Efficient Parameterized Algorithms for Data Packing

Transcription

1 Effiient Prmeterized Algorithms for Dt Pking Krishnendu Chtterjee, Amir Kfshdr Gohrshdy, Nstrn Okti, Andres Pvloginnis To ite this version: Krishnendu Chtterjee, Amir Kfshdr Gohrshdy, Nstrn Okti, Andres Pvloginnis. Effiient Prmeterized Algorithms for Dt Pking. ACM Symposium on Priniples of Progrmming Lnguges (POPL 2019), Jn 2019, Lison, Portugl. <hl > HAL Id: hl Sumitted on 17 Ot 2018 HAL is multi-disiplinry open ess rhive for the deposit nd dissemintion of sientifi reserh douments, whether they re pulished or not. The douments my ome from tehing nd reserh institutions in Frne or rod, or from puli or privte reserh enters. L rhive ouverte pluridisiplinire HAL, est destinée u dépôt et à l diffusion de douments sientifiques de niveu reherhe, puliés ou non, émnnt des étlissements d enseignement et de reherhe frnçis ou étrngers, des lortoires pulis ou privés.

2 1 Effiient Prmeterized Algorithms for Dt Pking KRISHNENDU CHATTERJEE, IST Austri (Institute of Siene nd Tehnology Austri), Austri AMIR KAFSHDAR GOHARSHADY, IST Austri (Institute of Siene nd Tehnology Austri), Austri NASTARAN OKATI, Ferdowsi University of Mshhd, Irn ANDREAS PAVLOGIANNIS, Éole Polytehnique Fédérle de Lusnne (EPFL), Switzerlnd There is huge gp etween the speeds of modern hes nd min memories, nd therefore he misses ount for onsiderle loss of effiieny in progrms. The predominnt tehnique to ddress this issue hs een Dt Pking: dt elements tht re frequently essed within time proximity re pked into the sme he lok, therey minimizing esses to the min memory. We onsider the lgorithmi prolem of Dt Pking on two-level memory system. Given referene sequene R of esses to dt elements, the tsk is to prtition the elements into he loks suh tht the numer of he misses on R is minimized. The prolem is notoriously diffiult: it is NP-hrd even when the he hs size 1, nd is hrd to pproximte for ny he size lrger thn 4. Therefore, ll existing tehniques for Dt Pking re sed on heuristis nd lk theoretil gurntees. In this work, we present the first positive theoretil results for Dt Pking, long with new nd stronger negtive results. We onsider the prolem under the lens of the underlying ess hypergrphs, whih re hypergrphs of ffinities etween the dt elements, where the order of n ess hypergrph orresponds to the size of the ffinity group. We study the prolem prmeterized y the treewidth of ess hypergrphs, whih is stndrd notion in grph theory to mesure the loseness of grph to tree. Our min results re s follows: we show there is numer q depending on the he prmeters suh tht () if the ess hypergrph of order q hs onstnt treewidth, then there is liner-time lgorithm for Dt Pking; () the Dt Pking prolem remins NP-hrd even if the ess hypergrph of order q 1 hs onstnt treewidth. Thus, we estlish fine-grined dihotomy depending on single prmeter, nmely, the highest order mong ess hypegrphs tht hve onstnt treewidth; nd estlish the optiml vlue q of this prmeter. Finlly, we present n experimentl evlution of prototype implementtion of our lgorithm. Our results demonstrte tht, in prtie, ess hypergrphs of mny ommonly-used lgorithms hve smll treewidth. We ompre our pproh with severl stte-of-the-rt heuristi-sed lgorithms nd show tht our lgorithm leds to signifintly fewer he-misses. Additionl Key Words nd Phrses: ompilers, dt pking, he mngement, prmeterized lgorithms, dt lolity 1 INTRODUCTION We onsider the prolem of Dt Pking over two-level memory system onsisting of smll he nd lrge min memory. Given referene sequene of memory esses to dt elements, the gol is to orgnize the dt elements into loks in order to minimize he misses. Intuitively, putting ontemporneously-essed elements in the sme lok redues the numer of he misses, ut existing heuristi-sed results do not present ny theoretil gurntees. In this pper, we onsider this prolem from theoretil perspetive nd estlish its omplexity y presenting ext lgorithms nd stronger hrdness results. We lso omplement our theoretil results with n experimentl evlution. Authors ddresses: Krishnendu Chtterjee, IST Austri (Institute of Siene nd Tehnology Austri), Austri, krishnendu. htterjee@ist..t; Amir Kfshdr Gohrshdy, IST Austri (Institute of Siene nd Tehnology Austri), Austri, mir.gohrshdy@ist..t; Nstrn Okti, Ferdowsi University of Mshhd, Irn, nstrn.okti@mil.um..ir; Andres Pvloginnis, Éole Polytehnique Fédérle de Lusnne (EPFL), Switzerlnd, ndres.pvloginnis@epfl.h /2019/1-ART1 $

3 1:2 K. Chtterjee, A.K. Gohrshdy, N. Okti nd A. Pvloginnis Che Mngement. Consider memory system with n ssoitive he nd min memory. Dt items re stored in the min memory nd orgnized into sets of smll size, whih re lled loks (or pges). All dt items hve the sme size nd ll loks n hold the sme numer of dt items. The he hs smll pity nd n hold few loks t ny given time. Whenever progrm needs to ess dt element, its orresponding lok must e present in the he efore the ess n hppen. Therefore, if the lok is not lredy in the he, it will e opied into the he from the min memory, potentilly y eviting nother lok. This opying proess is lled he miss, nd given the onsiderly slower speed of the min memory, he misses re very time-onsuming nd led to signifint overhed [Wulf nd MKee 1995]. Therefore, the prolem of he mngement, i.e., minimizing the numer of he misses, is of gret importne in ompilers nd operting systems. Che mngement n nturlly e divided in two prts [Clder et l. 1998]: (i) deiding on how to reple the loks in the he, i.e. whih lok to evit when the he is full nd miss ours nd (ii) deiding on the plement sheme of the dt items inside loks. These prolems re respetively lled Pging (or hoosing replement poliy) [Sletor nd Trjn 1985] nd Dt Pking [Lvee 2016; Thit 1982]. Pging (Replement Poliy). In pging, given dt plement sheme tht divides the dt items into loks nd so-lled referene sequene of esses to dt elements, the prolem is to hoose lok to e evited eh time he miss ours. The gol is to do this in wy tht minimizes the totl numer of he misses over the referene sequene [Pngiotou nd Souz 2006]. An lgorithm tht hooses the lok to e evited is lled replement poliy. Common replement poliies inlude FIFO, whih evits the oldest lok in the he, nd LRU, whih evits the lest reently used lok [Borodin et l. 1995; Lvee 2016]. Note tht oth FIFO nd LRU n lso e pplied in the online setting, i.e., when the lgorithm does not know the entire sequene in dvne nd n only oserve esses s they re mde. In the offline se, where the entire referene sequene is given in the eginning, the optiml replement poliy is to evit the lok whose first use is furthest in the future [Borodin et l. 1995]. This is lled the optiml offline poliy (OOP). We primrily fous on LRU s the replement poliy, euse it is the one tht is most ommonly used in prtie [Zhong et l. 2004]. Dt Pking. The other spet of he mngement, whih is the fous of this pper, is Dt Pking [Thit 1982]. Consider he with pity of m loks, where eh lok n store p dt items. Given referene sequene R of length N of esses to n distint dt items nd replement poliy, Dt Pking sks for the optiml plement of dt items into loks in order to minimize the numer of he misses. The prmeters m nd p re onsidered to e smll onstnts, nd the omplexity is studied wrt n nd N whih re lrge. Dt Pking is n extremely hrd prolem nd is known to e hrd to pproximte within ny non-trivil ftor, i.e., ny ftor signifintly less thn N, unless P=NP [Lvee 2016]. Relevne of Dt Pking in Progrmming Lnguges. Dt Pking is n importnt tehnique for performne optimiztion during ompiltion nd hs een widely studied y the progrmming lnguges ommunity (See [Clder et l. 1998; Ding nd Kennedy 1999; Lvee 2016; Petrnk nd Rwitz 2002; Zhng et l. 2006; Zhong et l. 2004]). The key relevne is two-fold: Limit studies: To test the performne of ompiler for dt plement, vrious inputs n e generted s enhmrks, nd the seline omprison of the performne n e performed ginst n optiml lgorithm [Petrnk nd Rwitz 2002]. Hene, n optiml dt pking lgorithm is neessry s the seline. This neessrily mens tht there is limit on the size of dt items nd lrge item should e roken into smller prts, eh of whih is onsidered distint dt item.

4 Effiient Prmeterized Algorithms for Dt Pking 1:3 Profiling: Progrms usully hve similr memory-ess ehviors over different inputs [Petrnk nd Rwitz 2002]. Hene, n effetive pproximte pproh for the online he mngement prolem is to onsider severl representtive inputs, then run n optiml offline lgorithm for profiling, nd then synthesize n nswer to the online prolem from optiml offline solutions [Clder et l. 1998; Petrnk nd Rwitz 2002]. Previous theoretil results on dt pking hve ll een negtive (hrdness) results. Heuristis nd Affinity. Given the hrdness of he mngement nd Dt Pking, the reserh in this re hs een mostly foused on developing heuristis. The intuition ehind mny of these heuristis is to exploit the underlying ffinities etween dt elements or loks y trying to ple elements tht re ommonly essed together in the sme lok or eviting the lok tht is less frequently essed in onjuntion with the rest of the loks in the he [Clder et l. 1998; Ding nd Kennedy 1999; Ding nd Kndemir 2014; Hn nd Tseng 2006; Zhong et l. 2004]. Some pprohes, suh s [Zhng et l. 2006], provide more sophistited heuristis nd onstrut hierrhy of ffinities. However, none of the existing heuristis provide ny theoretil gurntees. Aess Grphs. The onept of ess grph [Borodin et l. 1995] hs een introdued to model the ffinities etween dt elements or loks. An ess grph is simply grph in whih there is vertex orresponding to every dt item nd two verties re onneted y n edge if their respetive items pper onseutively in the referene sequene. Aess grphs might e weighted to model how mny times every pir of elements hve ppered onseutively. Similr strutures nd extensions of ess grphs to ess hypergrphs hve een introdued in [Lvee 2016; Thit 1982] where they re lled proximity (hyper)grphs. Moreover, most of the heuristi-sed pprohes lso onsider vrints of the notion of ess grphs. Che Misses vs Che Hits. We onsider the Dt Pking prolem, whih sks to minimize the he misses. Its nturl dul prolem is to mximize he hits. While the two prolems re equivlent in se of ext lgorithms, n pproximtion lgorithm for mximum he hits does not neessrily led to n pproximtion for minimum he misses [Lvee 2016]. For exmple, if in n ess sequene of length N we hve N N he hits nd N he misses, n pproximtion of N N hits n led to n ritrrily d pproximtion of N he misses. In prtie, he misses our muh less frequently thn he hits, ut ontriute signifintly to the overhed. Thus, pproximtion of he misses is more importnt thn pproximtion of he hits, nd the Dt Pking prolem is defined in terms of he misses. Previous Results on Che Mngement. To the est of our knowledge, ll theoretil results on minimizing he misses re negtive or hrdness results. We summrize some of the min results in this re. Given referene sequene R of length N nd he with pity of m loks, the following results hve een shown: (i) In [Petrnk nd Rwitz 2002], the uthors onsidered the prolem of Che-onsious Dt Plement, whih is somewht different formultion of he-miss minimiztion nd is intuitively similr nd relted to Dt Pking. In Che-onsious Dt Plement, he onsists of m lines, eh ple of holding up to p dt items t ny given time. The prolem is to ssign eh dt item d to he line l d. When the progrm wnts to ess the dt item d, it should e present in he line l d, otherwise he miss ours nd d is opied to l d, potentilly y eviting nother dt item from l d. Given n evition strtegy, the gol is to ssign dt items to he lines in mnner tht minimizes he misses. In [Petrnk nd Rwitz 2002] it ws shown tht the prolem is NP-hrd nd unless P=NP, it nnot even e pproximted within ftor of O(N 1 ϵ ).

5 1:4 K. Chtterjee, A.K. Gohrshdy, N. Okti nd A. Pvloginnis (ii) In the sme pper, it ws shown tht ny lgorithm tht does not proess the entire sequene, ut insted relies on pirwise ffinity informtion on dt items, suh s the ess grph, nnot find solution within ftor of m 3 from the optiml, even with unounded time. (iii) In [Lvee 2016], the uthor showed tht Dt Pking is NP-hrd for ny he size nd hrd to pproximte within ftor of O(N 1 ϵ ) unless P = NP. Given these hrdness results, Dt Pking is usully hndled y heuristi-sed lgorithms tht do not provide ny theoretil gurntee. The only positive theoretil result dels with pproximting mximum he hits: (iv) In [Lvee 2016] it ws estlished tht the dul prolem of Dt Pking with the gol of mximizing he hits, insted of minimizing he misses, is pproximle within onstnt ftor. However, this does not pproximte the optiml numer of he misses. Exploiting Struturl Properties. Dt Pking is notoriously diffiult omputtionl prolem. In deling with this prolem, diretion tht hs not een pursued is to exploit struturl properties of the ess grphs. In mny ses, struturl properties of grphs help in otining effiient prmetrized lgorithms for omputtionlly-hrd prolems. Speifilly, well-studied struturl property in grph theory, whih is frequently pplied to omputtionlly-hrd grph prolems, is the notion of treewidth. We present this notion elow. Treewidth. Treewidth [Roertson nd Seymour 1984] is well-known nd extensively-studied prmeter in grph theory. The treewidth of grph is mesure of how tree-like the grph is. Speifilly, trees nd forests re the only grphs with treewidth of 1. The importne of treewidth in lgorithm design stems from the ft tht mny NP-hrd grph prolems (e.g., Vertex Cover nd Hmiltonin Cyle) n e solved in polynomil time if the input grphs hve onstnt treewidth nd moreover, mny other grph prolems n e solved in lower omplexity [Aoud et l. 2016; Bodlender 1997; Chtterjee et l. 2018; Cygn et l. 2015; Fomin et l. 2017; Roertson nd Seymour 1986]. The forml definition of treewidth is presented in the next setion. Treewidth in Progrm Anlysis. Mny importnt fmilies of grphs tht rise ommonly in lgorithm design re shown to hve onstnt treewidth, e.g., series-prllel nd outer-plnr grphs [Bodlender 1998]. Perhps the most importnt exmple in progrm nlysis is tht the ontrol-flow grphs of strutured goto-free progrms in mny lnguges suh s Psl nd C++ hve onstnt treewidth [Thorup 1998]. The sme result ws lso shown experimentlly for most Jv progrms [Gustedt et l. 2002]. This led to lgorithmi dvnes in verifition nd progrm nlysis [Chtterjee et l. 2015]. Moreover, treewidth hs lso een exploited to otin fster lgorithms for stti nlysis of reursive stte mhines [Chtterjee et l. 2015] nd onurrent systems [Chtterjee et l. 2016, 2017]. Treewidth in Dt Pking. In this work we show tht Dt Pking n e redued to grph prolem. In mny ses when grph rises from strutured proess, the treewidth of the grph is not very lrge [Bodlender 1998]. For Dt Pking, the ess grphs rise from strutured progrm essing dt from well-defined dt struture. Thus, it is nturl to study the prolem of Dt Pking in terms of the treewidth property of the rising grphs, s we do in this work. Our Contriutions. Our ontriutions inlude () polynomil lgorithms for Dt Pking in onstnt treewidth ess (hyper)grphs, () stronger hrdness results, nd () experimentl results demonstrting tht our pproh leds to onsiderly fewer he misses in omprison with previously-known heuristi-sed pprohes. Conretely, onsider tht the he hs size m, every lok n hold p dt items nd the referene sequene is of length N with n distint items. We onsider the ess hypergrph of order q, where eh vertex of the grph is dt item, nd n edge onnets set of q distint dt items if they pper ontiguously in the referene sequene. We show tht the Dt Pking prolem n e redued to grph prtitioning prolem of the

6 Effiient Prmeterized Algorithms for Dt Pking 1:5 q Liner-time Theorem Liner-time Theorem 3.1 NP-hrd Theorem (m 1)p + 2 NP-hrd Theorem (m 5)p + 1 Hrd to Approximte Theorem 4.3 Hrd to Approximte Theorem m Fig. 1. The omplexity of Dt Pking for p 3. Here m is the he size nd q is the highest order for whih the ess hypergrph hs onstnt treewidth. For forml definition of order see Setion 2.1. Theorem 2.1 ws estlished in [Lvee 2016]. The rest of the piture is filled y this pper. Our results re shown in old. ess (hyper)grph nd we study whether the onstnt treewidth property n e exploited for polynomil-time lgorithms. Our min results, ssuming onstnt m nd p, re s follows: (1) Results on Aess Grphs. We first onsider q = 2. Note tht order-2 ess hypergrphs re silly ess grphs. We estlish the following results: Liner-time lgorithm. We present liner-time lgorithm for Dt Pking when the ess grph is of onstnt treewidth nd m = 1 (Theorem 3.1). Hrdness of the ext prolem. The Dt Pking prolem remins NP-hrd for m 2 nd p 3 even if the underlying ess grph is tree (whih hs treewidth 1) (Theorem 3.2). Hrdness of pproximtion. Unless P=NP, for ny m 6,p 2 nd ny onstnt ϵ > 0, the Dt Pking prolem is hrd to pproximte within ftor of O(N 1 ϵ ) even if the underlying ess grph is tree (Theorem 3.2). (2) Results on Aess Hypergrphs. We define ess hypergrphs of higher orders nd onsider their treewidth. Let q = (m 1)p + 2. Note tht q depends only on the he prmeters, nd not on n or N. We onsider the ess hypergrph of order q. Intuitively, every edge of this ess hypergrph ontins ll the neessry historil he dt for determining whether miss ours t orresponding memory ess. Formlly, we estlish the following results: Liner-time lgorithm. We present liner-time lgorithm for Dt Pking when the ess hypergrph of order q hs onstnt treewidth (Theorem 4.1). Hrdness of the ext prolem. For m 2 nd p 3, the Dt Pking prolem remins NP-hrd even if the ess hypergrph of order q 1 hs onstnt treewidth (Theorem 4.2). Hrdness of pproximtion. Unless P=NP, for m 6 nd p 2 nd ny onstnt ϵ > 0, the Dt Pking prolem is hrd to pproximte within ftor of O(N 1 ϵ ) even if the ess hypergrph of order q 4p 1 hs onstnt treewidth (Theorem 4.3). Note tht while onstnt treewidth hs een exploited to otin polynomil-time lgorithms for NP-omplete grph prolems suh s Vertex Cover nd Hmiltonin Cyle, we show tht for Dt Pking the onstnt treewidth property does not lwys help, nd the prolem remins hrd even when the ess hypergrph of order q 1 hs onstnt treewidth. Our hrdness result nd liner-time lgorithm present shrp oundry (or fine-grined dihotomy) tht shows when the treewidth n e exploited. Conretely, the hrdness of

7 1:6 K. Chtterjee, A.K. Gohrshdy, N. Okti nd A. Pvloginnis the Dt Pking prolem n e ptured y single prmeter, nmely, the highest order mongst ess hypergrphs tht hve onstnt treewidth. We estlish the optiml vlue q of this prmeter whih is the neessry nd suffiient ondition for existene of effiient prmeterized lgorithms tht exploit treewidth. (3) Experimentl results. We present n experimentl evlution of prototype implementtion of our lgorithm on vriety of enhmrks from liner lger, sorting lgorithms, dynmi progrmming, reursive lgorithms, string mthing, omputtionl geometry nd lgorithms on tree dt-strutures. Our results show tht the ess hypergrphs of most of the enhmrks hve smll treewidth. We ompre our pproh with severl stte-of-thert heuristi-sed lgorithms. The experimentl results show tht on verge our optiml lgorithms otin 15-30% imporvement over the previous heuristi-sed pprohes. Novelty nd Signifine. In this pper, we define novel nd rih struturl property of progrms, i.e. ess hypergrphs nd their treewidth, nd show tht it n e exploited to otin fster lgorithms for Dt Pking. We present the first positive theoretil results for Dt Pking, i.e., for he-miss minimiztion. We lso enrih the omplexity lndspe s shown in Figure 1. Only the results of Theorem 2.1 were known efore, nd ll other results (whih re shown in old) re estlished in the present work. 2 PRELIMINARIES 2.1 Dt Pking In this setion, we define the prolem of dt pking nd fix our nottion. We lso present severl previously-known results. The prolem ws first studied in [Thit 1982]. Here, we present n dpttion of its definition s formlized in [Lvee 2016]. Nottion. We use Z to denote the set of integers nd N to denote the set of positive integers. Let G = (V, E) e (hyper)grph, nd X V, then we denote y G[X], the indued sugrph of G over X, i.e. G[X ] = (X, {e E e X }). Given two (hyper)grphs G 1 = (V 1, E 1 ) nd G 2 = (V 2, E 2 ), we define their union nd intersetion in the nturl wy, i.e. G 1 G 2 = (V 1 V 2, E 1 E 2 ) nd G 1 G 2 = (V 1 V 2, E 1 E 2 ). If F is fmily of sets, we write F (resp. F ) to denote A F A (resp. A F A). Given two funtions f,д : A Z, equlity nd summtion re defined in pointwise mnner, i.e. f д A; f () = д() nd for ny A, we hve (f + д)() = f () + д(). Given funtion f : A B nd suset A A, we use f A to denote the restrition of f to A. This restrition is funtion of the form f A : A B tht grees with f on every point in A. For set X, we write P(X ) to denote the power set of X, i.e., the set of ll susets of X. Dt Plement Shemes. Given set D of size n of dt items nd positive integer p, dt plement sheme σ is prtitioning of D into loks of size t most p. We ll p the pking ftor. It is often useful to think of σ s n equivlene reltion on D whose equivlene lsses re the loks. Hene, following the usul nottion, we write xσy to denote tht x nd y re in the sme lok, [x] σ to denote the lok of σ tht ontins the dt element x nd D/σ to denote the set of loks or equivlene lsses of σ. Replement Poliies. Given set D of n dt items, he of size m, dt plement sheme σ, nd sequene R D N of esses to dt items, replement poliy is funtion tht deides whih lok must e evited from the he t eh time. Formlly, replement poliy is funtion π : {0, 1, 2,..., N } P(D/σ) tht ssigns to eh time point i, the set of loks tht re present in the he right fter the ess R[i]. Any suh poliy must stisfy the following: π(0) =, i.e. the he must e empty t the eginning; For ll 1 i N, π(i) m, i.e. there re t most m loks in the he t eh time;

8 Effiient Prmeterized Algorithms for Dt Pking 1:7 For ll 1 i N, π(i) \ π(i 1) 1 nd π(i 1) \ π(i) 1, i.e. t most one lok n e dded to the he nd t most one lok n e evited t eh step; For ll 1 i N, R[i] π(i), i.e. the lok ontining n ess R[i] must e in the he right fter tht ess. Remrk 2.1. Note tht the replement poliy only mtters when the he hs size of t lest 2. When the he hs unit size, there is lwys unique hoie for the lok tht must e evited. Che Misses. Given dt plement sheme σ nd replement poliy π s ove, the numer of he misses used y σ nd π over R is defined s the numer of times new lok is loded into the he. Formlly, misses(σ, π) = {i 1 i N, π(i) \ π(i 1) }. The LRU Poliy. Due to its populrity, we ssume throughout this pper tht the replement poliy is LRU, i.e. the Lest-Reently-Used lok is lwys evited from the he. However, most of our results rry over to First-In-First-Out (FIFO) nd the Optiml Offline Poliy (OOP), s well. Rell tht FIFO evits the oldest lok in the he nd OOP evits the lok tht is going to e used furthest in the future. The Dt Pking Optimiztion Prolem. Consider memory susystem tht onsists of n distint dt elements nd fully-ssoitive he with pity of m loks nd pking ftor of p. Given sequene R of length N of referenes to dt elements, the Dt Pking prolem sks for dt plement sheme σ tht minimizes the numer of he misses inurred y the referene sequene R, using LRU s the replement poliy. We denote n instne of the Dt Pking prolem y I = (n,m,p, R). Prmeters. In the sequel, we onsider the prmeters m nd p to e smll onstnts nd try to find polynomil lgorithms in terms of N nd n. We now define the onepts of ess grph nd ess hypergrph. Vrious similr notions hve een defined in the pst, nd re sometimes lled ffinity grphs or proximity grphs. These hypergrphs will lter serve s sis for reduing the Dt Pking prolem to grph prolem. Aess Grph. Given sequene R of length N of esses to dt elements from set D of size n, the ess grph of R is simple grph G R = (V, E) in whih V onsists of n verties, eh orresponding to one of the dt elements in D, nd there is n edge etween two distint verties iff their orresponding dt elements pper onseutively somewhere in R. More formlly, {u, v} E iff u v nd there exists n index i, suh tht {R[i], R[i + 1]} = {u,v}. Intuitively, one n think of the grph G R s the struture on dt elements tht is respeted y the ess sequene R, in the sense tht R n only go from vertex in G R to one of its neighors. Moreover, G R is the sprsest grph over whih R is (non-simple) pth. Exmple 2.1. Consider the ess sequene R =<,,,,,, d,, d, e,,, f >. There re 6 dt elements in this sequene nd its ess grph G R is shown in Figure 2. Note tht R is pth on this grph nd every edge ppers somewhere long R, hene no sugrph of G R hs the sme property. f d e Fig. 2. The ess grph G R of R =<,,,,,,d,,d, e,,, f >

9 1:8 K. Chtterjee, A.K. Gohrshdy, N. Okti nd A. Pvloginnis We now extend the onept of ess grphs to higher order ffinity reltions etween dt items, resulting in ess hypergrphs. Hypergrphs nd Ordered Hypergrphs. A hypergrph G = (V, E) onsists of setv of verties nd multiset E of hyperedges. Eh hyperedge e E is in turn suset of the verties of G. An ordered hypergrph G = (V, E) onsists of set V of verties nd set E of ordered hyperedges. Eh ordered hyperedge e E is sequene of distint verties of G, i.e. hyperedge together with n order on its verties. Intuitively, hypergrphs re nturl extensions of grphs, where eh edge n onnet more thn two verties. Given hypergrph G, its priml grph G p is grph on the sme set V of verties, where two verties u nd v re onneted y n edge iff there exists hyperedge e E ontining oth u nd v. We shll simply refer to hypergrphs nd hyperedges s grphs nd edges when there is no fer of onfusion. Aess Hypergrph. Given nturl numer q nd n ess sequene R s ove, the ess hypergrph G q R = (V, E) is hypergrph defined s follows: There re n verties in V, eh orresponding to one dt element; For eh dt ess R[i], there is orresponding hyperedge e i in E. The hyperedge e i onsists of R[i] nd the q 1 distint dt elements tht re essed right efore R[i]. If there re less thn q 1 suh elements, e i will inlude ll of them. Conretely, e i is defined s follows: e i := {R[j] j i {R[j], R[j + 1],..., R[i]} q}. We ll q the order of the ess hypergrph. It is esy to verify tht removing repeted edges from the ess hypergrph GR 2 leds to the ess grph G R. Exmple 2.2. Consider the ess sequene R =<,,,,,,d,,d, e,,, f >. Letting q = 3, the orresponding ess hypergrph GR 3 of order 3 onsists of the following hyperedges (sometimes there re multiple opies of the sme hyperedge, s shown elow. We onsider these to e distint hyperedges): {}, {,}, {,,} 4, {,,d} 3, {,d, e}, {,d, e}, {,, e}, {,, f }. Figure 3 shows the segments of the sequene tht orrespond to edges in GR 3.,,,,,, d,, d, e,,, f Fig. 3. Segments of R orresponding to edges in the hypergrph G 3 R Ordered Aess Hypergrphs. Given n ess sequene R s ove, the ordered ess hypergrph Ĝ q R is defined similrly to Gq R, exept tht eh hyperedge is ordered in the nturl wy, i.e. in the order of pperne of its orresponding dt elements in R. Formlly, for every ess R[i], there is orresponding ordered hyperedge e i in Ĝ q R. The ordered hyperedge e i is sequene < v 1,v 2,...,v l > of verties of Ĝ q R suh tht v l = R[i], v l 1 is the first distint dt element essed efore R[i], v l 2 is the seond distint element, et. Moreover, l is the mximum etween q nd the numer of distint elements essed up until R[i]. Exmple 2.3. Consider the ess sequene R =<,,,,,,d,,d, e,,, f >. The ess hypergrph GR 3 ws shown in Exmple 2.2. We now onstrut the ordered hyperedges of Ĝ3 R. Intuitively, we strt from ny ess R[i] in R nd go k until we see 3 different dt elements. These dt elements will form the ordered hyperedge e i orresponding to R[i]. This is illustrted in Figure 4. Note tht the elements in n ordered hyperedge e i re ordered y their lst ess time efore or t R[i], e.g. see the hyeperedge <, d, > in Figure 4.

10 Effiient Prmeterized Algorithms for Dt Pking 1:9 R: <,, > <,, > <,, > <> <, > <,, > <,, f> <e,, > <d, e, > <, d, e>,,,,,, d,, d, e,,, f <,, d> <, d, > <,, d> Fig. 4. Ordered Hyperedges of G nd the segments in R to whih they orrespond 2.2 Tree Deompositions nd Treewidth In prmeterized omplexity, treewidth is one of the most widely-used prmeters for grph prolems. It is mesure of how tree-like given grph is. In this setion, we provide quik overview of tree deompositions nd treewidth. For n in-depth tretment see [Cygn et l. 2015]. Tree Deomposition. Given (hyper)grph G = (V, E), tree deomposition of G is pir (T, {X t t T }) where T is tree nd eh node t of T is ssoited with suset X t V of verties of G, suh tht the following onditions re met: (i) Every vertex ppers in some X t, i.e. t T X t = V ; (ii) Every (hyper)edge ppers in some X t, i.e. e E X t e X t ; (iii) For every vertex v V, the set T v = {t T v X t } of ll nodes of the tree T tht ontin v in their orresponding X t, forms onneted sutree of T. It is evident from the definition tht (T, {X t }) is tree deomposition of hypergrph G iff it is tree deomposition of its priml grph G p. To void onfusion, we reserve the word vertex for verties of G nd use the word node for verties of T. Moreover, we ll eh X t g. Treewidth. The width of tree deomposition (T, {X t }) is the size of its lrgest g minus 1, i.e. min t T X t - 1. The treewidth of grph G is the smllest width mong ll tree deompositions of G nd is denoted tw(g). Exmple 2.4. Figure 5 shows the grph G R (s in Figure 2) nd tree deomposition of G R. This tree deomposition hs width of 2 nd is n optiml tree deomposition. Hene, the treewidth of G R is 2. f {,,} d {,,d} {, f } e {,d, e} Fig. 5. A grph G R (left) nd one of its optiml tree deompositions (T, {X t }) (right). To simplify the lgorithms tht exploit tree deompositions, we now define the notions of leling nd nie tree deomposition. Nie Tree Deompositions. A nie tree deomposition [Cygn et l. 2015] of (hyper)grph G is tree deomposition (T, {X t }) in whih speifi node is designted s the root nd every node t T is leled y sugrph G t of G, suh tht the following rules re oeyed: (1) If t is lef in T, then X t = nd G t = (, ). (2) Otherwise, t stisfies one of the following ses: Join Node. The node t hs two hildren, t 1 nd t 2, X t = X t1 = X t2 nd G t = G t1 G t2.

11 1:10 K. Chtterjee, A.K. Gohrshdy, N. Okti nd A. Pvloginnis Introdue Vertex Node. The node t hs single hild t 1 nd X t = X t1 {v} for some vertex v X t1. In this se, we sy tht t introdues v. Moreover, G t = G t1 {v}, i.e. G t is defined s the grph resulting from dding v s n isolted vertex to G t1. Introdue Edge Node. Similr to the previous se, t hs single hild t 1. This time, X t = X t1, ut G t is defined s the grph resulting from dding new edge e to G t1. All verties of e must e present in X t. We sy tht t introdues e. Forget Vertex Node. The node t hs single hild t 1 nd X t = X t1 \ {v} for some vertex v X t1. We sy tht t forgets v. Moreover, G t = G t1. (3) Eh edge is introdued extly one. Intuitively, the lel grph G t is the sugrph of G onsisting of ll the verties nd edges tht re introdued in the sutree of T rooted t t. Remrk 2.2. Note tht in our (ordered) hypergrphs in this pper, we might hve multiple opies of the sme (ordered) hyperedge. We tret these s distint edges nd require tht eh of them e introdued seprtely in nie tree deompositions. Remrk 2.3. The notion of lel grphs G t is solely defined for theoretil purposes nd used in our proofs of orretness. In prtie, our implementtion voids the overhed of onstruting G t s. Exmple 2.5. Figure 6 shows nie tree deomposition of the grph G of Figure 2. In eh node t of the tree, its lel sugrph G t is illustrted nd the verties of the g X t re shown in red. Intuitively, nie tree deomposition onstruts the grph in smll inrements nd the g X t ontins the verties tht n prtiipte in the inrementl hnge. Fig. 6. A nie tree deomposition of the grph in Figure 2. The leftmost node is the root. The grph G t is illustrted in eh node t. The verties of the gs X t re shown in red. 2.3 Existing Results We now formlly present known results regrding Dt Pking nd Tree Deompositions tht will e used in the sequel. The Hrdness of Dt Pking. Note tht we re onsidering the prolem of minimizing he misses, not tht of mximizing he hits. While the two prolems re equivlent in terms of ext lgorithms, pproximting the miniml numer of he misses is muh hrder thn pproximting the mximl numer of he hits. The ltter prolem dmits polynomil-time onstnt-ftor pproximtion [Lvee 2016]. In ontrst, the following theorem shows tht the former prolem is hrd to even pproximte. Theorem 2.1 ([Lvee 2016]). Assuming either LRU, FIFO or OOP s the replement poliy, we hve the following hrdness results: For ny m nd ny p 3, Dt Pking is NP-hrd. Unless P=NP, for ny m 5, p 2 nd ny onstnt ϵ > 0, there is no polynomil lgorithm tht n pproximte the Dt Pking prolem within ftor of O(N 1 ϵ ).

12 Effiient Prmeterized Algorithms for Dt Pking 1:11 We now turn to tree deompositions. Wht mkes tree deompositions very useful tool is the ft tht one n perform ottom-up dynmi progrmming on them in mnner similr to trees. This is due to n property of tree deompositions, lled the seprtion lemm. Seprtors. Given (hyper)grph G = (V, E), nd two sets of verties A, B V, we sy tht the pir (A, B) is seprtion of G if A B = V nd no (hyper)edge in E ontins verties of oth A \ B nd B \ A. We ll A B the seprtor orresponding to the seprtion (A, B) nd the order of the seprtion (A, B) is the size of its seprtor A B. Lemm 2.1 (Seprtion Lemm, [Bodlender 1988; Cygn et l. 2015]). Let (T, {X t }) e tree deomposition of G, where G is grph or hypergrph, nd let {,} e n edge of T. By removing the edge {,}, T reks into two onneted omponents T nd T, respetively ontining nd. Let A = t T X t nd B = t T X t. Then (A, B) is seprtion of G with seprtor X X. In our lgorithms in the rest of this pper, we ssume tht whenever (hyper)grph G of onstnt treewidth ppers s n input to n lgorithm, the input lso ontins n optiml nie tree deomposition (T, {X t }) of G. This is justified y the following two lemms tht show one n otin (T, {X t }) from G in liner time. Lemm 2.2 ([Bodlender 1996] ). There is n lgorithm tht given (hyper)grph G = (V, E) nd onstnt k, deides in liner time whether G hs treewidth t most k nd if so, produes tree deomposition of G with optiml width nd O(k V ) nodes. Lemm 2.3 ([Cygn et l. 2015]). There is liner-time lgorithm tht given grph G = (V, E) nd tree deomposition (T, {X t }) of G of width k with O(k V ) nodes, produes nie tree deomposition (T, {X t }) of G with the sme width k nd O(k V ) nodes. This lgorithm n lso e pplied if G is hypergrph, in whih se the output tree deomposition (T, {X t }) will hve width k nd O(k V + E ) nodes. 3 DATA PACKING ON CONSTANT-TREEWIDTH ACCESS GRAPHS We now onsider the prolem of Dt Pking when prmeterized y the treewidth of the underlying ess grph. In Setion 3.1, we provide liner-time lgorithm when m = 1 nd the ess grph hs onstnt treewidth. Note tht this prolem is NP-hrd for generl ess grphs, s demonstrted y Theorem 2.1. Then, in Setion 3.2 we show tht for m 2 the prolem remins NP-hrd nd hrd-to-pproximte even when the ess grph is tree, i.e. hs treewidth Algorithm for m = 1 nd Constnt-treewidth Aess Grph We re given Dt Pking instne I = (n, 1,p, R), its ess grph G R nd nie tree deomposition (T, X t ) of the ess grph with width k nd O(n k) nodes. We first redue the prolem of Dt Pking to grph prolem over G R nd then provide liner-time fixed-prmeter lgorithm for solving the grph prolem. We strt y defining the minimum-weight p-prtitioning prolem. p-prtitionings. Given n integer p > 0 nd grph G = (V, E), p-prtitioning of G is prtitioning ψ of the set V of verties suh tht eh prtition set hs size of t most p. In other words, p-prtitioning of G is dt plement sheme where the verties of G re the dt elements nd p is the pking ftor. Cross Edges. Given p-prtitioning ψ of the grph G = (V, E), n edge e = {u,v} E is lled ross edge if its two endpoints re in different prtition sets, i.e. if [u] ψ [v] ψ. Minimum-weight p-prtitioning. Given simple grph G = (V, E), weight funtion w : E N nd positive integer p, the minimum-weight p-prtitioning prolem sks for p-prtitioning of G in whih the totl weight of ross edges is minimized.

13 1:12 K. Chtterjee, A.K. Gohrshdy, N. Okti nd A. Pvloginnis 2 1 f d 1 e Fig. 7. An optiml 2-prtitioning Redution of Dt Pking to Minimum-weight p-prtitioning. We now redue the Dt Pking prolem to minimum-weight p-prtitioning. Given n instne I = (n, 1, p, R) of Dt Pking, we onsider the ess grph G R = (V, E) nd define the weight funtion w R : E N s w R ({u,v}) := {i {R[i], R[i + 1]} = {u,v}}. Informlly, the weight of n edge is the numer of times its two endpoints hve ppered onseutively in R. The redution is now omplete. Lemm 3.1. The optiml numer of he misses in Dt Pking instne I = (n, 1,p, R) is 1 plus the totl weight of ross edges in minimum-weight p-prtitioning of G R with weight funtion w R. Proof. Every p-prtitioning ψ of G R is dt plement sheme for I nd vie vers. Given tht m = 1, the replement poliy does not mtter (Remrk 2.1) nd he miss ours eh time R esses new lok. If we onsider R s pth on G R, he miss ours t the very eginning nd then eh time this pth goes from one equivlene lss of ψ to nother. Therefore, the numer of he misses of ψ is 1 plus the totl weight of ross edges in ψ. Exmple 3.1. Consider the ess sequene R =<,,,,,,d,,d, e,,, f > of Exmple 2.1 nd the Dt Pking instne I = (6, 1, 2, R), i.e. eh lok n store up to 2 dt elements. Figure 7 shows the grph G R in whih every edge is weighted y the numer of times it is trversed in R. An optiml 2-prtitioning of G R is shown in whih verties of the sme olor re in the sme prtition. The totl weight of ross edges in this prtitioning is 7. The orresponding dt plement sheme is {{,}, {,d}, {e}, {f }} whih leds to 8 he misses on R. The he misses re underlined. We will provide n lgorithm for solving the Minimum-weight p-prtitioning prolem on grph G using n optiml nie tree deomposition of G. Our lgorithm employs ottom-up dynmi progrmming tehnique. We first need severl si onepts to define the lgorithm. Sttes over Set of Verties. Given grph G = (V, E), nturl numer p nd suset A V of verties, stte over A is pir s = (φ, sz) suh tht (i) φ is prtitioning of A in whih every equivlene lss hs size of t most p, nd (ii) sz is size enlrgement funtion sz : A/φ {0,...,p 1} tht mps eh equivlene lss [v] φ to numer whih is t most p [v] φ. Intuitively, the ide is to tke A to e one of the gs in the tree deomposition nd lter extend stte over A to p-prtitioning of G y dding the verties in V \ A. So, stte over A prtitions the verties of A into sets of size t most p nd for eh prtition [v] φ fixes the ext numer sz([v] φ ) of verties from V \ A tht should e dded to [v] φ. We denote the set of ll sttes over A y S A,p or simply S A when p is ler from the ontext. Reliztion. We sy tht p-prtitioning ψ relizes the stte s = (φ, sz) over A, if (i) the restrition ofψ to A is equl to φ, i.e.ψ A = φ nd (ii) for ll verties v A, sz([v] φ ) = [v] ψ [v] φ. Intuitively, ψ relizes s if (i) ψ prtitions the verties in A in the sme mnner s φ nd (ii) if prtition [v] ψ of ψ intersets A, then [v] ψ ontins s mny verties from outside of A s fixed y sz. Exmple 3.2. Figure 8 shows ll 14 possile sttes over the set A = {,,} of verties with p = 2. In eh se, eh row denotes one prtition set nd hene the order of rows nd the order of squres in row does not mtter. Empty squres orrespond to the possiility of extension of the set, s defined

14 Effiient Prmeterized Algorithms for Dt Pking 1:13 Fig. 8. All possile sttes over A = {,,} with p = 2 d e d e f f Fig. 9. Two omptile sttes over A = {,,} nd A = {d, e, f } y sz. The optiml 2-prtitioning ψ presented in Figure 7 relizes the highlighted stte in Figure 8, euse ψ puts nd in the sme prtition nd puts in prtition of size 2, whose other memer, d, omes from outside the set {,,}. Comptiility. We sy tht two sttes s nd s, respetively over the sets A nd A, re omptile if there exists p-prtitioning tht relizes oth of them. We write s s to show omptiility. Exmple 3.3. Intuitively, two sttes re omptile if they n fit into eh other. Figure 9 shows the sttes relized y the 2-prtitioning of Figure 7 ove over the sets A = {,,} nd A = {d, e, f } nd how they n e fitted together to rete the entire 2-prtitioning. Algorithm 1. We re now redy to desrie our lgorithm in detil. Given grph G, weight funtion w nd n optiml nie tree deomposition T of G, our lgorithm performs ottom-up dynmi progrmming on T. This is roken into three steps. Step 0: Initiliztion. We define severl vriles t eh node of our tree T. These vriles re ment to e omputed in ottom-up mnner. Conretely, for every t T nd every stte s over the g X t, we define vrile dp[t, s] nd initilize it to +. Invrint. Formlly, our lgorithm stisfies the following invrint for every dp vrile right fter the end of its omputtion: dp[t, s] = The minimum totl weight of ross edges over ll p-prtitionings of G t tht relize s. Intuitively, we re onsidering the sttes over the g X t nd extending them y dding verties tht were introdued in the sutree of t in T. Step 1: Computtion of dp. The lgorithm strts from the ottom of the tree T nd omputes the dp vriles ottom up, i.e. with n order suh tht for every node t T the dp vriles t its hildren re omputed efore the dp vriles of t. For every node t T nd stte s = (φ, sz) S Xt, we show how dp[t, s] is omputed sed on the type of the node t: (1.1) if t is Lef: dp[t, s] = 0; (1.2) if t is Join node with hildren t 1 nd t 2 : dp[t, s] = min dp[t 1, (φ, sz 1 )] + dp[t 2, (φ, sz 2 )]; sz 1 +sz 2 sz

15 1:14 K. Chtterjee, A.K. Gohrshdy, N. Okti nd A. Pvloginnis Note tht the summtion nd equlity ove re pointwise. (1.3) if t is n Introdue Vertex node, introduing v, with single hild t 1 : dp[t, s] = dp[t 1, (φ Xt1, sz Xt1 )]; (1.4) if t is n Introdue Edge node, introduing e, with single hild t 1 : dp[t, s] = dp[t 1, s] + w(e, φ), where w(e, φ) is equl to w(e) if e is ross edge in φ nd zero otherwise; (1.5) if t is Forget Vertex node, forgetting v, with single hild t 1 : dp[t, s] = Rell tht denotes omptiility. min dp[t 1, s ]. s S X t s s 1 Step 2: Computing the Output. The lgorithm omputes the output, i.e. the optiml weight of p-prtitioning, using the vlues stored t dp vriles. If r is the root node of T, then the lgorithm outputs the following vlue: min s SXr dp[r, s]. This onludes Algorithm 1. We now prove the orretness of our lgorithm. Lemm 3.2. Algorithm 1 orretly omputes the totl weight of ross edges in minimum-weight p-prtitioning. Proof. We prove this lemm in two steps. First, we show tht the invrint defined ove holds fter omputing dp[t, s] ssuming tht it ws stisfied for ll dp vriles in the hildren of t (Corretness of Step 1). Then, ssuming tht the invrint holds for dp vriles t the root, we show tht the output is the totl weight of n optiml p-prtitioning (Corretness of Step 2). Intuitively, the invrint sys tht if we only onsider the grph G t, i.e. the prt of G tht ws introdued in the sutree of T rooted t t, nd those p-prtitionings of G t tht relize the stte s, then dp[t, s] holds the minimum totl weight of ross edges mong these p-prtitionings. Corretness of Step 1. As in the lgorithm, we rek this prt into severl ses: (1.1) Computtions t Leves. The node t is lef in T, hene G t is the empty grph nd X t is the empty set. Therefore, S Xt ontins single trivil stte s nd we hve dp[t, s ] = 0 euse the totl weight of ross edges in n empty grph is zero. (1.2) Computtions t Join Nodes. The node t is join node with hildren t 1 nd t 2. We wnt to ompute dp[t, s] where s = (φ, sz). Therefore, we only onsider those p-prtitionings tht relize s. Given tht X t = X t1 = X t2, φ imposes itself on oth X t1 nd X t2. However, eh prtition in φ must e extended y numer of verties s defined y sz. These verties must ome from either G t1 or G t2 nd must not lredy e present in X t. Aording to the seprtion lemm (Lemm 2.1), the only verties tht re in oth G t1 nd G t2 re preisely those of X t. Hene, eh new vertex omes either from G t1 or G t2 ut not from oth. Therefore, we should minimize our totl ross edge weights wrt dp vriles of the form dp[t 1, (φ, sz 1 )] nd dp[t 2, (φ, sz 2 )] where sz 1 +sz 2 sz. The funtion sz 1 defines the numer of verties tht should e dded from G t1 X t to eh prtition of φ nd sz 2 does the sme for G t2 X t. Formlly, if we let w(φ) e the totl weight of ross edges used y φ in G t1 G t2 = G t1 [X t ] G t2 [X t ], then we should let: dp[t, s] = dp[t, (φ, sz)] = min dp[t 1, (φ, sz 1 )] + dp[t 2, (φ, sz 2 )] w(φ). sz 1 +sz 2 sz The reson we re sutrting w(φ) t the end is tht the weights of its orresponding edges re tken into ount twie, i.e. one in eh of dp[t 1, (φ, sz 1 )] nd dp[t 2, (φ, sz 2 )].

16 Effiient Prmeterized Algorithms for Dt Pking 1:15 We now show it is lwys the se tht w(φ) = 0. If n edge ontriutes to w(φ), then it must e present in oth G t1 nd G t2. However, y property (3) of nie tree-deomposition, eh edge is introdued extly one. Hene, G t1 nd G t2 do not shre ny edges nd w(φ) = 0. Therefore, y setting dp[t, s] = dp[t, (φ, sz)] = min sz1 +sz 2 sz dp[t 1, (φ, sz 1 )] + dp[t 2, (φ, sz 2 )], we stisfy the invrint. t t 1 t 2 Fig. 10. In join node t, G t1 nd G t2 do not shre ny edges nd their shred verties re in X t. (1.3) Computtions t Introdue Vertex Nodes. The node t is n introdue vertex node. So, it hs single hild t 1 nd X t = X t1 {v} for some v X t1. We know tht the vertex v nnot possily pper in G t1 euse every vertex ppers in onneted sutree of T nd v X t1. Hene, G t is otined y dding v s n isolted vertex to G t1. Agin, we wnt to ompute dp[t, s] nd should hene only onsider the p-prtitionings tht relize s. Given tht X t1 X t, s imposes unique omptile stte on X t1. Moreover, G t hs no new edges in omprison with G t1, so the totl weight of ross edges should only e omputed in G t1. Hene, we let dp[t, s] = dp[t, (φ, sz)] = dp[t 1, (φ Xt1, sz Xt1 )]. Intuitively, this is equivlent to removing v from its prtition nd then omputing the dp in t 1. t t 1 Fig. 11. In n introdue vertex node t, the newly introdued vertex is isolted nd there re no new edges. (1.4) Computtions t Introdue Edge Nodes. The node t hs single hild t 1 nd X t = X t1. Moreover, the only differene etween G t nd G t1 is in single edge e. When omputing dp[t, s], the stte s fores itself on X t1 = X t. Hene we should let dp[t, s] = dp[t 1, s] +w(e, s), where w(e, s) is the ontriution of the edge e to the totl weight of ross edges in s. It is zero if the two sides of e re put in the sme prtition set y s nd is equl to w(e) otherwise. t t 1 Fig. 12. A new edge is introdued in the node t. The sttes re only dependent on verties nd hene re the sme over X t nd X t1. However, we hve to ount for the weight of the new edge. (1.5) Computtions t Forget Vertex Nodes. In this se the node t hs single hild t 1 nd X t = X t1 \ {v} for some v X t1. However, G t = G t1. Hene, when omputing dp[t, s], it is suffiient

17 1:16 K. Chtterjee, A.K. Gohrshdy, N. Okti nd A. Pvloginnis to tke the minimum mong the vlues of dp vriles of ll sttes s over X t1 omptile with s. More preisely, we let dp[t, s] = min s S X t s s dp[t 1, s ]. 1 tht re t t 1 Fig. 13. When vertex v is forgotten y t, we hve G t = G t1, ut X t = X t1 \ {v}. Corretness of Step 2. Given tht r is the root node of T, we hve G r = G. Sine every p- prtitioning of G relizes some stte over X r, it follows tht the optiml weight of p-prtitioning is min s SXr dp[r, s]. This onludes the proof. Remrk 3.1. Algorithm 1 omputes the totl weight of ross edges in minimum-weight p- prtitioning. As is ommon with dynmi progrmming lgorithms, n optiml p-prtitioning itself n e otined y keeping trk of the hoies mde during the omputtion of dp vriles, i.e. keeping trk of the ses tht led to the miniml vlues in eh omputtion. We now estlish the omplexity of our pproh nd present the min theorem of this setion. Numer of Sttes. For fixed p, let C p denote the numer of different possile sttes over set of k size k, i.e. C p k := S {1,2,...,k }. We write C k insted of C p when p n e inferred from the ontext. k Appendix A estlishes ounds on the vlue of C k. Note tht this vlue only depends on p nd k. Theorem 3.1. Given Dt Pking instne I = (n, 1,p, R) s input, where n is the numer of distint dt elements, p is the pking ftor, R is the referene sequene with length of N nd the he hs unit size, the Dt Pking prolem, i.e. finding the miniml numer of he misses, n e solved in liner time, i.e. in time O(N + n k 2 C k p k ), when the underlying ess grph G R hs treewidth k 1. Proof. Given Dt Pking instne I = (n, 1,p, R), we first pply the redution of Lemm 3.1 whih tkes O(N ). We then use Algorithm 1 to solve the resulting minimum-weight p-prtitioning prolem. The orretness of this lgorithm ws estlished in Lemm 3.2. The only remining prt is to find the runtime of Algorithm 1. Note tht the time spent for omputing nie tree deomposition, s in Lemms 2.2 nd 2.3 re liner nd dominted y the rest of our runtime. The lgorithm needs vlues of dp vriles for ll nodes of the tree deomposition whih re t most O(n k). We otin upper-ounds for the runtime of our lgorithm on eh type of node: Leves. There is single stte t eh lef nd its dp is zero. Hene we spend O(1) t eh lef. Join Nodes. At join node t, there re t most C k sttes nd for eh stte s = (φ, sz) we hve to look into the sttes orresponding to every possile size enlrgement funtion sz 1 sz. As in the proof of Lemm A.2, there re t most p k suh funtions. Creting eh orresponding stte tkes O(k). Hene, we spend O(k C k p k ) t eh join node. Introdue Vertex Nodes. At node t, there re C k sttes nd we spend O(k) omputing the unique orresponding stte over X t1. Thus, eh introdue vertex node tkes O(k C k ). Introdue Edge Nodes. This se is similr to the previous one nd tkes O(k C k ). Forget Vertex Nodes. At node t, there re C k sttes nd for eh of them we hve to look into ll its omptile sttes over X t1. Note tht suh omptile sttes n e otined either y putting the vertex v in its own prtition set, whih n hve ny size etween 1 nd p, or

18 Effiient Prmeterized Algorithms for Dt Pking 1:17 y dding it to the prtition set of nother vertex in X t. Hene, there re t most p + k suh sttes nd the totl proessing time of forget vertex node is O(k (p + k) C k ). Note tht the runtime for join nodes domintes the rest. Given tht there re O(n k) nodes in totl, the whole omputtion tkes O(n k 2 C k p k ) time. Finlly, the lgorithm spends O(C k ) time omputing the finl result using the dp vlues t the root. Remrk 3.2. By exploiting treewidth, we provided liner-time lgorithm for finding the ext solution to the Dt Pking prolem when m = 1. Note tht in the generl se, i.e. without onsidering prmeteriztion y treewidth, this prolem is NP-hrd s mentioned in Theorem 2.1. Remrk 3.3. We ssumed LRU s the replement poliy. However, given tht the replement poliy does not mtter when the he hs unit size (Remrk 2.1), our lgorithm is pplile to ny replement poliy, inluding FIFO nd OOP. 3.2 Hrdness of Dt Pking on Trees In this setion, we provide redution from the generl prolem of Dt Pking to the speil se where the ess grph is tree, i.e. hs treewidth 1. This redution leds to hrdness results tht enhne those of [Lvee 2016] y showing tht the prolem remins hrd even on trees. This indites tht lthough onsidering onstnt treewidth ess grphs led to effiient lgorithms for the se of m = 1, onstnt treewidth ess grphs lone re not suffiient for m 2. Theorem 3.2 (Hrdness of Dt Pking on Trees). Given Dt Pking instne I = (n,m,p, R), we hve the following hrdness results: Hrdness of the Ext Prolem. For ny m 2 nd ny p 3, Dt Pking is NP-hrd even if the underlying ess grph G R is tree. Hrdness of Approximtion. Unless P=NP, for ny m 6,p 2 nd ny onstnt ϵ > 0, there is no polynomil pproximtion lgorithm for the Dt Pking prolem with n pproximtion ftor of O(N 1 ϵ ) even if the ess grph G R is tree. Proof. We provide liner-time redution tht trnsforms Dt Pking instne I = (n,m,p, R) to nother instne I = (n + (m + 1)p,m + 1,p, R ) suh tht the ess grph G R is tree. Both hrdness results n then e otined y pplying this redution to the hrdness results of Setion 2.1. Given I, we introdue (m + 1)p new dt elements d 1,d 2,...,d (m+1)p. Let X e the sequene d 1,d 2,...,d (m+1)p,d (m+1)p 1,...,d 1. We form the sequene R s follows: d 1, R[1],d 1, R[2],d 1,...,d 1, R[N ],d 1, X, X,..., X, } {{ } 2N +m+2 times i.e. we tke R nd dd d 1 t its eginning, end nd etween every two elements of it, then we ontente the result with 2N + m + 2 opies of X. We let I = (n + (m + 1)p,m + 1,p, R ). Note tht the he in I hs one spot more thn the he of I. By onstrution, G R is tree, euse it onsists of pth d 1,...,d (m+1)p nd every other vertex of the grph is only onneted to d 1. We now show tht the optiml numer of he misses in I is extly m + 1 plus the optiml numer of he misses in I. Let σ e n optiml dt plement sheme for I, then σ must neessrily put the d i s in extly m + 1 loks, otherwise eh X in the sequene R will led to t lest one he miss for totl of t lest 2N + m + 2. On the other hnd, putting the d i s in m + 1 loks leds to t most 2N + m + 1 he misses, even if ll esses efore the X s re missed. In prtiulr, σ does not put ny element of R in the sme lok s d 1. Therefore, σ first leds to he miss on the first ess to d 1, then keeps d 1 in the he forever. Hene, σ fills one spot of the he with the lok of d 1 nd hs m spots left for sheduling R. Finlly, σ lods the other m loks tht ontin some d i s ut not d 1.