A Dynamic-Programming Heuristic for Regular Grid-Graph Partitioning

Transcription

1 A Dynmic-Progrmming Huristic for Rgulr Grid-Grph Prtitioning W.W. Donldson nd R.R. Myr Dprtmnt of Computr Scincs, Univrsity of Wisconsin - Mdison, WI Octobr 27, 2000 Abstrct Prvious rsrchrs hv dmonstrtd tht striping huristics produc vry good (nd, in som css, symptoticlly optiml) prtitions for rgulr grid grphs. Ths rlir mthods diffrd in th domins of ppliction nd th strip-hight slction procss, nd did not hv polynomil run-tim gurnts. In this ppr, w trnsform th strip slction problm for gnrl grid grphs into shortst pth problm. Th running tim for th ntir procss of trnsforming th problm nd solving for th shortst pth is polynomil with rspct to th lngth of input for th originl problm. Computtionl rsults r prsntd tht dmonstrt improvd solution qulity for gnrl domins. 1 Introduction 1.1 Sttmnt of Problm Givn grph G = (V,E) nd numbr of componnts P, th grph prtitioning problm (with uniform nod nd dg wights) rquirs dividing th vrtics into P groups of spcifid siz such tht th numbr of dgs conncting vrtics in diffrnt groups (cut dgs) is minimizd. (Typiclly, ths problm constrints r motivtd by lod-blncing considrtions tht rquir th P groups to b qul or nrly-qul-sizd.) This problm is known to b NP-Complt [8]. In this ppr, rstrictd clss of grphs is studid, nmly rgulr grid grphs. Figur 1 shows n xmpl of rgulr grid grph. Th vrtics li t lttic points of rctngulr grid nd r connctd only to points djcnt on th lttic. (Not tht th ovrll domin nd not b rctngulr; it 1

2 my, for xmpl, b grid pproximtion to torus.) Grph prtitioning of lrg rgulr grid grphs riss in th contxt of minimizing intrprocssor communiction subjct to lod blncing in prlll computtion for vrity of problm clsss including th solution of PDEs using finit diffrnc schms [15], computr vision [13], nd dtbs pplictions [9]. Givn this ppliction contxt, w cn think of th problm s tht of optiml ssignmnt of tsks rprsntd by th nods or, in th trnsformd problm blow, ssignmnt of clls to procssors Figur 1: A grid grph 1.2 Trnsformtion to Cll Domin Formt Christou-Myr [3] considr nothr wy of formulting this problm, which is usful in trms of gnrting th lowr bounds on th optiml vlu considrd blow. Figur 2 givs n xmpl of how th originl grph is trnsformd into domin of squr clls. Ech nod is mppd into unit squr nd ch dg is mppd into n dg of th squr. Additionl boundry dgs r ddd s ndd to complt squrs. (In pplictions, it is ssumd tht gomtriclly djcnt nods r connctd by n dg; w ssum this proprty blow.) For th trnsformd problm, instd of counting cut dgs, th sum of th primtrs for th ssocitd componnts is minimizd. Formlly, th problm to b solvd is: minimiz i Pr(C i ) i = 1... P s.t. ch cll is ssignd to on componnt, nd ch componnt is ssignd givn numbr of clls, whr Pr(C i ) quls th primtr for componnt C i. (Although in th grph litrtur componnts r normlly considrd to b connctd, w do not ssum this connctdnss proprty in this ppr.) In most of th discussion blow, w ssum tht P divids th totl numbr of clls, 2

3 Figur 2: Th originl grph nd th corrsponding cll domin. nd this rtio is th givn r for ch componnt. (This rflcts th dsird lod blncing in prlll computing pplictions.) Th rltionship btwn cut dgs in prtition of th grph nd th totl primtr of th corrsponding prtition of th cll domin is: cut dgs = (totl primtr - primtr of th boundry of th domin)/2 This follows from th obsrvtions tht th domin boundry dgs r lwys primtr dgs but do not corrspond to grph dgs; furthrmor, ch cut dg contributs two to th totl primtr. Thus minimizing primtr is quivlnt to minimizing cut dgs. Figur 3 shows n xmpl of this rltionship. In this cs, P is 6 nd th spcifid componnt sizs r 3,3,3,3,2,3. (In this xmpl, P dos not divid th numbr of clls, so not vry componnt is th sm siz; w considr blow xtnsions to our bsic pproch tht llow this gnrliztion.) Th numbr of cut dgs in th grph quls 14. Th sum of th primtrs for th componnts in th cll domin is 46. Th primtr for th boundry of th cll domin is 18, so th numbr of cut dgs is (46-18)/2. Th bottom prtition is lso n xmpl of strip-form solution (to b dfind formlly blow; informlly this mns tht componnts r confind to horizontl bnds xcpt possibly for ovrflows t th nds of th bnds, which do not occur in this instnc). This prtition is lso n optiml solution sinc it is sily shown to yild th lowr bound vilbl from [16]. 3

4 A A B C C D A B B C D D F F F E E A A B C C D A B B C D D F F F E E Figur 3: Th top prtition of th figur shows th cut dgs of prtition of th originl grph. Th bottom figur shows th corrsponding cll-domin prtition. 1.3 Orgniztion of th Ppr W dvlop blow huristic for prtitioning rgulr grid grphs. This huristic is bsd on th construction of n optiml st of strip hights for strip-form solutions. Both thory nd computtion hv stblishd tht high qulity (nd, in som css, symptoticlly optiml) solutions cn b constructd vi striping. Topics r dvlopd in th following ordr: - Dfinitions - A mthod for prtitioning th problm into subproblms - Showing tht th numbr of strip-bsd solutions is gnrlly supr-polynomil function of th problm dimnsion, hnc brut forc mthod is not fsibl - Trnsforming th problm into shortst-pth problm - Numricl rsults - Concluding rmrks nd dirctions for furthr rsrch 4

5 2 Trms nd Dfinitions Th following xmpl, figur 4, is prsntd in ordr to dmonstrt concpts nd dfinitions tht will ppr in th ppr d d d d d d d d d d d d c c b b b b b b b b b b A b b b b b c c c c c c b c c c c c c b b b b d d d d d d d d d d d d d d c c c c c c c c c c c c b b b b b b b b b 2 11 d d d d d d d c c c c c c c b b b b b D d d d d d d f f f f f f f c c c c c c c b b b b b b g g g g g g G h h h h h f f f f f f f f f f f f f f f f f f f f f f f f f f f g g g g g g g g g g g g g g g g g g g g g g g g g g f f f f f f g g g g g g g I i i i i i i i i h h h h i i i i i i i i h h h h i i i i i i i i i i i i i i i i i i i i i i i h h h h h h h h h h h h h h h h h h h h h h h h h h h 3 4 Figur 4: An ssignmnt of grid grph In th grid, thr r 360 clls, which r to b dividd qully mong 9 procssors (in th fsibl solution shown, th corrsponding componnts r lblld -i). (If diffrnt rs (numbr of clls) r spcifid for th componnts (s in figur 3) w ssum tht th ssignmnt procdur complts th ssignmnt of componnts squntilly in th ordr in which sizs r spcifid. As discussd blow, it is possibl to modify th lgorithm to tk dvntg of diffrnt componnt sizs.) To obtin th fsibl solution shown, th 26 rows of th grid (numbrd 0 to 25) wr first prtitiond into th four strips indictd in th figur. (As will b shown blow, thr r 5

6 bttr slctions of strip hights, but this fsibl solution srvs to illustrt in simpl mnnr som bsic concpts.) For illustrtiv purposs, simpl fill procdur, for ssigning clls to componnts, ws thn usd tht llowd t most on componnt to ovrflow from on strip into th nxt strip. Clls r ssignd column-wis, top to bottom within strip i, nd thn th ssignmnt procdur is continud (still column-wis, but going in th othr horizontl dirction) in strip i+1. This procss ws trmd snk fill in [3], bcus of th mnnr in which it snks through th domin. In contrst to th simpl xmpl in figur 3, strip hights must b chosn crfully in this instnc in ordr to obtin good solution. Th totl primtr of this prtition is 294. Dfinitions - A row in grid is sid to b n nd-row of strip if it is th lst row in th strip (in figur 4, rows 6, 12, 18, nd 25 r nd-rows). A bgin-row is dfind similrly. A bgin-nd (row) pir dfins strip s th collction of clls with corrsponding row indics (in figur 4, strip 1 is dfind by th bgin-nd-row pir (0,6)). An optiml strip st (with rspct to givn fill procdur) is squnc of strips tht yilds minimum totl primtr rltiv to othr fsibl sts of strips, ssuming th givn fill procdur is utilizd for ch strip st. (Implicit in this dfinition is th proprty th combintion of fill procdur with st of strip hights uniquly dtrmins fsibl prtition. This proprty will b discussd furthr blow in conjunction with rstrictions on th fill procdurs tht w considr.) In sction 8, figur 13 shows four-strip prtition tht is optiml (with rspct to mor complx fill procdur) for th grid in figur 4. This lttr solution hs primtr 282. Dfinitions - A subst of th grid is sid to b filld if ll of its clls hv bn ssignd. Within strip, dividr cll is th lst cll ssignd by th fill procdur to complt prticulr lst componnt ssocitd with tht strip. (In 4, th upprcs A, D, G, nd I r dividr clls. For this xmpl, th fill procdur is such tht th dividr clls corrspond to th lst componnts tht cn fit ntirly within ch strip. Howvr, this nd not b th cs, nd in th implmnttion blow, w utiliz mor complx fill procdurs in which th dividr clls occur in othr componnts.) Th dividr cll concpt is dt structur tht limints th nd to kp trck of th individul clls in th strip U-turn rgion (dfind blow) in which componnts my cross strip boundris. (Th mthodologicl ids dvlopd blow r sily xtndd to mor gnrl pprochs in which th U-turn rgion is llowd to hv n rbitrry configurtion.) A bgin-nd-row pir is vlid if th corrsponding strip (trmd vlid strip) contins dividr cll. (Not tht th vlidity of bgin-nd pir is quivlnt to hving sufficint numbr 6

7 of clls ddd by th ppnding strip to llow th compltion of th ssignmnt of t lst on componnt. In th discussion blow, (b i, i ) will dsignt vlid bgin-nd pir unlss sttd othrwis.) A st of nd-rows (r 1, r 2,..., r k ) is sid to b fsibl squnc of nd-rows if: for i 0, (r i +1, r i+1 ) is vlid strip (whr r 0 = -1). Th corrsponding squnc of bgin-nd-row pirs is ((0,r 1 ), (r 1 +1,r 2 ),..., (r k 1 +1,r k )) (not: th us of vlid-strip squncs implis th xistnc of dividr cll within ch strip). W now dfin th ky gnric proprty tht w ssum for fill procdurs. This proprty mks it possibl to construct n optiml strip st for such fill procdur vi polynomil-tim mthod. Dfinition - A fill procdur is sid to b strip qusi-indpndnt (SQI) if, for ch strip i implid by fsibl squnc of nd-rows, th dividr cll position for th strip is uniquly dtrmind by th givn fill procdur nd th bgin-nd pir for th strip, nd is indpndnt of prior strips in th squnc. Morovr, w ssum tht th fill procdur complts th ssignmnt of th clls up to nd including th dividr cll in strip i, bfor ssigning ny othr clls in th domin. (S figur 7.) W will sy tht n ssignmnt is of strip-form if it my b obtind by pplying n SQI fill procdur to fsibl squnc of nd-rows. Th U-turn rgion for strip i consists of thos clls within strip i tht follow th dividr cll nd hnc r not ssignd until strip i+1 is considrd. (Givn strip nd dirction of ssignmnt, w cn numbr th clls in th strip squntilly, column-wis, procding from top to bottom in ch column. With such numbring, th U-turn rgion (if non-null) will b comprisd of th lst k clls of th strip for n pproprit k. In th simplst fill procdurs, k will lwys b lss thn th siz of componnt. Howvr, bttr rsults my b obtind by using mor complx fill procdurs tht llow vlus of k lrgr thn th componnt siz. For xmpl, not tht U-turn rgion consisting of singl spik column my ld to componnt with poor primtr whn th componnt ssignmnt is compltd in th nxt strip, sinc th totl primtr for th componnt will includ lrg contribution from th spik. This sitution my b llvitd by hving lrgr U-turn rgion consisting of th spik column plus th clls of th prvious componnt. In this cs, th fill procdur my thn hndl th U-turn rgion by using row-fill mthod within th U-turn rgion for th componnt tht fits ntirly within th U-turn rgion, in ssnc moving ssignmnts from wht would hv bn spik positions to clls in th row (or rows) immditly bov th nxt strip so tht th nxt componnt my b spik-fr nd hv good primtr.) Thos clls rquird in strip i+1 to complt th 7

8 ssignmnts for componnts only prtilly ssignd within th U-turn rgion of strip i r clld ovrflow. In figur 4, th uppr-strip rgions corrsponding to th clls ssignd to componnts b,, nd h r th U-turn rgions. W nxt considr dfinitions rlvnt to th stt grph usd in th shortst-pth problm dvlopd blow. Ths dfinitions provid insight into th ky ids ndd to construct optiml strip sts. Dfinition - A vrtx in th stt grph is uniquly dfind by bgin row, n nd row, nd dirction of ssignmnt within tht strip (lft-to-right or right-to-lft). In th figurs to follow, such vrtx is dfind by (b i, i, d i ). An ordrd tripl is vlid vrtx lbl if nd only if b i nd i corrspond to strip dtrmind within fsibl squnc of nd rows. For strip-form solution, dfin prim (b i 1,b i, i,d i ) to b th primtr of th componnts flling within th rgion comprisd of: (1) th U-turn rgion of strip i-1 (if ny), nd (2) th non-u-turn rgion of strip i. (In th construction of th stt grph, prim(b i 1, b i, i, d i ) is th lngth ssignd to th dg conncting (b i 1, i 1, d i 1 ) nd (b i, i, d i ) nd rprsnts th primtr incrmnt corrsponding to ppnding strip i. Th SQI proprty gurnts tht this lngth is dtrmind solly by nods (b i 1, i 1, d i 1 ) nd (b i, i, d i ) nd is indpndnt of th othr nods (strips). Not lso tht th primtr incrmnt corrsponds to th primtr of t lst on complt componnt nd only involvs primtr contributions corrsponding to complt componnts.) 3 An Ovrviw of Striping Algorithms Yckl-Myr [17], Christou-Myr [3], nd Mrtin [12] showd tht striping tchniqus producd vry good (nd, for som problm clsss, symptoticlly optiml) prtitions whn pplid to rgulr grid grphs. Any lgorithm bsd on striping cn b brokn into two min prts: 1) slcting strip-hights, nd 2) ssigning th clls within strip ( fill procdurs ). Yckl-Myr nd Christou-Myr us gntic lgorithms for gnrting strip hights for gnrl clss of grid grphs. Mrtin considrs rctngulr domins nd trnsforms th originl 8

9 problm into knpsck problm. Both mthods for slcting strip hights hv limittions. In gnrl, in th worst cs Christou-Myr rquirs t lst supr-polynomil ( constnt grtr thn on whos xponnt is risd to frctionl powr) tim for discovring n optiml solution. Mrtin s pproch cn only b pplid to rstrictd st of problms with rctngulr domins nd qul-sizd componnts. (It is not known wht th computtionl complxity is for Mrtin s mthod; but, sinc th problm is formultd s knpsck, it is xpctd to b NP-Complt.) Th Donldson-Myr mthod dscribd blow limints both of th brrirs whn slcting strip hights. Within-strip ssignmnt fill procdurs hv hs bn ddrssd xtnsivly. For rctngulr grids, w hv shown undr crtin conditions tht th Christou-Myr fill procdur [3] producs loclly optiml solutions. W hv lso dvlopd mor complx ssignmnt procdurs tht produc loclly optiml solutions undr mor rlxd conditions. Proofs of ths lst two proprtis cn b found in [6] nd [7]. Th bsic id of strip-bsd fill procdur is tht componnts with nrly miniml primtr cn b obtind by pplying column-wis ssignmnt in th intrior of strips of pproprit hight, nd thn using spcil procdurs in th rmindr of th strip (th U-turn nd ovrflow rgions) to dl with componnts tht cross strip boundris. (Strips tht r too tll or too short will rsult in componnts with lrg primtrs whn such fill procdurs r usd.) This ppr focuss on th dvlopmnt of mthods tht gurnt optiml choics of strip hights ssuming tht n SQI fill procdur is utilizd. 4 Dfining Subproblms 4.1 Idntifying Subproblms In this sction, mthod for brking up th originl problm into similr but smllr subproblms will b prsntd. This is th ky stp in th lgorithms tht will b prsntd ltr nd corrsponds to th construction of th nods in th stt grph. Dfinition - Lt (b i, i ) b vlid bgin-nd pir, lt d i b th dirction of ssignmnt within this strip, nd lt c i b th dividr cll ppring within this strip. W sy tht b i, i, nd d i dfin th vlid subproblm of dtrmining n optiml prtition of th domin whos lst cll is c i. In figur 5, which illustrts th domin of subproblm, th dfinition of dividr cll implis 9

10 tht n intgrl numbr of componnts cn b ssignd to th clls in th union of rgions A nd B, whr c i dnots th dividr cll in strip i. A A B b_i _i B c_i Domin of Originl Problm Domin of Subproblm Figur 5: Subproblm xmpl (If th grph is symmtric, thn only b i nd i r ndd to uniquly dfin subproblm; othrwis dirction of ssignmnt bcoms fctor. W focus on th non-symmtric cs in this ppr.) In th non-symmtric cs in figur 6, not tht th ssignmnts for th lst strips r in opposit dirctions. Although both subproblms in th figur corrspond to th sm bginrow nd-row pir, th configurtions (U nd U ) of rmining U-turn rgions r diffrnt. This diffrnc will gnrlly ld to diffrnt primtr incrmnts whn th nxt strip is ddd to th subproblm, hnc it is ncssry to includ dirction informtion to idntify subproblm. In th discussion to follow, w considr only strips tht contin dividr cll so tht thy corrspond to vlid subproblms. 4.2 Slcting Dividr Cll In th rgumnts to follow, w ssum n SQI fill procdur so tht th choic of dividr cll c i for givn strip (b i, i,d i ) is indpndnt of th hight of prvious strips. Figur 7 indicts how bgin-row, nd-row, nd dirction of ssignmnt dtrmin domin tht is indpndnt of prvious strips nd subproblms. This dtrministic wy of uniquly dfining subproblms vi th tripl (b i, i, d i ) llows th us of fficint dynmic-progrmming tchniqus for coordinting subproblm rsults nd limints rdundnt clcultions. This pproch to dfining subproblms lso provids usful wy of idntifying vrtics nd clculting dg wights in th corrsponding shortst-pth problm (s 10

11 b_i U _i U c_i c _i Figur 6: Dirction of ssignmnt ffcts th configurtion of U-turn rgions. Diffrnt U-turn configurtions b_i _i Lst cll ssignd (dividr cll) for both strip squncs is th sm. Figur 7: Th finl bgin-row, nd-row, nd dirction uniquly dtrmin domin tht is indpndnt of th prcding strips. 11

12 Subproblms: Primtrs: Sub. No. b i i d i Strip Subproblm Incrmnt Primtr Tbl 1: subst of th dynmic-progrmming dt for figur 4 sction 9.2, Dscription of Stt Grph ). Considr th prtitiond grid in figur 4. Exmpls of th two wys tht portion of th subproblm dt cn b prsntd r shown in tbl 1 nd figur 8. Tbl 1 contins th primtr dt s orgnizd by th dynmic-progrmming implmnttion. Figur 8 prsnts th sm dt in th form of wightd grph, with vrtics corrsponding to subproblms nd dg wights qul to th primtrs for th ppnding strips. Th totl lngth of th pth is th totl primtr of th corrsponding prtition, which is 294 (this is lso th finl subproblm primtr shown in th fr right-hnd column in tbl 1). This is not n optiml prtition. An optiml strip-bsd prtition, with primtr qul to 282, is shown in figur 13. This lttr prtition would thus corrspond to lrgr DP tbl nd nothr pth from th unssignd nod with diffrnt intrmdit nods nd totl lngth Unssignd Grid (0,6,0) (7,12,1) (13,18,0) (19,25,1) Assignd Grid Figur 8: Pth dt corrsponding tbl 1 5 Grid Prtitioning s Shortst-Pth Problm Constructing fsibl squnc of nd-rows tht prtitions grid is quivlnt to dtrmining pth from n origin (unssignd grid) to dstintion (ssignd grid) in stt grph (to b formlly dfind ltr). Th grph in figur 8 shows portion of stt grph. For givn squnc of nd-rows (0, r 1, r 2,...,r n ), w know for ny conscutiv nd-rows, r i, r i+1, tht (r i +1,r i+1,d i+1 ) 12

13 is rchbl from (r i 1,r i,d i ) if nd only if th bgin-nd-row pir (r i + 1, r i+1 ) is vlid strip nd d i d i+1. This pproch should b contrstd with th ltrntiv in which th nods of th stt grph qul twic th numbr of vlid nd-row squncs. In this ltrntiv stt grph, vrtics would hv th form (d 1, r 1, r 2,...,r i ), whr d 1 is th initil fill dirction nd th {r i } form vlid nd-row squnc. It will b provd undr vry limitd ssumptions tht th numbr of vrtics in th stt grph corrsponding to this scond mthod is t lst supr-polynomil. Th Donldson- Myr (DM) mthod uss th SQI proprty to rduc th numbr of vrtics in th stt grph to twic th numbr of vlid bgin-nd pirs, which is polynomil with rspct to input siz. 6 Proprtis of Subproblms 6.1 Rltionship btwn Subproblms nd Strips A strip is dfind by bgin-nd row pir nd dirction of ssignmnt. As sttd rlir, such tripl (b i 1, i 1,d i 1 ) lso uniquly dfins subproblm nd th U-turn rgion ppring within strip i-1. By th sm rsoning, (b i, i,d i ) dtrmins th U-turn rgion for strip i. W now hv mchnism for dfining th clls to b considrd in th primtr incrmnt computtion nmly thos clls tht r ppndd to subproblm (b i 1, i 1,d i 1 ) in ordr to gt subproblm (b i, i,d i ) (not: this rgion dos not includ th clls in th U-turn rgion for strip i ). Thos clls ppring within this ppnding strip rgion tht corrspond to n intgrl numbr of procssors my b ssignd indpndntly of ll othr clls. A fsibl solution of th problm cn thn b thought of s squnc of ppnding strips. 6.2 Construction of Strip-bsd Solution Primtr is computd in our pproch incrmntlly, strip-by-strip s follows (s figur 9): primtr in th first strip (0, 1,d 1 ) (not including its U-turn rgion in U 1 ) is computd, thn th primtr for (b 2, 2,d 2 ) + U 1 - U 2, is ddd, tc. Figur 9 shows fourth itrtion of this procss. 6.3 Th Existnc of Common Subproblms Figur 10 shows two diffrnt squncs of strip-hights usd to prtition th givn grid. Both css contin th subproblm nding with th strip (b 5, 5,d 5 ). 13

14 b_3 3 3 U_3 b_4 U_3 4 _4 4 Figur 9: Incrmntlly computing primtr 14

15 Solution Solution b_i _i _i _i+1 Figur 10: Fsibl solutions with th sm subproblm (strips 1-5). 15

16 Whn th sm subproblm pprs in svrl othr lrgr problms, th ovrll problm is sid to possss th proprty of common subproblms. With rspct to th stt-grph viwpoint, subproblm is nod nd this nod my ppr in multipl pths. 6.4 Optiml Substructur within Solution Cormn, Lisrson, nd Rivst [5] stt tht problm xhibits optiml substructur if n optiml solution cn b dcomposd into subproblm solutions ch of which is optiml. O p t i m l P r t i t i o n Strip prtition for th ith subproblm _1 _2. _i... _t Figur 11: Optiml Substructur Givn grid, ssum tht th squnc of nd-rows ( 1, 2,... t 1, t ) producs n optiml solution. Considr th subproblm nding t strip (b i, i,d i ) (s figur 11). For this subproblm, if th subsqunc ( 1, 2,... i 1, i ) ws not optiml for th corrsponding subproblm, thn bttr strip prtition for this subproblm ( 1, 2,..., i 2, i 1, i ) (notic tht th lst strip must 16

17 hv th sm bgin-row nd-row pir for ny solution for th subproblm) could b substitutd into th lrgr solution. Bcus of th SQI proprty, this nw squnc would produc bttr ovrll solution, which would contrdict th fct tht w prviously hd n optiml solution. From stt-grph viwpoint, th shortst pth hs th proprty tht ch sub-pth (bginning t th th strt nod) is lso shortst pth. 7 Strip-Hight Slction In this sction, w focus on th lowr bound of supr-polynomility on th numbr of stripform solutions. Givn n MxN rctngulr grid, Mrtin [12] convrts th originl problm into knpsck problm nd uss knpsck softwr [11] for solving th rformultd problm. This pproch will produc th bst strip-bsd solution, undr th following ssumptions. In ddition to th rctngulr grid ssumption, Mrtin [12] ssums tht MN/P is n intgr nd tht only strips contining n intgr numbr of componnts r llowd. Undr ths constrints, thr is no ovrflow into th nxt strip nd th primtr in ch strip is indpndnt of th othr strips, nd indpndnt of its position within th grid (s figur 12). Thrfor, for strip of givn hight, thr will b uniqu primtr ssocitd with tht hight, ssuming tht it is prmissibl hight corrsponding to n intgrl numbr of componnts in th strip. Th problm cn thn b thought of s th slction of st of hights totling to M with minimum totl primtr. With rspct to th knpsck problm, th strip hights corrspond to th constrint cofficints nd th strip-componnt primtr totls corrspond to th objctiv cofficints. Optiml solutions of th knpsck problm corrspond to optiml strip prtitions (ssuming column-wis ssignmnt nd prmissibl strips hights) of th rctngulr domin. For th CM mthod of [3], th uthors rlx mny of ths rstrictions (intgrl numbr of componnts pr strip nd rctngulr grids). Sinc CM is gntic lgorithm (GA), it dosn t in gnrl produc gurntd optiml strip prtitions, but dos gnrt high-qulity prtitions. Th numbr of diffrnt fsibl solutions tht r trid by th GA cn b limitd to gurnt polynomil running tim, but no qulity gurnt cn in gnrl b ssocitd with th rsult. Whrs th typ of problm tht Mrtin [12] studid could b rprsntd by thr numbrs (th lngth of th input bing th logs of th dimnsions of th rctngulr domin nd P), w ssum tht full djcncy mtrix or djcncy list is rquird to rprsnt gnrl grid grph. This ssumption implis tht tht th lngth of input is boundd blow by (V + E). Sinc E is 17

18 h Sm squnc of componnts in nothr strip. h Sm squnc of componnts in strip 1. Figur 12: Rctngulr domin cs illustrting Mrtin ssumption tht strip primtr dpnds only on strip hight (not position within domin). 18

19 O(V), w hv tht th lngth of input Ω(V) = Ω(numbr of clls in th grid). Tht riss th qustion of whthr or not it is possibl, in rsonbl mount of tim (i.., polynomil tim), to vlut th primtr for ll possibl strip-hight squncs (ssuming givn fill procdur). In this nd ltr sctions, it is ssumd tht th strips will run prpndiculr to th mjor xis of th grid, ssumd to b th lngth M. (In prctic, striping should b prformd rltiv to both domin orinttions, with th bttr rsults thn bing chosn.) By mking this ssumption, bounds r thn xprssd in trms of th siz of th originl problm. Th Christou-Myr lgorithm [3] uss th notion of bs hights. Th bs hight, k, for givn grid is qul to th floor of th squr root of th r to b ssignd to procssor. For this discussion, w will mk th following ssumptions: 1. Th numbr of rows in th grid, M, is grtr thn (k+1) k Strips r prpndiculr to th mjor xis of th grid. 4. Any st of conscutiv k-1 rows contins t lst on dividr cll. Th first ssumption is strongr thn tht of Christou-Myr [3] nd is ndd soly for th bound in this sction (not for th lgorithm). Th scond nd fourth ssumptions r rquird to gurnt rng of vlid strip hights in th proof. Th third ssumption is ndd to bound th clls in th grid by function of M. Christou nd Myr [3] showd tht thr xist non-ngtiv intgrs α nd β such tht M = α(k) + β(k+1) This squnc of α k s nd β (k+1) s will now b clld bs fsibl solution. In ordr to discovr lowr bound on th totl numbr of vlid strip-hight squncs, w nd only considr squncs for which vry strip hight flls within th following rng: [ k-1, k, k+1, k+2]. With ths ssumptions, w now hv th following thorm: Thorm 1 Th numbr of fsibl strip-form solutions is Ω (3 M 0.5 /4 ). Proof In ordr to prov this rsult, w nd only look t thos bs fsibl solutions tht contin S = α + β strips. 19

20 Assum tht α > β (if β > α similr rgumnt with β rplcing α cn b givn). Th strips ssocitd with α r of hight k. Lt s furthr divid th α strips into two pproximtly qul groups, so tht th smllst group is of siz t lst α/2-1. Cll this group of strips th indpndnt strips. Th rmining strips of hight k long with ll th β strips will b clld th dpndnt strips. For ny strip within th st of indpndnt strips, th hight my b incrsd/dcrsd by on or my rmin th sm ( totl of thr possibl hights), with rspct to th bs hight, providd th opposit chng is md to corrsponding dpndnt strip. Thus th numbr of fsibl solutions is t lst 3 α/2 /3 3 S/4 /3 3 M/(k+1)4 /3 3 M 0.5 /4 /3 Th scond inqulity follows from th fct tht w hv: M M/(k+1) = α(k) + β(k+1) = α(k)/(k+1) + β(k+1)/(k+1) = α(k)/(k+1) + β S Th third inqulity follows from th ssumption tht M (k+1) 2 To convrt this bound into th units of th input, not tht M (numbr of clls in grid) M 2 Th numbr of strips is supr-polynomil with rspct to th numbr of clls in th grid, sinc 3 M 0.5 /4 /3 3 (no.ofclls)0.25 /4 /3 From this rsult, w know tht thr r t lst supr polynomil numbr of fsibl solutions nd tht n xhustiv srch it is not prcticl. In [2], n uppr bound of 2 M 1 is givn, but this bound llows strips tht my not contin nough clls to complt th ssignmnt of procssor, nd hnc do not conform to th strip-form solutions considrd hr. 20

21 8 An Exmpl Illustrting th Mthods Th xmpl in figur 13 will b usd to dmonstrt both th th shortst-pth pproch nd th dynmic-progrmming pproch. Th solution shown is strip-fill optiml, in contrst to th non-optiml solution of figur 4, n ltrntiv strip-bsd prtition of th sm domin. (It is lso sy to s tht this solution is loclly optiml (rltiv to two-cll swps) in th contxt of gnrl (not ncssrily strip-form) prtitions. Howvr, it is not globlly optiml, sinc th positions of th 1 s in th cntrl column my b intrchngd with th 2 s to improv th totl primtr. This illustrts th potntil for improvd solutions bsd upon vn mor sophistictd fill procdurs or post-procssing procdurs tht focus on improving componnt boundris. This topic is discussd furthr in th concluding sction.) Figur 13: A strip-fill optiml solution 21

22 Subproblm Dt Prvious Subproblm Strip Totl b i i d i Prim Prim b i 1 i Tbl 2: Subst of primtr dt for strip prtitioning th grid in figur 13 22

23 9 A Stt-Grph Rprsnttion of Grid Grph Prtitioning (0,8,0) 34 (0,8,1) 120 (9,14,0) (15,18,0) (19,25,0) Unssignd Grid 0,9,0) (9,14,1) (15,18,1) (19,25,1) (0,9,1) 34 (10,14,0) (15,19,0) 94 (20,25,0) Assignd Grid (0,10,0) (10,14,1) (15,19,1) (20,25,1) Shortst Pth Lngth = 282 (0,10,1) (11,19,0) (11,19,1) Figur 14: Prtil stt-grph (shortst pth shown vi htchd dgs nd dg lngths circld) 9.1 Bckground An optiml prtition of grid grph cn b thought of s squnc of vlid strips, whr ch strip is ddd incrmntlly. Aftr ch strip is ddd, nw vlid subproblm is gnrtd. Th discovry of n optiml solution cn b thought of s going from th initil unssignd stt to th finl fully ssignd stt in n optiml fshion. If solution for (sub)problm is rprsntd by th squnc of strip-dfining nd-rows (r 1, r 2,...,r n ), th prvious thorm showd tht th numbr of such stts is supr-polynomil. Howvr, it ws shown bov tht th SQI fill proprty implis tht bgin-nd-row pir nd dirction of ssignmnt cn b usd to dfin subproblm. Thus, instd of th squnc (r 1, r 2,...,r n ), ll tht is rquird is th bgin-nd pir corrsponding to th currnt strip. Whn this 23

24 rprsnttion is usd, th siz of th stt grph bcoms polynomil. Figur 14 shows th stt grph for th dt contind in tbl 2. Th unssignd grid (sourc nod) is connctd to nods tht corrspond to n initil vlid strip hight. Ech of ths nods is thn connctd to nods whos corrsponding subproblms rprsnt vlid strip xtnsion, tc. Th pth contining th cross-htchd dgs corrsponds to n optiml prtition of th originl grid grph. 9.2 Proprtis of th Stt Grph In th stt grph, th numbr of vrtics quls th totl numbr of vlid subproblms. An dg conncts two vrtics v i nd v i+1 if nd only if vlid strip cn b ppndd to th subproblm dfind by v i to gt th subproblm dfind by v i+1. Lt th ordrd tripl (b i, i, d i ) rprsnt givn subproblm(vrtx). At most, thr r M diffrnt choics for b i. Onc b i hs bn dtrmind, th totl numbr of vlid strip hights (cll this numbr h(b i ), whr h(b i ) M) is n uppr bound on th numbr of diffrnt possibl vlus tht i my ssum. Lt h = mx(h(b i )), ovr ll ligibl vlus of b i (not tht h my lso b boundd by th siz of th lrgst componnt, sinc no strip should b tllr thn tht vlu). For ny givn strip, thr r only two dirctions of ssignmnt, hnc w gt th following bound: totl numbr of vrtics (subproblms) 2Mh 2M 2 Th inqulity coms from th fct tht undr crtin circumstncs not ll M rows my b bgin rows nd not ll bgin-nd-row pirs, (b i, b i + k), k = 0,..., h-1, rprsnt vlid subproblm. In th shortst-pth problm, thr will b two dditionl vrtics: sourc (unssignd grid) nd sink (ssignd grid). For givn subproblm, n uppr bound on th numbr of diffrnt strips tht my b ppndd to tht subproblm quls th mximum numbr of vlid strip hights, h. tht: It follows totl numbr of dgs th totl numbr of vrtics * h 2Mh 2 Whn constructing th stt grph, th numbr of dgs cn b rducd by th obsrvtion tht only th optiml prcding dg nds to b kpt for ch nod. Rcll tht th incrmnt to th primtr for th componnts ssignd in U-turn rgion nd n ppnding strip bcoms th wight of th dg conncting vrtics (b i 1, i 1,d i 1 ) nd (b i, i, 24

25 d i ), dnotd s prim(b i 1, b i, i, d i ). (In slight bus of nottion, w us prim(0, 0, 1, d 1 ) to dnot th primtr of th first strip (not including th first U-turn rgion).) Figur 15 shows portion of stt grph, undr th ssumption tht th initil strip lwys is ssignd lft to right. W mk this ssumption blow to rduc th complxity of th proofs, but th xtnsion to th bi-dirctionl cs illustrtd in figur 14 is strightforwrd. (0,h1,0) (*,M,*) (0,h2,0) prim(*,*,m,*) prim(0,0,h1,0) 0 Prim(0,0,h2,0) prim(*,*,m,*) Intrmdit (*,M,*) 0 Strips Unssignd Grid (*,M,*) 0 Assignd Grid prim(0,0,hn,0) (0,hn,0) prim(*,*,m,*) 0 prim(*,*,m,*) (*,M,*) First Strip Finl Strip Figur 15: Stt grph 9.3 Proof of Optimlity W must now show tht th pth corrsponding to n optiml striping is contind within th grph. Thorm 2 Th prviously dfind grph contins pth corrsponding to vry fsibl squnc of nd-rows. nd-rows. Convrsly, vry pth bginning t nod 0 corrsponds to fsibl squnc of 25

26 Proof - (by induction) Bs Cs - A squnc of lngth on. Givn tht ( 1 ) is fsibl squnc, (0, 1 ) is vlid strip. It follows tht vrtx (0, 1,0) nd th corrsponding dg (sourc vrtx)-(0, 1,0) xists within th grph. Induction Hypothsis - Givn fsibl squnc of nd-rows of lngth n, thr xists pth in th grph from th unssignd grid nod to th nod corrsponding to th lst strip in th squnc. Induction Stp - Givn squnc of n+1 vlid strips, by th induction hypothsis, thr xists pth from th unssignd grid nod to th nod corrsponding to th nth strip. Sinc ll n+1 strip hights r vlid, thr must b n dg from th nod corrsponding to th nth strip to th nod corrsponding to th lst strip. Th convrs follows dirctly from th wy th stt grph ws constructd. Lmm 1 Th shortst pth in th trnsformd grph corrsponds to n optiml strip-bsd solution. Proof (by contrdiction) - From thorm 2, w know tht trnsformd grph contins th pth for n optiml solution. Th vlu of th primtr for th optiml striping must qul th shortst pth in th grph. For if this wr not th cs, thn th strip hights corrsponding to th shortr pth would produc bttr solution in th originl grid grph, which would b contrdiction. Evry pth within stt grph rprsnts vlid squnc of strip hights. Any shortst pth for stt grph thn rprsnts strip-fill optiml solution(sfo) tht is optiml with rspct to ll strip-form solutions ssocitd with th givn fill procdur. 10 A Dynmic-Progrmming Approch 10.1 Bckground Erlir, it ws shown tht n optiml strip-bsd prtition of grid grph possssd two qulitis: optiml substructur nd common subproblms. Ths r two proprtis tht r prsnt whn 26

27 dynmic progrmming is pplicbl. For th rmindr of this this sction, n optiml solution will b dissctd to provid intuition s to why rcurrnc cn modl th optiml strip-bsd solution. Tbl 2 contins th striping informtion for n optiml solution for n xmpl with givn smll st of strip hights nd givn strip ssignmnt (fill) procss. Thr r four diffrnt lst strips in this cs. Th optiml solution corrsponds to th bst of ths four diffrnt possibilitis, nod (20, 25, 1). Th primtr ssocitd with this nod thrfor stisfis: optiml primtr Prim(20, 25, 1) = incrmntl primtr corrsponding to lst strip in n optiml solution + primtr for subproblm with this lst strip rmovd) = 94 + Prim(15,19,0) Not tht th vlu of Prim(15,19,0) is lso optiml for th corrsponding subproblm, sinc it rprsnts th optiml solution of tht subproblm obtind by considring ll possibl dgs lding to th nod (15,19,0). Similr rgumnts pply to th othr nods on th pth lding bck to th unssignd grid nod 10.2 A Rcurrnc Rltion for Grid-Grph Prtitioning This sction is dvotd to formlly dfining dynmic progrmming rcurrnc tht modls th optiml solution. Th proof tht th rcurrnc modls n optiml prtition closly follows th proof for th shortst-pth solution nd cn b found in [7]. To simplify th prsnttion, w ssum tht th first strip is ssignd lft to right. It is sy to gnrliz th rcurrnc to llow th first strip to b ssignd in ithr dirction in ordr to mtch. W will now dfin rcurrnc tht my b usd to obtin th primtr for subproblm dfind by (i,j,d), givn th primtr of smllr subproblms. This vlu will b stord in n rry Prim(i,j,d). W nd th following dfinitions whn dfining th rcurrnc. Dfinitions d = dirction of ssignmnt V = numbr of clls in th grid VALID-STRIPE = th st of bgin-nd-row pirs, (b i, i ), such tht ( i - b i + 1) is lss thn or qul to h (th mximum llowbl strip hight) nd th corrsponding rows contin 27

28 dividr cll. P rim(i, j, d) = 5V prim(0, i, j, d) min b (P rim(b, i 1, (d 1)mod2)+ f(b, i, j, d)) if (i,j) is not in VALID-STRIPE if (i = 0) nd ((i,j) is in VALID-STRIPE) b s.t. (b,i-1) is in VALID-STRIPE whr 1 if Prim(b,i-1,(d-1) mod 2) 5 * V f(b, i, j, d) = prim(b, i, j, d) othrwis Tim Anlysis - Th dynmic-progrmming rry, Prim, is M x M, whr M is th numbr of rows in th grid. For this structur, crtin fcts follow: 1. For ch row of Prim, th numbr of clls tht contin non-trivil strip dt is lss thn or qul to th mximum vlid strip hight,h. 2. For th ntir rry Prim, it thn follows tht th totl numbr of clls which will contin non-trivil strip dt is lss thn or qul to h*m. Ech cll in th dynmic-progrmming rry rprsnts uniqu subproblm within th grid. In ordr to dtrmin th corrsponding ntry in th dynmic-progrmming rry th following must b don. (This tim nlysis contins dtils of th ctul implmnttion. In prticulr, it rfrncs th nin ltrntiv ssignmnt tchniqus (bsd on vrious combintions of column-wis nd row-wis ssignmnts) tht r vlutd pr U-turn rgion. Aftr ll nin r vlutd, th bst is slctd, s [7] for mor dtils. An ltrntiv tchniqu bsd on rcursiv ppliction of th gnrl pproch to th U-turn rgions is considrd in th finl sction of this ppr. Also, w will dfin th vribl N to b th lngth of th minor xis nd V = min(v, hn).) 28

29 1. Crt th rry of clls to b ssignd whn ppnding strip. Thr will b diffrnt rry for ch llowd b i. Tim = O(h V ) 2. For ch strip-subproblm combintion, comput nin ltrntiv ssignmnts of th clls of th U-turn/ovrflow rgion. Tim = O(h V ) 3. For ch strip-subproblm nd ch strip ssignmnt, clcult th totl primtr. (This cn b don by looking t ch ssignd cll nd dding this mount to th primtr for th subproblm.) Tim = O(h V ) Th totl work for ch lmnt in Prim is thus O(h V ). Th totl mount of work to fill in th non-trivil clls for th dynmic-progrmming rry is O(Mh 2 V ). Th mount of tim to initiliz th dynmic-progrmming rry is O(M 2 ). Th ovrll tim bcoms O(M 2 + Mh 2 V ). How dos this rlt to th lngth of input? It ws ssumd tht to rprsnt this problm tht n djcncy mtrix or list is rquird. This would forc th lngth of input to b t lst qul to th numbr of vrtics in th grid. Assuming binry rprsnttion, tht would forc th lngth of input to b Ω(V). 11 Implmnttion 11.1 Softwr nd Hrdwr Issus All coding ws don using th C progrmming lngug, which ws compild using g++, GNU projct C++ compilr [1]. All softwr dvlopmnt ws don on Pntium Pro 200Mhz mchin with 64MB of RAM. All rsults wr gnrtd using this Pntium mchin nd n Ultr Entrpris 6000 Srvr. 12 Rsults Tbl 3 contins th rsults for svrl non-rctngulr grids compring METIS/Pmtis/Kmtis (s [7]), CM (th gntic lgorithm vrsion, gin s [7]), nd th DM lgorithm. Not tht DM substntilly outprforms METIS in ll css, nd tis or outprforms CM xcpt for th first tst problm. For this problm CM did bttr for th following rson: undr crtin conditions, CM 29

30 Shp Clls Comps METIS CM DM DM - CM rror rror rror bound % (h bound, Rltiv bound % bound % mx obsrvd hight) Improvmnt(%) Dimond (30,20) Dimond (20,15) Ellips (20,7) 0 Ellips (20,4) Torus (40,28) 8.61 Torus (20,14) Tbl 3: Comprison of prformnc: DM, CM nd METIS. hndls ssignmnts for th ovrflow from th U-turn rgions diffrntly thn by DM. W discuss blow rcursiv procdur for hndling ths ssignmnts tht should outprform th huristics of both CM nd DM. For th ctul prtitions producd by th DM mthodology, s figurs 16, 17, nd 18. Uppr bounds (h) on strip hights usd md to rduc th mount of computing. Th bounds wr st highr thn th mximum obsrvd hights in th solutions. Figur 16: Th DM prtition of th 823-cll llips into 64 prts. 30

31 Figur 17: Th DM prtition of th 7696-cll torus into 64 prts. 31

32 Figur 18: Th DM prtition of th 4019-cll dimond into 64 prts. 32

33 13 Conclusions From this rsrch thr min rsults hv mrgd. Thorm 1 dmonstrts tht brut-forc pproch hs running tim which is t lst supr-polynomil. Th scond mjor contribution of this rsrch is th dcomposition of th originl problm into smllr problms bsd on th principl of optimum substructur. From this dcomposition, two concpts hv mrgd. Th first is th id of strip-qusi- indpndnt(sqi) fill procdur. A SQI fill procdur llows th primtr vlu for subproblm ( i 1, i ) to b computd s th primtr vlu for th subproblm ( i 2, i 1 ) plus th primtr for th componnts ppring in th corrsponding u-turn rgion for strips i-1 plus th primtr for th componnts ssignd in strip i tht fll outsid of th u-turn rgion for strip i. Th scond concpt is strip-fill optiml solution(sf). A SF optiml solution is on tht is optiml with rspct to th st of fsibl solutions obtind by: 2) spcifying n SQI fill procdur. 2) pplying th fill procdur to vlid squncs of nd-rows. Th third mjor contribution of this rsrch is th orgniztion of th solution st. W hv dvlopd dynmic progrmming huristic tht xploits th proprty of common substructur. Through th us of dynmic progrmming, th st of solutions for th subproblms cn b orgnizd in such mnnr so s to limint ny dupliction of work for solving subproblms. As rsult of this pproch, w hv discovrd huristic tht will produc strip-fill optiml solution in polynomil tim. Futur dirctions for rsrch includ vrious rfinmnts nd xtnsions of th pproch in th 2-D cs s wll s gnrliztion to th 3-D cs. Th most troublsom subrgions with rspct to primtr contributions r th U-turn nd ovrflow rgions. In ordr to improv on th primtr contributions from ths rgions, rcursiv pproch could b tkn. Tht is, th strip-bsd pproch usd for th domin s whol could b dirctly pplid to thos subdomins, in ffct llowing compltly diffrnt striping mod to b usd for th subdomins. (Th striping for subdomin could vn b don vrticlly s opposd to horizontlly.) Anothr lvl of rcursion could vn b prformd for nw U-turn rgions gnrtd within ths subdomins by thir modifid striping. Anothr pproch for voiding bd componnts in th U-turn rgion would b to kp rcord of th componnt sizs usd thus fr, nd to dtrmin th mix of componnt sizs tht bst fits (in th sns of minimizing ovrflow) nwly ddd strip ( rcord of usd or 33

34 rmining componnt sizs could b crrid long with th othr nod informtion in th shortst pth pproch). A vrity of huristics for post-procssing th optiml strip-form solution r undr considrtion. On mthod would us modifiction of Krnighn-Lin huristic focussing on th primtr clls in U-turn rgions nd ny strips whos componnts r not of good qulity, mploying swp slction prioritis dtrmind by primtr contributions. To xtnd th stripslction procss to th 3-D grid grphs only rquirs modifiction s to how subproblm is to b dfind. Th bsic notion of focussing on strip-form solutions gnrlizs in n obvious wy to solutions obtind by ssigning within horizontl slics of ppropritly chosn widths, with spcil pprochs for U-turn nd ovrflow rgions t th nds of slics. Rfrncs [1] ANONYMOUS. [2] I. Christou. Distributd Gntic Algorithms for Prtitioning Uniform Grids. PhD thsis, Univrsity of Wisconsin - Mdison, August [3] I. T. Christou nd R. R. Myr. Optiml qui-prtition of rctngulr domins for prlll computtion. Journl of Globl Optimiztion, 8:15 34, Jnury [4] Ann Condon. Clss nots for CS 577 t th Univrsity of Wisconsin-Mdison. [5] T. H. Cormn, C. E. Lisrson, nd Ronld L. Rivst. Introduction to Algorithms. McGrw- Hill, Nw York, [6] W.W. Donldson. Loclly Optiml Striping for Rctngulr Grids. Mnuscript, [7] W.W. Donldson. Grid-Grph Prtitioning. PhD thsis, Univrsity of Wisconsin - Mdison, My [8] M. R. Gry nd D. S. Johnson. Computrs nd Intrctbility: A Guid to th Thory of NP-Compltnss. W.H. Frmn, [9] S. Ghndhrizdh, R.R. Myr, G. Schultz, nd J.Yckl. Optiml blncd prtitions nd prlll dtbs ppliction. ORSA Journl on Computing, 4: , [10] Udi Mnbr. Introduction to Algorithms: A Crtiv Approch. Addison-Wsly,

35 [11] S. Mrtllo nd P. Toth. Knpsck Problms: Algorithms nd Computr Implmnttions. Wily, [12] W. Mrtin. Fst qui-prtitioning of rctngulr domins using strip dcomposition. Discrt Applid Mthmtics, 82: , [13] R. J. Schlkoff. Digitl Img Procssing nd Computr Vision. John Wily & Sons, [14] Donld F. Stnt nd Dvid F. McAllistr. Discrt Mthmtics in Computr Scinc. Prntic-Hll, [15] J. Strikwrd. Finit Diffrnc Schms nd Prtil Diffrntil Equtions. Wdsworth & Brooks, [16] J. Yckl, R. R. Myr, nd I. Christou. Minimum-primtr domin ssignmnt. Mthmticl Progrmming, 78: , [17] J. Yckl, R. R. Myr, nd I. Christou. Minimum primtr domin ssignmnt. Mthmticl Progrmming, 78: ,