 2 5 6 1 2 5 6 1 2 5 6 1 2 5 6 4 3 1 2 4 3 1 2 4 3 1 2 4 3 1 2 4 3 1 2 4 3 1 2 4 3 1 2 4 3 1 2 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 Figure 5: The figure on the left shows a sc hedule for a non - resp ectable tiling with tw o tetrominos ( i.e. , p rototiles with 4 elements). The tiling contains tw o Z -shap ed tet rominos that are surround ed by S -shap ed tetrominos ( rotating the figure clockwise by 90 ◦ migh t h elp identifying the S and Z shap es). The sc hedule was determined with the algorithm giv en in the pro of of Theorem 2. I t is not difficult (though tedious) to sho w that this sc hedu le with m = 6 t ime step s is optimal. How ever, if the lattice is tiled in the symmetric fas hion sho wn in the right fi gu re, then the optimal schedule has m = 4 time steps. Therefore, in th e non-resp ectable case the num b er of time steps of an optimal schedule dep end s on the chosen tiling. 5 Conclusions W e ha ve int ro duce d a deterministic p erio dic s chedule for senso rs us ing wireless communication that are placed on the points of a lattice. W e have s hown that the schedule is optimal a ssuming that there exis ts a resp ectable prototile. A natural question is whether the schedule r emains optimal if one res tricts the schedule from the lattice L to a finite subset D of L . This question has a n affirmative answer if D co nt ains a translate of the set N 1 + N 1 , as the latter set consists of the resp ectable proto tile N 1 and its neig hbors, in which case our optimality pro of carries o ver without c hange . Another natural question is whether o ne can ex tend the metho d to the case o f mobile sensors . This questio n has an affirmative a nswer. Indeed, one str a ightforw ard way is to use our schedule to a ssign time slots to the lo cations r ather than to the sensors. Let us assume that the lattice p oints are spac e d fine enough to ensure that only one sensor is within a V o ronoi reg ion of a lattice p oint. If the time s lo t k is assigned to a lattice p o in t p , then a sensor s within the op en V or onoi re gion ab o ut p can send a t time t if and only if t ≡ k (mod m ) and the 8 int erfer e nce ra nge of s fits within the tile of p . Clearly , this yie lds a collis io n-free schedule for mobile sensor s. How e ver, it should b e stressed tha t there are many o ther solutio ns p ossible, but a comparison of such metho ds is beyond the sco pe of this pa p er . References [1] D. Bea uquier and M. Niv at. On translating one p olyomino to tile the pla ne. Discr ete Comput. Ge om. , 6(6):575– 592, 1991 . [2] I. Ga mb ini and L. V uillon. An a lgorithm for deciding if a p olyomino tiles the plane. The or. Inform. 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