Binary Dissection: Variants & Applications

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1 Binay Dissection: Vaiants & Applications Shahid H. Bokhai Depatment of Electical Engineeing Univesity of Engineeing & Technology Lahoe, Pakistan Thomas W. Cockett ICASE NASA Langley Reseach Cente Hampton, Viginia David M. Nicol Depatment of Compute Science Datmouth College Hanove, New Hampshie Abstact Patitioning is an impotant issue in a vaiety of applications. Two examples ae domain decomposition fo paallel computing and colo image quantization. In the fome we need to patition a computational task ove many pocessos; in the latte we need to patition a high esolution colo space into a small numbe of epesentative colos. In both cases, patitioning must be done in a manne that yields good esults as defined by an application-specific metic. Binay dissection is a technique that has been widely used to patition non-unifom domains ove paallel computes. It poceeds by ecusively patitioning the given domain into two pats, such that each pat has appoximately equal computational load. The basic dissection algoithm does not conside the peimete, suface aea o aspect atio of the two sub-egions geneated at each step and can thus yield decompositions that have poo communication to computation atios. We have developed and implemented seveal vaiants of the binay dissection appoach that attempt to emedy this limitation, ae faste than the basic algoithm, can be applied to a vaiety of poblems, and ae amenable to paallelization. The Paametic Binay Dissection (PBD) algoithm ties to minimize the diffeence between volume + λ (suface) fo each of the two subegions it geneates at each step. When applied to paallel computing, volume epesents the amount of computation equied while suface is popotional to intepocesso communication. The paamete λ pemits us to tade off load imbalance against communication ovehead. When λ is zeo, the algoithm educes to simple binay dissection. The Fast Adaptive Dissection (FAD) algoithm is used fo colo image quantization, whee samples in a high esolution colo space ae mapped into a lowe esolution space in a way that minimizes colo eo. In this case the algoithm ties to minimize the poduct populaity colo eo fo the esulting sub-egions. populaity is the numbe of colos in a egion while colo eo is the maximum distance between points within a egion. We descibe the pefomance of PBD and FAD on a vaiety of epesentative poblems and pesent ways of paallelizing the PBD algoithm on 2- o 3-d meshes and on hypecubes. This eseach was suppoted by the National Aeonautics and Space Administation unde NASA Contact NAS while the authos wee esident at the Institute fo Compute Applications in Science & Engineeing, NASA Langley Reseach Cente, Hampton, Viginia. Shahid Bokhai was suppoted in addition by a gant fom the Diectoate of Reseach Extension and Advisoy Sevices, Univesity of Engineeing & Technology, Lahoe. David Nicol was suppoted in addition by NSF gant CCR Authos addesses: shahid@icase.edu, tom@icase.edu, nicol@cs.datmouth.edu. Poduction note: Due to limitations in the epoduction pocess, hadcopy vesions of this epot may not adequately convey the subtle vaiations in image quality in Figue 14. Fo best esults, we ecommend obtaining the digital vesion of this epot (ftp://ftp.icase.edu/pub/techepots/97/97-29.pdf), and eithe viewing the images on a high-esolution 24-bit colo monito, o pinting them on a photogaphic-quality output device.

2 1 Intoduction The patitioning of poblems ove the pocessos of a paallel compute system emains the subject of consideable eseach. This poblem is paticulaly difficult when the domain o egion being patitioned has nonunifom computational equiements. Fo example, in a climate model, some aeas of the eath s suface may equie geate computational effot than othes. We would like to appotion pats of the poblem domain ove the pocessos of the system in such a way as to put equal computational load on all pocessos, theeby minimizing the total computational time. Anothe example is the solution of aeodynamic poblems using unstuctued meshes which ae gaphs embedded in 2- o 3-dimensional space [3]. Computation is caied out on the nodes of the gaph, altenating with communication ove the edges. Poblems of this type equie huge amounts of computational powe and ae at the limits of the memoy capacities of the lagest paallel pocessos. Thee is a pessing need fo techniques to impove the unning time of such poblems, because they equie scace and expensive esouces fo thei solution and also because the solution itself has geat economic value. The binay dissection o othogonal ecusive patition algoithm was developed by Bege & Bokhai [1] as a means fo patitioning non-unifom domains. This appoach pemits a vey fast solution to the patitioning poblem and has been successfully applied in pactice [4]. This algoithm makes a seies of bisections, along othogonal diections, minimizing the load imbalance at each step. Since the patitioning attempts only to balance the load, the subegions geneated may have poo communication to computation atios. This is because the simple patitioning citeion does not conside the peimete, suface aea, o aspect atio of the subegions being geneated. In the cuent pape we pesent two vaiants of the binay dissection algoithm. These ae Paametic Binay Dissection (PBD) and Fast Adaptive Dissection (FAD). In Paametic Binay Dissection each ecusive cut is chosen to minimize volume + λ (suface). When the domain is a 2-dimensional egion, suface could efe to the peimete of a subegion. In the 3-d case it could efe to the suface aea. When PBD is used to patition embedded gaphs, volume efes to the numbe of vetices in a egion and suface the numbe of edges leaving a egion. In geneal, suface is a measue of the communication ovehead and the paamete λ pemits us to tade off load imbalance against communication ovehead. We can sacifice some amount of load balance fo bette communication balance in ode to obtain faste oveall computation time. When λ is zeo, the new algoithm educes to simple binay dissection. 1

3 Fast Adaptive Dissection finds application in colo image quantization, inwhichsamplesina high-esolution colo space ae mapped onto a lowe esolution space in a way that minimizes the colo eo. In ou fomulation of this poblem, one is given a 3-dimensional gid in which some points ae occupied and othes ae vacant. The objective is to patition this gid into egions such that (1) the total numbe of egions is bounded by some given maximum, and (2) a single value can be computed fo each egion which is a satisfactoy epesentative of the enclosed points subject to some global eo metic. We have found that a patitioning pocess which attempts to minimize the poduct populaity colo eo yields good esults at high speed. Hee populaity is the numbe of colos in a egion, while colo eo is the maximum distance between two points in the egion. Mesh patitioning is one of the poblems to which we apply ou new algoithms. A numbe of othe patitioning stategies have been poposed fo this poblem, and it is wothwhile to compae ou appoach with existing wok. The pevious wok, e.g. [8], is built aound the notion of gaph sepaatos. In such a fomulation a mesh is viewed as an undiected gaph. An edge-sepaato is a set of edges that disconnects the gaph into two nealy equal sized pieces. The goal of sepaato based appoaches is to find sepaatos of small size, theeby educing the communication ovehead. Thee ae two pincipal diffeences between paametic binay dissection, and sepaato-based algoithms. PBD/FAD constain all cuts to be staight lines, a constaint not imposed on the othe methods. As a consequence, fo cetain poblems and anges of paamete values, the patitions poduced by PBD/FAD on this application ae almost cetainly infeio. This deficiency is balanced bythefactthat PBD/FAD ae moe geneal in thei application (e.g. we see no easy way to use gaph sepaatos fo the colo quantization poblem), linea cut constaints aise natually in a numbe of applications, and PBD/FAD ae undoubtedly the simplest, and likely fastest, methods among the altenatives. Thus the quality of patitions poduced by PBD on the specific poblem of mesh patitioning is not the sole measue of its value. Futhemoe, PBD is tivial to extend to gaphs with weighted nodes and edges. 2

4 2 Binay Dissection The oiginal binay dissection algoithm can be applied to a vaiety of situations. In the pesent pape we ae concened with the patitioning of 2, 3 (o possibly highe) dimensional domains containing n points specified by thei x,y,z,... coodinates. These points ae bisected along the x diection by soting the x coodinates and finding the mid-point. This pocess is accomplished in O(n log n) time fo soting and O(n) time fo splitting the list of points. The bisection pocess is then epeated along the y diection fo the two subdomains and so on. If the depth of patitioning (the numbe of times the bisection is caied out) is given by d, then the entie pocess takes time d 1 O( i=0 (2 i (n/2 i log n/2 i )+n)) = O(dn log n). (1) Since the depth of patitioning d log n, this esults in O(n log 2 n) in the case of poblems whee the patitioning is caied out to lage depths. Howeve, in many poblems of inteest the depth of patition d is small compaed to log n and it is moe meaningful to use expession (1). The basic bisection step descibed above can also be caied out using a fast (O(n)) median finding algoithm [2]. This eliminates soting and we ae left with O(dn) time. Howevethe constants involved in the median finding algoithm ae lage and this method emains of theoetical inteest only. Binay dissection patitions only on the basis of numbes of points and ignoes thei spatial distibution. As a esult it can yield patitions that have poo aspect atio (the atio of lagest to smallest sides), which may be undesiable in specific applications. When binay dissection is applied to the patitioning of gaphs embedded in 2 o 3 dimensional space, as is the case in many impotant aeodynamic poblems, the edge infomation (which detemines the amount of infomation that needs to be communicated between points) is ignoed. Thus while binay dissection can be (and has been) applied to such poblems, the patitions obtained can sometimes be poo as fa as the compute/communicate atio is concened. 3 Paametic Dissection Paametic binay dissection emedies one of the shotcomings of the basic algoithm by explicitly taking sufaces of egions into account. Thus, if the poblem is to patition a thee dimensional egion that contains a numbe of points, we minimize at each bisection step volume + λ (suface) fo the two subegions. 3

5 By volume we mean the numbe of points in each egion this is the quantity that simple binay dissection minimizes. Suface can efe to a vaiety of egion popeties. Fo example, if the poblem is to patition a 2-dimensional egion into subegions such that the esulting subegions ae as squae as possible, we may wish to use the peimetes of the esulting ectangles as ou suface popety. At each bisection step we would minimize the numbe of points in each ectangle plus λ times thei peimetes. The paamete λ pemits us to tade off volume against suface by sacificing some amount of volume balance, we can impove the suface balance. The peceding example can be extended in an obvious fashion to 3 o highe dimensions. Vaious suface popeties can be used. In the following discussion we shall assume that the suface popety can be computed easily, so that the analysis of time complexity of the algoithm is not affected by it. We can use a complicated suface popety if we ae willing to pay fo the time equied to compute it while caying out binay dissection. The discussion so fa has been in tems of point poblems, whee we ae given a collection of points in 2, 3 o highe dimensional space. A moe complicated situation aises when we ae given a gaph embedded in 2, 3 o highe dimensions. Each point o node has associated with it a set of coodinates and an adjacency list. The objective hee is staightfowad: each bisection minimizes nodes + λ (edges cut). Gaph patitioning poblems aise in many envionments, most notably in the analysis of unstuctued meshes. When such meshes ae patitioned and mapped onto paallel computes, the unning time is modeled by max [nodes in egion + λ (edges leaving egion)]. (2) all egions Hee λ coesponds to the communicate to compute atio fo the given paallel compute system, i.e. the atio of time equied to fetch a datum fom a emote pocesso to the time to compute on a datum on the local pocesso. The time given by (2) is nomalized to the time equied to compute on one point, assuming a unifom computation cost fo each point. The density of edges in unstuctued meshes can vay enomously fom one pat of the domain to anothe and it is easy to see that taking edges into account can esult in bette patitionings. This would not be the case fo embedded gaphs in which the edges ae moe o less unifomly distibuted. We ae assuming that the communication ovehead is popotional to the amount of data tansmitted. This holds tue fo moden paallel machines with high pefomance communication hadwae. Figue 1 gives a simple example of how paametic dissection can lead to bette patitions when 4

6 applied to gaphs. Fo the case of point poblems the complexity of paametic binay dissection is unchanged at O(dn log n), whee d is the depth of patitioning. Fo gaph poblems, we have to look at all edges befoe splitting, at evey depth of the patition. Hence, the complexity becomes O(max[dn log n, de]), whee e is the numbe of edges in the gaph. 4 Fast Paametic Dissection A majo facto contibuting to the time complexity of the binay dissection algoithms pesented in Sections 2 and 3 is epeated soting at each depth. We now show how paametic binay dissection can be accomplished by soting only once pe dimension by using a well known technique [12]. The fast algoithm we pesent also impoves the time equied fo plain binay dissection. A sepaate index list is ceated fo each dimension. Element i of this list gives the index of the of the data point that is ith in soted ode. When a egion is patitioned, all indices ae split, so that the sublists coesponding to each subegion emain soted. Fo puposes of exposition, we assume a 3-d gaph patitioning poblem and patition on the basis of expession (2) of Section 3. Assume that the index lists fo the x, y and z dimensions ae stoed in aays xlist[],ylist[] and zlist[]. The subegion to be patitioned is stoed in aay positions L...U. This means that the x dimension index list extends fom xlist[l] to xlist[u] and so on. The cuent depth of patitioning is depth. This vaiable is initialized to be the maximum depth to which patitioning is to be caied out and is decemented at each level. The coodinates of point i ae stoed in x[i], y[i], z[i]. The paametic bisection is computed as follows. 5

7 (a) (b) Figue 1: The benefit of paametic dissection. In pat (a) plain binay dissection is applied to give pefect node balance. The numbes in the fou egions give the sum of nodes in egion and edges leaving egion (i.e., in the expession fo time to execute (2) in main text, λ is assumed to be one). The maximum sum is 6. In pat (b) paametic dissection is employed and esults in maximum sum 5, as the new algoithm avoids cutting though egions with high edge density. 6

8 pocedue paametic cut(depth,l,u,x,y,z,xlist,ylist,zlist); 1. Sweep fowad fom i=l to U counting the edges that would leave the left hand egion, if the left hand egion was L to i (inclusive). Stoe the esult in leftvec[i]. 2. Sweep backwads fom i=u down to L counting the edges that would leave the ight hand egion, if the ight hand egion was i to U (inclusive). Stoe the esult in ightvec[i]. 3. Sweep fowad again fom i=l to U to find the optimal split point: the left hand egion compises L to i, the ight hand egion compises i+1 to U the optimal split point splitplace is the value of i fo which the objective max((i L+1) + λ (leftvec[i]), (U i) +λ ( ightvec[i+ 1])) is minimum. The value x[splitplace] is splitvalue. 4. xlist has now been split into two pats, L to splitplace and splitplace+1 to U. Thex coodinates of these points ae aleady soted since the oiginal soted index list is undistubed. 5. Split the ylist: sweep fowad fom i=l to U moving successive values of ylist[i] fo which x[ylist[i]] splitvalue to the fist pat of the list (Figue 3 illustates this fo a 2-d poblem). The emaining values ae moved to the second pat of the list. 6. Similaly split the zlist. 7. All thee indices xlist, ylist and zlist have now been split so that elements [L..splitplace] of these lists contain the points in one of the subegions and [splitplace+1..u] thoseinthe othe. When accessed though these lists the x, y and z coodinates of points ae in soted ode. 8. Recusively cut fo next depth but along next dimension: if(depth>1) then{ paametic cut(depth 1,L,splitplace,y,z,x,ylist,zlist,xlist) paametic cut(depth 1,splitplace+1,U,y,z,x,ylist,zlist,xlist)} end paametic cut; 7

9 leftvec[i] ightvec[i] i i Figue 2: Computation of leftvec and ightvec in steps 1 and 2 of pocedue paametic cut. The domain to be patitioned (along the x-diection) is given by the top ectangle. 8

10 ylist : XXy C X CC 456 XXXXXXXXX C 3 CCW,,,C HY -, C,B - H H 6 B 2 : Qk C B Q C CCW B Q BB 1 Q HY H H B H H BN 0 H xlist 7 6 ylist Qk C CC Q Q 2 Q 6 7 C CCW 5 1,,C, C,B - 6 B,,, TK Qk C C T B Q B Q CCW T BB 0 Q T T B T BN T xlist 76 Figue 3: When splitting index lists about the vetical cut, xlist (thin ed aows) is split in constant time. ylist (thick blue aows) takes time popotional to the numbe of points, since each point may have to be moved. 9

11 Figue 2 claifies how leftvec and ightvec ae computed. The vetical dashed lines in this figue show one possible splitting location, splitplace. Thevalueofleftvec fo this splitplace is 5. This is because if the ight hand egion was chosen to be up to and including the node though which this dashed line passes, the numbe of edges leaving the left hand egion (shown in blue) would be 5. Similaly, if the ight hand egion was chosen to stat fom this point onwads (i.e., including the node though which the dashed line passes), the numbe of edges leaving the ight hand egion (shown in ed) would be 8. Note that the edge lying wholly between points outside the egion has no impact on the computation. Figue 3 shows how the index lists ae split. Assuming a fixed numbe of dimensions, the sots take O(n log n) time. Fo point poblems, each patition o split takes O(n) time. We theefoe get O(n log n)+o(dn) =O(n log n) fopoint poblems. Fo gaph poblems the soting time is unchanged. The time to split is now O(e) pe level, as we have to look at evey edge at evey level esulting in O(max[n log n, de]) time. Howeve in this case it is impotant to emembe that the gaphs coesponding to unstuctued gids fom 2-d aeodynamic poblems ae plana and thus have e = O(n). Typical 3-d aeodynamic gids have bounded degee and again have e = O(n). Thus we again obtain O(n log n). The time fo fast paametic dissection is thus an impovement ove the O(n log 2 n) time fo simple binay dissection, even though paametic dissection uses a moe complex patitioning citeion. 5 A Simple Paallel Algoithm We now discuss a paallel vesion of the paametic dissection algoithm. This is a simple algoithm that does not utilize the available pocessos well: its untime is O(n) independent of the numbe of pocessos, assuming that the data points ae supplied in soted fom. Howeve its exteme simplicity is likely to make its implementation easy and its measued un times may well be competitive with the moe complex algoithm pesented in Section 6. We stat by consideing point poblems and discuss gaph poblems (which ae only slightly moe complicated to implement) at the end of this Section. We make the easonable assumption that the patitioning is to be caied out on the same paallel machine on which the poblem is to be solved. Thus 2- and 3-d poblems ae computed on 2- and 3-d meshes, espectively. Altenatively, since a lage enough hypecube can have any lowe dimensional mesh embedded in it, we may choose to un ou poblems on hypecubes. Discussion of a paallel implementation is complicated by the issue of mapping. Wheeasinthe seial algoithm we wee only concened with the patitioning, in the paallel algoithm we would 10

12 like to patition ou domain and at the same time delive the esulting subdomains to the coect pocessos. This can esult in substantial savings in time, as discussed below. The question that now aises is how we ae to map the 2 d subdomains that aise afte a depth d patitioning onto a p =2 d pocesso system. The mapping that we choose is the natual mapping descibed by Bege & Bokhai[1]. When the fist bisection is made, dividing the domain into, say, a left half and a ight half, then the left subdomain is associated with the left half of the mesh and the ight subdomain with the ight half. This pocess is epeated until the subdomains at the dth level ae eached these ae associated with individual pocessos. 5.1 Basic bisection step Suppose that we have a p =2 d pocesso chain-connected paallel machine. We shall descibe how the basic bisection step is caied out on this chain and then late show how this chain is mapped onto the taget paallel machine. 1 Fo puposes of illustation, we shall assume that we have a 2-d point poblem with n points and that the point data (compising <x, y> coodinates) has been duplicated and two soted lists pepaed, one fo each coodinate. These lists ae loaded into ou chain in a linea ode, with 2n/p points pe pocesso. Sweep-x Sweep though each point of the x-list sequentially fom left to ight, in ode to identify the optimal split point. The x-coodinate of the split point is splitvalue. Migate-x Move all points of the x-list with x-coodinate splitvalue to the left half of chain and emaining points to the ight half. Mak-y Sweep though the y-list, making with the label left, thosepointswhosex-coodinates ae splitvalue and all othes with ight. Migate-y Move all points of the y-list maked left (ight) to the left (ight) half of the chain. Each of the above fou steps takes time popotional to n. The basic bisection step can now be epeated on the two halves of the chain, with the oles of x and y intechanged. If at each bisection step the numbe of points is exactly halved, the time equied is popotional to n + n 2 + n 4 + n 8 + < 2n. 1 Which could be a 2 d/2 2 d/2 2-d mesh, a 2 d/3 2 d/3 2 d/3 3-d mesh, o a dimension d hypecube. 11

13 Figue 4: Constucting Bisectionable Chain Embeddings (BCEs). Paametic binay dissection does not guaantee that each dissection step will exactly halve the numbe of points. We shall assume that the maximum numbe of points at evey step of the patitioning is a constant times the ideal balance at that step. Thus the O(n) esult obtained above holds fo simple as well as paametic binay dissection. 5.2 Bisectionable Chain Embedding The bisection pocedue descibed above only seves to patition the domain ove a chain of pocessos. When caying out computations on 2-d o 3-d domains we would natually pefe to use 2- o 3-d meshes fo ou computation. We now descibe embeddings of chains in 2- o 3-d meshes which have the inteesting popety that when the basic bisection step of Section 5.1 is successively applied to such chains, then the points migate to the pocessos on which they should be mapped accoding to the natual mapping. No explicit outing of data blocks is equied. This popety eliminates an expensive outing step. Figue 4 shows how a Bisectionable Chain Embedding (BCE) is constucted by combining two smalle BCEs. These cuves ae simila to Peano s space filling cuves. A fomal desciption fo the pocedue fo geneating these cuves is given in [13]. Figue 5 shows a bisectionable chain embedding of size Ou soted x and y-lists ae mapped onto this chain stating at and ending at. Application of the basic bisection step (vetical cut) esults two sets of sublists, one set stating at and ending at 2; the othe stating at and ending at. The pocedue is epeated with 2 hoizontal cuts. The key popety of BCEs is that at this stage the left half of the mesh chain will contain only the points of the oiginal lists that should be mapped onto the left half of the mesh and similaly fo the ight half of the chain. Thus when the bisection pocedue is caied out ecusively on a BCE, 12

14 d t Figue 5: A16 16 BCE. The soted x and y lists ae mapped onto this chain stating at and ending at. PP P PP P PP PP P P PP P PP P PP P PP P u PP P z PP 6 P y P P * PPi x Figue 6: A3-dBCEofsize x, y and z lists ae mapped onto this chain stating at and ending at. The fist bisection (with a plane pependicula to the x axis) will cut the dashed segment. This is epeated ecusively fo the y and z diections (spacing along the x axis is distoted). 13

15 the data points move to thei espective pats of the mesh, so that at the end of the pocedue each pocesso contains its natually mapped points. The concept of Bisectionable Chain Embeddings is easily extended to highe dimensions (Figue 6). 5.3 Gaph Poblems We pesent ou analysis fo the case of degee constained gaphs embedded in 3-space, and assume that 3 copies of the gaph ae available to us, soted by each of the dimensions 2. Each item in the x-list, fo example, contains the <x,y,z> coodinates of the point and the coodinates of all points adjacent to this point. These lists ae mapped onto a chain of pocessos as befoe and the chain of pocessos embedded in a 3-d mesh. The basic bisection step fo gaph poblems equies visiting each pocesso sequentially, and within each pocesso, tavesing the x-list. As each point is visited, we count the numbe of edges that would be cut if this point wee the exteme point in the bisection. This pocess is epeated in the evese diection and then the point whee the minimum of nodes + λ (edges cut) occus is found along the lines of the seial pocedue of Section 4. This is followed by list migation. This step is then epeated successively in the y and z diections. Of couse, at each point we must visit the nodes adjacent to that point, and list migation involves moving not only each point, but also its adjacent points (i.e., the complete list enty). Ou time and space complexity is unchanged at O(n) because we have assumed a constant degee constaint. 6 Fast Paallel Algoithm The esults of the peceding Section suggest an O(n) algoithm fo dissection. This is a simple algoithm that does not utilize the available pocessos well: its untime is independent of the numbe of pocessos available, assuming that the data points ae supplied in soted fom. Howeve its exteme simplicity is likely to make its implementation easy and its measued un times may well be competitive with the moe complex algoithm pesented below. The ideal algoithm fo paametic dissection of an n node poblem on a p pocesso system would have complexity O(n/p log n/p), which is the same as if each pocesso wee solving the 2 Applications to highe o lowe dimensions ae immediate, although it is to be kept in mind that the space equied by this algoithm (on each pocesso) is popotional to the numbe of dimensions of the poblem. It should be ecalled that the ultimate objective of the patitioning is to pemit a complex aeodynamic computation to take place. The patitioning is caied out befoe the computation. The actual computation equies a lage numbe of vaiables fo each point to stoe, fo example, the velocity vectos, pessue, density etc. Typically fom 50 to 100 locations ae equied fo each point [3]. This space can thus feely be used fo the binay dissection. 14

16 Achitectue Run Time 2-d mesh O( n + p 1/2 log p) p 1/2 3-d mesh O( n + p 1/3 log p) p 2/3 hypecube O( n p log3 p) Table 1: Algoithmic complexity of paallel paametic dissection on vaious achitectues. subpoblem esident on it in isolation. This lowe bound is difficult to achieve because of the ovehead of intepocesso communication. Nevetheless we have succeeded in developing good algoithms fo 2 and 3-d meshes and hypecubes. The details of these algoithms ae involved and may be found in ou ealie technical epot [13]. Table 1 summaizes ou esults. 7 Applications to Unstuctued Meshes A potion of a 2-d unstuctued mesh is shown in Figue 7. It can be seen that this mesh has a vey lage vaiation in node density. The objective, in geneating the mesh, is to have a highe density of nodes in the egions whee thee is geate need fo accuacy. It is this vaiation in density that makes such meshes difficult to patition. Thee-dimensional unstuctued meshes ae an obvious extension but ae difficult to potay on a 2-d page. We show in Figue 8 a 3-dimensional mesh suounding the wing, fuselage and engine of an aeoplane. We have implemented the Fast Paametic Dissection algoithm of Section 4, using equation (2) of Section 3. This algoithm has been used to patition seveal vey lage 3-d unstuctued gids taken fom aeodynamic poblems. When applying paametic dissection on such gids, it is often the case that the fist cut is badly imbalanced as fa as the numbe of nodes is concened. This is because binay dissection consides the gaph to be embedded in a ectangle o cuboid, with edges extending to the sides of the ectangle o cuboid (as shown in Figue 7). The mesh eally occupies a oughly ellipsoidal egion of 2 o 3-d space (which cannot be depicted in Figue 7 as it is vey lage compaed to the wing coss-section shown). When λ is non zeo, the fist cut is likely to slice off a small tip of the ellipsoid, so as to minimize the numbe of edges cut. Thus we have a tiny numbe of nodes in one egion and most of the nodes in the othe egion. The objective (2) is coectly minimized and the patitioning obtained is supeio to a plain patitioning, but only fo depth 1. Beyond depth 1 o 2 this poo initial cut leads to bad patitions. This phenomenon is vey simila to that descibed by Stone [10] in connection with the patitioning of andom gaphs. Ou solution to this poblem is to cay out the fist 1, 2 o 3 cuts with λ =0andswitchoveto 15

17 Figue 7: A 2-d unstuctued mesh suounding the coss section of an aeoplane wing with extended flaps and slat. 16

18 Figue 8: A 3-dimensional unstuctued mesh suounding the wing, fuselage and engine of an aeoplane. 17

19 the desied value of λ only afte these initial cuts have balanced the numbe of nodes in the initial 2, 4 o 8 subegions. In ode to evaluate the speedup that would be obtained if a paametic binay dissection wee used, compaed to plain binay dissection, we caied out an expeiment with a 3-d mesh of size 106,064 nodes and 697,992 edges (patly shown in Figue 8). This mesh is deived fom a poblem involving a wing and engine pod and half a fuselage. Measued un time on a 50 MHz MIPS R4000 pocesso fo a depth 15 patition of this mesh is 83 seconds (excluding time to input the mesh). The following evaluation pocedue was epeated fo depths = Run the paametic dissection algoithm fo vaious values of λ, stating with 0. Fo each un obtain maxnodes(λ) andmaxedges(λ), the maximum numbe of edges and nodes ove all egions. The nomalized un time fo a dissection is t paametic (λ) =maxnodes(λ)+λ maxedges(λ). This assumes ideal communications on the taget paallel pocesso. maxnodes(0) and maxedges(0) ae the values that would have obtained if plain binay dissection had been used, since fo λ = 0 paametic dissection educes to plain dissection. Thus fo this poblem the time taken by a plain dissection is t plain = maxnodes(0) + λ maxedges(0). Fo a given value of λ the pefomance advantage of the paametic algoithm is Impovement(λ) = t plain t paametic (λ). The esults of the above expeiment ae summaized in the plots of Figue 9 which show the pefomance impovement of paametic dissection ove plain dissection. Since paametic dissection educes to plain dissection fo λ = 0, the cuve coesponding to this λ is constant at Thee is no impovement fo depth=1 and 2 because plain dissection is used fo these depths, as discussed above. 18

20 Figue 9: Impovement of paametic binay dissection ove plain binay dissection, when applied to a 3-d aeodynamic mesh with 0.1 million nodes and 0.7 million edges, fo depths 1-15 (coesponding to 1, 2, 4,...,32768 pocessos). Fo λ = 0 paametic dissection educes to plain binay dissection and thee is no speedup. 19

21 Figue 10: Pefomance impovement of paametic binay dissection on a andom gaph with 0.1 million nodes and 0.5 million edges. Pefomance impovements ae obtained ove plain binay dissection fo all but the smallest values. In pactical applications, λ is likely to be geate than 1. 20

22 Figue 11: Pefomance impovement of paametic binay dissection on a andom gaph with 0.1 million nodes and 2.5 million edges. Fo this elatively dense gaph, pefomance impovement satuates at about λ =

23 It can be seen that paametic dissection gives good impovements ove plain dissection fo λ>1 ove most of the ange of depths. Fo λ 1, thee is a steep incease in impovement beyond depth 10. Fo λ = 4 the impovement is geate than 20% fo depth 15. We note that in pactical machines the value of λ (the emote/local access time atio) is usually 1 o geate. Figues 10 and 11 show the pefomance of paametic binay dissection on andom gaphs. Hee too the pefomance impovement is substantial fo λ Colo Image Quantization with Fast Adaptive Dissection We now come to the second vaiant of binay dissection, namely Fast Adaptive Dissection (FAD). This algoithm finds application in colo image quantization, whee samples in a high-esolution colo space ae mapped onto a lowe esolution space in a way that minimizes the colo eo [7]. Moe fomally, we ae given a digital image whose pixels ae chosen fom a palette containing 2 m colos, and we wish to geneate an acceptable epoduction using a palette of 2 k colos, whee k<m. Typical values fo m un fom 15 24, while k is usually in the ange fom Colo quantization is commonly used to convet full-colo images into colomapped o pseudocolo images in which each pixel is a k-bit index into a colo lookup table, o colomap. The esulting images ae moe compact than the oiginals and ae suitable fo display using the inexpensive video systems found in most pesonal computes and many wokstations. In full-colo images, the m bits of colo infomation ae typically divided into thee distinct colo components, each using appoximately m/3 bits. If we assume a ed-geen-blue (RGB) colo model, then each component epesents an axis in a thee-dimensional colo gid. The component values at each pixel can be thought of as indices into this gid. The poblem then becomes one of patitioning the gid such that (1) the total numbe of egions is bounded by 2 k, and (2) the single colo value computed fo each egion seves as a satisfactoy epesentative of all the colos within the egion. 8.1 Fast Adaptive Dissection A vaiety of heuistic techniques have been poposed fo patitioning the colo space. Ou appoach most closely esembles Heckbet s median cut algoithm [7], but uses a modified vesion of Fast Paametic Dissection (Section 4) to speed up opeations involving the colo samples within egions. These include seaching fo split points, detemination of bounding boxes, and computation of epesentative colos. 22

24 The fist step is to scan the oiginal image and ecod which points in the colo space ae epesented, and how fequently. We stoe this infomation in a 3-d histogam matix (Figue 12). The histogam is then scanned to poduce a list of the colos which occu. The colo list is eplicated and soted fo each colo component, as equied by the Fast Dissection algoithm. Since ou colo components equie only a few bits each, we can avoid the level of indiection equied by the paametic cut algoithm of Section 4. Instead, each colo component is stoed diectly as a bit field within a list item, educing both memoy and computation costs. We next need a stategy fo patitioning the lists. In the context of this poblem, the two most impotant citeia ae populaity, defined as the numbe of pixels epesented by the colos within a egion, and colo eo, defined as the maximum distance between points within a egion. We have found that a multiplicative elationship between these two paametes poduces bette images than eithe of them alone, and is also supeio to an additive elationship. Thus ou objective function in FAD is populaity colo eo. In ou ealie desciptions of Paametic Binay Dissection and Fast Dissection, we have assumed a ecusive patitioning pocess which descends until the maximum numbe of subegions is poduced, o until a egion cannot be subdivided futhe. Fo Fast Adaptive Dissection, we follow Heckbet s lead and modify this stategy to utilize adaptive patitioning. With adaptive patitioning, the diections of the cuts ae not pedetemined by the depth of the ecusion, but ae chosen dynamically based on popeties of the data. We discuss this point in moe detail below. Anothe disadvantage of ou oiginal ecusive fomulation is that the patitioning pocess can bottom out pematuely one o moe banches of the ecusion tee may encounte egions which cannot be futhe subdivided, even though othe banches may offe ample oppotunity fo subdivision. The net esult is that some of the available colomap enties go unused. To ovecome this poblem, we use an iteative vaiant of Fast Dissection. Afte each cut is made, the objective function is evaluated fo the esulting subegions, and they ae placed on a global subegion list, soted by descending value of the objective function. At each step of the iteation, the fist subegion on the list is patitioned. This pocedue guaantees that evey available colomap enty will be used (assuming the oiginal image contains at least 2 k colos), and gives pioity to splitting egions with the lagest deviations fom the ideal. Ou colo quantization algoithm is theefoe adaptive in two ways: the diection of the cuts is data-dependent, as is the choice of egions to be split. We call this modified appoach Fast Adaptive Dissection (o FAD), and efe to colo quantization using this technique as FAD quantization. 23

25 We have found that, fo colo quantization, the objective function is needed only to detemine which egion to split next. To find the split point within each egion, a simple and faste heuistic woks well: egions ae split at the midpoint along thei longest edge, i.e., in the diection of lagest colo eo (Heckbet s median cut stategy). With the FAD algoithm the midpoint can be found quickly using a binay seach on the coesponding soted sublist. The populaity value fo each subegion is conveniently tallied duing the e-odeing pass on eithe of the emaining sublists, with the colo fields in each list item seving as indices into the oiginal histogam matix. To detemine colo eo, we use a simple estimate. Rathe than seaching fo the two most exteme points in each egion o explicitly computing the eo elative to the oiginal image, we use the squae of the length of the diagonal of the bounding box fo the egion. With Fast Dissection, finding the bounding box is tivial we simply obtain the espective maximum and minimum colo components fom each of the thee sot lists. At each patitioning step, these ae diectly available via the L and U list indices. A dynamic view of the patitioning pocess can be found in [14]. When the patitioning phase is complete, the epesentative colo of each egion is set to the aveage of the colo values within that egion, weighted by the fequency of thei occuence (Figue 13). The collection of epesentative colos foms the new colomap fo the image. All of the histogam enties ae then eplaced by indices to thei epesentative colos. The sot lists speed up both of these steps, since empty cells in the histogam ae not epesented in the lists and theefoe do not have to be examined. To complete the pocess, the oiginal image is e-scanned, and the value of each pixel is eplaced with its colomap index, using the histogam matix as a lookup table. 8.2 Expeimental Results Many colo quantization techniques have been developed peviously [5][6][7][9][11]; ous is of inteest because it poduces good esults at high speed. methods ae difficult because Unfotunately, diect compaisons with ealie pevious esults have been epoted ove a peiod of 15 yeas, spanning seveal geneations of pocesso technology; diffeent methods wee tested against diffeent sets of input images; and souce code fom pevious implementations is not eadily available. 24

26 Figue 12: RGB histogam aay fo the 15-bit image in Figue 14b. Non-zeo elements ae epesented by thei coesponding colos. Figue 13: The colo space in Figue 12 patitioned into 256 egions using Fast Adaptive Dissection. The influence of the populaity paamete gives ise to lage vaiations in patition size. 25

27 Pocesso Type IBM POWER2 SGI/MIPS R4400 Sun UltaSPARC Algoithm Step 67 MHz 250 MHz 167 MHz 1. Pe-quantize & histogam Build sot lists Patition colo space Build colomap Remap image Total time Table 2: Runtimes (in seconds) fo FAD quantization. Howeve, we believe that ou method is competitive with existing techniques. While highe image quality may be obtained with moe computational effot, and faste esults can be achieved at lowe image quality, the FAD algoithm povides a good balance fo applications in which both speed and quality ae impotant. We have implemented FAD quantization in the C language and tested it on seveal wokstation platfoms. Table 2 shows execution times fo the test image shown in Figue 14a. As pat of the histogamming step, we educe the colo pecision of the oiginal image fom 24 to 15 bits. This simplification, employed by many colo quantization algoithms, deceases memoy and pocessing equiements significantly while degading the oiginal image only slightly (Figue 14b). As the timings indicate, the pixel-level opeations on the image data (steps 1 and 5) dominate the execution time, compising fom 78 85% of the total. The patitione (steps 2 and 3) equies 12 19% of the total time, while computation of the epesentative colos and constuction of the mapping table (step 4) equies a miniscule 3%. Although the fast dissection method patitions the colo space apidly (step 3), we must take cae that the ovehead fo the initial sots (step 2) doesn t outweigh the benefits. We have found that the fastest way to sot the lists is to simply scan the histogam aay in the desied ode, appending non-zeo enties to the list as we encounte them. This is easily accomplished by using a tiply-nested loop (one fo each dimension) with the appopiate soting index (R, G, o B) at the outemost level. Fo example, the following C code fagment poduces a list of colos soted 26

28 (a) Oiginal 24-bit image ( colos). (b) Unifom quantization to 15 bits (6980 colos). QRMSE = 4.2. Figue 14: Test image ( pixels). 27

29 (c) Unifom quantization to 8 bits. QRMSE = Only 140 colos ae used; the emaining 116 lie in egions of the colo space which ae not epesented in the input image. Note the sevee colo banding in the flesh tones and backgound, and loss of highlight detail in the t-shits. (d) FAD quantization to 256 colos. QRMSE = 9.2. Some colo banding pesists, but details e-emege in the backgound and on the shits. Figue 14 (cont d): Test image ( pixels). 28

30 along the G axis: const int DIM = 32, MAXCOLORS = DIM * DIM * DIM; unsigned histogam[dim][dim][dim]; int, g, b, i; unsigned list[maxcolors], glist[maxcolors], blist[maxcolors];... fo (i = 0, g = 0; g < DIM; g++) fo (b = 0; b < DIM; b++) fo ( = 0; < DIM; ++) if (histogam[][g][b]) glist[i++] = ( << 10) (g << 5) b; A simila loop is used to sot in the B diection. Fo the R sot, we can optimize futhe by taking advantage of C s ow-majo stoage ode fo aays: unsigned *h; int j;... h = (unsigned *) histogam; fo (i = j = 0; i < MAXCOLORS; i++) if (h[i]) list[j++] = i; Ou tests indicate that this soting stategy is an ode of magnitude faste than a tuned quicksot, and is theefoe essential in ealizing the pefomance gains of the FAD appoach. 29

31 A fast colo quantize is useful only if it poduces satisfactoy images. To measue image quality, we compae the oiginal full-colo image to the quantized esult using the quantization oot mean squae eo (QRMSE) metic defined in [9]. Fo compaison puposes, Figue 14c shows the esults obtained by simply deceasing the colo esolution of the oiginal image to 8 bits (3 bits each fo ed and geen, 2 bits fo blue), an opeation which uns in about one-fifth the time of the FAD algoithm. Figue 14d shows the impovement obtained using FAD quantization. 9 Conclusions We have pesented two vaiants of the binay dissection algoithm fo patitioning non-unifom domains. The fist of these, Paametic Binay Dissection, o PBD, finds applications in paallel pocessing. We have analyzed the un time of PBD fo seial and paallel machines and pesented some measued pefomance figues. The paametic dissection algoithm is seen to povide bette pefomance than the oiginal binay dissection algoithm fo lage depths of patitioning. A fast algoithm fo paametic dissection was pesented in Section 4. This algoithm has un time O(n log n) as opposed to the oiginal O(n log 2 n) dissection algoithm. The time fo dissection is thus completely masked by the time equied to sot the input data. We also pesented two paallel algoithms fo paametic dissection. The O(n) algoithm is simple to implement and will likely be useful in situations whee the mesh is being input seially to the pocesso, as in this case the dissection time is masked by the time to load. Ou moe elaboate algoithm has time O((n/p 1/2 )+p 1/2 log p), O((n/p 2/3 )+p 1/3 log p) ando((n/p)log 3 p) fo 2-d meshes, 3-d meshes and hypecubes, espectively. This algoithm pefoms well fo poblems in which the numbe of nodes n is lage compaed to the numbe of pocessos, a case that is of consideable pactical inteest. The second vaiant is Fast Adaptive Dissection, o FAD. This algoithm has been applied to the poblem of colo image quantization, whee it yields good esults at high speeds. The sot lists used in ou fast dissection appoach ae key in educing execution times fo opeations on data values within subegions. 30

32 Futue wok in this aea shall develop along the following lines: 1. Impovements in the paallel PBD algoithm. Communication ovehead shows up pominently in the expessions fo un time of ou algoithm. Whethe this can be educed significantly is an open question. 2. Implementations of the paallel vesions of the dissection algoithms fo mesh achitectues such as the Intel Paagon and ASCI Red machines. 3. Evaluation of the pefomance of PBD on a lage set of unstuctued meshes. 4. Use of these dissections fo actual computation, especially fo aeodynamic poblems. 5. Applications of PBD/FAD to othe aeas, such as patitioning poblems in cicuit and VLSI design. Acknowledgments We ae gateful to Dimiti Maviplis fo many useful discussions. We thank Clyde Gumbet fo poviding us with a lage 3-d mesh fo expeimentation. Caig Gotsman gaciously povided us with images fo compaison testing. We ae gateful to M. Y. Hussaini, M. Salas and K. E. Duani fo thei encouagement of this eseach. Refeences [1] Masha J. Bege and Shahid H. Bokhai, A patitioning stategy fo non-unifom poblems acoss multipocessos. IEEE Tansactions on Computes, C-36: , May [2] Thomas H. Comen, Chales E. Leiseson, and Ronald L. Rivest, Intoduction to Algoithms. MIT Pess, Cambidge, Massachusetts, [3] D. J. Maviplis, Thee dimensional unstuctued multigid fo the Eule equations. AIAA Jounal 30(7): , July [4] Kaen M. Dagon and John L. Gustafson, A low-cost hypecube load-balance algoithm. In The Fifth Confeence on Hypecubes, Concuent Computes and Applications, Ma. 1989, pp [5] J. L. Fulani, L. McMillan, and L. Westove. Adaptive colomap selection algoithm fo motion sequences. Poceedings ACM Multimedia 94, Oct. 1994, pp [6] Michael Gevautz and Wene Pugathofe. A simple method fo colo quantization: octee quantization. In Gaphics Gems, A. Glassne, ed., Academic Pess, 1990, pp ,. [7] P. Heckbet, Colo image quantization fo fame buffe display. Compute Gaphics, 16(3): , July [8] A. Pothen, H. D. Simon and K. P. Liou, Patitioning spase matices with eigenvectos of gaphs. SIAM Jounal of Mathematical Analysis and Applications, 11: ,

33 [9] E. Roytman and C. Gotsman, Dynamic colo quantization of video sequences. IEEE Tansactions on Visualization and Compute Gaphics, 1(3): , Sept [10] Haold S. Stone, High Pefomance Compute Achitectue. Addison-Wesley, Reading, Massachusetts, 1990, pp [11] Xiaolin Wu, Efficient statistical computations fo optimal colo quantization. In Gaphics Gems II, J. Avo, ed., Academic Pess, 1991, pp [12] F. P. Pepaata and M. I. Shamos, Computational Geomety: An intoduction. Spinge-Velag, [13] S. H. Bokhai, T. W. Cockett and D. M. Nicol, Paametic Binay Dissection. ICASE Repot No , NASA Contacto Repot , July Available at ftp://ftp.icase.edu/pub/techepots/93/93-39.ps.z. [14] T. W. Cockett, S. H. Bokhai and D. M. Nicol, Colo Image Quantization using Fast Adaptive Dissection, (MPEG animation), June

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