Pushing squares around

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1 Pshing sqares arond Adrian Dmitresc János Pach Ý Abstract We stdy dynamic self-reconfigration of modlar metamorphic systems. We garantee the feasibility of motion planning in a rectanglar model consisting of sqare modles that are allowed to slide along or rotate abot one another. That is, we show that any two connected configrations of the same nmber of modles can be transformed into each other by a seqence of moes so that all intermediate configrations are connected. This settles a conjectre formlated in [6]. 1 Introdction A modlar metamorphic system consists of a nmber of identical modles that can connect to, disconnect from, and relocate relatie to adjacent modles (see, for example, [2, 7, 9, 10, 12]). While indiidal modles are not capable of moing by themseles, the entire system may be able to reconfigre or moe to a new position, when its members repeatedly change their positions relatie to their neighbors, by rotating or sliding arond other modles [1, 7, 11], or by expansion and contraction [10]. It is sally assmed that the entire system mst remain connected dring reconfigration. The motion planning problem for sch a system is that of compting a seqence of modle motions that brings the system from a gien initial configration Á into a desired goal configration. Depending on the existence of sch a seqence of motions, we say that the problem is feasible or respectiely, infeasible. For instance, in [2, 9] pper and lower bonds on the nmber of moes needed to change Á to are discssed for a system of hexagonal modles. In [8] it is shown that in sch a system any two connected configrations are mtally reachable as long as they do not contain a certain prohibited pattern. Demaine et. al. [3] hae considered a family of one-player games, inoling the moement of coins from one configration to another. Moes are restricted so that a coin can only be placed in a free position adjacent to at least two other coins, in contrast to or motion rles that are reqired to maintain oerall connectedness throghot the reconfigration process. In Section 2, we present the rectanglar model, and in Section 3 we proe that its two motion rles (sliding and rotation) garantee the feasibility of motion planning for any pair of connected configrations haing the same nmber of modles. This settles a conjectre formlated in [6]. We conclde (Section 4) with the presentation of another rectanglar model for which the same property holds. 2 Rectanglar metamorphic systems Consider a plane that is partitioned into a rectanglar integer grid of sqare cells indexed by their center coordinates in the nderlying Ü-Ý coordinate system. Of the eight adjacent cells of cell Ü Ý in the Uniersity of Wisconsin Milwakee, Milwakee, WI , USA; ad@cs.wm.ed Ý Corant Institte of Mathematical Sciences, 251 Mercer Street, New York, NY , USA; pach@cims.ny.ed 1

2 ( Ü), Ï ( Ü), Æ ( Ý), Ë ( Ý), Æ, Ë, ÆÏ and ËÏ directions, the for in the, Ï, Æ and Ë directions are said to be side-adjacent to, while the other for in the Æ, Ë, ÆÏ and ËÏ directions are said to be corner-adjacent to. We denote by Æ µ (resp. Æ µ) the cell side-adjacent to in the Æ direction (resp. the cell corner-adjacent to in the Æ direction). Similar notation is sed to denote the cells side-adjacent or corner-adjacent to in the other axis and diagonal directions. At any time each cell may be empty or occpied by a modle. The reconfigration of a metamorphic system consisting of Ò modles is a seqence of configrations (distribtions) of the modles in the grid at discrete time steps Ø ¼ ½ ¾, see below. Let Î Ø be the configration of the modles at time Ø, where we often identify Î Ø with the set of cells occpied by the modles or with the set of their centers. We are only interested in configrations that are connected, i.e., for each Ø, the graph Ø Î Ø Ø µ mst be connected, where for any Ø, Ø is the set of edges connecting pairs of cells in Î Ø that are side-adjacent. Î Ø yields Î Ø ½ when one modle Ñ moes from its crrent location to new location in step Ø. In this paper we restrict orseles to seqential reconfigration, in which only one modle moes at each discrete time step, as explained aboe. Note that, according to the aboe definition, the pattern (set of cells or set of integer points) Î Ø niqely determines the edge set Ø so that the graph Ø can be characterized by its ertex set Î Ø. The nion of all closed sqares belonging to Î Ø is a connected point set of area Î Ø, which will be denoted by Ë Î Ø µ. The following two generic motion rles (Fig. 1) define the rectanglar model. These are to be nderstood as possible in all axis parallel orientations, in fact generating ½ rles, eight for rotation and eight for sliding. Rotation: A modle Ñ side-adjacent to a stationary modle rotates throgh an angle of ¼ Æ arond either clockwise or conterclockwise. Fig. 1(a) shows a clockwise Æ rotation. For rotation to take place, both the target cell and the cell at the corresponding corner of that Ñ passes throgh (ÆÏ in the gien example) hae to be empty. Sliding: Let ½ and ¾ be stationary cells that are side-adjacent. A modle Ñ that is side-adjacent to ½ and adjacent to ¾ slides along the sides of ½ and ¾ into the cell that is adjacent to ½ and side-adjacent to ¾. Fig. 1(b) shows a sliding moe in the direction. For sliding to take place, the target cell has to be empty. In order to ensre motion precision, each moe is gided by one or two modles that are stationary dring the same step. The two motion rles of this model also appear in [5, 6]. A somewhat similar model is presented in [1]. m m f (a) f1 f2 (b) Figre 1: (a) Clockwise Æ rotation and (b) sliding in the direction. Fixed modles are shaded. The cells in which the moes take place are otlined in the figre. Theorem 1 The set of motion rles of the rectanglar model garantees the feasibility of motion planning for any pair of connected configrations Î and Î ¼ haing the same nmber of modles. That is, following the aboe rles, Î and Î ¼ can always be transformed into each other so that all intermediate configrations are connected. 2

3 We refer to a set of modles that form a straight line chain in the grid, as a straight chain. It is easy to constrct examples so that neither sliding nor rotation alone can reconfigre them to straight chains. Howeer, in Section 3 we proe that the motion rles of the rectanglar model (rotation and sliding, Fig. 1) are sfficient to garantee reachability, while maintaining the system connected at each discrete time step. In [6], it was proed that this is tre for a special class of systems, called horizontally conex, where also a distribted algorithm was gien for this task in a setting where concrrent moes are allowed. 3 Proof of Theorem Algorithm otline Clearly, it is enogh to proe that any configration can be transformed into a straight chain. Then one can append to the seqence of moes that transforms the start configration into a straight chain, the reersed seqence of moes that transforms the goal configration into a straight chain to obtain the desired effect. In addition, we make se of the easy fact that the two motion rles permit the relocation of a straight chain from a gien initial location to any target location. Assme withot loss of generality that the maximm Ü-coordinate of a cell in the configration is ¼. Arbitrarily select one of the occpied cells haing Ü ¼ as the base, say. The algorithm relocates all the modles except, one by one, to extend a horizontal chain, whose leftmost cell is. Denote by Þ the rightmost cell of (initially, Þ). Dring the exection of the algorithm, all modles except those in Ò hae non-positie Ü coordinates. In each iteration of the algorithm, one modle extends the chain in the direction by one cell. 3.2 Preliminaries Sometimes we refer to the cell occpied by a modle Ñ by ÐРѵ. A pair of modles Ù Ú is said to form a critical pair if Ù and Ú are corner-adjacent, say at point Ô, and the other two cells corner-adjacent at point Ô are empty. See Fig. 2. p Figre 2: A critical pair Ù Ú. The other two cells corner-adjacent to point Ô are empty. Consider the grid graph Ø Î Ø Ø µ, where the time-sbscript will be omitted for conenience. Recall that is a connected graph, so that Ë Î µ, the nion of the occpied cells (closed sqares), is an arc-wise connected set in the plane. The complement Ë Î µ of Ë Î µ consists of one or more connected components, called holes. They are denoted by À ¼ À, ¼, where À ¼ denotes the niqe nbonded component, the oter hole, while eery other hole À, ½, is said to be an inner hole. Each hole is bonded by a simple orthogonal polygon, which is called the contor of the hole. The set of all cells in Î side-adjacent or corner-adjacent to at least one cell in À ¼, denoted ÓÙØ, is called the oter bondary of the configration. Similarly, we define the inner bondary Ò of the configration to be the set of all elements of Î side or corner-adjacent to at least one empty cell belonging to an inner hole. Note that a cell may belong to both the oter bondary and the inner bondary of the configration, and it may be adjacent to empty cells in more than one of the inner holes À, ½. 3

4 A hole À, ¼, is said to be critical if it contains an empty cell side-adjacent to a pair of modles that form a critical pair. See Fig. 3. A hole which is not critical is said to be perfect. It follows from the connectedness of or configrations that a critical pair is always associated with exactly two holes. In what follows, we consider the set of simple (i.e., non-self-intersecting) cycles in. For any simple cycle ¾, let Ê µ denote the set of cells (empty or occpied) belonging to or enclosed by. Define the area of as Ê µ. Each simple cycle in the graph corresponds to a simple closed cre obtained by connecting the centers of the cells belonging to in the cyclic order. The area of is the nmber of cells enclosed or crossed by this cre. The contents of a cycle ¾, denoted by É µ, is defined as É µ Ê µ Î. That is, É µ is the set of occpied cells belonging to or enclosed by. See Fig. 3 for an illstration. H1 H2 z s H3 H0 H4 Figre 3: A configration Î in which the minimm degree in Î Ò is at least two. There are three maximal cycles, haing areas of ½ ½,, and, and contents of size,, and. A polygonal line delineates the large cycle with area ½ ½. There are fie holes, ot of which À ¼ and À ¾ are critical, and the other three are perfect. We say that a cycle is a maximal cycle, if Ê µ is maximal with respect to inclsion, i.e., there is no cycle ¼, sch that Ê µ Ê ¼ µ. Denote by Å the set of maximal cycles in. Note that if, Å. The next three lemmas gie some sefl properties of the system of maximal cycles, similar to those of the block decomposition of graphs [4]. Lemma 1 If and ¼ are two maximal cycles, then Ê µ Ê ¼ µ ½. Proof. Assme to the contrary that Ê µ Ê ¼ µ ¾. Then ¼ indces a (simple) cycle ¼¼ in so that Ê µ Ê ¼¼ µ (and Ê ¼ µ Ê ¼¼ µ ), contradicting the maximality of (and ¼ ). ¾ Assme now that in the graph the degree of eery ertex except Þ is at least two at some point dring the reconfigration algorithm. Since is connected, for any two maximal cycles, and ¼, there exists a simple path È Ú ½ Ú, ½, in, connecting a ertex Ú ½ ¾ with Ú ¾ ¼, none of whose intermediate ertices belong to or ¼. The ertex Ú ½ (resp. Ú ) is said to be a connector for (resp. for 4

5 ¼ ). (It is possible that Ú ½ and Ú coincide.) Notice that, althogh there may exist more than one sch paths È connecting and ¼, the connectors (their endpoints) are niqely determined. See Fig. 4. Similarly, for any maximal cycle ¾ Å, consider a simple path connecting to (the base of ), whose intermediate ertices do not belong to, and let the corresponding ertex of be also called a connector (that is, if ¾, is a connector). By the maximality of, the connector is also niqe in this case. H0 H 1 c d a b s z H2 H3 H4 Figre 4: A configration with fie maximal cycles, with areas of ½¾,,, and. Connectors are shaded in the figre. The configration has fie holes, for of which are critical (À ¼, À ¾, À, À ). For example, is a critical pair associated with holes À ¼ and À. is a rightmost edge of the maximal cycle with area, as specified in STEP 2 of the algorithm. Lemma 2 Assme that in the degree of eery ertex except of Þ is at least two. If, there exists a maximal cycle haing exactly one connector. Frthermore, eery ertex of not in Ê µ can be connected to by a path, all of whose intermediate ertices lie otside of Ê µ. Proof. By definition, each maximal cycle has at least one connector. Assme for contradiction that each maximal cycle has at least two connectors. It follows from the degree condition that and so Å. If Å ½, i.e., Å, has exactly one connector, as noted aboe. Assme therefore that Å ¾. There exist two maximal cycles, ½ and ¾, and a simple path in between them that does not pass throgh any ertex belonging to another maximal cycle. Since ¾ has at least two connectors, there exist two maximal cycles, ¾ and, and a simple path between them that ses another connector of ¾ and does not inclde any point belonging to another maximal cycle. Any new cycle that we may reach, has at least two connectors, so we can repeat this procedre. Finally, we mst either isit a ertex belonging to some path already isited before or reach one of the preiosly considered cycles. In either case, we obtain a simple cycle ¼ with Ê µ Ê ¼ µ for some ¾ Å, which contradicts the maximality of. To erify the second part of the lemma, it is sfficient to obsere that eery ertex of not in Ê µ lies either in Ê ¼ µ for some other element ¼ ¾ Å or along a simple path connecting to another maximal cycle ¼ ¾ Å, whose intermediate points do not lie in Ê µ or Ê ¼ µ. ¾ 5

6 Lemma 3 Sppose that in the degree of eery ertex except of Þ is at least two, and let be a maximal cycle with precisely one connector. Then no ertex of other than its connector is adjacent in to any other ertex that occpies a cell not in Ê µ. Proof. Sppose to the contrary that some ertex Ú ¼ of, different from its connector, is adjacent to another ertex Ú ½ ¾ Î Ò that lies in the exterior of. Let Ú ¾ denote a neighbor of Ú ½ different from Ú ¼. In general, if Ú has already been determined for some ¾, then let Ú ½ be any neighbor of Ú in, different from Ú ½. Using the fact that Ú ¼ is not a connector, and sing the maximality of, we can arge that Ú ½ cannot belong to, and cannot be identical to or to any other ertex Ú µ. This procedre can be contined foreer, which is impossible. ¾ Lemma 4 Consider two empty cells, and ¼, belonging to a hole À ¼µ, each sharing at least one segment (side) with its contor. Place a new modle Ñ in cell. Then there exists a seqence of moes throgh which Ñ relocates to ¼ so that at each step Ñ remains adjacent to the contor of À. Proof. The assertion follows by analyzing all possible local configrations and by checking that the motion rles permit eery single step of Ñ making a fll cycle along the contor of À. See Fig 5. ¾ Figre 5: Moes along the contor of a hole. Stationary modles are shown shaded. 3.3 Algorithm description Next, we describe one iteration of the algorithm, clminating in the extension of the chain by one modle in the -direction. For any hole À, let À µ denote the set of all (empty) cells in À that contribte at least one segment to the contor of À. STEP 1 Assme that there exists at least one ertex in Î Ò, whose degree in is one, else go to STEP 2. STEP 1.A: If there exists a ertex of degree one in ÓÙØ Ò (i.e., on the oter bondary), select one sch modle, say Ñ. Clearly, the remoal of Ñ does not disconnect, and once Ñ is remoed, ÐРѵ ¾ À ¼ µ, where À ¼ is the new nbonded hole. By Lemma 4, if Ñ is placed in ÐРѵ, there exists a seqence of moes throgh which Ñ relocates to the empty cell Þµ, so that at each step Ñ ¾ À ¼ µ. Ths, Ñ extends by one cell. Then start a new iteration. STEP 1.B: If no ertex of degree one in Î Ò belongs to the oter bondary ÓÙØ, select a ertex Ñ of degree one that belongs to the inner bondary Ò. The remoal of Ñ does not disconnect, and once Ñ is remoed, ÐРѵ ¾ À µ for some ½. By Lemma 4, if Ñ is placed back in ÐРѵ, there is a seqence of moes taking Ñ into a position where its degree in is at least two. (The existence of sch a position follows from the fact that À is an inner hole.) If now there exists a ertex of degree one in ÓÙØ Ò, go to STEP 1.A. If there exists a ertex of degree one in Ò, repeat STEP 1.B. 6

7 STEP 2 Assme that in the degree of eery ertex belonging to Î Ò is at least two. Then we hae so that Å. By Lemma 2, there exists a maximal cycle haing exactly one connector. Consider all ertical edges of with the smallest and with the largest Ü-coordinates. Assme withot loss of generality that the connector of does not belong to the highest ertical edge haing the largest Ü-coordinate. Sppose frther that lies aboe. (See Fig. 6, where Ù Ú is the clockwise order of ertices of in the neighborhood of. The case when the connector does not belong to the highest ertical edge of haing the smallest Ü-coordinate, can be treated similarly.) By Lemma 3, µ, µ, and Æ µ mst be empty, and the remoal of does not disconnect the graph. Set Ñ. There are two cases, depending on whether Ñ belongs to the oter bondary a b a b or not. Figre 6: Illstration of possible moes in STEP 2 of the algorithm. STEP 2.A: If Ñ ¾ ÓÙØ Ò, proceed as in STEP 1.A: relocate Ñ to the empty cell Þµ so that at each step Ñ ¾ À ¼ µ. Ths Ñ extends by one cell. Then start a new iteration. STEP 2.B: If Ñ ¾ ÓÙØ then we hae Ñ ¾ Ò since Ñ is adjacent to at least two empty cells, µ and Æ µ. Consider the holes À À ½µ inclding the -side and the Æ-side of, respectiely. (It may happen that.) It follows from the maximality of that À and À cannot be perfect. Once Ñ is remoed, becomes part of a larger hole À À À Let ½ ½ µ ¾ ¾ µ µ be the circlar seqence of critical pairs arond À, listed in clockwise order, and following. That is, the cells ¾ Î are corner-adjacent to each other and sideadjacent to an empty cell ¾ Àµ. See Fig 4. Obiosly, Ò Ñ remains connected. Claim. There is exists an index for which, and so that in Ò Ñ, there are two simple paths, µ and µ, connecting, the niqe connector of, to and, respectiely, satisfying the following condition: µ,, and µ, together with a piece of Ò Ñ indce a cycle in with Ê µ Ê µ and É µ É µ. Let denote the index whose existence is garanteed by the Claim. In iew of Lemma 4, if Ñ is placed back in ÐРѵ, then by a sitable seqence of moes it can be taken to cell, where it is side-adjacent to both and. Therefore, in the final position, the area of is larger than that of and the contents of is larger than that of (since ). Then go to STEP 1. Proof of Claim. Assme withot loss of generality that the cells ½ ½ ¾ ¾ follow arond the bondary of À in clockwise order, and set Ì ½ ½ ¾ ¾. Since the remoal of Ñ does not disconnect, for any Ü ¾ Ì, there exists a simple path ܵ in Ò Ñ connecting to Ü. We can represent this path in the plane by a polygonal line throgh the centers of the corresponding cells. Let ܵ denote a simple oriented Jordan arc rnning in the interior of the hole À from the ËÏ -corner of to the bondary of Ü. Connecting the endpoints of ܵ and ܵ by a simple arc in the nion of the cells in Ò and by a segment in Ü, we obtain a closed Jordan region  ܵ. Eery cell Ü ¾ Ì satisfies at least one of the following three possibilities: 7

8 1. Ü belongs to, 2.  ܵ lies on the right-hand side of ܵ, 3.  ܵ lies on the left-hand side of ܵ. It follows from the maximality of that, for any ½ the cells and ½ mst be of the same type, where the indices are taken mod. For similar reasons, as we go arond the bondary of the hole À in clockwise direction, the first (resp. the last) cell that participates in a critical pair mst be of type 1 or type 3 (resp. type 1 or type 2). By the definition of, it cannot happen that both elements of eery critical pair are of type 1. Ths, there exists a critical pair µ whose elements hae different types, and they will meet the reqirements of the Claim. See Fig. 7 for an illstration. ¾ c a b s z Figre 7: Illstration to the proof of Claim. A configration with nine maximal cycles, with areas of, ½,,,,,, and. Connectors are shaded in the figre. is a rightmost edge of the maximal cycle with area, as specified in STEP 2 of the algorithm. Its niqe connector is. The critical pairs of the hole À created by the remoal of are: ½ ¾µ, µ, µ and µ. Cells ½ throgh hae type 3, while cell has types 1 and 2. Critical pair µ meets the reqirements of the Claim. 3.4 Algorithm analysis Each of the steps 1.A, 1.B, 2.A, or 2.B consists of at most Ò moes by a single modle. We proe that after performing at most Ò steps of type 1.B and at most Ò steps of type 2.B, we mst make a step of type 1.A or 2.A, which completes one iteration of the algorithm. First, note that the seqence of relocations in STEP 1.B (if not empty) strictly increases the total nmber of edges in the graph. As remains connected throghot the algorithm, its nmber of edges is always at least Ò ½. On the other hand, is a sbgraph of the infinite grid with Ò ¾ ertices, so its nmber of edges neer exceeds ¾Ò. Taking into accont that the nmber of edges does not decrease dring STEP 2.B, we can conclde that each iteration ses fewer than Ò steps of type 1.B. 8

9 Recall that the contents of a cycle, denoted É µ, is the set of occpied cells belonging to or enclosed by. Consider the following weight fnction characterizing the configration: µ Õ Ý, where Õ ¾Å É µ is the total contents of maximal cycles, and Ý is the nmber of maximal cycles. Obsere that ¼ µ Ò. Note that STEP 1.B preseres all the existent cycles, and obsere that the aboe weight fnction strictly increases dring each step of type 2.B, and it does not decrease dring any step of type 1.B. Consider first a step of type 1.B: Ý can only increase by ½, and if Ý increases by ½, then Õ also increases by at least one (by the maximality condition). Consider now a step of type 2.B: Since ¾. Denote by Õ ¼ and Ý ¼ the ales of Õ and Ý respectiely at, assme for simplicity that the end of the step. If ¾ É, we hae Õ ¼ Õ and Ý ¼ Ý; whereas if ¾ É, we hae Õ ¼ Õ and Ý ¼ Ý. In either case, µ stricly increases. Therefore, each iteration ses at most Ò steps of type 2.B. In conclsion, after fewer than Ò Ò ¾Ò steps of type 1.B or 2.B, we mst perform at least one step of type 1.A or 2.A, and complete an iteration of the algorithm. After each iteration, the horizontal chain is extended by one cell in the -direction, so after Ò ½ iterations, the reconfigration to a straight chain is complete. Each step consists of at most Ò moes, so the entire algorithm takes fewer than ¾Ò moes. 4 Another rectanglar model The following two generic motion rles (Fig. 8) define the weak rectanglar model. These are to be nderstood as possible in all axis-parallel orientations, in fact generating eight rles, for diagonal moes and for side moes (axis-parallel ones). The only imposed condition is that the configration mst remain connected at each discrete time step. Diagonal moe: A modle Ñ moes diagonally to an empty cell corner-adjacent to ÐРѵ. Side moe: A modle Ñ moes to an empty cell side-adjacent to ÐРѵ. m m (a) (b) Figre 8: (a) Æ diagonal moe and (b) side moe in the direction. The cells in which the moes take place are otlined in the figre. The same reslt as in Theorem 1 holds for this second model, bt its proof is mch easier. Theorem 2 The set of motion rles of the weak rectanglar model garantees the feasibility of motion planning for any pair of connected configrations haing the same nmber of modles. Proof. Assme withot loss of generality that the lowest cell of the configration has Ý ¼. Consider the following weight fnction characterizing the configration: µ Ý µ ¾Î where Ý µ is the Ý-coordinate of (the center of) cell. Notice that is inariant with respect to horizontal translation. If µ ¼, then mst be a straight horizontal chain. We show that there exists a seqence 9

10 of moes dring which µ monotone decreases to ¼, with the additional condition that at each time step the Ý-coordinate of eery ertex is nonnegatie. Consider the top row of Î ; if its Ý-coordinate is eqal to ¼, there is nothing to proe (the reconfigration is complete). Else consider its rightmost cell Ù. Notice that Æ Ùµ, ÆÏ Ùµ and Æ Ùµ, as well as Ùµ, are all empty. Since is connected, at least one of the cells Ï Ùµ or Ë Ùµ is nonempty. We hae seen cases, fie for the first alternatie and two for the second. See Fig. 9. w w Figre 9: Case analysis in the proof of Theorem 2. In cases 1 throgh 5 (pper row), the cell Ï Ùµ is nonempty; in cases 6 and 7 (bottom row), the cell Ï Ùµ is empty. The horizontal line indicates in each case the top row of the configration. For the first fie cases, Ú Ï Ùµ is nonempty. Case 1: Ë Úµ is nonempty and Ë Ùµ is empty. Then Ù makes a side moe in the Ë direction. Case 2: Ë Úµ and Ë Ùµ are nonempty, and Ë Ùµ is empty. Then Ù makes a side moe in the Ë direction. Case 3: Ë Úµ, Ë Ùµ and Ë Ùµ are nonempty. Then Ù makes a side moe in the direction. Case 4: Ë Úµ is empty and Ë Ùµ is nonempty. Then Ù makes a diagonal moe in the ËÏ direction. Case 5: both Ë Ùµ and Ë Úµ are empty. Then Ù makes a diagonal moe in the ËÏ direction. For the last two cases, Ï Ùµ is empty and Û Ë Ùµ is nonempty. Case 6: Ë Ùµ is empty. Then Ù makes a diagonal moe in the Ë direction. Case 7: Ë Ùµ is nonempty. Then Ù makes a side moe in the direction. It is easy to check that the configration remains connected after each moe. The reader shold notice that the key is case 4 (a moe which is prohibited in the preios model). Note that each of the cases 1, 2, 4, 5, and 6 redces the weight fnction by one nit. Cases 3 or 7 can occr in a seqence at most Ò times after which one of the other fie cases (1, 2, 4, 5, or 6) mst occr. Therefore, µ ¼ after at most Ò ¾ steps (the initial ale of µ is at most È Ò ½ ½ Ò¾ ¾). A more carefl calclation shows that in fact not more than Ç Ò ¾ µ moes are made. Among all modles with minimm Ý-coordinate, the one whose Ü-coordinate is the smallest is called the reference modle of the configration. Consider the reference modle, say Ö, of the initial configration. Obsere that the reconfigration procedre otlined aboe does not moe any of the modles in the lowest row, so in particlar Ö remains fixed. In the only cases when Ý Ùµ is not redced (remains constant), 3 and 7, Ü Ùµ is increased by one. Note also that Ü Ùµ is decreased by one only if Ý Ùµ is decreased by one (cases 4 and 5). By the connectedness of each intermediate configration, Ü Öµ Ò Ü Ñµ Ü Öµ Ò, for each modle Ñ. Conseqently, the nmber of moes of the modle from cell of the initial configration that leae the ale of Ý nchanged, is not more than ¾Ò Ý µ. Since the nmber of moes of the same modle, dring which Ý gets redced is Ý µ, the total nmber of moes in the reconfigration process does not exceed ¾Î ¾Ò Ý µ Ý µµ ¾Ò ¾ ¾ Ý µ Ò ¾ 10 ¾Î

11 This bond is tight p to a mltiplicatie constant, see Section 5. It is ery easy to improe the last bond to ¾Ò ¾. ¾ As for the preios model, it is not hard to constrct examples that cannot be reconfigred to straight chains sing only diagonal moes or only side moes. There are, in fact, small examples that simltaneosly work for both models. 5 Conclding remarks 1. In or original model, the reconfigration of a ertical chain into a horizontal one reqires only Ò ¾ µ moes, and we beliee that no other pair of configrations reqires more. We hae shown that this holds in the weak rectanglar model. 2. A somewhat different model can be obtained if, instead of the connectedness reqirement at each time step, one imposes the following so-called single backbone condition [6]. Denote by Ø Î Ø Ò Ñ the set of modles that do not moe in step Ø (i.e., fixed). We refer to Ø as the backbone. The initial configration Î ¼ mst be connected: i.e., the graph ¼ Î ¼ ¼ µ mst be connected. At any time Ø, the backbone Ø mst be connected with respect to side-adjacency, i.e., the graph Ø Ø ¼ Ø µ mst be connected, where ¼ Ø Ø is the set of edges connecting pairs of cells in Ø that are side-adjacent. (The backbone cold howeer hae holes.) Together with the connectedness of the modles at time ¼, it ensres that the modles remain connected at any time step Ø. (If concrrent moes are allowed, additional conditions hae to be imposed, as in [6].) A sbtle difference exists between reqiring the configration to be connected at each discrete time step and reqiring the existence of a connected backbone along which a modle slides or rotates [6]. A one step motion that does not satisfy the single backbone condition appears in Fig. 10: the initial connected configration practically disconnects dring the moe and reconnects at the end of it. Figre 10: A rotation moe which temporarily disconnects the configration. Notice that at each step of or algorithm, the element Ñ selected to moe has the property that its remoal does not disconnect. This property is also maintained throghot the seqence of moes ntil another element is selected to moe. Therefore, the single backbone condition remains satisfied dring the whole procedre. 3. A related qestion is the following. A configration consisting of nit cbes of integer coordinates in -space is called an animal if the bondary of their nion is homeomorphic to a ½µ-sphere. It is easy to see that in the plane, any animal can be transformed into any other by adding or remoing one sqare at a time so that all intermediate configrations are animals. The corresponding statement in higher dimensions is not known to be tre. There exist, howeer, relatiely small animals in 3-space with the property that no cbe can be remoed from them withot iolating the condition. This is in sharp contrast to the sitation in the plane. 11

12 References [1] Z. Btler, K. Kotay, D. Rs and K. Tomita, Generic decentralized control for a class of selfreconfigrable robots, Proceedings of the 2002 IEEE International Conference on Robotics and Atomation (ICRA 02), Washington, May 2002, [2] G. Chirikjian, A. Pamecha and I. Ebert-Uphoff, Ealating efficiency of self-reconfigration in a class of modlar robots, Jornal of Robotic Systems, 13(5) (1996), [3] E. Demaine, M. Demaine and H. Verrill, Coin-moing pzzles, in More Games of No Chance, edited by R. J. Nowakowski, pp , Cambridge Uniersity Press, [4] R. Diestel, Graph theory (2nd ed.), Gradate Texts in Mathematics 173, Springer-Verlag, New York, [5] A. Dmitresc, I. Szki and M. Yamashita, Formations for fast locomotion of metamorphic robotic systems, International Jornal of Robotics Research, to appear. A preliminary ersion in Proceedings of the 2002 IEEE International Conference on Robotics and Atomation (ICRA 02), Washington, May 2002, [6] A. Dmitresc, I. Szki and M. Yamashita, Motion planning for metamorphic systems: feasibility, decidability and distribted reconfigration, IEEE Transactions on Robotics and Atomation, to appear. [7] S. Mrata, H. Krokawa and S. Kokaji, Self-assembling machine, Proceedings of IEEE International Conference on Robotics and Atomation, (1994), [8] A. Ngyen, L. J. Gibas and M. Yim, Controlled modle density helps reconfigration planning, Proceedings of IEEE International Workshop on Algorithmic Fondations of Robotics, [9] A. Pamecha, I. Ebert-Uphoff and G. Chirikjian, Usefl metrics for modlar robot motion planning, IEEE Transactions on Robotics and Atomation, 13(4) (1997), [10] D. Rs and M. Vona, Crystalline robots: self-reconfigration with compressible nit modles, Atonomos Robots, 10 (2001), [11] M. Yim, Y. Zhang, J. Lamping and E. Mao, Distribted control for 3D metamorphosis, Atonomos Robots, 10 (2001), [12] E. Yoshida, S. Mrata, A. Kamimra, K. Tomita, H. Krokawa and S. Kokaji, A motion planning method for a self-reconfigrable modlar robot, in Proceedings of the 2001 IEEE/RSJ International Conference on Intelligent Robots and Systems,