Gaps in Wireless Ad-hoc Lattice Computers

Philippe Moore and Anil M. ShendeRoanoke College

Salem, Virginia 24153Email: {pmoore,shende}@roanoke.edu

I. INTRODUCTION

Scientific computing largely deals with the prediction ofattribute values of objects participating in physical phenomena.Usually the unfolding of these phenomena involves motionof the participating objects. There are no known analyticalsolutions for some physical phenomena, and one method ofprediction is the analogical simulation of the unfolding ofthe phenomenon in a cellular automaton-like architecture - alattice computer- representing the region of euclidean spacein which the phenomenon unfolds [1]. Lattice computers aremassively parallel machines where the processing elements arearranged in the form of a regular grid (lattice), and where thecomputational demand on each individual processing elementis quite low [2], [3], [4]. Each processing element represents apoint/region of euclidean space. In analogical simulations ona lattice computer, the motion of an object across euclideanspace is carried out as a sequence of uniform time steps. Ineach step the representation of the object may move fromone processing element to a neighbour, as defined by theunderlying lattice of the lattice computer [5].

In the near future, the proliferation of portable, wirelesscomputing devices (e.g., cell phones, PDAs) promises theavailability of a large number of computing devices in arelatively small geographic region. It has been proposed ([6],[7]) that such an ensemble of wireless devices could be usedto create a wireless ad-hoc lattice computer (WAdL). The goalof a WAdL is to harness the collective computing capabilitiesof the devices for the common cause of scientific computingvia analogical simulations. Clearly, the accuracy of the resultsof these simulations is directly dependent on the resolution ofthe underlying lattice of a WAdL.

In this paper, we focus on the problem of bridging "gaps"in the underlying lattice of a WAdL. Such gaps may arise dueto a variety of reasons: lack of mobile devices in a piece ofthe geographic region under consideration, failures in devices,devices moving out of the geographic region, etc.

II. WADL: A BRIEF OVERVIEW

The goal of the WAdL architecture is to use the participatingmobile devices in a geographic area to discretize a boundedregion of euclidean space, B, and then carry out analogicalsimulations of physical phenomena in this discretized repre-sentation of B using the methodology proposed in [3]. Weassume that each participating device has some, minimal com-putational and storage facilities, is equipped with some form

0-7803-9206-X/05/$20.00 ©2005 IEEE

of location service, e.g., GPS, and has some communicationcapabilities, e.g., Bluetooth.A WAdL consists of a single immobile device or a base-

station (denoted by I) designated as the manager, and acollection of mobile devices in B. I fixes a lattice, L, with afixed origin, in its region of influence, and then, each latticepoint, p, is assigned all mobile devices within the Voronoicell' around p. Note that several devices may be assigned tothe same lattice point (see [6] for details).Once a WAdL is formed, I initiates the simulation of a

physical phenomenon in the lattice computer formed from themobile devices in B. Since several devices may be assigned toone lattice point, p, they elect a leader primarily responsiblefor p; all the other devices assigned to p also carry out thecomputations performed by the leader, so that if and whenthe leader moves out of the Voronoi cell of p, another deviceassumes leadership and the overall simulation can continueseamlessly. For our purposes, we assume there is only onedevice d assigned to each lattice point p, and in the rest ofthe paper we will not make any distinction between d and p.[7] discusses the proposed mechanism for creating a WAdLand shows the viability of the use of a WAdL in scientificcomputing through a simulation of a WAdL computing the liftand drag on an airplane wing in flight. [6], [7] gives algorithmsfor the construction of the underlying lattice for a WAdL giventhe locations of the participating mobile devices.

III. BRIDGING GAPS IN THE UNDERLYING LATTICE

21 .

* EUUU

C4: ' C3

Fig. I. Piece of a WAdL with a gap

We consider the following problem: Suppose the underlyinglattice L of a WAdL has a k x k square piece that is missing.For example, Figure 1 shows a piece of a WAdL with a gapof 3 x 3 points. In the figure, the missing devices are shown asempty squares, whereas the devices present are shown as filled

The Voronoi cell around a lattice point p, by definition, is the set of pointst in euclidean space such that t is closer to p than to any other lattice point.

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squares. The problem is to find a mechanism to ensure that thepassage of information from lattice point to lattice point, asrequired by a WAdL application, continues seamlessly and intime proportional to the euclidean distance between the latticepoints. Below we propose three solutions to this problem, anddiscuss why the last solution is the most effective.We assume that the initiator I detects the gap, assumed to be

a k x k subset of the lattice, and broadcasts the information toall the devices in the WAdL. In one unit of time, each devicecan receive one message from each of its four neighbours,perform some bounded amount of processing, and send onemessage to each of its four neighbours.As our first solution, we propose that the WAdL as a whole

starts operating as if neighbours in the underlying lattice area distance of k + 1 apart. Thus, the four corner devices(corresponding to the devices labelled c1,c2,c3 and C4 inFigure 1) around the gap, form a smallest square in thiseffective lattice. The other devices in the WAdL are used forrouting messages. In this case, the effective resolution of thelattice is decreased by a factor of k + 1. Moreover, messagessent from, say c1, will reach c2 after k + 1 units of time, thusnecessitating a slow-down of the entire WAdL application bya factor of k + 1 as well.As our second solution, we assign to each missing device

inside the gap one of the border devices. For example, deviceb1 may be assigned to missing device Pl, and device b2 tomissing device P2. b, and b2 then perform all the functionson behalf of P1 and P2, respectively. When bl, on behalfof P1 needs to send a message to P2, it actually sends themessage to b2, the device responsible for the missing deviceP2. For all such routing between border devices, I is usedas the intermediary (e.g., b, sends the message to I whichthen forwards it to b2). One advantage of using I thus isthat each border device needs to know only about its ownassignment. We believe this solution is better than the firstbecause the resolution of the underlying lattice is not compro-mised. Nonetheless, each border device may be responsible forseveral devices, thus increasing the load on these devices, andeffectively slowing down the WAdL application. In addition,I becomes a bottleneck in this scheme, likely placing a severelimitation on the speed of the WAdL application.

In our third solution, we first make an assignment ofbordering devices to missing devices as the second solution.Then, messages from border devices to other border devicesare routed through devices along the bordering cycle. Timeis "dilated" by a factor of (k + 1) - k/4 to accomodate forthe dilation of the assignment and the load on the borderdevices (see Theorem 1 below). We believe this solution issuperior to the above two since it does not decrease theresolution of the lattice, nor is I a bottleneck. Furthermore,if the assignment scheme ensures a dilation of at most k + 1,the WAdL application will run slower than in the first solutiononly by a factor of k/4. However, we note that this solutionalso suffers from the border devices being overloaded as inthe second solution.

In this paper, we present a provably correct assignment

scheme (for all k > 3) to be used in the third solution. Weshow that our assignment scheme (1) has a dilation of k + 1,(2) is optimal with respect to the load on the border devices,and (3) can be computed in constant time by each borderdevice. Figure 2 shows our assignment scheme for k = 8.The border devices are labelled consecutively starting at 0,and each missing device (in the gap) is assigned a borderdevice. The dilation of the assignment is 9. In this case, since82 - 64 missing devices are to be assigned to 4 x 9 = 36border devices, the optimal load of an assignment is 2; ourassignment has a load of 2 as well.We note that our assignment problem shares some similarity

with the work on fault tolerance in parallel architectures - inboth cases the problem is to embed a mesh (square gap) intoa ring (bordering cycle). Nonetheless, our scenario has twodistinctive features that warrant our treatment of the problem:(1) for k > 4, our embedding is necessarily many-to-one, and(2) the elements of the ring are connected to the mesh border,and must maintain this connectivity with low dilation.

--00- 01 02 03 04 05 06 07- 08- 0935 1 0-2 03 040 6 071-I C134 34 02 --0 3 04 0:5 06 1 1 11- 1-1-33 33 33 35 0 07 08- 12 12-- 12--32 32 32 3 4 35 08 10 13 13 1-331- 31 31 2-26 177--1-6-14 14 14-3Y0 31 310 26 25 19 17 15- 15' 1529-29 29 2 4 23 2- 21 20 16 16-28 28 25 24 23 22 21 20- 19 1727 26 25 24 2-3 22 21 20 19 18

Fig. 2. Assignment for k = 8.

IV. THE ASSIGNMENT SCHEMES

In this section we present our schemes for assigning bor-dering devices to missing devices. For a k x k gap in theWAdL, without loss of generality, we will assume that theborder devices are sequentially numbered from 0 starting at thetop left corner (as shown in Figure 3). We will assume that theco-ordinates of the top left border device are (0, 0), where thefirst co-ordinate is the row number and the second co-ordinateis the column number. Thus a device with co-ordinates (i, j) isa border device if and only if i, j E {0, k + 1}. All devices atco-ordinates (i, j) for 0 < i, j < (k + 1) are missing devices.For the purposes of this paper, we are not concerned with anyof the other devices in the WAdL.We present two assignment schemes: one for odd k, and one

for even k. Our schemes have desirable properties as capturedby the following theorem.

Theorem 1: For each k > 3, there exists an assignmentscheme to assign border devices to missing devices in a k x kgap such that:

1) each missing device is assigned to exactly one borderdevice,

2) the maximum dilation of the assignment is k + 1, i.e.,no two neighbouring missing devices get assigned toborder devices that are at a (shortest) distance of morethan k + 1 along the border,

77

a

sc

ca

prth

A.

3) the load on the border devices is bounded above by k/4, the missing device at (i, j) in the gap. Then, for 1 < i, j < k,and j, if (i,j) E A;

4) for each (i,j) in the gap, the assignment can be com- 37-, if (i,j) E B;

puted in constant time. j, if (i, j) E C;Te first present our scheme for the odd case, and a sketch of 8H - if (i, j) E D;proof of the above theorem for this case to show that the 57, if (i, j) Eheme is desirable. We then present the scheme for the even 35H 1, if (i, j) E

F;ise. The proof that the scheme for the even case satisfies the A(i, j) = 41 + 2 j, if (i, j) E G;roperties enumerated in Theorem 1 is similar to the proof in 87 i, if (i, j) E H;e odd case. 771- L--i, if (i, j) E I;

57- - i, if (i, j) E J;

Scheme for Odd Values of k 57 -1 - j, if (i, j) e K;

It ic n1cffizl tn rronmriher the. mk6nor ripvirAz in hnrfkr(cQtc6t-j, if(i,j)EM.IL VO Ui51FIUI LU %,V1VO1UCbl tII; 1111,5bil1r, UUVILN III U1UM&b kNULnof co-ordinates) A through M as shown in Figure 3. We firstdefine each of these sets in terms of the following quantities:

= Fk/41,H = (k + 1)/2, and p = min{k,H + 2}.

A = {(i,j) I1.i< L,1<j<H}B = {(i,j) II<i<L,j= ±+I}C {(i,i) I1.< i < L,H + 2 < j < k}D = {(i, j) L+1 < i < A, 1 < j < HI}E {(i,j) I + 1 < i < fl,j =H+1}F = {(i,j) L+ 1 < i <7-,H+±2 < j <.}G {(i, j) L+ 1 < i <.7, + 3 < j < k}H = {(i, j) H + 1< i < k, I < j < L}I = {(i,j)IH+1<i< k-L,+1< j<N}J = {(i,j)I 1H+I<i< k-17-I+1<j.< +L}K = {(i,j) H +I1< i < k - L,- + C + 1 < j < k}L {(i,j) Ik-L+1 < i < k,L+1 < j<.-}M = {(i,j) k-L+1<i < k,1-+1 < j < k}

j12O 2 3

A B

D E F G

I K

H

3(k+1 )

k+l

k+2

k+3

2(k+1 )

Fig. 3. Blocks in the gap for odd k

Let A(i, j) be the number of the border device assigned to

Fig. 4. Assignment for k = 9.

Figure 4 shows the assignments for the missing devices ina 9 x 9 gap in a WAdL.

Proof: [Theorem 1, k odd]1) From the definition of A(i, j) above, the definitions of

the blocks A through M, the definitions of NH and L,and the fact that there are a total of 4(k + 1) borderdevices numbered 0 through 4(k + 1) - 1, it is clearthat each missing device is assigned at least one, and nomore than one, border device.

2) It is immediate from the definition of A(i,j) that anytwo missing devices in the same block are assigned toborder devices at a (shortest) distance at most k+ 1 alongthe border. We need to verify the assignments to pairs ofmissing devices across common block boundaries (e.g.,the the bottom row of block A and the top row of blockD), and also across a block boundary and the border(e.g., the right column of block M and the right part ofthe border). We will show one case for each of the abovetwo categories; proofs for the other cases are similar.Consider the bottom row of block A and the top row

of block D. From the definitions of the blocks, it isclear that both these rows have NH number of elements,and thus have NH pairs of neighbouring missing devicesacross this common border. Each such pair is given bythe co-ordinates (4, j) and (L +1, j) for < j <Then, from the definition of A(i, j), the two missing

78

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devices in this pair are assigned the border devicesnumbered j and 8H -j = 4(k+ 1)-j, respectively. The(shortest) distance between these two border devices is2j. Thus the largest (shortest) distance along the borderbetween two border devices assigned to any such pairis witnessed by j = H = (k + 1)/2, and the distance is(k + 1) in this case.Next, consider the right column of block Ml and thelower portion of the right border that is adjacent tothis column. Each pair of such adjacent devices has thecoordinates (i, k) (in block AM) and (i, k + 1) (on theright border), for k - L + 1 < i < k. The device onthe border at (i, k + 1) is numbered k + i + 1. Fromthe definition of A(i, j), the missing device at (i, k)is assigned to the border device numbered 6H - k =3(k+1)-k = 2k+3. Since i < k, the (shortest) distancealong the border between these two border devices is2k + 3 - (k + i + 1) = k + 2 - i. Within the range of i,the maximum for this value is witnessed by i = k-L+1,and this maximum value is L + 1, which is clearly lessthan k + 1.Note that blocks C, F, G, I, J, and K do not exist fork = 3. Similarly, for k = 5, blocks G, I, J, and K donot exist. For k > 5, all the blocks exist. Therefore, inthe cases k = 3 and k = 5, the appropriate inter-blockboundaries and the block-border boundaries need to beverified.

3) Using the identities

k 2H- 1, andL > H-L

it can be verified that the range of values assigned toeach block is as shown in the table in Figure 5 andthat the values assigned to devices in one block are notassigned to devices in any other block. (The table liststhe blocks so that the ranges of values are in ascendingorder.) It can now be easily verified that within each

Fig. 5. Ranges of values assigned to blocks

block no value is assigned more than h times. Since weare assigning k2 missing devices to 4k border devices

(we do not use the corner border devices), some borderdevice must be assigned to at least k/4 missing devices.Since k is odd, some border device must be assigned toat least [k/4] missing devices. As shown above, ourassignment scheme ensures that no border devices isassigned to more than Fk/41 missing devices, and isoptimal in this sense.

4) From the definition of A(i, j) is is clear that the as-

signment can be computed in constant for any givenco-ordinate (i, j).

B. Scheme for Even Values of k

Just as in the case of odd k, we will consider the missingdevices in this case in blocks A through L. (See Figure 6.)

A = {(i,j) 1<i < L,i<j < k-i}B _(i,j) k-L+1 <j < k,k-j+1 <i <j-1}C {(i,j) Ik- +1<2 i < k,k-i+2<j <i}D = {(i,j) 1 < j < L, j + 1< i < k - j +1}E = {(i,j) N- -C-+ 1 < i <. L4+ 1< j <.i}F = {(i, j) L + 1 < i < X - 1, L + i < j < '}G = {(i,j) H ±+ 1 < j < H+±L,

l + 1 < i < 2N- j + 1}H = {(i,j) NH+2 <j < k-L,k-j+2 < i < H}I = {(i,j) 7-H +1<i< 21t-L,i<j < 21--L}J {(i,j) -H+±2< i < k-4L, ±+1 < j < i-1}K {(i,j) N -t-L+ 1 < j < -H,

21t - j + 1 < i < k - L}L = {(i, j) L + 1 < j <.H-1,H±+ 1 < i < k-j}

Fig. 6. Blocks in the gap for even k

Let A(i, j) be the number of the border device assigned to

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Block A 1... HBlock C H+2...2H-1Block G 2H+3... 3H-1Block B 3HBlock F 3H + 1Block K 3H+2 ... 4H-LBlock J 3H + L + 1... 4H-1Block M 4H+ 1...5H-1Block E 5HBlock I 5H + 1... 6H-L-1Block L 6H-L ... 7H-2L-1Block H 6H + 1. .. 7H-1Block D 7H ... 8H - 1

the missing device at (i, j) in the gap. Then, for 1 < i,j < k,

j, if (i, j) E A;k+i+1, if (i,j) E B;3(k + 1)-j, if (i, j) E C;4(k + 1)-i, if (i, j) E D;4(k + 1) +j-i-1, if (i, j) E E;

A(ij) j -, if (i, j) E F;i +j - 1, if (i, j) E G;k-L+ i, if (i,j) EH;2(k + 1) + i -j-1, if (i, j) E I;2(k + 1) + i - j, if (i,j) E J;4(k + 1)-i-j, if (i,j) K;4(k + 1)-i - j, if (i,j) E L.

Figure 2 shows the assignments for the missing devices inan 8 x 8 gap in a WAdL.

V. SIMULATION RESULTSWe simulated a WAdL with a 7 x 7 gap, and then, within

this environment, simulated the simultaneous passage of a row

of messages moving from the bottom to the top, and a columnof messages moving from the right to the left. Our goals ofthis experiment were

1) to verify that with the appropriate time "dilation" all themessages are received at the other end of the gap at thesame time, despite having to be routed along the border,and

2) to compare the performance of solutions 2 and 3 pro-

posed in Section III above.Recall that in solution 2, we use our assignment scheme, butrather than route messages along the border devices (as insolution 3), the border devices use I, the initiator of the WAdL,as the one hop routing mechanism. Figure 7 shows the routealong the border that one of the messages takes when usingsolution 3.

00 01 02-00-04 -5-06 07 08I I0-92-03 4 t2 AD:& 071ost"I-Oeo 0,4 -12 o0 OR71 103X,1-3-292-24-29 I- 11-1

&_ M:w..... 4.-;vM..-G-2..-X>.-.-2_l27 21,7 2i1 21 5 -i' -14-

s216-256-28 --22 -i'9 -l 11R

_ 25 225 -28 -2 19 -Ris -17r

Fig. 7. Route taken by one of the messages in solution 3

As expected, in solution 2, when time was dilated by a

factor of 2(k + 1), the number of border devices processingmessages at any point in time, all the messages were receivedat the other end of the gap at the same time. Similarly, whenusing solution 3, dilating time by a factor of (k + 1) * k/4ensured the simultaneous receipt of messages.

Figure 8 shows the maximum, minimum and average num-

bers of messages received, sent and processed by each ofthe border devices and I in each of solutions 2 and 3. Amessage was said to be processed by a device if the messagewas destined for that device. As is evident from this data,

Fig. 8. Messages Sent, Processed and Received

the amount of communication activity of the border devicesdid not vary much between the two solutions. On the otherhand, I exhibited a dramatic increase in activity in solution2, supporting our belief that solution 2 creates a bottleneck atI, and showing that solution 3 is indeed the most effective ofthe three solutions presented.

VI. CONCLUSIONSIn this paper we have addressed the problem of carrying

out analogical simulations in a WAdL in the presence ofsquare gaps in the WAdL. We present three solutions, anddiscuss the superiority of the third solution presented. Ourmain contribution is the constant time assignment scheme toassign devices bordering the square gap in a WAdL to missingdevices; the effectiveness of the third solution depends on thisassignment scheme. Future work will focus on the study ofsimilar assignment schemes for arbitrary shaped gaps.

REFERENCES[1] D. Greenspan. "Deterministic computer physics." International Journal

of Theoretical Physics, vol. 21, no. 617, pp. 505-523, 1982.[2] W. D. Hillis, "The connection machine: A computer architecture based

on cellular automata:, Physica D, vol. 10, pp. 213-228, 1984.[3] J. Case, D. S. Rajan, and A. M. Shende, "Lattice computers for approxi-

mating euclidean space," Journal ofthe ACM, vol. 48, no. 1, pp. 110-144,2001.

[4] N. Margolus, "CAM-8: a computer architecture based on cellular au-tomata," in Pattern Formation and Lattice-Gas Automata, A. Lawniczakand R. Kapral, Eds., 1993.

[5] A. M. Shende, "Digital analog simulation of uniform motion in represen-

tations of physical n-space by lattice-work mimd computer architectures:'Ph.D. dissertation, SUNY, Buffalo, 1991.

[6] V. Gupta, G. Mathur, and A. M. Shende, "Lattice formation in a WAdL(Wireless Ad-hoc Lattice computer)," in Workshop on Algorithms forWireless and Mobile Networks, 2004.

[7] V. Gupta and G. Mathur, "Wireless Ad-hoc Lattice computers (WAdL)$'in 18th Annual Consortium for Computing Sciences in Colleges: South-eastern Conference, 2004.

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Solution 2Received Processed Sent

min/max/avg min/max/avg min/max/avgBorder 2/10/9 2/10/9 0/10/9I 206 0 235

Solution 3Received Processed Sent

min/max/avg min/max/avg min/max/avgBorder 6/24/17.69 0/10/7.88 4/22/14.5I 0 0 29