Optimal Embedding of Functions for In-Network Computation: Complexity Analysis and Algorithms

Transcription

1 1 Opimal Embedding of Funcions for In-Nework Compuaion: Complexiy Analysis and Algorihms Pooja Vyavahare, Nuan Limaye, and D. Manjunah, Senior Member, IEEE arxiv: v3 [cs.dc] 14 Jul 2015 Absrac We consider opimal disribued compuaion of a given funcion of disribued daa. The inpu (daa) nodes and he sink node ha receives he funcion form a conneced nework ha is described by an undireced weighed nework graph. The algorihm o compue he given funcion is described by a weighed direced acyclic graph and is called he compuaion graph. An embedding defines he compuaion communicaion sequence ha obains he funcion a he sink. Two kinds of opimal embeddings are sough, he embedding ha (1) minimizes delay in obaining funcion a sink, and (2) minimizes cos of one insance of compuaion of funcion. This absracion is moivaed by hree applicaions in-nework compuaion over sensor neworks, operaor placemen in disribued daabases, and module placemen in disribued compuing. We firs show ha obaining minimum-delay and minimumcos embeddings are boh NP-complee problems and ha cos minimizaion is acually MAX SNP-hard. Nex, we consider specific forms of he compuaion graph for which polynomial ime soluions are possible. When he compuaion graph is a ree, a polynomial ime algorihm o obain he minimum delay embedding is described. Nex, for he case when he funcion is described by a layered graph we describe an algorihm ha obains he minimum cos embedding in polynomial ime. This algorihm can also be used o obain an approximaion for delay minimizaion. We hen consider bounded reewidh compuaion graphs and give an algorihm o obain he minimum cos embedding in polynomial ime. Index Terms In-nework funcion compuaion; operaor placemen; graph embedding; A. Background I. INTRODUCTION Efficien compuing of funcions of disribued daa is of ineres in many applicaions, mos recenly in sensor neworks [1] and in programming models for disribued processing of large daa ses, e.g., MapReduce [2] and Dryad [3]. Consider a ypical sensor nework scenario where sensor nodes measure and sore environmen variables and form a disribued daabase. These nodes also have compuing and communicaion capabiliies and can form a conneced nework. A sink node, also called erminal or receiver, is ineresed in one or more funcions of his disribued daa raher han in he raw daa iself. Convenional examples for such funcions are maximum, minimum, mean, pariy, and hisogram [1]. More sophisicaed funcions are also easily moivaed, e.g., spaial Pooja Vyavahare and D. Manjunah are wih he Dep. of Elecl. Engg., and Nuan Limaye is wih he Dep. of Comp. Sc. & Engg. of IIT Bombay. addresses are {vpooja,dmanju}@ee.iib.ac.in and nuan@cse.iib.ac.in This work was suppored in par by DST projecs SR/SS/EECE/0139/2011, ITRA/15(64)/Mobile/USEAADWN/01, and was carried ou in he Bhari Cenre for Communicaion a IIT Bombay. and emporal correlaions, specral characerisics of he daa (ha may be obained by performing FFT on he daa), and filering operaions on he raw daa in sensor neworks. A naive approach o obain he required funcion(s) of he disribued daa a he sink would be o collec he raw daa a he sink and have i perform he compuaion. Alernaively, since he nodes have compuaion capabiliy, i could possibly be more efficien o push he compuaion ino he nework, i.e., use a disribued compuaion scheme over he communicaion nework. Our ineres is in he laer approach efficien innework compuaion of he required funcion. Several measures of efficiency of compuaion may be defined. Toal energy expended in obaining one sample of he funcion is a possible measure. Delay from he ime a which he daa is available a he sources o he ime a which he funcion value is available a he sink is a second possible measure. If he daa a each of he sources were o be a sream, hen he rae a which he funcion values are available a he sink is a hird possible measure. In his paper we consider only he firs wo measures above cos of compuaion and delay in compuaion. Thus our focus is on algorihms ha find an compuaion and communicaion (rouing) sequence o compue a arge funcion on a given nework ha minimize he delay or he cos. The arge funcions are assumed o belong o he class of funcions ha are compued using a scheme ha can be represened as direced acyclic graphs (DAGs). The following example illusraes our inen. Consider a nework of N nodes, K of which collec measuremen daa from heir environmen. Le x k () be daa sample available a node k (k = 1,..., K) a ime and le r() = 1 K 1 K 1 i=1 x i()x i+1 () be he funcion of ineres. r() can be compued using he schema of Fig. 1a. Each edge in his direced graph represens an inermediae value in he compuaion of r() and each node corresponds o an operaion ha is o be performed on is inpus. The communicaion nework over which r() is o be compued is shown in Fig. 1b as a weighed undireced graph. In he example all he edges have uni weigh. Two possible compuaion and communicaion schemes are shown in Figs. 1c,d. We see ha he scheme in Fig. 1c has a lower cos han ha in Fig. 1d. Several flavors of in-nework funcion compuaion exis in he lieraure. A randomly deployed mulihop wireless nework of noise free links is considered in [1], [4] [6]. They deermine asympoic achievable raes a which differen symmeric funcions like minimum, maximum and ype vecor may be compued. An idenical objecive is addressed in [7] [11] for single hop neworks wih noisy links and in [12], [13] for mulihop wireless neworks wih noisy links. When

2 2 s 1 s 2 s 3 + (a) s 1 s 2 (c) s 3 s 1 s 2 (b) s 1 s 2 Fig. 1: Compuaion of funcion r() = x1x2+x2x3 2. (a) A schema o compue r() (b) Communicaion Nework (c) Implemenaion 1 (d) Implemenaion 2 he nodes are in a n n grid and communicae over wireline or wireless links, [14] obains he ime and number of ransmissions required o compue a funcion. Randomized gossip algorihms [15], [16], where a random sequence of node-pairs exchange daa and perform a specific compuaion is used in [17] [19] for funcion compuaion. The ineres is in ime for all nodes o converge o he specified funcion. Anoher sream of work considers compuing of specific funcions using nework coding and designs he communicaion neworks o maximize he compuaion rae [20] [22]. None of he above consider minimum cos funcion compuaion. Furher, o he bes of our knowledge, only [23] considers compuaion of arbirary funcions in finie size neworks; he ineres here is on maximizing he rae of compuaion. Raher han arihmeic operaions o perform a funcion compuaion, he nodes in Fig. 1a could correspond o operaions required o execue a daabase query and he direced edges would represen he flow of he resuls of he operaions. In his case he graph of Fig. 1a is called a query graph. The inpu daa o his query graph could be from a disribued daabase, which in urn could be a sensor nework; in his case he graph in Fig. 1b would represen he inerconnecion among he unis of he disribued daabase and some oher nodes ha may be available for compuaion. In his conex performing an efficien query requires ha he operaor placemen be efficien. Efficien operaor placemen is addressed in [24] [29] all of which assume ha he query graph is a ree. While [24], [27], [29] develop heurisics based algorihms for efficien operaor placemen when he query graph is a ree, algorihms wih formal analyses are available in [25], [26], [28]. Alhough [30] considers a non-ree query graph, only heurisic algorihms are provided. (d) s 3 s 3 Going back in he lieraure, we see ha he module placemen problem for disribued processing from he 1980s has a flavor similar o in-nework compuaion and operaor placemen problems; see e.g., [31] [35]. In his case he nodes of Fig. 1a correspond o he modules of a program and a direced edge (u, v) implies ha module u calls module v during execuion. Furher, he graph of Fig. 1a is called a call graph. The nodes of Fig. 1b represen he processors on which he modules of he program need o be execued and he edges represen he iner processor links. The cos srucure for his problem is more complex. The execuion cos is specified for each module-processor pair and here is also an iner processor communicaion cos over an edge ha in urn depends on he modules placed a he ends of he edge. The objecive is o place he modules on he processors such ha he oal cos of execuion is minimized. The firs cenralized algorihm for opimal module placemen when he call graph is a direced ree was given in [32] and ha for k-rees was given in [35]. [31] showed ha he problem can be efficienly solved for a wo processor sysem by using nework flow algorihms and [33] uses a similar approach o develop heurisic algorihms for a general call graph. From he preceding discussion we see ha he objecives of, and hence he soluion echniques for, efficien in-nework compuaion, operaor placemen and module placemen all have a very similar heme embedding (a formal definiion is provided in Secion II) a graph represening a compuaion schema on a conneced weighed graph represening a nework of processors. Much of he lieraure on such problems assumes ha he graph describing he funcion is a ree. This is clearly very resricive because he compuaion schema for a very large class of useful compuable funcions canno be described by a ree and are more generally described by a DAG, e.g., fas Fourier ransform (FFT), soring, any polynomial funcion of inpu daa, and any funcion of Boolean daa. MapReduce, he popular cloud compuing paradigm, also has a DAG represenaion and is discussed in some deail in Example 3 in Secion V. B. Organizaion of he Paper Our ineres is in compuing arbirary funcions ha have a specific algorihmic represenaion and he communicaion nework is an arbirary nework and no a random wireless nework. (We reierae ha alhough we assume he in-nework compuaion scenario o anchor he discussion, our resuls are also direcly applicable o he operaor and module placemen scenarios.) Furher, we do no seek o specifically maximize he compuaion hroughpu; raher, our ineres is in minimizing he cos or delay in a one sho compuaion of he funcion. As we menioned before, wo measures of efficiency are used in his paper cos of compuaion and delay in compuaion. Our resuls on minimum cos embedding can be used o maximize compuaion hroughpu when used in he algorihms developed in [23]. Given an arbirary funcion via is DAG descripion and an arbirary nework over which his funcion is o be compued, our firs ineres is o analyze he complexiy of finding he

3 3 opimal compuaion and communicaion scheme ha will compue he funcion in he nework. While much of he lieraure claims ha his problem is NP-complee (for he case of minimizing cos), o he bes of our knowledge a formal proof is no available; he lieraure evenually leads o a ciaion o a privae communicaion in [32]. In Secion III we prove ha in general, boh he minimum cos and he minimum delay embedding problems are NP-complee. Some srucure in he DAG can be exploied o provide polynomial ime algorihms for our problems. If he DAG is a ree, which is he assumpion in much of he exan lieraure, he minimum cos embedding is similar o shores pah algorihms [23], [28]. To he bes of our knowledge, minimum delay embeddings are no considered in he lieraure and in Secion IV we provide a polynomial ime algorihm o find he minimum delay embedding when he compuaion schema is a ree. Nex we consider wo large classes of compuaion graphs (1) layered graphs, (2) bounded reewidh graphs. We derive he moivaion for layered graphs from disribued daa processing frameworks like MapReduce [2] and Dryad [3]. In Secion V we provide a polynomial ime algorihm o find he minimum cos embedding when he DAG is a layered graph. This same algorihm also obains an approximaion for he minimum delay embedding. In Secion VI, we show ha he algorihm from Secion V can find he minimum cos embedding when he DAG is a bounded reewidh graph. The noaion and he formal problem definiion is described in he nex secion. In Secion VII, we describe an updae mechanism when here is perurbaion in he DAG and we conclude wih a discussion in Secion VIII. II. PRELIMINARIES The communicaion nework is represened by an undireced conneced graph N = (V, E) wih V being he se of n nodes and E being he se of m edges. The elemens of V are denoed by {u 1,..., u n }. Each edge (u i, u j ) E has a non negaive weigh T (u i, u j ) associaed wih i. The weigh could, for example, correspond o ransmission ime of a bi on he link, or he energy required o ransmi one bi on he link or somehing more absrac. For a given T, and any u i, u j V le d ui,u j be he weigh of he minimum cos pah from u i o u j. Le [[D = d ui,u j ]] be he n n disance marix. Of he n nodes in N, here are K source nodes denoed by {s 1, s 2,..., s K } V. Source node s i generaes daa x i ; denoe x = {x 1,..., x K }. A sink node V requires o obain a funcion f(x 1, x 2,..., x K ) of he daa. We assume ha schema o compue f(x 1, x 2,..., x K ) is given and is represened by a direced acyclic graph G = (Ω, Γ), where Ω is he se of p nodes and Γ is he se of q edges. The nodes in G are denoed by {ω 1,..., ω p } and correspond o operaions ha need o be performed on he inpu daa o he node and he ougoing edges denoe he flow of he resul of hese operaions. Thus each edge in G represens a sub-funcion of he inpus. The sources in he compuaion graph are denoed by nodes {ω 1, ω 2,..., ω K } wih node ω i corresponding o source x i ; node ω p is he sink ha receives he funcion f(x 1, x 2,..., x K ). The direcion on he edges in G represen he direcion of he flow of he daa. Each edge (ω i, ω j ) Γ has a non negaive weigh W(ω i, ω j ) associaed wih i which could correspond o he number of bis used o represen he inermediae funcion. Since G is a direced acyclic graph here is a parial order associaed wih is verices. If (ω i, ω j ) Γ hen he funcion a ω j canno be compued unil he funcion a ω i is compued and he resul forwarded o ω j. Le Φ (ω) and Φ (ω) denoe, respecively, he immediae predecessors and successors of verex ω, i.e., Φ (ω) = {τ Ω (τ, ω) Γ} and Φ (ω) = {τ Ω (ω, τ) Γ}. A processing cos (delay) funcion P : Ω V R + is used o capure he cos (delay) of performing a paricular operaion on a paricular verex of he nework. Now we define he embedding of G on N as follows. Definiion 1: An embedding of a compuaion graph G on a communicaion nework N is a many-o-one funcion E : Ω V which saisfies he following condiions. 1) E(ω i ) = s i for i = {1,..., K} 2) E(ω p ) =. In his definiion of embedding each node in he compuaion graph is mapped o a single node in he nework graph and he edge (ω i, ω j ) Γ is mapped o he shores pah beween E(ω i ) and E(ω j ). Alernae definiions of an embedding are possible, e.g., an edge in G can be mapped o more han one pah in N while saisfying some coninuiy consrains; his is he definiion of an embedding in [23]. An embedding defines a compuaion and communicaion sequence in N o obain f a he sink. Le E be he se of all embeddings of G in N which follow Definiion 1. The weigh funcions T and P can be reaed as, respecively, he communicaion and he processing delays in N for compuing and communicaing he sub-funcions leading o compuaion of f. We can define he delay in compuing a sub-funcion by a node ω Ω in he embedding E as d(e(ω i )) := max [d(e(ω j)) + W(ω j, ω i )d E(ωj),E(ω ω j Φ (ω i)]+ i) P(ω i, E(ω i )). Recall ha d u,v is he lengh of he shores pah beween verices u, v V ; hus he firs erm here corresponds o he delay in obaining all he operands a node E(ω i ) and he second erm corresponds o he processing delay a he node. Seing he delay a he sources o zero, i.e., d(e(ω i )) = 0 for all i [1, K], we can recursively calculae he delay of each verices of G on N. The delay d(e) of an embedding E is defined as he delay of he sink, i.e., d(e) := d(e(ω p )). This leads us o he firs problem ha we consider in his paper: Find an embedding Eop d such ha he delay of he embedding is minimum among all he embeddings for a given G, N, T, W and P, i.e., solve he opimizaion problem Eop d := arg min d(e). E E (1) (MinDelay) Alernaively, considering he weigh funcions T, W, and P as cos of communicaion and compuaion, e.g., he energy

4 4 ω 1 ω 2 ω 3 + ω 4 + ω 5 ω 6 s 1 s 2 s a b c d 1 ω 1 ω 2 ω 3 α 1 ω 4 α 2 α 3 α 4 s 1 s 2 0.5a 0.1a a 0.7a u 1 v 1 ɛ ɛ s 3 ɛ ɛ s 4 u 2 v 2 s 2l 1 s 2l ω 7 (a) (b) ω 2l 1 ω 2l α 3l 4 α 3l 3 0.5a 0.1a a 0.7a s 1 s 2 s 3 ω 4 b ω 5 (c) ω 6 s 1 s 2 s 3 ω 4 ω 5 c Fig. 2: (a). Compuaion graph G for f = (x 1 + x 2 )(x 2 + x 3 ) (b). Communicaion Nework N. The numbers near he edges represen he weighs on hem. (c). Minimum delay embedding E 1. (d). Minimum cos embedding E 2. cos, he cos of an embedding can be defined as C(E) := P(ω, E(ω)) + ( ) W(ωi, ω j )d E(ωi)E(ω j). ω Ω (ω i,ω j) Γ (2) We can hen find an embedding Eop c such ha he cos of he embedding is minimum among all he embeddings for a given G, N, T, W and P, i.e., solve he opimizaion problem Eop c := arg min C(E). E E (d) ω 6 (MinCos) The following example illusraes he preceding problems and he sysem se up. Example 1: Consider a compuaion graph G = (Ω, Γ) and communicaion nework N = (V, E) shown in Fig. 2. The labels of each verex in boh he graphs are shown in Fig. 2. The processing cos (delay) for sources is assumed o be zero and for oher verices of G i is assumed o be uniy. An embedding E of G on N will have E(ω i ) = s i i [1, 3] and E(ω 7 ) =. Now consider wo embeddings E 1, E 2 such ha E 1 (ω 4 ) = a, E 1 (ω 5 ) = c, E 1 (ω 6 ) = d and E 2 (ω 4 ) = a, E 2 (ω 5 ) = b, E 2 (ω 6 ) = d. These are shown in Figs. 2c and 2d. Using (2), i is easy o verify ha C(E 1 ) = 36 and C(E 2 ) = 34. The delays in he embedding E 1 are: d(e 1 (ω 4 )) = max(10, 2) + 1 = 11, d(e 1 (ω 5 )) = max(8, 10) + 1 = 11, d(e 1 (ω 6 )) = max(12, 12) + 1 = 13 and finally d(e 1 ) = d(e 1 (ω 7 )) = = 14. Similarly, d(e 2 ) = 16. Observe ha he delay of E 1 is lower among he wo bu is cos is higher han ha of E 2. Now we presen an example which shows ha he difference beween he delay obained by he soluion of MinCos problem and ha of he MinDelay problem can be of he order of he number of sources in he nework. ω p (a) α 3l 2 u l+1 v l+1 ɛ ɛ Fig. 3: Illusraing Example 2. (a). Compuaion graph wih 2l sources and a sink. (b). Communicaion nework. Example 2: Consider he compuaion graph G = (Ω, Γ) and a nework graph N = (V, E) as shown in Figs. 3a and b respecively. The labels of he verices are shown in he figure and he numbers near he edges of N represen he weigh of ha edge. Noe ha he srucure beween s 1 o u 2 and s 2 o v 2 is repeaed in he nework graph l imes. Le us call his srucure as A. We assume ha he weighs on he edges of G are all one and he processing coss are zero for sources and for all oher verices i is assumed uniy. Any embedding E of G on N will have E(ω i ) = s i i [1, 2l] and E(ω p ) =. Consider an embedding E 1 such ha E 1 (α 1 ) = u 1, E 1 (α 2 ) = u 2, E 1 (α 3 ) = v 2,..., E(α 3l 2 ) = u l+1. Le us compue he cos of his embedding. Noe ha he oal cos of he embedding is l imes he cos coming from he srucure A plus he weigh of edge u l+1. The cos due o embedding of A is he sum of he weighs of edges s 1 u 1, s 2 u 1, u 1 u 2, u 1 v 2 and he processing coss a u 1, u 2, v 2. Hence he cos of he embedding is C(E 1 ) = l( a + 2ɛ) + ɛ. Similarly he delay of his embedding is d(e 1 ) = (a+ɛ+2)l+ɛ. Now consider anoher embedding E 2 such ha E 2 (α 1 ) = v 1, E 2 (α 2 ) = u 2, E 2 (α 3 ) = v 2,..., E 2 (α 3l 2 ) = v l+1. The cos and delay for his embedding can be compued in a similar fashion o obain C(E 2 ) = l( a + 2ɛ) + ɛ and d(e 2 ) = l(0.7a + ɛ + 2)l + ɛ. I can be shown ha he firs embedding E 1 is he soluion of MinCos where as E 2 is he soluion of MinDelay problem of G on N. If we use he soluion of MinCos o ge he soluion of MinDelay hen he difference would be d(e 1 ) d(e 2 ) = 0.3al which is of he order of he number of sources in he graph. (b) III. HARDNESS OF EMBEDDING We begin by considering MinDelay. Firs consider he case when here is no processing delay, i.e., P(ω, E(ω)) = 0 in he nework and he compuaion graph is unweighed, i.e., W(ω i, ω j ) = 1 (ω i, ω j ) Γ. The delay of he embedding in his case is he delay of he longes embedded pah from any source o he sink in N. Le d si be he delay of he minimum delay pah

5 5 beween source s i and he sink in N. Then he delay of any embedding of G on N which follows he condiions of Definiion 1 has o be more han he delay on he longes of all he minimum delay pahs from sources o sink. In oher words, d(e) max i [1,K] (d si). Now consider an embedding E which maps all he inermediae verices of G o he sink in N. The delay of his embedding will be d(e ) = max i [1,K] (d si). Comparing i wih d(e) ells us ha he embedding E minimizes he delay. Hence MinDelay is easy o solve if here are no processing delays and G is unweighed. Now we analyze he hardness of MinDelay and MinCos for arbirary G and N and show ha he boh he opimizaion problems are NP-hard. We prove he hardness of he opimizaion problems by proving ha he corresponding decision versions are NP-hard. The decision versions of he MinDelay and MinCos are defined as follows: Definiion 2: For a given G, N, T, W, P and a posiive number L he decision version of he MinDelay problem oupus yes if here exiss an embedding E of G on N such ha d(e) L and oupus no if no such embedding exiss. Definiion 3: For a given G, N, T, W, P and a posiive number L he decision version of he MinCos problem oupus yes if here exiss an embedding E of G on N such ha C(E) L and oupus no if no such embedding exiss. Noe ha if one can solve he opimizaion version of MinDelay (resp. MinCos) problem in ime, say, hen using he soluion of ha problem we can solve he corresponding decision version in ime O(). Hence he original opimizaion problem is aleas as hard as is decision version. This implies ha if we jus prove ha he decision version of he MinDelay(resp. MinCos) is NP-hard hen he opimizaion version is also NP-hard. We now proceed o prove ha he decision version of our opimizaion problems are indeed NP-complee. Theorem 1: The decision version of MinDelay problem is NP-complee. Proof: We firs prove ha he decision version of MinDelay is NP-hard by giving a reducion from he NP-complee problem Precedence Consrain Scheduling wih fixed mapping (PCS-FM) [36]. PCS-FM problem is defined as follows. Given a se of asks H wih a parial order on i, ask h having lengh l(h) = 1 h H, a se τ H, and m Z + processors, find a schedule σ of asks on he processors which mees an overall deadline D, maps each τ i τ o a paricular processor p(τ i ) and obeys he precedence consrain ha if h i h j hen σ(h j ) σ(h i )+1. To he bes of our knowledge he hardness of PCS-FM problem has no been proved in he lieraure and we provide he proof of NP-compleeness of PCS-FM in Appendix A. We firs give a reducion from an insance φ = (H,, τ, p(τ), m, l, D ) of PCS-FM o an insance of Min- Delay ψ = (G, N, γ, λ, T, P, D), where G = (Ω, Γ) and N = (V, E) are he compuaion and communicaion graphs respecively. The se γ Ω is o be mapped o λ V under any embedding E and T, P are he communicaion and processing delay of he nework. Noe ha he se of asks H along wih he parial order creaes a DAG and we define G =. We define N o be a complee graph on m processors. Le γ = τ and λ = p(τ). The ransmission delay T beween any wo verices of N is aken as ɛ = 1 H 2 and P(ω, u) = 1 for all ω Ω, u V. Finally D = D + 1. We have o prove ha here is a schedule σ of φ which finishes in ime D if and only if here is an embedding E of ψ wih delay D D D + 1. The forward direcion is easy o prove. If here is a schedule of φ which finishes in D ime and maps a ask h H o a processor q hen we can creae an embedding of ψ which maps he same verex h Ω o a verex q V. Noe ha because γ = τ and λ = p(τ) he condiions of Definiion 1 are me in his embedding. The delay of any verex u G in his embedding will be a mos he ime a which he ask u finishes in he schedule σ plus he number of imes edges in N are used because of he same precedence order. Hence he delay of his embedding will be D < D < D + H ɛ D + 1 H < D + 1. To complee he proof we need o prove ha if here is an embedding E of ψ of delay α R + hen here is a schedule σ of φ which finishes in ime α. We will creae a schedule σ from he embedding E. If a verex h Ω is mapped o a verex u V hen in he schedule σ also he ask h is execued by processor u. Because γ = τ and λ = p(τ) he asks in τ are mapped o p(τ) in his schedule also. Le he oal number of edge uses in his embedding be b. Noe ha b < H 2 in any embedding where H are he number of verices in graph G. The oal ransmission delay in he embedding is bɛ < 1. Recall ha he processing delays are all 1 hence he delay of he embedding can be wrien as α = α + bɛ. I is easy o verify ha he ime required by he schedule σ o complee is α. This proves ha he decision of MinDelay is NP-hard. Given an insance of MinDelay problem an embedding E can be guessed non-deerminisically and checked wheher d(e) L in polynomial ime. Thus he decision version of MinDelay problem is in NP and our reducion proves ha i is in fac NP-complee. We now look a he problem of approximaing he MinDelay problem. We say ha a polynomial ime approximaion algorihm has a performance guaranee α > 1 if i oupus a feasible soluion of he problem which has delay a mos α imes he opimal soluion. We prove he following: Theorem 2: Unless P=NP, an insance of he MinDelay problem wih (G, N, T, W, P) and wih uni processing delays and uni weighs on he edges of G, does no have a polynomial-ime approximaion algorihm wih performance guaranee sricly less han 100/99 if is soluion is greaer han 10. Proof: Noe ha while proving he hardness of MinDelay we firs reduced an insance of a PCS problem o an insance of a PCS-FM problem and hen we reduced PCS-FM o MinDelay problem. Le op 1 and sol 1 be he deadline achieved by he opimal and a feasible soluion of PCS problem respecively. Similarly, le op 2 and sol 2 be he deadline achieved by he opimal and a feasible soluion of he PCS-FM problem. Finally, le op 3 and sol 3 be he opimal and feasible soluions for MinDelay. While proving he NP-compleeness of PCS- FM (in Appendix A) we showed ha op 2 = op We also showed ha if a schedule of PCS problem achieves a deadline

6 6 sol 1, hen here is a schedule of PCS-FM wih sol 2 = sol Le PCS-FM have a polynomial-ime approximaion algorihm wih soluion sol 2 = x op 2. By Observaion 5.1 of [37] we know ha he PCS problem does no have polynomial-ime approximaion algorihm wih performance guaranee less han 4/3. Hence, sol 1 > 4 3 op 1. Subsiuing he relaion beween sol 1 and sol 2 in he above equaion we ge, sol 2 2 > 4 3 op 1. Thus, xop 2 > 4 3 op and x > 4 op 1 3 op op. We now 2 observe ha op 1 1 op 2 3 and hence op 1 > 1 3 op 2 we can wrie, x > 4 op 1 3 op + 1 op 1 2 op and finally x > Hence, unless P=NP, PCS-FM canno have a polynomial ime approximaion algorihm wih performance guaranee less han 10/9. While proving he hardness for MinDelay we showed ha for any insance of PCS-FM wih sol 2 we can ge an insance of MinDelay problem wih soluion sol 3 such ha sol 2 sol 3 sol Le MinDelay have a polynomial ime approximaion algorihm wih soluion sol 3 = y op 3. We know ha sol 2 > 10 9 op 2. Subsiuing he relaion beween sol 3 and sol 2 in he above expression we ge, sol 2 > 10 9 op 2 and sol 3 = yop 3 > 10 9 op 2. This implies ha y > 10 op 2 9 op. 3 Observe ha, if op 2 10 hen op 3 op (because of he reducion) implies ha op 2 > op 3. So we ge y > 100/99. This complees he proof. We now consider he decision version of MinCos. Remark 1: Recall ha he cos of an embedding E is compued using (2) which does no consider he direcion on he edges of G. I only considers he weigh W on any edge of G. Thus he cos of an embedding does no depend on he direcions of he edges and he soluion of MinCos problem is same irrespecive of wheher he compuaion graph G has direcions or no. Theorem 3: The decision version of he MinCos is NPcomplee. Proof: We acually prove ha he decision version of MinCos is NP-hard even when he processing coss are zero and he coss on he edges of G and N are all one. In his case he cos of he embedding E is given by C(E) := (ω d i,ω j) Γ E(ω i)e(ω j), where d uv is he shores disance beween he verices u, v V. We prove he hardness of he decision version of MinCos by giving a reducion from he unweighed version of Mulierminal Cu problem which is NP-complee [38]. Mulierminal Cu problem is defined as follows: Given an arbirary graph = (V, E ) and a se S = {s 1, s 2,..., s k } V of k specified verices, find he minimum number of edges E s E such ha he removal of E s from E disconnecs each verex in S from all he oher verices in S. The cos of an embedding can be compued in polynomial ime using (2). Hence, given an insance of he decision version of he MinCos problem, one can guess an embedding in a non-deerminisic way and check wheher is cos is less han L or no in polynomial ime. Thus he decision version of he MinCos problem is in NP. To prove he NP-hardness of he problem we will firs show a ransformaion of an insance of Mulierminal Cu problem ψ = (N, S, D) o an insance of he decision version of MinCos φ = (N, G, γ, λ, D). Then we will show ha here exiss a se of edges in N of size D N which separaes all he verices of S from all oher verices in S if and only if here is an embedding E such ha a verex γ i γ is mapped o a verex λ i λ and has cos equal o D. From ψ define an insance of decision version of MinCos φ as follows: Le N be a complee graph on V = {u 1,..., u k } where k = S, and G = N. Noe ha in his case G is undireced. Define γ = S and λ = V such ha a verex γ i is mapped o u i V. In oher words, k disinc verices of Ω are mapped o disinc verices of V. Now we prove ha here is an embedding E of cos D if and only if ψ has a Mulierminal Cu of size equal o D. The if par is easy o see. If here is a minimum Mulierminal Cu E s of size D which divides he verex se V = V 1... V k ino k disjoin subses hen define E such ha each verex in V i is mapped o u i V. Then he cos of he embedding is he oal number of edges which go from V i o V j for i j. This is nohing bu he size of he se E s which is equal o D. To complee he proof we need o show ha if here is no minimum Mulierminal Cu of ψ of size D hen here is no embedding (which maps he verices of γ o verices of λ) of φ wih cos D. Le us assume ha here is no minimum Mulierminal Cu of ψ of size D bu here is an embedding for φ wih cos D. I implies ha here is a mapping of Ω on k differen verices of V such ha γ i = s i is mapped o u i. Le us denoe all he verices of Ω ha are mapped o u i by Ω i. The cos of he embedding is equal o he number of edges beween ses Ω i and Ω j for i j which is equal o D. Now i is easy o see ha if we divide he verices of N in k disjoin subses such ha all he verices in Ω i are in he same se hen we can creae a Mulierminal Cu of he graph N which has cos exacly D. Bu his conradics our assumpion. Hence if here is an embedding funcion E for φ of cos D hen here is a Mulierminal Cu of ψ of same size. This proves ha he decision version of MinCos is NP-hard when he compuaion graph is undireced. From Remark 1, hus he decision version of MinCos wih direced compuaion graph is also NP-hard. So far, we have no considered any weigh funcions. From [38] we know ha he Mulierminal Cu problem for weighed graphs is also NP-complee. And a simple modificaion in our reducion will prove ha decision version of MinCos problem wih weigh funcions is also NP-hard. I is also shown in [38] ha Mulierminal Cu problem is MAX SNP-hard. To prove any problem being MAX SNP-hard i is sufficien o give a linear reducion o i from a known MAX SNP-hard problem. The linear reducion is defined as follows: Definiion 4: Le Π and Π be wo opimizaion problems. Then we say ha Π linearly reduces o Π if here are wo polynomial ime algorihms A, B and wo consans α, β > 0 such ha 1) Given an insance π of Π wih an opimal cos op(π) an algorihm A produces an insance π = A(π) of Π such ha he cos of an opimal soluion for π (op(π )) is a mos α op(π), i.e., op(π ) α op(π). 2) Given π, π = A(π) and any soluion y of π here is an algorihm B which produces a soluion x of π such ha cos(x) op(π) β cos(y) op(π ).

7 7 Noe ha in he proof of Theorem 3 we use polynomial ime algorihms o reduce an insance ψ of he Mulierminal Cu problem o an insance φ of he MinCos problem and we proved ha cos(ψ) = cos(φ). We can also ge a soluion of φ from a soluion of ψ and vice versa wih parameers α = β = 1. Hence he reducion we used o prove ha MinCos is NP-complee is in fac a linear reducion of Mulierminal Cu problem o MinCos. We hus have he following corollary. Corollary 1: Problem MinCos is MAX SNP-hard and hence does no have any polynomial ime approximaion scheme unless P=NP [39]. The NP-hardness of MinCos can also be proved by reducion from anoher well known NP-complee problem k-clique [36]. The k-clique problem is defined as follows: Given an arbirary graph N and a posiive ineger k check wheher N = (V, E ) has a clique (or a complee subgraph) of size k. Since we know ha he k-clique problem is W [1] complee [40], he reducion from i also implies ha MinCos does no have any fixed-parameer racable algorihm and is also hard for W [1]. A variaion of MinCos is considered in [35] in ha he communicaion coss (compued as W(ω i, ω j ) d E(ωi)E(ω j) in his paper) are no necessarily zero if E(ω i ) = E(ω j ). Furher, he source and he sink nodes are no fixed like in his paper. The key complexiy resul is Lemma 2.1 which shows ha heir variaion of MinCos wih zero-one communicaion coss is NP-complee by reducing i from he planar 3-SAT problem. Their proof echnique does no allow he communicaion o be necessarily zero if E(ω i ) = E(ω j ). For non zeroone communicaion coss, [35] also gives a polynomial ime algorihms for heir variaion when G is a parial k-ree and almos rees wih parameer k. As is he case wih many NP-complee problems our problems also become racable when G has special srucures. We consider hree such srucures ha have wide applicaions he ree, layered graphs and bounded reewidh graphs. These are considered nex. IV. G IS A TREE Many funcions ha are useful on sensor neworks, e.g., average, maximum, minimum ec., can be represened by direced ree graphs. Operaions ha are required o resolve many daabase queries can also be represened as direced ree srucures. The rees represening a G is from a class of rees ha have a se of leaf verices whose in-degree is zero and a roo verex whose ou-degree is zero. We consider a ree srucured G such ha all he leaf verices represen he sources of daa and he roo acs as he sink which wans o know he final funcion value. Recall ha we label all he verices in Ω as {ω 1,..., ω p }. Also Φ (ω i ) and Φ (ω i ) represen he predecessor and successor verices of he verex ω i Ω. I is easy o verify ha in his ype of ree srucured compuaion graph, every verex (excep he roo) has only one successor verex, i.e., for any ω Ω (excep he roo) he se Φ (ω) is a singleon se. The se Φ (ω p ) is null, where ω p is he roo and here is a unique pah from each source o he sink. As we have menioned earlier, [28] and [23] adap, respecively, he Bellman-Ford and he Dijksra shores pah algorihms o solve MinCos when G is a ree. In he following we will describe Algorihm 1 ha solves MinDelay when G is a ree. Algorihm overview: Algorihm 1 is a cenralized algorihm which assumes knowledge of he all-pair shores pah delay marix D of he communicaion nework N. The opimal embedding is compued by ieraing hrough all he edges of he compuaion graph G. For an edge (ω i, ω j ) of G, he algorihm compues he opimal delay of he pah leading o he verex ω j from sources via ω i for all possible mappings of he verex ω j in he nework. The delay ill any verex ω j is he maximum of he opimal delays of all he pahs reaching o ω j plus he processing delay a ha verex. Once he opimal delay for he sink node is compued, he algorihm backracks o find he opimal mapping of all he oher verices. Algorihm Descripion: In each ieraion he Algorihm 1 mainains he following daa srucures: Algorihm 1: Opimal embedding algorihm o solve Min- Delay for ree graphs Inpu: Nework graph N = (V, E), V = n, E = m, Weigh funcion T : E R +, Tree compuaion graph G = (Ω, Γ), Ω = p, Γ = p 1, Weigh funcion W : Γ R +, Cos funcion P : Ω V R +. Oupu: Embedding E wih minimum delay under T, W, and P. 1: [[D = d uv ]] // n n disance marix for N. // Iniializaion of ables 2: h l (v) := 0 v V, l [1, p 1] ; 3: f l (u, v) := 0 u, v V, l [1, p 1] ; 4: for l = 1 o K do 5: for all v V do 6: f l (s l, v) := W(ω l, Φ (ω l ))d uv ; 7: h l (v) f l (s l, v) ; 8: x l (v) s l ; 9: end for 10: end for 11: for l = K + 1 o (p 1) do 12: for all v V do 13: for all u V do 14: f l (u, v) := P(ω l, u) + W(ω l, Φ (ω l ))d uv ; 15: end for 16: h l (v) min {max[h i (u) ω i Φ (ω l )] + f l (u, v)} u ; 17: x l (v) arg min {max[h i (u) ω i Φ (ω l )] + f l (u, v)} ; u 18: end for 19: end for 20: d(e) := max[h i () ω i Φ (ω p )] + P(ω p, ) ; 21: E(ω p ) = ; // Backracking 22: for l = (p 1) o 1 do 23: E(ω l ) = x l (E(Φ (ω l ))) ; 24: end for 1) f l (u, v) : I is he delay associaed wih edge

8 8 (ω l, Φ (ω l )) and verex ω l when ω l and Φ (ω l ) are mapped o verex u and v, respecively. 2) h l (v) : I is he opimal delay of he pah leading o verex Φ (ω l ) (via ω l ) when i is mapped o v N. The algorihm also sores he mapping of verex ω l in x l (v) corresponding o his value. Afer iniializing hese daa srucures o zero (lines 2, 3) he algorihm complees in he following wo seps. Lines 4 10 : This is he ieraion over all he sources in G. As he mapping of source ω i G is fixed o source s i N, here we jus calculae he minimum delay required o reach o all he verices from he source s i. Noe ha he processing delay a he source is zero, i.e., P(ω i, s i ) = 0. Lines : This is he main loop of he algorihm which runs over all he remaining verices of G saring from ω K+1 o ω p 1. In each ieraion f l (u, v) is updaed for all possible mappings of verices ω l, Φ (ω l ) (lines 13 15). The funcion f l (u, v) is compued by adding he following delay erms. P(ω l, u) : This is he processing delay associaed wih he verex ω l when i is performed a verex u V. W(ω l, Φ (ω l ))d uv : This is he communicaion delay associaed wih he edge (ω l, Φ (ω l )) when mapped o u and v respecively. Then h l (v) is updaed in line 16. Noe ha max[h i (u) ω i Φ (ω l )] is he minimum delay ill he verex ω l when i is mapped o u. This is equivalen o he firs erm in righ hand side of (1) (Secion II, Page 3). This along wih he processing delay P(ω l, u) gives he delay of he verex ω l when mapped o u. As menioned earlier he algorihm also sores he mapping u in x l (v) which minimizes he h l (v). Lines : Once all he delays are compued he algorihm compues he delay a he sink verex (as he mapping of ω p is fixed o in embedding E) and finds he mapping of verices of G on N which gives his value by backracking from he sink o he sources. 1) Analysis of Algorihm 1: Theorem 4: Algorihm 1 solves MinDelay when G is a ree and runs in O(pn 2 ) ime. Recall ha p and n are he number of verices in G and N respecively. Proof: We give he proof of correcness of Algorihm 1 only when he compuaion graph is unweighed. The proof can easily be exended o he case when here are weighs on he edges of G. Recall ha he delay of an embedding is defined recursively over all he verices of G saring from he sink verex. I is sufficien o prove ha a any ieraion l he algorihm compues he opimal delay of embedding he pah from any source ω i o an inermediae verex Φ (ω l ) via ω l for all possible embeddings of Φ (ω l ). And hen a he end i chooses he fixed mapping of he sink ω p and races back he opimal pahs from sink o all he sources via he inermediae verices. We will prove his inducively. Le x i be he assignmen of ω i in N and a any ieraion Φ (ω l ) = ω j for some j [1, p]. The opimal pah from any source ω i for i [1, K] o is successor verex Φ (ω i ) = ω j is jus he shores pah disance beween s i and he verex o which Φ (ω i ) will evenually be mapped. This is equal o d si x j. I is easy o verify ha in he algorihm (line 11) his value is sored in h l (v) l [1, K] daa srucure for all v V. Assuming ha he opimal delays ill he (l 1) h run are calculaed by he algorihm and sored in h i s we will show ha a l h run he algorihm compues he opimal delay. The opimal delay of he pah { from sources } o Φ (ω l ) via ω l is given by g l ( x j ) := min xl d( xl ) + d xl x j, where, d( xl ) is he opimal delay of he pah ill ω l. The opimal delay d( x l ) can furher be expanded and wrien{ in erms of he delay of is predecessors } as: d( x l ) := min xl max [d( x i) + d xi x l ] + P(ω l, x l ). ω i Φ (ω l ) Subsiuing value of d( x l ) in g l ( x j ) we ge { g l ( x j ) = min x l max ω i Φ (ω l ) g i( x l ) + P(ω l, x l ) + d xl x j }. (3) Recall he line 14 of Algorihm 1 which compues f l (u, v) = P(ω l, u) + d uv. This is nohing bu he las wo erms of he righ hand side of (3) when x l = u and x j = v. Now observe he line 16 of Algorihm 1 which compues h l (v) = min {max[h i (u) ω i Φ (ω l )] + f l (u, v)}, where u h i (u) is he opimal delay of he pah leading o ω l via is predecessor ω i. The opimal delay of he pah h i (u) = g i ( x l ) for x l = u V. Hence Algorihm 1 indeed compues he opimal delay of he pah leading o Φ (ω l ) via ω l for all possible mappings v V of Φ (ω l ) and sores i in h l (v) a ieraion l. Recall ha G is a ree hus he oal number of edges in i are p 1 and Algorihm 1 is execued once for each edge. In each ieraion i compues he delay of an edge for all possible mappings of is end poins which requires n 2 (line 14) ime where n is he number of verices in N. Then i adds he delay o he delay of is predecessors and chooses he one wih minimum value (h l (v) in line 16) which requires O(n) ime. Hence he ime o complee one ieraion is O(n 2 +n) = O(n 2 ). The oal ime complexiy of Algorihm 1 is O(pn 2 ). V. G IS A LAYERED GRAPH We now consider he case when G is a layered graph. An example of a layered compuaion graph is shown in Fig. 4. We assume ha here are r layers and number of verices in each layer is a mos k. The verices a layer l are labelled ω l1, ω l2,..., ω lk. The direced graph has edge (ω ai, ω bj ) only if b = a or a + 1. Here we also assume ha all he sources are a layer one and here is only one sink on he las layer. We derive our moivaion for hese kind of compuaion graphs from he MapReduce applicaion framework. In MapReduce framework each user comes o a nework of processors wih a se of Map and Reduce asks. There is a precedence order beween Map and Reduce asks. Each Reduce ask canno be sared unless he processing of corresponding se of Map asks is finished. Each ask akes predefined ime o finish and he oupus of he Map asks are used by he corresponding Reduce asks. This dependency can be represened by a direced graph wih edges showing he dependency beween he wo asks. The aim in his seing is o embed he Map and Reduce asks on he processors such ha he oal ime of compuaion and communicaion

9 9 Number of layers = r File A abc xyz pqr abc File B xyz abc def pqr abc xyz pqr abc xyz abc def pqr Spliing Widh of a layer = k Fig. 4: Layered compuaion graph is minimized. We explain our moivaion wih he following example from [2]. Example 3: Consider a ypical daabase query by a server (call i sink) o check he number of occurrences of differen words in wo large files which are available a wo separae servers. The ask of calculaing he number of occurrences of each word inside he files can be divided ino he following sub-asks. Spliing: Firs he files are spli ino smaller sub-files such ha each sub-file can be processed by a processor in he nework. Mapping: Each sub-file is hen parsed o ge he number of imes each word occurred in i. Shuffling and reducing: Once he coun from each sub-file is available hen he couns of one word are ranspored o one processor o compue he final coun of ha word. Each processor adds all he individual couns and resuls he final coun of each word. Final resul: Finally he resul is ranspored o he node which asked his query. Fig. 5 represens a ypical MapReduce daa flow diagram for his problem. The aim is o deermine he processors for each of he sub-asks such ha he ime o answer he query a he sink is minimized. The whole process can be represened by a direced layered graph wih each layer represening one sub-ask and a verex a a layer represening a paricular subask. Observe ha he edges in his graph are only beween he consecuive layers and he operaions a a verex canno sar unil he daa from all is predecessors is available. Now we presen an algorihm which solves he MinCos problem for layered graphs in polynomial ime. Algorihm overview: Like Algorihm 1, Algorihm 2 also has wo phases a forward pah and a backward racking. The forward pah is a dynamic program ha ieraes over all he layers of he compuaion graph. In he firs ieraion i compues he opimal mappings of all he verices of layer 1 corresponding o a possible mapping of he verices of layer 2. If a verex is a source verex hen is mapping is always fixed o he corresponding source verex in N. Similarly, in ieraion l i compues he opimal mapping of verices of layer l each corresponding o a possible mapping of verices of layer l+1. I also compues he opimal cos ill layer l+1 for every possible mapping of verices of layer l+1. Once i reaches he abc 1 xyz 1 pqr 1 abc 1 xyz 1 abc 1 def 1 pqr 1 Mapping abc 3 xyz 2 pqr 2 def 1 Reducing abc 3; xyz 2 pqr 2; def 1 Final oupu on a server Fig. 5: Sub-asks and daa flow diagram of a ypical daabase query in MapReduce framework las layer he algorihm chooses he mapping of he verices of las layer which minimizes he overall cos and backracks o ge he corresponding mappings of all he previous layers. Algorihm Descripion: Algorihm 2 ieraes over he layers in G and in each ieraion i mainains he following wo daa srucures: 1) f l (X i, Y j ), he cos of embedding all he verices ill layer l when he verices a layer (l + 1) are placed a Y j and verices of layer l are placed a X i, where X i, Y j V of size k. 2) h l (Y j ), he opimal cos of embedding all he verices (and corresponding edges) ill layer l when he verices a layer (l +1) are mapped o an ordered subse Y j V of size k. Afer iniializing hese daa srucures o 0 (in line 5, 6) he algorihm complees in he following hree seps. Lines 7 15: This is he main loop of he algorihm which runs for he firs layer (from sources) o las bu one layer. A each layer l he daa srucure f l (X i, Y j ) is updaed for all possible combinaions of k size subses X i and Y j of V (lines 7 11). The following cos erms are added ogeher o calculae f l (X i, Y j ) along wih he opimal cos ill layer l 1, h l 1 (X i ). X i P(ω lu, a u ) : Cos of puing compuaion node u=1 a u X i ω lu Ω a a u V for each node a he curren layer. W(ω lu, ω (l+1)v )d aub v : Toal communica- a u X i,b v Y j (ω lu,ω (l+1)v ) Γ ion cos, when node ω lu Ω is placed a node a u V and ω (l+1)v Ω is placed a b v V, is he muliplicaion of corresponding coss in compuaion graph (weigh W) and communicaion graph (weigh T ). This erm capures he cos for all he edges beween layer l and layer l + 1. a u,b v X i (ω lu,ω lv ) Γ W(ω lu, ω lv )d aub v : Toal communicaion cos, when node ω lu Ω is placed a node a u V and ω lv Ω is placed a b v V, is he muliplicaion of corresponding coss in compuaion graph (weigh W) and

10 10 Algorihm 2: Opimal embedding algorihm for layered graphs Inpu: Nework graph N = (V, E), V = n, E = m, Weigh funcion T : E R +, Layered compuaion graph G = (Ω, Γ), Ω = p, Γ = q, Weigh funcion W : Γ R +, Cos funcion P : Ω V R +. Oupu: Embedding E wih minimum cos 1: [[D = d ui,u j ]] // n n disance marix for N. 2: X := {X i : X i V and X i = k} // X = n k, X i = {a 1, a 2,..., a k } 3: Y := {Y i : Y i V and Y i = k} // Y = n k, Y i = {b 1, b 2,..., b k } 4: Z i := [z i1, z i2,..., z ik ] // z ij V, is he assignmen of node ω ij Ω in opimal embedding // Iniializaion of ables 5: h l (X i ) := 0 X i X, l [1, r] ; 6: f l (X i, Y j ) := 0 X i X, Y j Y, l [1, r] ; 7: for l = 1 o (r 1) do 8: for j = 1 o Y do 9: for i = 1 o X do 10: f l (X i, Y j ) := X i P(ω lu, a u ) + u=1 a u X i,b v Y j a u X i (ω lu,ω (l+1)v ) Γ W(ω lu, ω lv )d aub v + h l 1 (X i ) ; a u,b v X i (ω lu,ω lv ) Γ 11: end for 12: h l (Y j ) min l(x i, Y j ) ; X i X 13: x l (Y j ) arg min f l (X i, Y j ) ; X i X 14: end for 15: end for 16: for i = 1 o X do 17: h r (X i ) := a u,b v X i,(ω ru,ω rv) Γ 18: end for 19: C(E) := min h r(x j ) ; X j X 20: Z r = {z r1,..., z rk } = arg min h r (X j ) ; X j X // Backracking 21: for l = r 1 o 1 do 22: Z l = x l (Z l+1 ) ; 23: end for W(ω lu, ω (l+1)v )d aub v + d aub v + h r 1 (X i ) ; communicaion graph (weigh T ). This erm capures he cos for all he edges a layer l. Lines 16 20: Here he algorihm finally compues he oal cos of he embedding he graph G by adding he cos of he edges beween he verices of las layer r, if any, when he verices of las layer are placed a X i. And compues he opimal cos of embedding E, C(E), by choosing he placemen of las layer which minimizes he overall cos (line 19 20). The vecor Z r sores he mapping of verices a layer r under he embedding E. Lines 21 23: Afer finding he opimal mapping for he verices a layer r he algorihm races back he corresponding opimal mapping for verices a layer r 1 all he way upo he firs layer. To simplify he descripion of he algorihm we do no show he fixed mapping of sources and sink ino he nework graph in Algorihm 2. I is easy o see ha o incorporae he fixed mapping of sources and sink he algorihm needs o only consider hose subses X i, Y j of V which map he source ω i (sink ν) o he corresponding source s i (sink ) while calculaing he daa srucures of he layer on which ω i is presen (layer r). A. Analysis of Algorihm 2 Theorem 5: Algorihm 2 solves MinCos and he ime complexiy of he algorihm is O(rn 2k ) when G is a layered graph wih r layers and a mos k nodes per layer. Proof: We prove he correcness of 2 when G is an unweighed graph and he processing coss are all zero. We also assume ha here are no edges beween he verices of a layer. The proof can easily be exended wih weigh funcions W and P. I is sufficien o show ha a each ieraion l he algorihm compues he opimal cos of embedding he compuaion graph ill layer l for all possible embeddings of every node of layer (l + 1) given ha he cos compued ill layer (l 1) is opimal. Le x i be a vecor of size 1 k whose l h elemen represens he assignmen of a nework node for ω li Ω. In oher words, x i is a k size subse of V. Le us define, d( x i, x i+1 ) := d aub v. (4) a u x i,b v x i+1 (ω iu,ω (i+1)v ) Γ This represens he sum of disance beween all he adjacen nodes in layer i and (i + 1) when he nodes of layer i are embedded o x i and nodes of layer (i + 1) are embedded o x i+1. Toal cos of any embedding can hen be wrien as C := r 1 d( x i, x i+1 ). To obain he opimal embedding i=1 we have o minimize he above equaion wih respec o all he possible mappings x i. Therefore ( he opimal cos can be r 1 ) wrien as C op = min x1,..., x r d( x i, x i+1 ). Separaing he erms wih x 1 and some algebraic manipulaion will give us ( r 1 C op = min g 1 ( x 2 ) + min x 2 x 2,..., x r ) d( x i, x i+1 ), (5) i=1 i=2 where, g 1 ( x 2 ) = min x1 d( x 1, x 2 ). Similarly afer minimizing wih respec o x l we can wrie he cos as, ( ) C op = min x l+1,..., x r g l ( x l+1 ) + r 1 i=l+1 d( x i, x i+1 ), (6) where g l ( x l+1 ) = min x l g l 1 ( x l ). Now i suffices o show ha he algorihm indeed calculaes g l a l h ieraion. Recall ha x l and x l+1 are k size subses of V which represen he mapping of nodes of layer l and l +1 respecively. In he algorihm, X i represens a k size subse of V o which he nodes of he layer