EQUI-PARTITIONING OF HIGHER-DIMENSIONAL HYPER-RECTANGULAR GRID GRAPHS

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1 EQUI-PARTITIONING OF HIGHER-DIMENSIONAL HYPER-RECTANGULAR GRID GRAPHS ATHULA GUNAWARDENA AND ROBERT R MEYER Abstact A d-dimensional gid gaph G is the gaph on a finite subset in the intege lattice Z d in which a vetex x = (x 1, x 2,, x n) is joined to anothe vetex y = (y 1, y 2,, y n) if fo some i we have x i y i = 1 and x j = y j fo all j i G is hype-ectangula if its set of vetices foms [K 1 ] [K 2 ] [K d ], whee each K i is a nonnegative intege, [K i ] = {0, 1,, K i 1} The suface aea of G is the numbe of edges between G and its complement in the intege gid Z d We conside the Minimum Suface Aea poblem, MSA(G, V ), of patitioning G into subsets of cadinality V so that the total suface aea of the subgaphs coesponding to these subsets is a minimum Although seveal efficient 2-dimensional heuistics which exploit the geomety of the gid ae available in the liteatue, vey little pogess has been made in constucting simila algoithms fo patitioning highe dimensional (d > 2) gid gaphs We pesent an equi-patitioning algoithm fo highe dimensional hype-ectangles and establish elated asymptotic optimality popeties Ou algoithm genealizes the two dimensional algoithm due to Matin [ Discete Appl Math, 82 (1998), pp ] It uns in linea time in the numbe of nodes (O(n), n = G ) when each K i is O(n 1/d ) Utilizing a esult due to Bollabas and Leade [Combinatoica, 114 (1991), pp ], we constuct an algoithm to calculate a useful lowe bound fo the suface aea of an equi-patition Ou computational esults eithe achieve this lowe bound (ie, ae optimal) o stay within a few pecent of the bound Key wods highe dimensional gid gaph, hype-ectangle, gaph patitioning, knapsack poblem AMS subject classifications 05A18, 68R01, 68W01 1 Intoduction Gid gaph equi-patition aises in the context of minimizing intepocesso communication subject to load balancing in paallel computation fo a vaiety of poblem classes including the solution of PDEs using finite diffeence schemes [11], compute vision [10], and database applications [7] Simila to the geneal gaph patitioning poblem [6], abitay gid gaph patitioning poblem is NP-complete [12], and at pesent one has to depend on effective heuistic algoithms In this pape, we focus on patitioning gaph analogs of hype-ectangles and heeafte call ou poblem the Minimum Suface Aea poblem fo ectilinea gid gaphs, denoted by MSA(G,V ), in which the d-dimensional gid gaph G = [K 1 ] [K 2 ] [K d ] is patitioned into subgaphs of cadinality V so that the total suface aea of the subgaphs is a minimum We assume that V divides K 1 K 2 K d Figue 11 shows a 25-component equi-patition fo the MSA([5] 3,5) poblem geneated by ou equipatitioning algoithm pesented in Section 4 Since the suface aea of this patition does not equal the lowe bound deived in Section 3, its optimality is not guaanteed but its elative optimality gap [[aea-lowe bound](=2)/lowe bound(=500)] is 004%, and only one component allows any possibility fo suface aea impovement Seveal effective algoithms fo patitioning a 2-dimensional gid gaph ae available in the liteatue [2, 4, 5, 8, 14] Ou algoithm genealizes the fast two dimensional algoithm due to Matin [8] With this genealization, we educe the poblem to seveal knapsack poblems and use dynamic pogamming to solve them Depatment of Compute Science, Univesity of Wisconsin-Madison, Wisconsin (agunawadena@wiscedu), and Depatment of Mathematical and Compute Sciences, Univesity of Wisconsin-Whitewate, Whitewate, Wisconsin (gunawaa@uwwedu) Depatment of Compute Science, Univesity of Wisconsin-Madison, Wisconsin (m@cswiscedu) The eseach of this autho was suppoted in pat by NSF gant DMI

2 2 ATHULA GUNAWARDENA AND ROBERT R MEYER V V T R P V V T R P L L J H F L L J H F D D B B A V W T R P W W T R P L M J H F M M J H F D D B B A X X U S Q W W T R P N N K I G M M J H F D E B C A X X U S Q X Y U S Q N N K I G N O K I G E E C C A Y Y U S Q Y Y U S Q O O K I G O O K I G E E C C A Fig 11 An equi-patition fo MSA([5] 3, 5) with bound gap 004% Only component A does not have minimum suface aea Those gid gaphs that meet the minimum suface aea fo a given numbe of vetices daw special inteest fom eseach in Combinatoics (ie, study of Polyominoes) [1], and in Physics (ie, study of the metastable behavio of the stochastic ising model) [9] In 1991, Bollabas and Leade [3] deived a class of gid gaphs with minimum suface aea In Section 3, utilizing thei esult, we povide a fomula fo the minimum suface aea fo gid gaphs with a given cadinality and use this esult to find a lowe bound fo the suface aea of a given equi-patition We pesent ou equi-patitioning algoithm in Section 4 and give computational esults in Section 5 fo some 3 and 4-dimensional equi-patitions We also compae ou esults with the lowe bound given in Section 3 2 Peliminaies We follow the notation and definitions given in [3] Fo the eades convenience, we estate them hee A d-dimensional gid gaph is the gaph on a finite subset in the intege lattice Z d in which x = (x 1,x 2,,x n ) is joined to y = (y 1,y 2,,y n ) if fo some i we have x i y i = 1 and x j = y j fo all j i Since gid gaphs ae detemined by thei nodes, in much of the discussion below we identify the gaph with its nodes Let P([n]) be the powe set of [n] = {0, 1,, n 1} The binay ode on P([n]) is a total ode on P([n]) s 2 n elements such that fo some S,T P, S < T if and only if the geatest element of [n] which is in one of S and T but not the othe is actually in T (ie, max(s T) T) The cube ode on [k] n is a total ode such that fo some x = (x 1,x 2,,x n ) and y = (y 1,y 2,,y n ), x < y if and only if fo some s [k], {i : x i = s} < {i : y i = s} in the binay ode on P([n]) with {i x i = t} = {i y i = t} fo all t > s Fo S [k] n, the cube ode on S is the estiction to S of the cube ode on [k] n We wite x + [k] to epesent {x,x + 1,,x + k 1} Let G be a gid gaph and E be the set of edges in Z n The edge bounday of G, denoted by σ(g), is the set {(x,y) E x G,y G} The suface aea of G, denoted by A(G), is the cadinality of σ(g) ( σ(g) ) (In thee dimensional space this matches the exposed suface aea of the set of unit hypecubes with centes at the nodes of G) The following poposition states that the cube ode may be used to ceate gaphs with minimum suface aea This is a diect coollay to Theoem 15 in [3] Othe vesions of poofs of this esult can be found in [9], [1] (d = 3), and [13] Poposition 21 Let G be a d-dimensional gid gaph such that G [k] d fo some lage enough positive intege k, and J be the set of fist G elements in the cube ode on [k] d Then A(G) A(J) Let d be a positive intege and n d be a nonnegative intege Let k d be the unique intege such that k d d n d < (k d + 1) d, and m d be the unique intege in [d] such that (k d +1) m d k d m d d n d < (k d +1) m d+1 (k d ) d m d 1 Obseve that k 1 = n 1 and m 1 = 0

3 PARTITIONING OF HIGHER DIMENSIONAL GRID GRAPHS 3 Fo an intege n d, we define the following special gid gaph, called the quasi-cube ecusively, and we show in Section 3 that quasi-cubes ae geneated by the cube ode Definition 22 (Quasi-Cube) The d dimensional quasi-cube coesponding to n d is denoted by C d (n d ) Fo d = 1, C 1 (n 1 ) = [n 1 ] Fo d > 1, C d (n d ) is the set [k d + 1] m d [k d ] d m d P(C d 1 (n d 1 )), whee k d and m d ae unique nonnegative integes as mentioned above, n d 1 = n d (k d + 1) m d (k d ) d m d, and P(C d 1 (n d 1 )) is the set of points, (x 1,x 2,,x md,k d,x md +1,,x d 1 ), such that (x 1,x 2,,x d 1 ) C d 1 (n d 1 ) Since k i and m i (d i 1) ae unique integes, the quasi-cube C d (n d ) is uniquely defined 3 A Lowe Bound fo the Total Suface Aea of an Equi-Patition We use the quasi-cube to obtain ou esults in this section The following lemmas descibe some of the popeties of the quasi-cube Lemma 31 The quasi-cube has the popeties: d (a) C d (n d ) = n d = (k + 1) m k m (b) A(C d (n d )) = 2 =1 d [( m )(k + 1) m k m 1 =1 + m (k + 1) m 1 k m ] Poof By Definition 22, C d (n d ) = S d P(C d 1 (n d 1 )) whee S d = [k d + 1] m d [k d ] d m d It is clea that S d and P(C d 1 (n d 1 )) ae disjoint and P(C d 1 (n d 1 )) = C d 1 (n d 1 ) So we have C d (n d ) = S d + C d 1 (n d 1 ), and get (a) by applying induction on d To pove (b), we conside the suface aea of C d (n d ) We have A(C d (n d )) = A(S d ) + A(P(C d 1 (n d 1 ))) 2 σ(s d ) σ(p(c d 1 (n d 1 )) Since k d 1 k d, we get σ(p(c d 1 (n d 1 ))) σ(s d ) Using A(S d ) = 2(d m d )(k d +1) m d k d m d 1 d +2m d (k d + 1) md 1 k d m d d, A(P(C d 1 (n d 1 ))) = A(C d 1 (n d 1 )) + 2 C d 1 (n d 1 ), and σ(s d ) σ(p(c d 1 (n d 1 )) = C d 1 (n d 1 ), we get the ecuence elation A(C d (n d )) = A(C d 1 (n d 1 )) + 2(d m d )(k d + 1) m d k d m d 1 d + 2m d (k d + 1) md 1 k d m d d, and the initial condition A(C 1 (n 1 )) = 2 if n 1 > 0 and, 0 if n 1 = 0 It is staightfowad to solve this ecuence elation and get ou esult in (b) Lemma 32 The quasi-cube, C d (n d ), has the minimum suface aea among the d-dimensional gid gaphs with n d vetices Poof Fist we pove that the quasi-cube C d (n d ) foms an initial segment of the cube ode in dimension d We pove this by induction on d Clealy C 1 (n 1 ) = [n 1 ] is an initial segment of the cube ode Assume C d 1 (n d 1 ) is an initial segment of the cube ode By the definition, C d (n d ) = S d P(C d 1 (n d 1 )) whee S d = [k d ] m d [k d + 1] d m d Let x = (x 1,x 2,,x d ), whee x i = k d 1 if i m d and x i = k d if i > m d Clealy x S d and any element y in the cube ode such that y < x is in S d Since thee ae S d elements which ae less than o equal to x in cube ode, S d is an initial segment of the cube ode We may assume that C d 1 (n d 1 ) φ Othewise C d (n d ) = S d is an initial segment of the cube ode Let y be the element in the cube ode such that y > x and thee is no othe element between x and y Then y i = 0 fo i m d + 1, y md +1 = k d Clealy y is the lowest element in P(C d 1 (n d 1 ) in the cube ode Ou assumption that C d 1 (n d 1 ) is an initial segment of the cube ode guaantees that P(C d 1 (n d 1 )) foms a complete

4 4 ATHULA GUNAWARDENA AND ROBERT R MEYER segment of the cube ode stating fom y Theefoe C d (n d ) is an initial segment of the cube ode By Poposition 21, we obtain the esult Theoem 33 Let C be a d-dimensional gid gaph with n vetices and n d be a positive intege such that n d divides n Then the total suface aea of an equi-patition of C into subgaphs of cadinality n d is at least 2n n d d [( m )(k + 1) m k m 1 =1 + m (k + 1) m 1 k m ] whee k i and m i, i = 1,2,,d ae calculated as mentioned in the definition of the quasi cube Poof Since the equi-patition has n/n d gid gaphs and each gaph is having n d vetices, the esult follows fom Lemma 31 and Lemma Lowe Bound Algoithm Using the above esults, this algoithm calculates a lowe bound fo the total suface aea of an equi-patition of a gid gaph with cadinality n into subgaphs of cadinality n d begin { Lowe Bound } fo = 1 to d aea 0 k 0 m 0 end fo s n d d while (s > 0) Select k so that k s < (k + 1) Select m so that (k + 1) m k m s s (k + 1) m k m aea aea + ( m )(k + 1) m k m 1 1 end while etun 2n n d aea end {Lowe Bound } < s (k + 1) m+1 k m 1 + m (k + 1) m 1 k m When n = n d, Lowe Bound Algoithm etuns the minimum suface aea of a gid gaph with cadinality n The gid gaphs that meet the minimum suface aea ae of special inteest in many fields as mentioned in Section 1 Table 31 shows the minimum suface aea of a gid gaph fo some selected values of n and d 4 Equi-Patitioning Algoithm Let K 1,K 2,,K d,v be positive integes and V divides K 1 K 2 K d Hee we pesent a heuistic algoithm fo MSA([K 1 ] [K 2 ] [K d ],V ) poblem The key idea of the algoithm is to decompose the given hype-ectangle into a collection of well chosen towes of the fom [k 1 ] [k 2 ] [k d 1 ] [K d ], in which the towe bases [k 1 ] [k 2 ] [k d 1 ] tile the [K 1 ] [K 2 ] [K d 1 ] base of the given hype-ectangle Associated with each towe is the aea of a paticula equi-patition of the towe In theoy, one could constuct an optimal tiling of the given base by minimizing the total of the suface aeas of the coesponding towes, but this poblem is compaable in

5 PARTITIONING OF HIGHER DIMENSIONAL GRID GRAPHS 5 Table 31 The minimum suface aea of a d-dimensional gid gaph with n vetices n d = 2 d = 3 d = 4 d = 5 d = 6 d = 7 d = difficulty to the oiginal poblem Ou heuistic builds up the base tiling via easily computed patitions of lowe dimensions The main task of the algoithm involves solving 1 + K 1 + K 1 K K 1 K d 2,d 2 knapsack poblems that epesent total aeas associated with tilings by towes Fo d = 2, thee is only one knapsack poblem and this case was consideed by Matin [8] The algoithm is implemented in d + 1 phases In Phase-0, we patition each hype-ectangle, [k 1 ] [k 2 ] [k d 1 ] [K d ] whee k i [K i ] (1 i d 1), into components of size V and calculate the total suface aea fo the patition If V < k 1 k 2 k d 1 (in which case a component ceated by the algoithm would be badly shaped by having size 1 in dimension d) o V does not divide k 1 k 2 k d 1 K d, we will conside the component as an invalid component and assign as the suface aea Othewise we calculate the aea of the patition of the towe assuming the vetices of each component ae assigned by the lexicogaphical ode on Z d which can be descibed as follows Let x = (x 1,x 2,,x d ) and y = (y 1,y 2,,y d ) be two d-tuples in [k 1 ] [k 2 ] [k d 1 ] [K d ] We say x > y if j = max{i : x i y i 0} and x j y j > 0 If x > y then we assign y befoe x If the component size is V, we may assign the fist V cells in the above total ode to the fist component and the next V cells to the second component and cay on the pocess until we fill the tageted egion Fo a tall towe whose base is appopiately chosen, lexicogaphic assignment geneates components that ae good appoximations to quasi-cubes and hence neaoptimal suface aeas ae obtained This popety is addessed in the following two theoems Theoem 41 pesents an expession that may be used to compute the total suface aea esulting fom lexicogaphic assignment of the nodes in a towe unde the assumptions that the towe has an integal numbe of components and that each component has a height of at least one Theoem 42 establishes unde some mild assumptions that the aveage suface aea of the components poduced by lexicogaphic assignment of nodes in tall towes with well-chosen bases asymptotically appoaches minimum suface aea The computational esults pesented in Section 5 demonstate this asymptotic optimality popety of lexicogaphic assignment Theoem 41 (Total aea fo lexicogaphic assignment) Assume that V divides k 1 k 2 k d 1 K d, V k 1 k 2 k d 1, and let p 1 be the numbe of components of cadinality V in the d-dimensional egion [k 1 ] [k 2 ] [k d 1 ] [K d ] Let s j = k 1 k 2 k j whee 1 j d 1 Let ij be the j-dimensional emainde of the ith component which is calculated by ij = iv mods j, whee 1 i p 1, and 1 j d 1 Then the total suface aea of an equi-patition of [k 1 ] [k 2 ] [k d 1 ] [K d ]

6 6 ATHULA GUNAWARDENA AND ROBERT R MEYER into p components of cadinality V using the lexicogaphical ode is (41) d 1 2[ps d 1 + j=1 s d 1 k j K d + p 1 d 1 a ij ], i=1 j=1 whee a ij is calculated as follows a i1 = { 1 if i1 0, 0 othewise When j > 1, a ij = 0 if k j = 1, ij if k j > 1 and ij < s j 1, s j ij if k j > 1 and s j ij < s j 1, s j 1 othewise Poof It is clea that the oute suface aea of the gid gaph [k 1 ] [k 2 ] [k d 1 ] [K d ] is 2[s d 1 + d 1 j=1 K d ] To find the aea of the inne boundaies, we s d 1 k j need to add the aea between the ith component and the (i+1)th component fo each 1 i p 1 Since V >= s d 1, thee ae p 1 inne boundaies and each bounday is shaed by exactly 2 components So the total aea of the pojections of the inne boundaies of the components in the dth diection is 2(p 1)s d 1 Fo 1 j d 1, let a ij be the common bounday aea between ith component and the i+1 th component in the jth diection The possible cases fo a ij ae mentioned in the statement of Theoem 41 Hee we explain two cases and leave the moe staightfowad cases fo the eade s veification We can visualize [k 1 ] [k 2 ] [k j ] as k j coss sections of [k 1 ] [k 2 ] [k j 1 ] in the jth diection These coss sections ae assigned to the ith component in inceasing ode of j When ij < s j 1, the ith component is assigned ij vetices fom the fist [k1] [k 2 ] [k j 1 ] coss section in the jth diection, and hence a ij = ij When s j ij < s j 1, the ith component is assigned ij vetices fom the last [k1] [k 2 ] [k j 1 ] coss section in the jth diection, and hence a ij = s j ij Summing up the contibutions in each diection, we get the aea between the ith component and the (i + 1)th component as 2 d 1 j=1 a ij So the total d 1 j=1 a ij Adding suface aea of inne boundaies is equal to 2(p 1)s d p 1 i=1 the oute and inne bounday aeas gives us ou esult Theoem 42 Conside the family of d-dimensional towes T(N) of the fom [k 1 (N)] [k 2 (N)] [k d 1 (N)] [K d (N)], whee k i (N) N C fo some constant C fo 1 i d 1 and all N = 1,2, and K d (N) = N z whee z > 1 Let V = N d and let a l (N d ) be the aveage suface aea of the components of T(N) geneated lexicogaphically unde the assumptions of Theoem 41 Then the aveage suface aea a l (N d ) asymptotically appoaches the minimum suface aea (fo volumes of size N d ), ie, a l (N d ) lim N a (N d ) = 1 whee a (N d ) is the minimum suface aea (2dN d 1 ) fo volume V Poof By Theoem 41, a l (N d ) = 2s d 1 (N) + 2 p(n) [ d 1 S d 1 (N) j=1 k K j(n) d(n) + p(n) 1 d 1 i=1 j=1 a ij(n)] whee

7 PARTITIONING OF HIGHER DIMENSIONAL GRID GRAPHS 7 s d 1 (N) = k 1 (N)k 2 (N)k d 1 (N) and p(n) = s d 1 (N) N z /N d Since z > 1, note that p(n) when N Fo sufficiently lage N, it is easily veified that a l (N d ) 2(N +C) d 1 +2 d 1 j=1 [ Nd N C +d(n +C)d 2 a ] Thus lim sup l (N d ) N a (N d ) 1, but since a l(n d ) 1 fo all N, the conclusion follows a (N d ) 41 Phase-0 (Suface aea computation fo towes) Suface aea computations using fomula (41) ae pefomed fo all valid towes begin {Phase-0 } Define a (d 1)-dimensional aay A d 1 [K 1 ][K 2 ][K d 1 ] of suface aeas as follows: V the cadinality of a component fo k 1 = 1 to K 1 fo k 2 = 1 to K 2 fo k d 1 = 1 to K d 1 begin if V does not divide k 1 k 2 k d 1 K d o V < k 1 k 2 k d 1 A d 1 [k 1 ][k 2 ][k d 1 ] INFINITY else A d 1 [k 1 ][k 2 ][k d 1 ] The total suface aea given by (41) fo the patition of k 1 k 2 k d 1 K d into components of cadinality V using the lexicogaphical ode end if end fo end {Phase-0 } 42 Phase-i (1 i d 1) (Detemine optimal base decompositions in successively highe dimensions) In the ith (1 i d 1) phase, we solve K 0 K 1 K 2 K d i 1 ( whee K 0 = 1) knapsack poblems using dynamic pogamming Fo each k i [K i ], We define the k 1 k 2 k d i 1 th knapsack poblem as follows minimize K d i j=1 c jx j subject to K d i j=1 h jx j = K d i whee each x j is a nonnegative intege vaiable, h j = j, and c j = A d i [k 1 ][k 2 ][k d i 1 ][j] (whee A d i is a (d i)-dimensional matix (ie, i = 1 case is used in Phase-0)) contains the optimal objective value of the k 1 k 2 k d i 1 jth knapsack poblem when i > 1, and the aea of the equi-patition of subdomain [k 1 ] [k 2 ] [k d i 1 ] [j] [K d ] when i=1 (fom Phase 0) The above knapsack poblem detemines the combination of hype-ectangula subdomain heights in diection (d i) that minimizes the total suface aea as expessed by the objective function The value of x j epesents how many sub domains of height j in the (d i) diection (ie, [k 1 ] [k 2 ] [k d i 1 ] [j] [K d i+1 ] [K d ]) ae in the optimal solution The eade can obseve that Phase-i knapsack poblems use the esults of Phase-(i 1) knapsack poblems as thei aea measue objective

8 8 ATHULA GUNAWARDENA AND ROBERT R MEYER coefficients c j Initially, Phase-0 aea values ae used as c j values fo Phase-1 begin { Phase-i (1 i d 1)} if(i= d-1) (ie the final knapsack poblem) A 0 The solution of the final knapsack poblem (ie the optimal objective value and a coesponding solution {(h j,x j ) x j 0} call Phase-d etun else Define a d i 1 dimensional aay A d i 1 [K 1 ][K 2 ][K d i 1 ] fo k 1 = 1 to K 1 fo k 2 = 1 to K 2 fo k d i 1 = 1 to K d i 1 begin A d i 1 [k 1 ][k 2 ][k d i 1 ] The solution of the (k 1 k 2 k d i 1 )th Knapsack poblem (ie the optimal objective value and a coesponding solution {(h j,x j ) x j 0} end fo end if end { Phase-i } Ou equi-patitioning algoithm is feasible when the above Phase-d 1 algoithm which solves the last knapsack poblem is feasible The following theoem guaantees the feasibility of Phase-(d 1) knapsack poblem Theoem 43 If V divides K 1 K 2 K d then Phase-(d 1) knapsack poblem is feasible Poof Since V divides K 1 K 2 K d, we can find positive integes p i (1 i d) such that p i divides K i and V = p 1 p 2 p d (ie, use the fundamental theoem of algeba) Now we can wite [K i ] = ni j=0 jp i + [p i ], whee n i = (K i /p i ) 1 We conside the p 1 p 2 p d i 1 th knapsack poblem in Phase i (i < d 1) whee c j = A d i [p 1 ][p 2 ][p d i 1 ][j] This poblem has a feasible solution x pd i = n d i, and x j = 0 if j p d i with the objective value c pd i n d i if c pd i is finite Now we show that c pd i = A d i [p 1 ][p 2 ][p d i 1 ][p d i ] is finite fo each i (1 i d 1) by induction on i In ou base case (i = 1), since V divides p 1 p 2 p d 1 K d and p 1 p 2 p d 1 V, we get a finite value fo c pd 1 = A d 1 [p 1 ][p 2 ][p d 1 ] (fom Phase 0) which is the total suface aea of the patition of the hype-ectangle [p 1 ] [p 2 ] [p d 1 ] K d We assume c pd i is finite fo (i < d 1) Then p 1 p 2 p d i 1 th knapsack poblem in Phase-i (i < d 1) is feasible as mentioned befoe and c pd i 1 gets its finite optimal objective value Theefoe, by induction, c pd i is finite fo all i (1 i d 1) Since c p1 is finite, Phase-(d 1) knapsack poblem is feasible with a feasible solution of x p1 = n 1, and x j = 0 if j p 1 with the objective value c p1 n 1 43 Phase-d (Assignment) Let x = (x 1,x 2,,x nj ) and h = (h 1,h 2,,h nj ) epesent the solution of a knapsack poblem that patitions the j th diection such that n j i=1 h ix i = K j and each x i > 0 Now we can patition [K j ] such that [K j ] = mj i=1 k ji+[h ji ] whee m j = n j i=1 x i, k ji = q 1 p=1 x ph p +(i 1)h q and H ji = h q

9 PARTITIONING OF HIGHER DIMENSIONAL GRID GRAPHS 9 if 1 + q 1 p=1 x i i q p=1 x i begin {Phase-d } count 0 poc 1 fo i d = 0 to K d 1 begin Select the solution (x,h) stoed in A 0 fom Phase-(d 1) which patitions [K 1 ] fo i 1 = 1 to m 1 begin Select the solution (x,h) stoed in A 1 [H 1i1 ] fom Phase-(d 2) which patitions [K 2 ] fo i 2 = 1 to m 2 begin Select the solution (x,h) stoed in A d 2 [H 1i1 ][H 2i2 ][H (d 2)i(d 2) ] fom Phase-1 which patitions [K d 1 ] fo i d 1 = 1 to m d 1 begin {assignment } fo j d = 0 to K d 1 fo j d 1 = k (d 1)i(d 1) ) to k (d 1)i(d 1) + H (d 1)i(d 1) 1 fo j 1 = k 1i1 to k 1i1 + H 1i1 1 gid[j 1 ][j 2 ][j d ] poc count = count + 1 if(count = V ) then poc poc + 1 count 0 end if end fo (all j) end {assignment} end fo (all i) end {Phase-d} 44 Analysis of the Equi-patitioning Algoithm Fist we show that ou algoithm solves the MSA([K 1 ] [K 2 ] [K d ],V ) poblem in O(n) (n = K 1 K 2 K d )) time if each K i is in O(n 1/d ) In Phase 0, we do n/k d assignments and each assigned value takes at most O(n 1/d ) time to calculate (Theoem 41) Since K d = O(n 1/d ), we need O(n) time fo Phase 0 In Phase-i (0 i d 1), we solve K 0 K 1 K d i 1,(K 0 = 1), knapsack poblems using dynamic pogamming which takes O(K 2 d i ) = O(n2/d ) time fo each poblem Hence we solve Phase-i in O(n (d i+1)/d ) time with the wost case i = 1 giving O(n) It is clea that the final assignment phase (Phase-d) also takes O(n) time Theefoe the algoithm uns in O(n) time when each K i is O(n 1/d ) Let us define a gap of a feasible solution fo the MSA poblem as the diffeence between a lowe bound and an uppe bound fo the total aea of the patition The

10 10 ATHULA GUNAWARDENA AND ROBERT R MEYER % Gap The numbe of vetices x 10 8 Fig 51 Gap% vesus the numbe of vetices (M 3 ) fo MSA([M] 3, M) (1 M 1000) Time (Seconds) The numbe of vetices x 10 8 Fig 52 Execution time vesus the numbe of vetices (M 3 ) fo MSA([M] 3, M) (1 M 1000) pecent gap (gap%) is the gap as a pecentage of the given lowe bound Although the above algoithm is always feasible when V divides K 1 K 2 K d, thee ae examples whee the algoithm may not pefom well (ie, lage gap%) Fo example, conside MSA([P] 4,P 2 ) poblem whee P is a pime In this poblem evey valid domain in Phase-0, [k 1 ] [k 2 ] [k 3 ] [P], has exactly one of k 1, k 2 o k 3 equal to P So the aea of each component is Ω(P 4/3 ) The lowe bound given in Theoem 32 fo the aea of a component is O(P) Hence gap% is Ω(P 1/3 ) which is undesiable fo lage P We can modify Phase-0 and Phase-d of ou algoithm to avoid these undesiable gaps In Phase-0, when V does not divide k 1 k 2 k d 1 K d (ie, patition contains a patial component), the appoach of Theoem 41 can be used to obtain an uppe bound fo the aea of the patition Instead of assigning to A d 1 [k 1 ][k 2 ][k d 1 ], we assign U + 2s d 1 whee U is the total aea expession in Theoem 41 assuming p = k 1 k 2 k d 1 K d /V In the final phase, Phase-d, we fist assign labels fo the full components and then wok with the emaining patial components We get at most one inne bounday inside a patial component justifying ou uppe bound U +2s d 1 in Phase-0 5 Computational Results The algoithms wee coded in C and an in 30GHz PCs with Linux OS We pesent computational esults fo selected thee and fou dimensional poblems Figue 51 shows the esults fo MSA([M] 3,M) (1 M 1000) The pecent-

11 PARTITIONING OF HIGHER DIMENSIONAL GRID GRAPHS % Gap The numbe of vetices x Fig 53 Gap% vesus the numbe of vetices (M 4 ) fo MSA([M] 4, M 2 ) (1 M 400) Time (Seconds) The numbe of vetices x Fig 54 Execution time vesus the numbe of vetices (M 3 ) fo MSA([M] 4, M 2 ) (1 M 400) age of solutions that ae at the lowe bound (ie, optimal solutions) is 103%, and within 4% of gap is 953% Figue 52 shows the execution times fo the cases given in Figue 51 Due to memoy limitation fo lage poblems, we have excluded the final assignment phase So the unning times in the gaph ae fo obtaining the solutions without actual assignments Figue 53 shows the esults fo MSA([M] 4,M 2 ) (1 M 400) The pecentage of solutions that ae at the lowe bound (ie, optimal solutions) is 150%, and within 4% of gap is 7775% Hee we used the modified algoithm as mentioned in Section 44 Figue 54 shows the execution times fo the cases given in Figue 53 6 Conclusions and Futue Reseach We have developed a heuistic that uns in linea time in the size of the gaph( fo well shaped hype-ectangles) that geneates asymptotically optimal equi-patitions of hype-ectangles, and demonstated this pefomance fo gaphs of size up to When the aea of an equi-patition does not meet the lowe bound, it is vey challenging to decide whethe the patition is optimal As an example, one may conside the equi-patition fo the MSA([5] 3,5) poblem shown in Figue 11 which is the smallest non tivial poblem in this family This patition has a suface aea of 502, bound gap of 2, and with only one non-optimal component which is shown with the lette A Is this an optimal solution? Constaint pogamming may allow

12 12 ATHULA GUNAWARDENA AND ROBERT R MEYER optimality veification fo smalle poblems such as this Othe inteesting extensions to this eseach include elaxing the divisibility assumptions (ie, modified algoithm), and impoving the assignments obtained by moe complex pocedues fo bad components as suggested in [5] fo the two-dimensional case Acknowledgments The authos thank Winston Yang fo binging the efeences [1] and [3] to thei attention REFERENCES [1] L Alanzo, and R Cef, The thee dimensional polyominoes of minimal aea, Electonic Jounal of Combinatoics, 31 (1996), R27 pp 1-39 [2] S L Bezukov, and B Rovan, On patitioning gids into equal pats, Computes and Atif Intell, 16 (1997), pp [3] B Bollabas, and M Leade, Edge Isometic Inequalities in the Gid, Combinatoica, 114 (1991), pp [4] I T Chistou, and R R Meye, Optimal equi-patition of ectangula domains fo paallel computation, Jounal of Global Optimization, 8 (1996), pp [5] W W Donaldson, Gid-gaph patitioning, PhD thesis, Depatment of Compute Science, Univesity of Wisconsin-Madison, Madison, WI, 2000 [6] M R Gaey and D S Johnson, Computes and Intactability: A Guide to the Theoy of NP-Completeness, W H Feeman, 1979 [7] S Ghandehaizadeh, R R Meye, G Schultz, and J Yackel, Optimal balanced patitions and a paallel database application, ORSA Jounal on Computing, 4 (1993), pp [8] W Matin, Fast equi-patitioning of ectangula domains using stipe decomposition, Discete Appl Math, 82 (1998), pp [9] E J Neves, A discete vaiational poblem elated to ising doplets at low tempeatues, Jounal of Statistical Physics, 80 (1995), pp [10] R J Schalkoff, Digital Image Pocessing and Compute Vision, John Wiley & Sons, Inc, 1989 [11] J Stikweda, Finite Diffeence Schemes and Patial Diffeence Equations, Wadswoth & Books, 1989 [12] J Yackel, Minimum-peimete tiling fo optimization of paallel computation, Ph D thesis, Depatment of Compute Science, Univesity of Wisconsin-Madison, Madison, WI, 1993 [13] J Yackel, Patitioning gid gaphs with patial cubes, pivate communication [14] J Yackel, R R Meye, and I Chistou, Minimum peimete domain assignment, Mathematical Pogamming, 78 (1997), pp