The Importance of Ordering the Number of Lattice Points Inside a Rational Polyhedron Using Generating Functions

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1 Ieraioal Joural of Copuer Sciece ad Elecroics Egieerig (IJCSEE Volue Issue ( ISSN 48 (Olie he Iporace of Orderig he Nuber of Laice ois Iside a Raioal olyhedro Usig Geeraig Fucios Halil Sopce Absrac I pure ad applied aheaics here are differe probles ha ivolve couig he uber of laice pois i covex bouded polyhedro shor called he polyope. his paper provides he ehology for obaiig he raioal geeraig fucio fro which is possible o fid he uber of laice pois iside a raioal polyhedro. We offer wo differe varias of fidig he geeraig fucio. he secod uses he so called MacMaho s oega calculus. Usig his heory a give polyhedro is augeed io he syse of liear equaliies ad he he ools of his heory offers possibiliy o fid he correspodig raioal geeraig fucio. o ephasis he iporace of his ehology we prese a ieresig applicaio i parallel processig. his ca be e usig he equivalece aog orderig he uber of laice pois i polyopes ad he lower boud of processor elees for parallel soluio of he esed loop probles. We used his o fid he lower boud i he case of algorih for arix-arix uliplicaio. I is e ruig he progra Diophaie Gf which gives he correspodig bioial forula obaied fro he geeraig fucio. eywords geeraig fucio uber of laice pois polyopes syse of Diophaie equaliies raioal coe lower boud of processors. I. INRODUCION WO coo quesios for a give polyhedro are: How ay ieger pois es coai? Ad how ay ieger pois es coais where is a give posiive ieger. More cocerig he aalysis fro he aheaical poi of view ad how he rasforaios are e he reader ca fid i [-6]. A. Defiiio Le a R adb R. he a hyperplae cosiss of he se { x R \ a x b} ad a halfspace cosiss of he se { x R \ a x b}. If A is a x arix ad b R he a polyhedro cosiss of he se { x R \ Ax b} i oher words a polyhedro is he iersecio of fiiely ay halfspaces. A discree se of pois i R such is Z fors a laice. he discree volue of a -diesioal polyhedro is he Halil Sopce Souh Eas Europia Uiversiy Ilideska bb. eovo R. Maceia (phoe: ;e-ail: h.sopce@seeu.edu.k. uber of laice pois coaied i he polyhedro. Sayig i ore siple way Z is a discree volue. he coiuous volue is obaied by he discree ad i is give wih: Vol( li( Z ( he ai is o fid he geeraig fucio correspodig o + ( Z x. B. Defiiio Le v ω ω... ωk R he a coe is a se { v + ω + ω kωk / i } where ωi are he geeraors ad v is called he apex. he coe is said o be siple if i has exacly liearly idepede geeraors. Le R be a polyhedro ad x. If here exis soe c R such ha for all y c x < c y he we say x is a verex of. Le v v... v k R be verices. ω v ω v... ω v as a pois k k akig ( ( ( + i R he usig hese pois as geeraors a coe + p R will be obaied. For a siple coe R le ( where d Z ad (... R ( Z Wihou provig we give wo heores ake fro [9]. C. heore Le be a -diesioal raioal siple coe wih geeraorsω ω... ω Z. he: ( where ( ω ω ω ( (... ( ( 5

2 Ieraioal Joural of Copuer Sciece ad Elecroics Egieerig (IJCSEE Volue Issue ( ISSN 48 (Olie + { ω + ω + + ω /... } D. Lea... < Le be a polyhedro ad ( Z + (... + he coe over he: (... + ( + ( + E. Exaple Le {( x y R / x + y ; y ; x }. Verices of his polyhedro are he pois v ( v ( v ( (skechig he graph obaied by iequaliies i s o difficul o cofir his. New geeraors will be: ω ω ω ( ( ( v ( ; ω ( v ( ( v ( v If ( is a laice poi i he: ad he correspodig arix equaio of he syse (4 is: Iverig he arix equaio we have: Because of he fac ha < i s o difficul o coclude ha he oly ieger pois of ha saisfies he iverig arix equaio (6 are ( ad (. As a cosequece of his we have: ( Z + + (7 (4 (5 (6 Usig heore he geeraig fucio will be: ( Fro where: ( ω ω ω ( (... ( + ( Z (8 ( ( ( Ad fially usig he lea we ca obai he fial for of geeraig fucio for : ( + (9 ( II. GENERAING FUNCION USING MACMAHON S HEORY he geeral idea is ha if here is a polyope { x R \ Ax b} he for each defiig halfspace a i x b i we ebed ( a i x b i io a crude geeraig fucio. Because of cosiderig oly he posiive cosrais we ebed y i he crude geeraig fucio o obai: ( axb ( axb ( a xb y ( F y... ( xi A. Defiiio: For a uliple Laure series A ν.. operaor Ω is defied by: ν A... k ν... ν.. A ν ν... ν.. ν ν... k k We give wo of he ay ideiies preseed i [8]. Lea : For ay ieger s y ( x s Lea : s ( x( x y... he + xy y ( ( x( xy x Le s discuss agai for he sae exaple 54

3 Ieraioal Joural of Copuer Sciece ad Elecroics Egieerig (IJCSEE Volue Issue ( ISSN 48 (Olie {( x y R / x + y ; y ; } x. he he uber of ieger pois coaied i is equivale o he uber of ieger soluios of he syse: y + x y ( x Algorih for k o for o a b c for i o a ; b( i ; c( i k + a b he correspodig crude geeraig fucio o his syse will be: x + x x x y ( x i he for i ( ay be rasfored io: xi x x y ( his correspods o he followig crude raioal geeraig fucio: \ y (4 Usig lea for s lea for s ad fially lea he fial geeraig fucio will be obaied. his fucio is: + y ( y he fucio (5 is he sae as he geeraig fucio (9. III. USING GENERAING FUNCION FOR ORDERING HE NUMBER OF LAICE OINS (5 I is ieresig o poi ou ha here is a special progra called Diophaie Gf. which copues he uber of laice pois i polyopes. I [] i is show ha his uber is equivale wih he lower boud of processor elees for he give esed loop algorih for i s parallel ipleeaio. We will use his idea o order he lower boud i he algorih of arix-arix uliplicaio. A ad ( a ik Le ( B be wo x arices. A b k esed loop algorih (suiable for sysolic arrays for copuig heir produc C A* B is give wih he followig algorih [7]: Where a aik ; b( bk ; c ; c i c by he idex space { / i k }. he copuaioal srucure is characeried i.he array copuaio for he algorih above ( esh is give by G ( i A where: A { ( i k / i ( i k i ad i i + k k or + i i k k or k k + i i } I his case ( i are laice pois iside -diesioal covex polyhedro whose faces are defied by he iequaliies which are he cosequece of he algorih. We raslae he loop akig i k as opposed o i k because i is iplici. For differe cosrucios of arrays for he proble of algorih ad abou perforaces like uber of processor elees for each of he see [7]. We cover he geoerical ierpreaio of he proble explaied above io a cobiaorial ierpreaio exacly io fidig he soluios of he syse of Diophaie equaios. We have hree iequaliies i k (we o specify he case i k. We rasfor his io he syse of equaliies puig he slack variables s s s. We auge his by he codiio of liear schedule for he correspodig dag which is give wih i + + k. his rages fro o. We will ake he halfway poi i his schedule which easi + + k. herefore we have his syse of Diophaie equaliies where he uber of soluios is a lower boud for he uber of processors: i + + k i + s + s k + s (6 he correspodig geeraig fucio for he above obaied syse of equaliies is: 55

4 Ieraioal Joural of Copuer Sciece ad Elecroics Egieerig (IJCSEE Volue Issue ( ISSN 48 (Olie ( + ( + ( + (7 Ruig he aheaica progra Diophaie Gf. he uber of laice pois as a fucio of is obaied by he bioial forula: -/6 ( C[+/ (-9+]-7 C[+/ (- 7+]+5 C[+/ (-5+]-8 C[+/ (- 4+]+5 C[+/ (-+]-8 C[+/ (- +]- C[-4+]+9 C[-+]- C[- +]+9 C[-+]-8 C[] Siplifyig he bioial coeffiies above we ge he lower boud which is. 4 IV. HE LOWER BOUND OF HE ROCESSOR ELEMENS OF HE SYSOLIC ARRAY FOR DISCREE FOURIER RANSFORM (DF BASED ON MARIX MULILICAION he algorih for wriig he diesioal DF which is used o desig he correspodig sysolic array is give below (ake fro [7]: for o for o for o ( ( + ω x ( ( ; for o for o for o + y( y( + ω + Oupu copuaios ( ( ; [ y ] y( [ ] ; x x (8 (9 Fro above we coclude ha he copuaioal srucure is characeried by he idex space {( } Ζ i where +. he daa depedece vecors for variables fro (8 ad fro (9 are ( ( + ( ad ( ( ( respecively. I his case ( are laice pois iside - diesioal covex polyhedro whose faces are defied by he iequaliies which are he cosequece of he algorih. Coverig he geoerical io a cobiaorial ierpreaio he followig iequaliies will be obaied: +. We rasfor his io he syse of equaliies puig he slack variables s s s. We auge his by he codiio of liear schedule for he correspodig dag which is give wih (akig. his rages fro o 4 akig he halfway poi we have he correspodig syse of Diophaie equaliies will be: s + s + s Fro he syse ( we have: A b Ruig he progra DiophaieGF. we have: I[]:<<DiophaieGF. I[]:a{{} {} {} {}}; I[]:b{};c{----}; I[4]:DiophaieGF[abc] Forula ( + Ou[]: ( + Bioial Forula:C[]+C[+] ower Forula : c ( his eas ha he lower boud for he uber of processig elees (Es of sysolic array for diesioal DF is. I [7] is give a able of uber of processors elees i sysolic arrays. here we ca see ha he array 56

5 Ieraioal Joural of Copuer Sciece ad Elecroics Egieerig (IJCSEE Volue Issue ( ISSN 48 (Olie obaied alog he proecio direcio ( i.e. alog axis is opial i ers of uber of Es. his uber is which is he sae wih our resul for. If we copare hese wo algorihs we ca coclude ha i he firs case he uber of Es is beer bu i s oral because i his case he uber of loops is bigger. We us have applied he firs coclusio o he special case for obaiig he uber of Es o DF. hese wo resuls are i fac he opial kow resuls. herefore his ehology ca be used for fidig he opial uber of Es i oher algorihs where he ipleeig of he described ehology is possible. V. CONCLUSION Alhough he idea of his paper was o poi ou he iporace of geeraig fucio for eueraig he laice pois i polyopes he applicaio of his is ore ipora. We gave a exaple for he case of he algorih of arix uliplicaio ad he algorih of DF based o arix uliplicaio. he sae ehology ca be used for oher algorihs as well. REFERENCES []. Cappello Oer Egecioglu rocessor lower boud forulas for array copuaios ad paraeric Diophaie syses I. J. Foud. Copu. Sci. 9 (4 ( []. Clauss V. Loecher araeric aalysis of polyhedral ieraio spaces J. VLSI Sigal rocess. 9 ( [] R.. Saley Liear Hoogeeous Diophaie equaios ad agic labeligs of graphs Duke Mah. J. 4 ( [4] G.E Adrews MacMaho s ariio Aalysis II: Fudaeal heores A. Cob. 4 ( 7-8. [5] George E. Adrews eer aule A. Riese MacMaho s ariio Aalysis: he Oega ackage Europea Joural of Cobiaorics Ocober vol. iss. 7 pp (8 Acadeic press. Lo. [6] Guouce Xi A Fas Algorih for MacMaho s ariio Aalysis. Elec. Joural of Cob. ( 4 R58 pp. [7] M.. Bekakos Highly arallel Copuaios-Algorihs ad Applicaios Deocrius Uiversiy of hrace Greece pp [8] Maor ercy A. Macaho F.R.S. D.Sc. LL.D. Cobiaory Aalysis volue II Cabridge Uiversiy ress 96 (Repried: Chelsea New York 96. [9] Mahias Back ad Siai Robbis Copuig he coiuous discreely spriger. 57