Grain Sorting in the One-dimensional Sand Pile Model. J r me Durand-Lose

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1 aoratoire Bordelais de Recherche en Informatique, ura cnrs 304, Universit Bordeaux I, 35, cours de la i ration, Talence Cedex, France. Rapport de Recherche Num ro 5797 Grain Sorting in the Onedimensional Sand Pile Model J r me Durandose lari, ura cnrs 304, Universit Bordeaux I, 35, cours de la i ration, F Talence Cedex, France. Astract We study the evolution of a onedimensional pile, empty at rst, which receives a grain in its rst stack at each iteration. The nal position of grains is singular: grains are sorted according to their parity. They are sorted on trapezoidal areas alternating on oth sides of a diagonal line of slope p 2. This is explained and proved y means of a local study. Each generated pile, encoded in height dierences, is the concatenation of four patterns: 22, 33, 0202, and. The relative length of the rst two patterns and the last two patterns converges to p 2. We make asymptotic expansions and prove that all the lengths of the pile are increasing proportionally to the square root of the numer of iterations. Introduction We consider an innite sequence of stacks. Each stack can hold any nite numer of grains, this numer is called the height of the stack. The Sand pile model (spm) and Chip ring games (cfg) are ased on local interactions. Both models conserve the total numer of grains. In spm, a grain goes to a neighor stack if the height dierence etween stacks is more than a given threshold; whereas in cfg a stack gives a chip to each of its neighors if its numer of chips is aove a certain value [3, 4]. In the onedimensional case, oth models are onedimensional cellular automata and they are equivalent via some simple encoding. Both spm and cfg, like Petri nets [], are used to model in parallel computing the ows of information in systems. spm is used to model gradiandriven dynamic load alancing. Grains model data or tasks, and stacks, a processor network [2]. The aim is to nd a simple, fast, and relatively inexpensive local rearrangement which ensures that all processors have almost the same amount of work. spm is important for granular ows in physics. It admits invariants, entropy like functions, and veries the socalled Selforganized criticality and is related to the =f phenomenon [2, 9, 0, 3]. The sequential onedimensional spm and the related cfg are studied in [6, 7, 8]. They have proved the uniqueness of the nal pile whatever the order of the iterations as well as descried the dynamics in various sequential cases. The prolem studied in this paper is the parallel evolution of a onedimensional spm, empty at rst, which receives a grain onto its rst stack at each iteration. It can also e seen as sand dripping in a thin ut long hourglass. In section 2, we dene the spm and cfg models and recall that they are equivalent in dimension one. We also dene the dripping process studied. jdurand@lari.uordeaux.fr,

2 In section 3, we prove that the generated piles, in height dierences, are made of four patterns: 22, 33, 0202 and. The frontiers etween patterns act like signals. The silhouette of each pile is made of two parts of dierent slopes: 2 then. In section 4, grains are marked depending on their parity. Even and odd grains are arranged in a very special way: they are located in trapezoidal areas alternating on oth sides of a diagonal line of slope p 2. We explain this y looking locally at the interactions etween moving grains and signals. In section 5, we give asymptotic approximations of the dierent parameters. We do this y making a continuous approximation of the pile and use a dierential resolution as in []. We prove that the length of the part of slope is p 2 times the length of the part of slope 2 and that the lengths of all the piles are increasing proportionally to the square root of the numer of iterations. 2 Denitions The onedimensional sand pile is an innite sequence of stacks. Each stack can hold any nite numer of grains. We use the notation from [8], the dierence is that our model is parallel. A pile is encoded y the sequence of the numer of grains, or height, of the stacks. It is then denoted with square rackets: = [[ 0 : : : k ]]. We call slope the dierence of height, i i, etween two consecutive stacks. If more stacks are considered, the slope is the average slope. If a stack is higher y at least two grains than the next stack, then one grain tumles down. This is depicted y the movement of the grains a to f in Figure. The starting pile is empty. At each iteration, a grain falls onto stack numer. Grains c to f in Figure are the newly arrived grains. The numer of grains is nite. Except for the grain added to the pile at each iteration, the numer of grains is constant. The numer of grains is then equal to the iteration numer. c a [[ ]] hh ii d c a [[ ]] hh ii e d c a [[ ]] hh 3 0 ii f e c d a [[ ]] hh 3 ii Figure : Iterations 4 to 7. Since grains are only moved to smaller stacks, a direct induction proves that only decreasing sequences are generated from the initial pile. A pile is now an element of NN decreasing to zero. This ensures that the height dierence etween any two consecutive stacks is always positive. Denition et ( n ) e the following threshold function: 8n 2 Z, ( n ) = if 0 n, otherwise 0. et e a pile. The dynamics of spm with dripping is driven y the following transition function F : F () 0 = 0 ( 0 2 ) + ; 0 < i; F () i = i ( i i+ 2 ) + ( i i 2 ) : The negative terms correspond to the possiility of giving a grain to the next stack, while the positive term corresponds to the possiility of getting a grain from the previous stack. All the stacks are updated at the same time, making this is a parallel process. Denition 2 A pile can e encoded y the list of the height dierences etween stacks: for any pile ' () = hh ( 0 ) ( 2 ) ( 2 3 ) : : : ii. With this encoding, the transition function ecomes: (x) 0 = x 0 2 ( x 0 2 ) + ( x i+ 2 ) + ; 8i; 0 < i; (x) i = x i + ( x i 2 ) 2 ( x i 2 ) + ( x i+ 2 ) : 2

3 We call the dierence of height of one etween a stack and the next one chip. Denition 2 is equivalent to a stack having more than two chips and ring one to oth neighors. This is the chip ring game (cfg). In a onedimensional lattice, spm and cfg are equivalent with this encoding Iterations Height Piles 0 Figure 2: Iterations 0 to 50. Figure 2 illustrates the rst 50 steps of this dynamic. The lengths and heights, as well as the slopes, exhiit some regularity. After some iterations, there are two straps of triangles drawn on the surface as depicted in Figure 3 for iteration 00 to Triangles and signals Piles are encoded in height dierences in Figure 4 (steps to 20). Triangles appear with patterns 22, 33, 0202, and. Those patterns are stale. It should e noted that for the second and third patterns, digits are alternating, like in a chessoard and the frontier etween them is either 2 or 30. et " e the empty word. The Kleene operator is denoted *; that is, (3) is " or 3 or 33 or 333 : : :. We use the theory of languages in the next proposition in order to get a synthetic expression. Proposition The pile, encoded in height dierences, is a word of the following language: 2 ( " j 3 ) ( 3 ) ( " j 2 ) ( 0 2 ) ( " j 0 ) : Proof. We prove proposition y induction. It is true for the rst 20 iterations, as can e seen in Figure 4. Interaction, as expressed in denitions and 2, only depends on the two closest neighors. It is enough to look locally at the interactions of the frontiers in Figure 4. Suppose that the nth pile is the concatenation of four parts with the patterns 22, 33, 0202, and respectively. We call frontier the limit etween two patterns and order the limits of the pile. We denote (left), M (middle), and R (right) the positions of the frontiers etween respectively; rst and second, second and third, third and fourth patterns. They are represented in Figure 4 where and R ehave like signals moving on oth sides of M. Geometric denitions are given in Figure 5. First we investigate each signal alone, from left to right: is going left (right) if it is equal to 2j (2j3) (lines 07 to 7); M is not moving (lines 96 to 02) and R is going left (right) if it is equal to 0j (2j) (lines 94 to 04). While the proposition is true, only the following encounters are possile, from left to right: on 3

4 22 50 Height Iterations Piles 00 Figure 3: Iterations 00 to 50. the left order, ounces (lines 59 to 65); when meets M, it ounces and M is moved one step to the right (lines 8 to 87); when R meets M, it ounces and M is moved one step to the left (lines 50 to 57). The order of the signals is kept and the only possile encounter with more than two frontiers is MR. The meeting can e exactly synchronous (lines 40 to 44) or not (lines 62 to 67 and 03 to 09). In all cases the order is kept and no other case arises. The dynamics are very simple except when signal or R reaches one of its limits; the rest are only linear displacements. When reaches the left order, it only ounces ack. When R reaches the right order, it ounces ack and the total length is increased y one. When R comes ack to the center, the total length has een increased y one. In height dierences, piles are the concatenation of four parts of patterns 22, 3, 02, and respectively. The two rst parts have a slope of 2 while the two last parts have a slope of. This explains the shape of piles (in heights) as depicted in Figure 5. 4 aeling grains Grains are laeled according to the iteration during which they enter the pile. In Figure 6, at iteration 5000, all odd grains are spotted in lack. Their localization is singular. The odd grains, like the even ones, are located on trapezoidal areas delimited y the axis, lines of slope and 2, and a diagonal line. These areas are alternating like in a chessoard. The diagonal separation seems to e a straight line. There is also some kind of relation etween the intersections of the line of slope and 2 with the axis and the edges in the middle as depicted in Figure 6. We do not have any explanation nor proof for this phenomenon yet. Nevertheless, if the diagonal separation p is a line, ecause of such coincidences, its slope would e: a= = ( + 2a)=(a + ) which leads to =a = 2. We prove in section 5 that indeed it is a straight line with slope p 2. 4

5 # M R M R M R Figure 4: Representation with height dierences :G +" 2 6 D +" 2 G D ", " 2 2 {, 0, } M R Figure 5: Geometric denitions of G, D,, and M. Theorem Odd and even grains are always sorted in trapezoid areas delimited y a diagonal, lines of slope and 2, and the axes. With gures 7 through, we prove that the grains are always on either side of the frontier, depending on their parity. In all these gures, grains are either lack or white depending on their parity. Grains for which parity is unknown are drawn with a little circle. The grains which do not move any more are represented y their silhouette. et us rst consider that signal is away from the left order. Even and odd grains come alternatly and go down the pile one after the other as depicted in Figure 7. Grains ehave like a wave of marles on stairs. From this, a direct induction proves that the pattern 22 corresponds to an evenodd wave of grains. et 5

6 2a a! Figure 6: Position of the odd grains (in lack).! ! ! Figure 7: Arrival of new grains us consider that signal is going right. As depicted in Figure 8, the wave is just going down with scarce grains running in front of it. Going right, the signal encounters the middle order M as depicted in Figure 9. The rst grain crosses the order and ecause of the height dierence, the second gets locked. The third passes over the second and restrains the fourth from passing, and so on. The phenomena of one grain getting locked and the next passing over it, one layer up, is the way the signal goes right as depicted in Figure 0. When reaches the left order, it ends uilding a layer and goes ack to the middle on the new layer as depicted in Figure 0. In comparison to Figure 7, we know now that the grains that are running in front of the wave are all of the same parity. The rst grain of the wave is of the same parity as the rst grain of the previous wave, which is also the parity of the grains the running scarce grains, so that the phenomena starts again and loops. We now consider signal R. When R is away from the middle M, it has no action whatsoever since the selection of grains is made efore. Signal R only orders the grains on layers in the right part. When or R meets M it only moves it and that does not change the dynamic of Figure 9. But, when all three signals 6

7 ! 2 2 j3 3! j j3 M 2j3 j2 0 Figure 8: Signal goes right.!!!! M 2 2j3j j j 2! M 2 2j 3j0 M 2j 3 j2! 2 2 2j 3 Figure 9: Signal reaches the middle order alone. 2 2j 3 3! 2j 3 3!! j 3 3 3! j j3 3 3 Figure 0: Signal goes left and reaches the left order., M, and R meet, things are dierent as depicted in Figure. This time, the fate of odd and even grains are switched. The changes in the destination of odd and even grains in Figure 6 are directly linked to the synchronous encounter of and R detailed in section 3.!!!! M R 2j3 j2 0j MR 2 2j3j0j 2 2 2j MR 2 2jj2j M R 2j 3j0 2j Figure : Signals and R exactly synchronized. In Figure 6, the separation lines represent the silhouettes of piles at some iterations and the diagonal separation is the trace of the middle order M. Since there are as even grains as odd grains, the two symmetric areas in Figure 6 have the same surface, 7

8 that is, they correspond to the same numer of grains. 5 Asymptotic ehavior All the results in this section can also e found in [5]. The proof of [5] is too long to t here, we give a shorter one that we feel is more like an a posteriori verication. Theorem 2 The diagonal separation is a line of slope p 2. The value of G increases (decreases) y one for each round trip of (R). The value of D decreases (increases) y one (two) for each loop of (R). The round trip delay for a signal is twice the length of the part it evolves in, up to a constant. Since every quantity can go to innity, when G and D are very ig, the equations can e extended to continuity as in []: 8 >< >: dg = dt 2:G dt 2:D ; dd = dt 2:G + 2 dt 2:D : These equations can e solved with the hypothesis D = section 4. It leads to: G:dG = With this hypothesis, the possile solutions are: p p 2 G which comes from the oservations of dt : G = D = s p 2 p 2 p 2 G = t + c ; r p 2 t + 2c : Where t is the time (or numer of fallen grains or numer of iterations) and c is a constant. From Figure 5, the numer of fallen grains n is also the total area, that is, of the two triangles and of the rectangle. We get the following approximations: n D2 2 + G:D + G2 (2 + r n G p ; 2 r+ 2 D p n 2 G p 2 + p 2) G 2 ; : () It should e noted that oth triangles of Figure 5 have almost the same area, G 2. This is coherent with the surface oservations of section 4. The rectangle is equally parted y the diagonal, and even and odd grains are equally parted on oth sides of the diagonal. 8

9 6 Conclusion A more random distriution of odd and even grains might have een expected, on the contrary grains are sorted. This is important, ecause if even and odd grains, or tasks, are very dierent, in a onedimensional processor array sequentially fed using a spm load alancing technique, disparities arise. When taken modulo 3, 5, or more, there is no such segregated location as efore, grains are more uniformly spread. The way grains spread as a wave and are xed in the silhouette is very interesting. It gives a physical meaning to the signals. When goes right it spreads grains. When it goes left, it makes a one over two selection. Signal is going right and left while the grains are always running to the right. The signal R is acting similarly. When it goes right it is spreading the grains on a new layer, opening it. When it goes left it xes them. When grains and signals are going in opposite directions, since they have speed one, signals only meet every other grain. These signals, from a physical point of view, are very interesting ecause they correspond to waves on a pile of sand that can e seen when you dig at the ottom. We have proved that the pile is expanding in the square root of the numer of fallen grains (or iterations). This is asolutely normal when one considers that the grains (linear) are lling a surface (quadratic). The relative length of the two parts is p 2. To compare with the work in [], on the one hand, they found a quadratic relaxation time for the cfg starting with the pile ::: 0 0 n 0 0 ::: and nal pile ::: 0 0 ::: 0 :::. But when considered as stacks of grains, they correspond to ::: n n n 0 0 ::: and ::: n n (n) (n2) ::: 2 0 ::: respectively. This is a very dierent case ecause of the inuence of the left order which is high and feeds grains to the right part. On the other hand, they also oserved geometric patterns and signal propagations. Acknowledgements This work was partially supported y ECOS and the French Cooperation in Chile. This research was done while the author was in the Departamento de Ingenier a Matem tica, Facultad de Ciencias F sicas y Matem ticas, Universidad de Chile, Santiago, Chile. References [] R. Anderson,. ov sz, P. Shor, J. Spencer, E. Tardos, and S. Winograd. Disks, alls and walls: Analysis of a cominatorial game. American Mathematical Monthly, 96:48493, 989. [2] P. Bak, T. Tang, and K. Wiesenfeld. Selforganized criticality: An explanation of =f noise. Physical Review etters, 59(4):38384, 987. [3] J. Bitar and E. Goles. Parallel chip ring games on graphs. Theoretical Computer Science, 92:29300, 992. [4] A. Bj rner,. ov sz, and P. W. Shor. Chipring games on graphs. European Journal of Cominatorics, 2:28329, 99. [5] J. O. Durandose. Automates Cellulaires, Automates Partitions et Tas de Sale. PhD thesis, lari, 996. In French. [6] E. Goles and M. Kiwi. Sandpile dynamics in onedimensional ounded lattice. In Boccara, Goles, Martinez, and Picco, editors, Cellular Automata and Cooperative Systems, pages Kluwer, 99. [7] E. Goles and M. Kiwi. Dynamics of sandpiles games on graphs. In latin'92, numer 583 in ecture Notes in Computer Science, pages SpringerVerlag, 992. [8] E. Goles and M. Kiwi. Games on line graphs and sand piles. Theoretical Computer Science, 5:32349,

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