Optimal Function Computation in Directed and Undirected Graphs

Optimal Function Computation in Dir ected and Undirected Graphs Hemant K o wshik CSL and Department of ECE Uni v ersity of Illinois Urbana- Champaign Email: ko wshik2 @illinois.edu P . R. K umar CSL and Department of ECE Uni v ersity of Illinois Urbana- Champaign Email: prkumar@illinois.edu Abstract W e consider the problem of info rmation aggregation in sensor networks, where one is interested in computin g a function of the sensor measurements. W e allow for block processing and study in-network function computation in dire cted graph s and und irected graph s. W e study how the structure of the function affects the en coding strategies, and the effect o f interactive infor mation exchan ge. Depen ding on the applicatio n, there could b e a designated collector n ode, or every nod e might want to compu te the fu nction. W e begin by considerin g a directed graph G = ( V , E ) on the sensor no des, where the goal is to determin e the op timal en coders o n each edge which achieve func tion com putation at the co llector node. Ou r goal is to character ize the rate region in R | E | , i.e., the set of p oints f or which there exist feasible en coders with g iv en rates wh ich achieve zero-e rror compu tation fo r asymptotica lly large blo ck length. W e d etermine the solutio n for d irected trees, spe cifying the optimal en coder and d ecoder for each edg e. For gener al directed acyclic gr aphs, we p rovide an ou ter bou nd on the rate region by finding the disamb iguation r equirem ents for each cu t, and d escribe examples where this o uter b ound is tigh t. Next, we address the scenar io wh ere nod es are connec ted in an undir ected tree n etwork, an d every node wishes to co mpute a given symmetr ic Boolean function of the sensor data. Undirected edge s permit interactive co mputation , an d we therefo re study the effect o f inter action o n th e a ggregation and commun ication strategies. W e f ocus on sum-thr eshold functions, and determine the minimum worst-case total numbe r o f bits to be exchanged on each ed ge. T he optimal strategy inv olves rec ursive in-network aggregation wh ich is reminiscen t o f message p assing. In the ca se of general gra phs, we pr esent a c ut- set lower bo und, and an achievable schem e based o n aggr egation along trees. For complete gr aphs, we prove that the complexity of this scheme is n o more tha n twice that of the optima l sch eme. This material is based upon work partially supported by AFOSR under Contract F A9550-09-0 121, NSF under Contract No. CNS -1035378 , Science & T echnology Center Grant CCF-0939370, Contract CNS-0905397, and Contract CNS-1035340, and USARO under C ontract Nos. W911NF-08-1-0238 and W -911-NF-0710287. Any opinions, fi ndings, and conclusions or recommendation s expressed i n this publication are those of the authors and do not necessarily reflect the views of t he above agencies. I . I N T RO D U C T I O N Sensor networks are composed of nodes with sensing, wireless commun ication a nd com- putation capabilities. These networks are designed for appli cations like fault m onitoring, data harvesting and en vironm ental mo nitoring; tasks which can be broadly classified as inform ation aggregation. In these applications , one is int erested o nly in comput ing some rele va nt f unction of the sensor measurements. For example, one might want to compute th e mean temperature for en vironm ental mo nitoring, or the maxim um tem perature in fire alarm s ystems. This sugg ests moving a way from a data- forwarding paradigm, a nd focusing on ef ficient in-network computation and communication strategies for the function of interest. This is parti cularly important since sensor nodes may be severely limit ed in terms of power and bandwidth, and can pot entially generate enormo us volumes o f data. There are two possi ble architectures for sensor networks that o ne might consider . First, one could desi gnate a single collector node/fusion center which seeks to com pute the function. This goal is m ore appropriate for data harvesting and centralized fault monitorin g. Alternately , one could supp ose that ever y nod e in th e network wants to compute the function. The latter g oal can be vi ewe d as providing situati onal awareness to each sens or node, which could be very useful in appli cations like distributed fault monito ring, adapt iv e sensing and sensor-actuator n etworks. For example, sens or nodes m ight want to modify their sampli ng rate depending on the value of the function. W e will consider both t hese problems. In order to m ake progress on the general problem of comp uting functions o f distributed data, we wi ll study specific network topolo gies and so me specific classes of fun ctions. In this paper , we abstract out the medium access control prob lem associated wit h a wi reless network, and view the n etwork as a graph with edges representing noiseless links. The fundamental challenge is to exploit t he structure of the particular function, so as to o ptimally combine transmissions at i ntermediate nodes. Thus, the p roblem of function comp utation could be regarded as being more general than findi ng the capacity of a wi reless network. In our probl em formulation, we consider the zero error block comput ation framework. W e allow for nodes to accumulate a block of measurements and realize greater efficienc y usi ng block coding strategies. Howe ver , we require the function to b e computed wit h zero error for the bl ock. T o so lve the problem under this framew ork, one needs to d etermine the optim al strategy for communication and comp utation, which inclu des determini ng the order i n which nodes s hould transmi t and the informat ion that nodes m ust con vey whenever t hey t ransmit. Th e strategy for comput ation may benefit from interactiv e inform ation exchange between nodes, whi ch presents an addition al degree of freedom vis-a-vis t he standard poin t-to-point commun ication set-up. In Section III , we view the network as a directed graph with edges representing noiseless links. W e thus cons ider the problem o f general function comput ation in a di rected graph G = ( V , E ) with a desi gnated coll ector . W e focus specifically on s trategies for com bining inform ation at intermediate nodes, and opt imal codes for transmi ssions on each edge. W e consider bot h the worst case and the a verage case com plexity for zero error block comput ation with a joint probability dis tribution on the node measurements. Our goal i s to characterize the rate region in R | E | , i .e., the set of points for which there exist feasible encoders wit h given rates which achie ve zero-error computatio n for large enough block leng th. In the case of tree graphs , we deriv e a necessary and s uffi cient condit ion for the encod er on each edge, which provides a complete characterization of t he rate region. The extension of these results to directed acyclic graphs is more diffi cult. Howe ver , we provide an o uter bound on the rate region by findi ng the disambiguati on requirements for each cut, and describe examples wh ere thi s out er bo und is t ight. In Section IV, we address the probl em of com puting symmetri c Boolean functions in undi- rected graphs. Th e key di ff erence from Section III is that we consid er bid irectional li nks and study the benefit o f interaction between nodes. W e show h ow the approach described in Section III, together wi th ideas from communication comp lexity theory , can be synthesized to dev elop a th eory of optimal comput ation of sym metric Boolean functions in undirected graphs. In t he case of tree net works, each edge i s a cut-edge, and this all ows us to derive a lower b ound on the number of bi ts exchanged on each edge, by considering an equiv alent two node probl em. Further , we show that a protocol of recursi ve in-network aggre gation along with a smart interactiv e coding strategy , achiev es t his lower b ound for the class of sum -threshold function s in tree networks. The optimal st rategy has a simple structure th at is reminiscent of message p assing, where messages flo w from t he l ea ves t ow ards an interior node, and then flow back from the interior node to the lea ves. In t he case of general graphs , we present a cut-set lower bound, and an achiev able scheme based on aggregation along t rees. For com plete graphs , we show that t he compl exity of this s cheme is no more than twice that of th e op timal scheme. I I . R E L A T E D W O R K In its si mplest form, the probl em of network function computation can be modeled as a problem of com putation on graphs obtained by abst racting out the medium access cont rol problem and channel noise. This problem is closely related to the network coding problem. Indeed, assuming independent measurements x i and the identity fun ction f ( x 1 , x 2 , . . . , x n ) = ( x 1 , x 2 , . . . , x n ) , we hav e the reverse of t he mult icast probl em stud ied in [1]. Computing a function of independent measurements is a net work computat ion prob lem as oppos ed t o a network coding problem. In [2], the min-cut upper bound on t he rate of com putation is shown to be tight for the computation of divisible functio ns on tree graphs. In th is paper , we generalize t his resul t using a differ ent approach. Further , t he si mplicity of the app roach presented allows extensions to the case of general graphs and collocated networks. The problem of worst-case block function computatio n was formulat ed i n [3]. The authors determine t he maxi mum rate at w hich a sy mmetric functi on can be com puted in a random network, given the constraints of the wireless medium. They identify t wo classes of symmetric functions namely t ype-sensitive functions exemplified by Mean and Median, and type-thr eshold functions, exemplified by M aximum and Mi nimum. The maximu m rates for computati on of type-sensitive and typ e-threshold fun ctions in random planar networks are shown to be Θ ( 1 log n ) and Θ ( 1 log log n ) respectiv ely , for a network of n nodes. A com munication complexity approach was used to establish upper b ounds on the rate of comput ation in collocated networks. Some extensions to t he case of finite degree graphs are presented in [4]. In the study of the com munication complexity of m ulti-party computatio n [5], on e seeks to minimize the number of bits that m ust be exchanged in t he worst case between two nodes to achiev e zero-error com putation of a function of t he node variables. T he commun ication complexity of Boolean functions h as been studied in [6], [7]. Further , o ne can consider the dir ect-sum pr oblem [8] where several instances of the problem are consi dered together to obtain savings. This block computation app roach is used to compute the exact complexity of the Boolean AND function in [9]. In this paper , we consid erably generalize this result, wh ich allows us to deriv e opt imal strategies for com puting more general classes o f symmetric Boolean functions in undirected tree networks. The optim al comm unication scheme is reminiscent of m essage passing algorithms which have been appl ied very ef fectiv ely to the problems of comput ing marginals and probabilistic i nference [10], [11]. An information-th eoretic formulation of this probl em com bines t he complexity o f s ource coding of correlated sources with rate di stortion, together with the compli cations int roduced by the functi on structure; see [3]. There is li ttle or no work that addresses t his most g eneral frame work. Th e problem of source coding wi th side information has been st udied for the vanishing error case in [12]. Th is has b een extended in [13] to the case where the receiv er desires to know a certain function f ( X , Y ) of the single source X and the si de information Y ; the authors determined the required capacity of the channel between the source and receive r to be th e condi tional graph entropy . Howe ver , the extension to larger net works has proved diffic ult. In Zero-error Informatio n Theory , the problem of source coding with side inform ation ensuring zero error for finite block length has been s tudied in [14] and [15]. The problem reduces to the task of coloring a probabi listic graph defined on the set of source s amples. The minimum entropy of such a coloring approaches t he graph entropy or Korner entropy , as the b lock length approaches infinity . Recently , the rate region for mul ti-round i nteractiv e function computation has been characterized for two n odes [16], and for collocated networks [17]. In thi s paper we do not address t he probl em of function computation in noisy networks. In [18], the problem of comput ing parity in a collo cated network in the presence of noise is considered. It is shown that O ( n log log n ) bits suffice to achieve correct com putation wi th high probability . This has been extended to random planar networks i n [19], where the sam e lo g l og n factor of redundancy is shown to be suffi cient. Remarkably , this factor was recently shown to be t ight in [20]. I I I . F U N C T I O N C O M P U T A T I O N I N D I R E C T E D G R A P H S In this section, we abst ract out the medi um access control problem ass ociated with a wi reless network, and view the network as a directed graph wit h edges representing essent ially noiseless wired links between nodes. W e formulate the prob lem of zero error functi on computation on graphs. W e suppose that there is a joint p robability distri bution on the node measurements, and allow nodes to realize greater ef ficiency by using b lock codes. W e will consider both the worst case and the av erage case complexity for zero error bl ock comput ation. Given a graph, the problem we address is to determine the set of rates on th e edges which will allow zero error function computation for a large enough block length. In essence, we are exploring the interaction between the function structure and the structure of t he graph; how information needs to be routed and combined at intermediate nodes to achie ve certain rate vectors. In Section III-A , we begin with t he two node problem. W e compute t he number of bits that node v X needs to comm unicate to node v Y so that t he latter can com pute a functi on f ( X , Y ) with zero error . For correct function computation, an encoder must disambi guate certain pairs of source s ymbols of node v X , on which the function di sagrees. W e show b y explicit construct ion of a code that this necessary condi tion is in fact sufficient. This y ields the optimal alphabet and we calculate the minim um worst case and av erage case complexity , with the latter obtained by Huff man coding over the op timal alphabet. In Section III-B, we extend thi s result to directed trees wi th the collector as root, exploiting the fact that each edge i s a cut-edge. T his yi elds the optimal alph abet for each edg e, and we separately opt imize the encoders for the worst case and the a verage case. Thus the rate region consists of all rate points dominatin g a singl e po int that is coordin ate-wise opti mal. In Section III-C , we consider directed acyclic graphs. A key difference from the tree case i s the presence of mult iple p aths to route the data, which present di ff erent opportunit ies to combine information at intermediat e nodes. W e arrive at an outer b ound to the rate region by findi ng the disambiguati on requirements for each cut of the directed graph. Th is outer bound i s not alwa ys tight, as we show in Example 3. Howe ver , for t he worst case computation of finit e field parity , and the maxi mum or minim um fun ctions, the out er bo und is shown to b e indeed tigh t. Further , the onl y extreme points of t he rate region are rate points corresponding t o activa ting only a tree subset o f edges. A. T wo Node Setti ng 1) W orst case comple xity: W e begin by considering the simple t wo node problem. Suppose nodes v X and v Y hav e measurements x ∈ X and y ∈ Y , where the alphabets X and Y are finite sets. Node v X needs to optimally commu nicate its i nformation to no de v Y so that a function f ( x , y ) , which takes values in D , can be computed at v Y with zero error . W e d o not consider the case where v X and v Y interactiv ely com pute the function. Th us node v X has an encoder C : X → { 0 , 1 } ∗ , w hich maps its measurement x to the codew ord C ( x ) , and node v Y has a decoder g : { 0 , 1 } ∗ × Y → D which maps the received codeword C ( x ) and its own measurement y to a functi on estim ate, g ( C ( x ) , y ) . The set of all possible code words is called the codebook, denoted by C ( X ) Definition 1 (F easi ble Encoder): An encoder C is feasible if there exists a decoder g : { 0 , 1 } ∗ × Y → D such t hat g ( C ( x ) , y ) = f ( x , y ) for all ( x , y ) ∈ X × Y . T hus, a feasible encoder is one that achieves error-free function com putation. Theor em 1 (Characterization of F easible Encoders): An encoder C is feasibl e if and only if giv en x 1 , x 2 ∈ X , C ( x 1 ) = C ( x 2 ) implies f ( x 1 , y ) = f ( x 2 , y ) for all y ∈ Y . Pr oof: By definition, if C i s a feasible encoder , then there exists a corresponding decoder g such that g ( C ( x 1 ) , y ) = f ( x 1 , y ) and g ( C ( x 2 ) , y ) = f ( x 2 , y ) , for al l y ∈ Y . Further , i f C ( x 1 ) = C ( x 2 ) , we have f ( x 1 , y ) = f ( x 2 , y ) for all y ∈ Y . T o prove the con verse, we need to cons truct a decoding functi on g : { 0 , 1 } ∗ × Y → D . For each codew ord C ∗ in the codebook, define C − 1 ( C ∗ ) : = { x ∈ X : C ( x ) = C ∗ } . For fixed y ∈ Y and fixed codew ord C ∗ ∈ C ( X ) , the decoder mapping is give n by g ( C ∗ , y ) : = f ( x nom ( C ∗ ) , y ) for any arbit rary x nom ( C ∗ ) ∈ C − 1 ( C ∗ ) . W e show t hat this decoder works for any fixed x and y . Indeed, g ( C ( x ) , y ) = f ( x nom , y ) where x nom ∈ C − 1 ( C ( x )) . Thus, C ( x nom ) = C ( x ) and by assumption f ( x nom , y ) = f ( x , y ) . Hence, g ( C ( x ) , y ) = f ( x nom , y ) = f ( x , y ) for all y ∈ Y . ✷ Any feasible encoder C can be viewed as partitio ning the set X into Π ( C ) : = { S 1 , S 2 , . . . , S k } such that for x 1 ∈ C i , x 2 ∈ C j , we have C ( x 1 ) = C ( x 2 ) i f and only if i = j . Define an equ iv alence relation “ ↔ ” between x 1 , x 2 ∈ X by: x 1 ↔ x 2 if and only if f ( x 1 , y ) = f ( x 2 , y ) for all y ∈ Y . Consider the encoder C OPT which assigns a di stinct code word to each resulting equiv alence class. Clearly , C OPT is a feasible encoder , since C OPT ( x 1 ) = C OPT ( x 2 ) implies x 1 ↔ x 2 , and hence f ( x 1 , y ) = f ( x 2 , y ) for all y ∈ Y . C OPT is opt imal in the sense that any other feasible encoder C must have at least as many codew ords as C OPT : Theor em 2 (Optimal ity of C OPT ): Let Π ( C OPT ) : = { S OPT 1 , S OPT 2 , . . . , S OPT k } b e the partition of X generated by C OPT , and let Π ( C ) : = { S 1 , S 2 , . . . , S l } be the partition of X generated by any other feasible encoder C . Then, (i) Π ( C ) must be a finer partitio n than Π ( C OPT ) . (ii) The min imum num ber of bits that node v X needs to comm unicate is ⌈ log | Π ( C OPT ) |⌉ . Pr oof: First we clai m that any subset S i ∈ Π ( C ) can have nonempty intersection with exactly one sub set S OPT j ∈ Π ( C OPT ) . Suppo se not . Then there exist x 1 , x 2 ∈ S i such t hat x 1 ∈ S OPT j 1 and x 2 ∈ S OPT j 2 . Since C ( x 1 ) = C ( x 2 ) , by Theorem 1, we must h a ve f ( x 1 , y ) = f ( x 2 , y ) for all y ∈ Y . Howe ver , by const ruction of C OPT , x 1 and x 2 must belong to distinct equiv alence classes i.e., x 1 = x 2 . Hence, there exists y ∗ such that f ( x 1 , y ∗ ) 6 = f ( x 2 , y ∗ ) , which is a contradiction . This shows th at the partitio n generated b y any encoder C m ust b e a further s ubdivision of the partition generated by C OPT , i.e., finer t han Π ( C OPT ) . So node v X needs to comm unicate at least ⌈ log | Π ( C OPT ) |⌉ bit s. ✷ W e can e xtend this to the case wh ere v X collects a block of N measurement s x = ( x 1 , x 2 , . . . , x N ) ∈ X N , and v Y collects a block of N m easurements y = ( y 1 , y 2 , . . . , y N ) ∈ Y N . W e want to find a block encoder C N : X N → { 0 , 1 } ∗ so that the vector fu nction f ( N ) ( x , y ) = ( f ( x 1 , y 1 ) , . . . , f ( x N , y N )) can be com puted without error , for all x ∈ X N , y ∈ Y N . All the above results carry over to t he error -free block com putation case. As before, we define an equiva lence ↔ between x 1 , x 2 ∈ X N if f ( N ) ( x 1 , y ) = f ( N ) ( x 2 , y ) for all y ∈ Y N . The o ptimal encoder C N , OPT is once again obtained by assigning disti nct code words to each equiv alence class. Since we are s tringing together N independent i nstances, we hav e | Π ( C N , OPT ) | = | Π ( C OPT ) | N . H ence the minimu m number of bits per comp utation that node v X needs to comm unicate is ⌈ N log | Π ( C OPT ) |⌉ N which con ver ges to log | Π ( C OPT ) | as N → ∞ . 2) A verage case complexity: Supp ose now that the measurements X , Y are drawn from the joint probability distribution p ( X , Y ) , with the goal being to m inimize t he av erage number of bits t hat need to be communicated, i.e., the averag e case complexity . Definition 2 (F easi ble Encoder): An encoder C : X → { 0 , 1 } ∗ is feasible if there exists a decoder g : { 0 , 1 } ∗ × Y → D such t hat g ( C ( x ) , y ) = f ( x , y ) for all { ( x , y ) ∈ X × Y : p ( x , y ) > 0 } . Theor em 3: An encoder C is feasibl e if and only i f, g iv en x 1 , x 2 ∈ X , C ( x 1 ) = C ( x 2 ) implies f ( x 1 , y ) = f ( x 2 , y ) for { y ∈ Y : p ( x 1 , y ) p ( x 2 , y ) > 0 } . Pr oof: By d efinition, if C i s a feasible encoder , t hen there exists a corresponding decoder g such that g ( C ( x 1 ) , y ) = f ( x 1 , y ) and g ( C ( x 2 ) , y ) = f ( x 2 , y ) , for all { y ∈ Y : p ( x 1 , y ) p ( x 2 , y ) > 0 } . Further , if C ( x 1 ) = C ( x 2 ) , we have f ( x 1 , y ) = f ( x 2 , y ) for { y ∈ Y : p ( x 1 , y ) p ( x 2 , y ) > 0 } . T o prove the con verse, we need to cons truct a decoding functi on g : { 0 , 1 } ∗ × Y → D . For each codeword C ∗ in the codebook, define C − 1 ( C ∗ ) : = { x ∈ X : C ( x ) = C ∗ } . For fixed y ∈ Y and fixed code word C ∗ ∈ C ( X ) , the decoder mapping is given by g ( C ∗ , y ) : = f ( x nom ( C ∗ , y ) , y ) for any arbitrary x nom ( C ∗ , y ) ∈ C − 1 ( C ∗ ) with p ( x nom ( C ∗ , y ) , y ) > 0. W e show that this d ecoder works for any fixed x and y with p ( x , y ) > 0. Indeed, g ( C ( x ) , y ) = f ( x nom , y ) where x nom ∈ C − 1 ( C ( x )) with p ( x nom , y ) > 0. Thus, C ( x nom ) = C ( x ) and by assum ption f ( x nom , y ) = f ( x , y ) since p ( x nom , y ) p ( x , y ) > 0. Hence, g ( C ( x ) , y ) = f ( x nom , y ) = f ( x , y ) . ✷ W e now define “ x 1 ↔ x 2 ” when f ( x 1 , y ) = f ( x 2 , y ) for { y ∈ Y : p ( x 1 , y ) p ( x 2 , y ) > 0 } . Now the ↔ relation is reflexi ve and s ymmetric, but not necessarily t ransitive. Howe ver , if p ( x , y ) > 0 for all ( x , y ) ∈ X × Y , then ↔ is an equivalence relation. W e can con struct an encoder C OPT which assigns a di stinct codeword to each equiva lence class. Let Π ( C OPT ) : = { S OPT 1 , S OPT 2 , . . . , S OPT k } be the partition of X generated by C OPT . Analogous to Theorem 2, we can sho w that the encoder C OPT has t he o ptimal alph abet A , with the probabilit y distribution vector q = { q 1 , q 2 , . . . , q k } where q i : = ∑ x ∈ S OPT i ∑ y ∈ Y p ( x , y ) . Once the op timal alphabet is fixed, the optimal code C OPT is the binary Huffman code for the probabili ty vector q . Since the Huffman code has an avera ge code length within one bit of the entropy , H ( q 1 , q 2 , . . . , q k ) ≤ E [ l ( C OPT )] ≤ H ( q 1 , q 2 , . . . , q k ) + 1 . The extension to the case where nodes v X , v Y collect a b lock of N i.i.d. measurements is straightforward. T he o ptimal alphabet is A N , wh ich has the product di stribution q N . T he op timal encoder is obtained via the Huffman code for the opt imal alphabet. Its expected length satisfies H ( q N ) N ≤ E [ l ( C N , OPT )] N ≤ H ( q N ) + 1 N . Hence the m inimum number of bits per computation t hat node v X needs to com municate con verges to H ( q ) as N → ∞ . B. Function Computa tion in Dire cted T r ees Let us now con sider compu tation on a t r ee graph . Consider a di rected t ree G = ( V , E ) with nodes V : = { v 1 , v 2 , . . . , v n } and root node v 1 . Ed ges represent communication links , so t hat node v j can transmi t to n ode v i if ( v j , v i ) ∈ E . Each node v i makes a measurement x i ∈ X i , and the collector node v 1 wants to compute a functi on f ( x 1 , x 2 , . . . , x n ) with no error . W e seek to minimize th e worst case com plexity on each edge. For each node i , let π ( v i ) be the uniq ue node to which node i has an outgoing edge, and let N − ( v i ) : = { v j ∈ V : ( v j , v i ) ∈ E } . The height of a node v i is the l ength of th e longest directed path from a leaf node to v i . Define the descendant set D ( v i ) to be the subset of nodes in V from which there exist di rected path s to node v i . Th e graph induced on D ( v i ) is a tree with node v i as root. Each nod e transm its exactly once and the computation proceeds in a bott om-up fashion, starting from the leaf nodes and proceeding up t he t ree. Each leaf nod e v i has an encoder C i : X i → { 0 , 1 } ∗ that maps its measurement x i to a codeword C i ( x i ) which is transmitt ed on the edge ( v i , π ( v i )) . Each non-l eaf node v j for j 6 = 1 has an encoder C j which maps its measurement x j as well as the codew ords recei ved from N − ( v j ) , to a codeword transmitted on t he edg e ( v j , π ( v j )) . Th us the computatio n proceeds in a bottom-up fashion. Let C i denote the codew ord transmi tted by node v i , and C S : = { C i : v i ∈ S } denote the set of codewords t ransmitted by nod es in S . Definition 3: A set of encoders { C i : 2 ≤ i ≤ n } is sai d to be feasible if there is a decod- ing functi on g 1 at t he collector node v 1 such that g ( x 1 , C N − ( v 1 ) ) = f ( x 1 , x 2 , . . . , x n ) for all ( x 1 , x 2 , . . . , x n ) ∈ X 1 × X 2 × . . . × X n . Lemma 1: If a set of encoders { C i : 2 ≤ i ≤ n } i s feasible, then the encoder C i at node v i must s eparate 1 x 1 D ( v i ) ∈ X D ( v i ) from x 2 D ( v i ) ∈ X D ( v i ) , i f t here exists an assignm ent x ∗ V \ D ( v i ) such that f ( x 1 D ( v i ) , x ∗ V \ D ( v i ) ) 6 = f ( x 2 D ( v i ) , x ∗ V \ D ( v i ) ) . Pr oof: The removal of edge ( v i , π ( v i )) separates the graph into two disconnected subtrees D ( v i ) and V \ D ( v i ) . W e combine all th e nodes in D ( v i ) int o a supernode v α , and all the no des in V \ D ( v i ) int o a supernode v β . The resul t now follows from Theorem 1. ✷ T o prove the con verse, we explicitly define the encoders C 2 , C 3 , . . . , C n and a decoding function g , and prove that it achie ves correct fun ction computation. Define the alphabet for encoder C i on edge ( v i , π ( v i )) as, A i : = { h i : X V \ D ( v i ) → D s. t. ∃ x ∗ D ( v i ) ∈ X D ( v i ) , h i ( x V \ D ( v i ) ) = f ( x ∗ D ( v i ) , x V \ D ( v i ) ) ∀ x V \ D ( v i ) ∈ X V \ D ( v i ) } . Thus code words s ent by node v i can be vie wed as normal forms on var iables X V \ D ( v i ) , or as partial functio ns on X V \ D ( v i ) . Encoder at node v i : On receiving the codew ord corresponding to h j : X V \ D ( v j ) → D , on incoming edge ( v j , v i ) , node v i assigns nominal values, x nom D ( v j ) to variables X D ( v j ) such that f ( x nom D ( v j ) , x V \ D ( v j ) ) = h j ( x V \ D ( v j ) ) ∀ x V \ D ( v j ) ∈ X V \ D ( v j ) . (1) Giv en nominal values for all nodes in D ( v i ) \ { v i } , and its own measurement x i , node v i substitut es 1 Node v i does not hav e access to x D ( v i ) directly but only the codew ords r ecei ved from N − ( v i ) . W hen we say that the encoder C i must separate x D ( v i ) , ˜ x D ( v i ) , we are considering C i as an implicit function of x D ( v i ) . these values t o obtain a function h i : X V \ D ( v i ) → D such that h i ( x V \ D ( v i ) ) = f ( x nom D ( v i ) \{ v i } , x i , x V \ D ( v i ) ) for al l x V \ D ( v i ) ∈ X V \ D ( v i ) . If v i 6 = v 1 , node v i then transmits the code word C i corresponding to function h i ∈ A i on the edge ( v i , π ( v i )) . Decoding function g : The collector node v 1 assigns nom inal values to t he variables X D ( v 1 ) \{ v 1 } . The decoding functi on g is given by g ( x 1 , C N − ( v 1 ) ) : = h 1 = f ( x 1 , x nom D ( v 1 ) \{ v 1 } ) . Theor em 4: Let x f ix 1 , x f ix 2 , . . . , x f ix n be any fixed assig nment of node values. Let the encoders at node v 2 , v 3 , . . . , v n be as above. Then function h i computed by node v i is, h i ( x V \ D ( v i ) ) = f ( x f ix D ( v ( i )) , x V \ D ( v i ) ) ∀ x V \ D ( v i ) ∈ X V \ D ( v i ) . Consequently t he decoding funct ion g satisfies g ( x f ix 1 , C N − ( v 1 ) ) = f ( x f ix 1 , x f ix 2 , . . . , x f ix n ) . Pr oof: The proof proceeds by induction. The theorem is trivially true for all l eaf nodes v i , si nce by assumptio n h i ( x V \ D ( v i ) ) = f ( x f ix v i , x V \ D ( v i ) ) for al l x V \ D ( v i ) ∈ X V \ D ( v i ) . Suppose it is true for all nodes with height less than κ . Cons ider a node v i with heig ht κ . Al l the nodes in N − ( v i ) must have height less than κ . On receiving the code word corresponding t o h j on edge ( v j , v i ) , node v i assigns nominal values to variables in X D ( v j ) so that (1) is sati sfied. From the indu ction assumption , we h a ve h j ( x V \ D ( v j ) ) = f ( x f ix D ( v j ) , x V \ D ( v j ) ) ∀ x V \ D ( v j ) ∈ X V \ D ( v j ) . (2) Since (2) i s t rue for all v j ∈ N − ( v i ) , we can si multaneousl y s ubstitut e the n ominal values x nom D ( v i ) \{ v i } for th e variables X D ( v i ) \{ v i } and the value x f ix i for th e variable X { v i } , to obtain a functi on h i satisfying h i ( x V \ D ( v i ) ) = f ( x nom D ( v ( i )) \{ v i } , x f ix v i , x V \ D ( v i ) ) ∀ x V \ D ( v i ) = f ( x f ix D ( v ( i )) , x V \ D ( v i ) ) ∀ x V \ D ( v i ) , (3) where (3) follows from (1) and (2). This establishes the induction step and completes the proof. For the s pecial case of the collecto r node v i , we have g ( x f ix 1 , C N − ( v 1 ) ) = h 1 = f ( x f ix D ( v 1 ) ) = f ( x f ix 1 , x f ix 2 , . . . , x f ix n ) . Since this is true for every fixed assignment of the nod e values, we can achiev e error-free computation of th e function. Hence the set of encoders described above is feasibl e. ✷ For node v i , consider the equiv alence relation “ ↔ i ” where x 1 D ( v i ) ↔ i x 2 D ( v i ) if f ( x 1 D ( v i ) , x V \ D ( v i ) ) = f ( x 2 D ( v i ) , x V \ D ( v i ) ) for all x V \ D ( v i ) ∈ X V \ D ( v i ) . It is easy to check t hat the equiv alence classes generated by ↔ i are captured exactly by th e alphabet A i . Thus th e above encoders use exactly the opti mal alp habet. Hence, the min imum worst case complexity for encoder C i is ⌈ log ( | A i | ) ⌉ on the edge ( v i , π ( v i )) . The extension to the case where node v i collects a block of N in dependent m easurements X i ∈ X N i , and the coll ector node v 1 wants to compu te the vector functi on f ( N ) ( X 1 , X 2 , . . . , X n ) , is straigh tforward. W e can thus achieve a minimum worst case com plexity arbitrarily close to log | A i | bits for encoder C i . It shoul d be noted that the m inimum worst case complexity of encoder C i does not d epend on the encoders of the other nodes. If there i s a probabilit y dist ribution p ( X 1 , X 2 , . . . , X n ) on the measurements , then we can obtain a necessary and sufficient condition by considering all edge cuts. Lemma 2: Consi der a cut which partition s the nodes into S and V \ S with v 1 ∈ V \ S . Let δ + ( S ) be the s et of all edges from n odes in S to nodes in V \ S . Then the s et of encoders { C i : 2 ≤ i ≤ n } is feasible if and only if for every cut, th e encoder on at least on e of the edges in δ + ( S ) separates x 1 S , x 2 S ∈ X S if there exists an assignment x ∗ V \ S such that f ( x 1 S , x ∗ V \ S ) 6 = f ( x 2 S , x ∗ V \ S ) and p ( x 1 S , x ∗ V \ S ) p ( x 2 S , x ∗ V \ S ) > 0. Pr oof: Necessity is as before. For the con verse, s uppose th e set of encoders is n ot feasible. Then there exist assig nments ( x ∗ 1 , x A V \ v 1 ) and ( x ∗ 1 , x B V \ v 1 ) such that f ( x ∗ 1 , x A V \ v 1 ) 6 = f ( x ∗ 1 , x B V \ v 1 ) and p ( x ∗ 1 , x A V \ v 1 ) p ( x ∗ 1 , x B V \ v 1 ) > 0. Howe ver , the codewords recei ved from nodes in N − ( v 1 ) are the same for both assignm ents. For t he cut which separates v 1 from V \ v 1 , there i s no encoder on δ + ( S ) which separates x A V \ v 1 and x B V \ v 1 . ✷ The above proof of the con verse is not constructive. The construction is mu ch harder now since the encoders are coupled, as shown by t he following example. Example 1: Consider t he three node network G = ( V , E ) with V = { v 1 , v 2 , v 3 } and E = { ( v 2 , v 1 ) , ( v 3 , v 1 ) } (see Fig ure 1(a)). Let X 1 = { x 1 a } , X 2 = { x 2 a , x 2 b } , X 3 = { x 3 a , x 3 b } . Suppose p ( x 1 a , x 2 a , x 3 a ) = p ( x 1 a , x 2 b , x 3 b ) = 1 2 . The function is given by f ( X 1 , X 2 , X 3 ) = ( X 1 , X 2 , X 3 ) . Con- sidering t he cut ( { v 2 , v 3 } , { v 1 } ) , either v 2 or v 3 needs t o separate it s two values. Thus the two encoders are no long er independent. (a) (b) Fig. 1. T wo simple networks of Examples 1 and 2 In general, we can trade of f bet ween the encoders on different edges. Howe ver , if we assum e that p ( x 1 , x 2 , . . . , x n ) > 0 for all ( x 1 , x 2 , . . . , x n ) , we can separately minimize the a verage description length of each encoder . The optimal encoder const ructs a Huffman code on th e opt imal alphabet A i . Suppose q i is the probability vector induced on the alphabet A i . Then, by taking long bl ocks of measurements, we can achie ve a mini mum ave rage case compl exity arbit rarily cl ose to H ( q i ) for encoder C i . C. Fu nction Compu tation in Dir ected Acyclic Graphs The extension from trees to directed acyclic graphs presents significant challenges, since there is no longer a uni que path from ev ery node to the collector . Consider a weakly conn ected directed acyclic graph (D A G) G = ( V , E ) , where each node v i collects a b lock o f N measurements X i ∈ X N i . The collector node v 1 is the uni que node with only incoming edges, which wants to compute t he vector function f N ( X 1 , X 2 , . . . , X n ) with zero error . Let the encoder m apping on edge ( v j , v i ) be denoted by C N j i , which maps t he measurement vector X j and the codewords received thus far , to a cod e word transmitted on edge ( v j , v i ) . Since th ere are no cycles in G , function comp utation proceeds in a bottom-up fashion. No de v i recei ves codew ords C N j i on each incoming edge ( v j , v i ) and then transmits a codew ord C ik on each outg oing edge ( v i , v k ) . A set of encoders i s said to be feasibl e if t here is a decodin g functio n at t he collector node v 1 which maps t he received codew ords to th e correct function value. Let l wc ( C N i j ) and l avg ( C N i j ) denote the worst case and a verage case com plexity , respectiv ely , of the encoder C N i j . The rate of encoder C N i j is R wc ( C N i j ) = l wc ( C N i j ) N and R avg ( C N i j ) = l avg ( C N i j ) N . Thus we can ass ign a rate vector in R | E | to ever y feasible set of encoders. Let R ( N ) wc in the worst case (or R ( N ) avg in the ave rage case) be t he s et o f feasible rate vectors for encoders of bl ock length N . Then the rate region R wc (or R avg ) i s given by the closure in R | E | of the finite block length rate vectors: R wc : = [ N ≥ 1 R ( N ) wc and R avg : = [ N ≥ 1 R ( N ) avg . Consider the following example. Example 2: W e have three nodes { v 1 , v 2 , v 3 } connected as shown in Figu re 1(b). Let X 1 = X 2 = X 3 = { 0 , 1 , 2 , 3 } , and s uppose node v 1 wants to com pute f ( X 1 , X 2 , X 3 ) = ( X 1 + X 2 + X 3 ) mod 4. It i s easy to check that ( 2 , 0 , 2 ) and ( 2 , 2 , 0 ) are feasible rate vectors fo r ( l 1 , l 2 , l 3 ) . These are rate vectors ass ociated with t he two tree subgraphs. Further , one can also check that ( 2 , 1 , 1 ) is 1) Outer bou nd on the rate r e gio n: Consider any cut of the graph G which parti tions nodes into subs ets S and V \ S wit h v 1 ∈ V \ S . Let δ + ( S ) be the set o f edges from s ome nod e in S to some no de i n V \ S . Lemma 3: Consi der a set of encoders which achieve error free block function computation with rate vector { R wc ( i , j ) } ( v i , v j ) ∈ E . Giv en any assign ments x 1 S and x 2 S of the nod es in S , if there exists an assignm ent x V \ S such that f ( N ) ( x V \ S , x 1 S ) 6 = f ( N ) ( x V \ S , x 2 S ) , then the encoders on at least one of t he edges in δ + ( S ) must s eparate x 1 S and x 2 S . (i) In th e worst case block com putation scenario, an outer bound on the rate region is given by ∑ ( v i , v j ) ∈ δ + ( S ) R i j ≥ log | Π ( C 1 S ) | for all cuts ( S , V \ S ) , where Π ( C 1 S ) is th e partitio n of X S into t he appropriate equiva lence classes. (ii) Suppose we have a probability dist ribution with p ( X 1 , X 2 , . . . , X n ) > 0. Given a cut ( S , V \ S ) , let R ⊂ V \ S be the su bset of nodes w hich have a directed path to some node i n S . In the a verage case block computation s cenario, an ou ter b ound on the rate region is given by ∑ ( v i , v j ) ∈ δ + ( S ) R i j ≥ H ([ X S ] | X R ) for all cuts ( S , V \ S ) , where [ X S ] | X R is t he equiv alence class to which X S belongs, given X R and a particul ar function. 2) Achievable r e gion: Lemma 4: Consi der any di rected tree subgraph G T with root node v 1 . Let us su ppose that only the edges in G T can be used for commu nication. Then we can construct encoders on each edge, w hich mi nimize worst case or average case com plexity . The rate vector corresponding to a tree G T is the li mit of the rate vectors for the optimal finite blo ck length encoders for G T . Thus, for a given tree G T : (i) The worst case rate vec tor corresponding to the tree G T is an extreme point of t he worst case rate region R wc . (ii) If p ( x 1 , x 2 , . . . , x n ) > 0 for all ( x 1 , x 2 , . . . , x n ) , the rate vector corresponding to t he tree G T is an extreme poi nt of the av erage case rate region R avg . The conv ex hull of the rate poi nts corresponding to trees i s achiev able. Howe ver , we do not know i f these are the only extreme points of t he rate region R . 3) Some examples: Example 3 (Arit hmetic Sum): Consider three nodes v 1 , v 2 , v 3 connected as in Figu re 1(b). Let X 2 = X 3 = { 0 , 1 } , wit h node v 1 having no m easurements. Suppose node v 1 wants to compute f ( X 1 , X 2 , X 3 ) = X 2 + X 3 . Let ( R 21 , R 31 , R 32 ) be the rate vector ass ociated with edg es ( l 1 , l 2 , l 3 ) . The outer bound on R wc is: R 21 ≥ 1; R 21 + R 31 ≥ log 3; R 32 + R 31 ≥ 1 . The subset of th e rate region achiev able by trees is: R 21 = λ + ( 1 − λ ) log 3 , R 31 = λ , R 32 = ( 1 − λ ) for 0 ≤ λ ≤ 1 . Suppose that X 1 , X 2 are i.i.d. with p ( X 1 = 0 ) = p ( X 1 = 1 ) = 0 . 5. Th e ou ter bound on R avg is: R 21 ≥ 1; R 21 + R 31 ≥ 3 2 ; R 32 + R 31 ≥ 1 . The subset of th e rate region achiev able by trees is: R 21 = λ + ( 1 − λ ) 3 2 , R 31 = λ , R 32 = ( 1 − λ ) for 0 ≤ λ ≤ 1 . Example 4 (F inite field par ity): Let X i = { 0 , 1 , . . . , D − 1 } for each node v i . Supp ose the collector node v 1 wants to compute th e functi on ( X 1 + X 2 + . . . + X n ) mod D . In t his case, the outer bound on t he worst case rate region described in Lemma 3 is t ight. Indeed, since t he set of al l outg oing links from a node is a valid cut, we have ∑ ( v i , v j ) ∈ E R i j ≥ log 2 D An obvious achiev able strategy i s for every leaf node v i to split its b lock and transmi t it on the outgoing edg es from v i . Next, we move t o a node at height 1. This n ode recei ves partial blo cks from various leaf nodes, and can hence comput e an int ermediate parity for some instances of the block. It then splits its block alon g the various outgoin g edges. The crucial po int i s that the worst case d escription length per instance remai ns log 2 D . Proceeding recursively up the D A G, we s ee that we can achieve the outer bound. Example 5 (Max/Min): Let X i = { 0 , 1 , . . . , D − 1 } for each node v i . Suppose the c ollector node v 1 wants to com pute ma x ( X 1 , X 2 , . . . , X n ) . The outer bound t o the worst case rate region described in Lem ma 3 i s ti ght. The achiev able st rategy i s simil ar to t he parity case, where nodes compute intermediate maximum v alues and split their blocks on the outgoi ng edges. Once again, we utilize the fact that the range of th e Max function remains const ant irrespectiv e of t he number of n odes. I V . C O M P U T I N G S Y M M E T R I C B O O L E A N F U N C T I O N S I N U N D I R E C T E D G R A P H S In t his s ection, we address the problem of symmet ric Boolean function computation in an undirected graph. Each node has a Boolean variable and all nod es want to compute a given symmetric Boolean function. As in Section III, we adopt a determi nistic framework and consider the problem of worst case block comput ation. Further , since t he graph is undi rected, t he set of admissible s trategies i ncludes al l int eractiv e strategies, where a node may exchange severa l messages with ot her nodes, with node i ’ s transmi ssion being allowed to depend on all previous transmissio ns heard by n ode i , and node i ’ s block o f measurements. This is in contrast with the problem s tudied i n Section III. W e begin by re viewing a toy problem from [9] where the exact comm unication com plexity of th e AND function of two variables is sh own to be log 2 3 b its, for block comput ation. In Section IV -A, we generalize t his approach to the two nod e problem, where each node i has an integer variable X i and b oth nod es want t o compute a function f ( X 1 , X 2 ) which only depends on X 1 + X 2 . W e deri ve an opt imal si ngle-round strategy for th e class of sum-threshold functions, which ev aluate t o 1 if X 1 + X 2 exceeds a t hreshold, and an approximate st rategy for the class of sum-interval functions, which ev aluate t o 1 if a ≤ X 1 + X 2 ≤ b , the upper and lower bounds do not match. The general achie vable strategy i n volves separation of the source alphabet, followed by coding , and can be used for any general function. In Section IV -B, we consider sy mmetric Boolean fun ction com putation on t rees. Since ev ery edge is a cut-edge, we can obtain a cut-set lower bo und for the number of bits that must be exchanged on an edge, by reducing it to a two no de problem with g eneral alph abets. For the class of sum-th reshold functi ons, we are able to m atch the cut-set boun d by constructing an achiev able strategy that is reminis cent of message passing algorithms. In Section IV -D, for general g raphs, we can stil l derive a cut -set l ower bound by cons idering all partitio ns of the vertices. W e also propose an achie vable scheme that consi sts of activa ting a subt ree of edges and using the optimal strategy for transmissi ons on the t ree. While the upper and lower bound s do not match ev en for very simple functi ons, for complete g raphs, we show that aggregation along trees provides a 2-OPT sol ution. A. The t wo no de pr oblem Consider two nodes 1 and 2 with variables X 1 ∈ { 0 , 1 , . . . , m 1 } and X 2 ∈ { 0 , 1 , . . . , m 2 } . Both nodes wish to com pute a function f ( X 1 , X 2 ) which onl y depends on the value of X 1 + X 2 . T o put thi s in context, one can suppos e there are m 1 Boolean variables collocated at node 1 and m 2 Boolean variables at nod e 2, and both nodes wish to com pute a symmetric Boolean function of the n : = m 1 + m 2 var iables. W e po se the problem in a bl ock computation setting, wh ere each node i has a blo ck of N ind ependent measurements, d enoted by X N i . W e consider the class of all interactiv e st rategies, w here nodes 1 and 2 transmit messages alternately wit h the value of each subsequent message bein g allowed to depend on all previous transmis sions, and the block of measurements av ailable at the transmitt ing node. W e define a round to include one t ransmission by each node. A strategy is s aid to achiev e correct block com putation if for every choi ce of input ( X N 1 , X N 2 ) , each node i can correctly decode the value o f the function bl ock f N ( X 1 , X 2 ) using the sequence of transmis sions b 1 , b 2 , . . . and its own m easurement block X N i . Let S N be the set of strategies for block length N , which achie ve zer o-error block comput ation, and let C ( f , S N , N ) be the worst-case total number of b its exchanged und er strategy S N ∈ S N . The worst-case per-instance complexity of computin g a function f ( X 1 , X 2 ) is defined as C ( f ) : = lim N → ∞ min S N ∈ S N C ( f , S N , N ) N . 1) Complexity of sum-thr esho ld fu nctions: In thi s paper , we are only interested in function s f ( X 1 , X 2 ) which only d epend o n X 1 + X 2 . L et u s suppose wi thout loss of generality t hat m 1 ≤ m 2 . W e define an interesting class of { 0 , 1 } -valued functions called sum-thresho ld functions. Definition 4 (sum-th r eshol d functions): A sum-threshold functio n Π θ ( X 1 , X 2 ) with threshold θ is defined as fol lows: Π θ ( X 1 , X 2 ) =    1 if X 1 + X 2 ≥ θ , 0 otherwise. For the special case where m 1 = 1 , m 2 = 1 and θ = 2 , w e recov er th e Boolean AND functi on, which was studied in [9]. It i s critical to understand this problem before we can address the general problem of computin g sym metric Boolean functions . Consider t wo nodes wit h measurement blocks X N 1 ∈ { 0 , 1 } N and X N 2 ∈ { 0 , 1 } N , which want to com pute the element-wise AND of t he two bl ocks, denot ed by ∧ N ( X 1 , X 2 ) . Theor em 5: Given any strategy S N for bl ock computation of X 1 ∧ X 2 , C ( X 1 ∧ X 2 , S N , N ) ≥ N log 2 3 . Further , there exists a strategy S ∗ N which satisfies C ( X 1 ∧ X 2 , S ∗ N , N ) ≤ ⌈ N log 2 3 ⌉ . Thus, t he com plexity of com puting X 1 ∧ X 2 is given by C ( X 1 ∧ X 2 ) = l og 2 3. Pr oof of achievability: Suppose node 1 transmits first u sing a prefix-free codebook. Let th e length of the codeword transmit ted be l ( X N 1 ) . At the end of this transmi ssion, bo th no des know the value of the functio n at the ins tances where X 1 = 0. Thus n ode 2 only n eeds t o indi cate i ts bits for the instances of the block where X 1 = 1. Th us t he total num ber of bit s exchanged under this scheme is l ( X N 1 ) + w ( X N 1 ) , where w ( X N 1 ) is the number of 1s in X N 1 . For a giv en scheme, let us define L : = max X N 1 ( l ( X N 1 ) + w ( X N 1 )) , to be the worst case to tal number o f bits exchanged. W e are interested in finding t he codebook which wil l result in the minim um worst-case nu mber of bits . Any prefix-free code must satisfy th e Kraft in equality giv en by ∑ X N 1 2 − l ( X N 1 ) ≤ 1. Consider a codebook with l ( X N 1 ) = ⌈ N log 2 3 ⌉ − w ( x N 1 ) . This sati sfies the Kraft inequalit y since ∑ X N 1 w ( X N 1 ) = 3 N . Hence th ere exists a valid prefix free code for whi ch the worst case num ber of bits exchanged is ⌈ N log 2 3 ⌉ , w hich establ ishes th at C ( X 1 ∧ X 2 ) ≤ log 2 3. The lower bo und i s sh own by constructing a foolin g set [5] of t he appropriate s ize. W e digress bri efly to i ntroduce the concept o f foo ling sets in t he context o f two-party com munication complexity [5]. Consider two nodes X and Y , each of which take values in finite sets X and Y , and b oth no des want to compute some function f ( X , Y ) with zero error . Definition 5 (F ool ing Set): A s et E ⊆ X × Y is said to b e a fool ing set, if for any two dist inct elements ( x 1 , y 1 ) , ( x 2 , y 2 ) in E , we have either • f ( x 1 , y 1 ) 6 = f ( x 2 , y 2 ) , or • f ( x 1 , y 1 ) = f ( x 2 , y 2 ) , but either f ( x 1 , y 2 ) 6 = f ( x 1 , y 1 ) or f ( x 2 , y 1 ) 6 = f ( x 1 , y 1 ) . Giv en a fooling set E for a fun ction f ( X 1 , X 2 ) , we have C ( f ( X 1 , X 2 )) ≥ log 2 | E | . W e have described t wo dimension al fooling sets above. The extension to multi-dim ensional fooling sets is straightforward and give s a lower boun d on the communi cation comp lexity of the function f ( X 1 , X 2 , . . . , X n ) . Lower bound for Theor em 5: W e define t he measurement matrix M to be t he matrix o btained by stacking the row X N 1 over t he row X N 2 . Thus we need to find a s ubset of the set of all measurement m atrices which forms a foolin g set. Let E the set of all measurement matrices which are made up of only t he colu mn vectors {   1 0   ,   0 1   ,   1 1   } . W e claim that E i s the appropriate fooling set. Consider two distin ct measurement matrices M 1 , M 2 ∈ E . Let f N ( M 1 ) and f N ( M 2 ) be the b lock function values obt ained from these two matrices. If f N ( M 1 ) 6 = f N ( M 2 ) , we are done. Let us su ppose f N ( M 1 ) = f N ( M 2 ) and since M 1 6 = M 2 , t here must exist o ne col umn where M 1 has   0 1   but M 2 has   1 0   . Now if we replace the first ro w of M 1 with the first row o f M 2 , t he result ing m easurement matrix, say M ∗ is such that f ( M ∗ ) 6 = f ( M 1 ) . Thus, the set E i s a valid fooling set. It is easy to verify t hat th e E has cardinalit y 3 N . Thus, for a ny strategy S N ∈ S N , we m ust hav e C ( X 1 ∧ X 2 , S N , N ) ≥ N log 2 3, imp lying that C ( X 1 ∧ X 2 ) ≥ l og 2 3. This concludes th e proo f o f Theorem 5. ✷ W e no w return to the general tw o node problem with X 1 ∈ { 0 , 1 , . . . , m 1 } and X 2 ∈ { 0 , 1 , . . . , m 2 } and t he sum-thresho ld function Π θ ( X 1 , X 2 ) . W e wi ll extend th e approach presented above to this general scenario. Theor em 6: Given any strategy S N for bl ock computation of the function Π θ ( X 1 , X 2 ) , C ( Π θ ( X 1 , X 2 ) , S N , N ) ≥ N log 2 { min ( 2 θ + 1 , 2 m 1 + 2 , 2 ( n − θ + 1 ) + 1 ) } . Further , t here exist singl e-round strategies S ∗ N and S ∗∗ N , starting with nodes 1 and 2 respectiv ely , which sati sfy C ( Π θ ( X 1 , X 2 ) , S ∗ N , N ) ≤ ⌈ N lo g 2 { min ( 2 θ + 1 , 2 m 1 + 2 , 2 ( n − θ + 1 ) + 1 ) }⌉ . C ( Π θ ( X 1 , X 2 ) , S ∗∗ N , N ) ≤ ⌈ N log 2 { min ( 2 θ + 1 , 2 m 1 + 2 , 2 ( n − θ + 1 ) + 1 ) }⌉ . Thus, the com plexity of com puting Π θ ( X 1 , X 2 ) is given by C ( Π θ ( X 1 , X 2 )) = log 2 { min ( 2 θ + 1 , 2 m 1 + 2 , 2 ( n − θ + 1 ) + 1 ) } . Pr oof of achiev ability: W e consider three cases: (a) Suppos e θ ≤ m 1 ≤ m 2 . W e specify a strategy S ∗ N in which node 1 transm its first. W e begin by observing that inputs X 1 = θ , X 1 = ( θ + 1 ) . . . , X 1 = m 1 need not be separated , since for each of these values of X 1 , Π θ ( X 1 , X 2 ) = 1 for all v alues of X 2 . Th us node 1 h as an ef fectiv e alphabet of { 0 , 1 , . . . , θ } . Suppose node 1 transm its using a prefix-free code word of length l ( X N 1 ) . At the end of t his transmis sion, node 2 only needs to i ndicate one bi t for the inst ances of the block where X 1 = 0 , 1 , . . . , ( θ − 1 ) . Thus the worst-case t otal num ber of b its is L : = max X N 1 ( l ( X N 1 ) + w 0 ( X N 1 ) + w 1 ( X N 1 ) + . . . + w θ − 1 ( X N 1 )) , where w j ( X N 1 ) is the num ber of i nstances in the b lock where X 1 = j . W e are in terested in finding the codebook w hich will result in the minimum worst-case number of bi ts. From the Kraft inequalit y for prefix-free codes we hav e ∑ X N 1 ∈{ 0 , 1 ,..., θ } N 2 − L + w 0 ( X N 1 )+ w 1 ( X N 1 )+ ... + w θ − 1 ( X N 1 )) ≤ 1 . Consider a codebook with l ( X N 1 ) = ⌈ N log 2 ( 2 θ + 1 ) ⌉ − w ( x N 1 ) . This satisfies the Kraft in- equality s ince ∑ X N 1 ∈{ 0 , 1 ,..., θ } N 2 w 0 ( X N 1 )+ w 1 ( X N 1 )+ ... + w θ − 1 ( X N 1 )) . 1 w θ ( X N 1 ) = ( 2 θ + 1 ) N . Hence there exists a prefix-free code f or which the worst-case total number of bits exchanged is ⌈ N log 2 ( 2 θ + 1 ) ⌉ . Since θ ≤ m 1 ≤ m 2 , we have C ( Π θ ( X 1 , X 2 ) , S ∗ N , N ) ≤ ⌈ N log 2 { min ( 2 θ + 1 , 2 m 1 + 2 , 2 ( n − θ + 1 ) + 1 ) }⌉ . The strategy S ∗∗ N starting at no de 2 can be sim ilarly derive d. Node 2 now has an ef fectiv e alphabet of { 0 , 1 , . . . , θ } , and we have C ( Π θ ( X 1 , X 2 ) , S ∗∗ N , N ) ≤ ⌈ N log 2 ( 2 θ + 1 ) ⌉ . (b) Supp ose m 1 ≤ m 2 < θ . Consider a strategy S ∗ N in which nod e 1 trans mits first. The i nputs X 1 = 0 , X 1 = 1 , . . . , X 1 = θ − m 2 − 1 need not be separated since for each of t hese values of X 1 , Π θ ( X 1 , X 2 ) = 0 for all values of X 2 . Thus node 1 has an ef fectiv e alphabet of { θ − m 2 − 1 , θ − m 2 , . . . , m 1 } . Upon hearing node 1’ s transmi ssion, node 2 only needs to indi cate one bit for the instances of th e block wh ere X 1 = θ − m 2 , . . . , m 1 . Consi der a codebook w ith l ( X N 1 ) = ⌈ N log 2 ( 2 ( m 1 + m 2 − θ + 1 ) + 1 ) ⌉ − w θ − m 2 ( X N 1 ) − . . . − w m 1 ( X N 1 ) . This satisfies the Kraft i nequality and we have L = ⌈ N log 2 ( 2 ( n − θ + 1 ) + 1 ) ⌉ . Since m 1 ≤ m 2 < θ , we have that C ( Π θ ( X 1 , X 2 ) , S ∗ N , N ) ≤ ⌈ N log 2 { min ( 2 θ + 1 , 2 m 1 + 2 , 2 ( n − θ + 1 ) + 1 ) }⌉ . The strategy S ∗∗ N starting at node 2 can be analogously deriv ed. (c) Suppos e m 1 < θ ≤ m 2 . For the case where node 1 t ransmits first, we construct a trivial strategy S ∗ N where node 1 uses a codeword of length ⌈ N log 2 ( m 1 + 1 ) ⌉ bi ts and node 2 replies with a string of N bits indicatin g the fun ction block. Thus we h a ve C ( Π θ ( X 1 , X 2 ) , S ∗ N , N ) ≤ ⌈ N log 2 ( 2 m 1 + 2 ) ⌉ . Now consi der a strategy S ∗∗ N where node 2 transmits first. Observe t hat the input s X 2 = 0 , X 2 = 1 , . . . , X 2 = θ − m 1 − 1 need no t be s eparated since for each of these values of X 2 , Π θ ( X 1 , X 2 ) = 0 for all values of X 2 . Further , t he inputs X 2 = θ , X 2 = θ + 1 , . . . , X 2 = m 2 need n ot be separated . Thus node 1 has an ef fectiv e alphabet of { θ − m 1 − 1 , θ − m 1 , . . . , θ } . Upon hearing n ode 2 ’ s transmissio n, node 1 only needs to indicate one bit for t he instances of the block w here X 2 = θ − m 1 , . . . , θ − 1 . Consider a codebook wi th l ( X N 2 ) = ⌈ N l og 2 ( 2 m 1 + 2 ) ⌉ − w θ − m 1 ( X N 1 ) − . . . − w θ − 1 ( X N 1 ) . This satisfies t he Kraft i nequality and we hav e L = ⌈ N log 2 ( 2 ( n − θ + 1 ) + 1 ) ⌉ . Sin ce m 1 < θ ≤ m 2 , we have t hat C ( Π θ ( X 1 , X 2 ) , S ∗∗ N , N ) ≤ ⌈ N log 2 { min ( 2 θ + 1 , 2 m 1 + 2 , 2 ( n − θ + 1 ) + 1 ) }⌉ . The l ower b ound is shown by constructing a f ooling set as before. Let E deno te t he s et o f all measurement m atrices which are made up only of th e col umn vectors from the set Z =      z 1 z 2   : 0 ≤ z 1 ≤ m 1 , 0 ≤ z 2 ≤ m 2 , ( θ − 1 ) ≤ z 1 + z 2 ≤ θ    . W e claim that E is t he appropriate fooling set. Consider two di stinct m easurement matrices M 1 , M 2 ∈ E . Let f N ( M 1 ) and f N ( M 2 ) be the block function values obtained from these t wo matrices. If f N ( M 1 ) 6 = f N ( M 2 ) , we are done. Let us su ppose f N ( M 1 ) = f N ( M 2 ) , and note that since M 1 6 = M 2 , there mus t exist one column where M 1 and M 2 diffe r . Suppose M 1 has   z 1 a z 2 a   while M 2 has   z 1 b z 2 b   , where z 1 a + z 2 a = z 1 b + z 2 b . Assume with out loss of generality that z 1 a < z 1 b and z 2 a > z 2 b . • If z 1 a + z 2 a = z 1 b + z 2 b = θ − 1, then the diagonal element f ( z 1 b , z 2 a ) = 1 si nce z 1 b + z 2 a ≥ θ . Thus, if we replace t he first row of M 1 with the first row of M 2 , the resul ting measurement matrix, s ay M ∗ , is such that f ( M ∗ ) 6 = f ( M 1 ) . • If z 1 a + z 2 a = z 1 b + z 2 b = θ , then the diagonal elem ent f ( z 1 a , z 2 b ) = 0 since z 1 b + z 2 a < θ . Thus, if we replace the second row of M 1 with the second row of M 2 , the resulti ng m atrix M ∗ is such t hat f ( M ∗ ) 6 = f ( M 1 ) . Thus, t he set E is a valid fooli ng set with cardinality | Z | N . For any strategy S N , we have C ( f , S N , N ) ≥ N log 2 | Z | . T he cardinality of Z can be modeled as th e sum of the coefficients of Y θ and Y θ − 1 in a carefully cons tructed pol ynomial: | Z | = h Y θ i + h Y θ − 1 i ( 1 + Y + . . . + Y m 1 )( 1 + Y + . . . + Y m 2 ) = h Y θ i + h Y θ − 1 i ( 1 − Y m 1 + 1 )( 1 − Y m 2 + 1 ) ( 1 − Y ) 2 . This i s sol ved using the bino mial expansion for 1 ( 1 − Y ) k [21]. | Z | = h Y θ i + h Y θ − 1 i ( 1 − Y m 1 + 1 )( 1 − Y m 2 + 1 ) ∞ ∑ k = 0   k + 1 1   Y k . (a) Suppos e θ ≤ m 1 ≤ m 2 . Then | Z | = θ + θ + 1. (b) Supp ose m 1 ≤ θ ≤ m 2 . Then | Z | = 2 m 1 + 2. (c) Suppos e m 1 ≤ m 2 ≤ θ . Then | Z | = 2 ( n − θ + 1 ) + 1. This com pletes t he proof of Theorem 6. ✷ 2) Complexity of sum-interval functio ns: Definition 6 (sum-in terval function s): A s um-interval function Π [ a , b ] ( X 1 , X 2 ) on the i nterval [ a , b ] is defined as follows: Π [ a , b ] ( X 1 , X 2 ) : =    1 if a ≤ X 1 + X 2 ≤ b , 0 otherwise. Theor em 7: Given any strategy S N for bl ock computation of Π [ a , b ] ( X 1 , X 2 ) where b ≤ n / 2, C ( Π [ a , b ] ( X 1 , X 2 ) , S N , N ) ≥ N log 2 { min ( 2 b − a + 3 , m 1 + 1 ) } . Further , there exists a sing le-round strategy S ∗ N which sati sfies C ( Π [ a , b ] ( X 1 , X 2 ) , S ∗ N , N ) ≤ ⌈ N log 2 { min ( 2 ( b + 1 ) + 1 , 2 m 1 + 2 ) }⌉ . Thus, we have obtained the comp lexity of computing Π θ ( X 1 , X 2 ) to wit hin one bit. Pr oof of Achiev ability: (a) Suppos e b ≤ m 1 ≤ m 2 . Node 1 has an effecti ve alphabet of { 0 , 1 , . . . , b + 1 } . Then the worst- case tot al num ber of bits exchanged is given by L : = max X N 1 ( l ( X N 1 ) + w 0 ( X N 1 ) + . . . + w b ( X N 1 )) . From the Kraft inequalit y , we can obtain a prefix free codebook with L = ⌈ N log 2 ( 2 b + 1 ) + 1 ) ⌉ . Thus we hav e C ( Π [ a , b ] ( X 1 , X 2 ) , S ∗ N , N ) ≤ ⌈ N log 2 ( 2 ( b + 1 ) + 1 ) ⌉ . (b) Supp ose m 1 ≤ a ≤ b ≤ m 2 or a ≤ m 1 ≤ b ≤ m 2 . In either of these scenarios, node 1 has an ef fectiv e alphabet of { 0 , 1 , . . . , m 1 } . Then the worst-case tot al numb er of bits exchanged is giv en by L : = max X N 1 ( l ( X N 1 ) + w a − m 2 ( X N 1 ) + . . . + w m 1 ( X N 1 )) From the Kraft i nequality , we can obtain a prefix free codebook wi th L = ⌈ N log 2 ( 2 m 1 + 2 ) ⌉ . Thus we have C ( Π [ a , b ] ( X 1 , X 2 ) , S ∗ N , N ) ≤ ⌈ N log 2 ( 2 m 1 + 2 ) ⌉ . Pr oof of Lower Bound: W e attem pt to find a fooling subset E o f the set of measurement matrices. Our first guess would be the set of m easurement matrices which are composed of only column vectors which sum up to b or b + 1. Ho wev er we see that this is not necessarily a fooling set, because if [ z 1 a , z 2 a ] T and [ z 1 b , z 2 b ] T are two columns which sum to b + 1, and if z 1 a ≤ z 1 b − ( b − a + 2 ) , then neither of the diagonal elements ev aluate to fun ction value 1. Thus, we can pick a maxim um of ( b − a + 2 ) consecutiv e elements alon g the line z 1 + z 2 = b + 1, and, as before, all the element s on the line z 1 + z 2 = b . It is easy to check that this mod ified set of columns indeed yield s a fooling set of measurement matrices. Now we need to compute the number of such colu mns. (a) Suppos e b ≤ m 1 ≤ m 2 . The numb er of colum ns which sum up to b i s equal to b + 1 . Thus the size of the foolin g set is g iv en by ( 2 b − a + 3 ) N . (b) Supp ose a ≤ m 1 ≤ b ≤ m 2 or m 1 ≤ a ≤ b ≤ m 2 . Th e number of columns which sum up to b is equ al to m 1 + 1 and the nu mber of columns wh ich sum up to b + 1 is equal t o m 1 + 1. Thus, t he size of the fooling set is given by { ( m 1 + 1 ) + min ( m 1 + 1 , b − a + 2 ) } N . 3) A general strate gy for achievability: The s trategy for achiev abil ity used in Theorems 6 and 7 s uggests an achiev able scheme for any general fun ction f ( X 1 , X 2 ) of variables X 1 ∈ X 1 and X 2 ∈ X 2 which depends only on the value of X 1 + X 2 . This i s done in two stages. • Separation: T wo inputs x 1 a and x 1 b need not be separated if f ( x 1 a , x 2 ) = f ( x 1 b , x 2 ) for all values x 2 . By checking this condi tion for each p air ( x 1 a , x 1 b ) , we can arrive at a partition of { 0 , 1 . . . , m 1 } into equiv alence classes, which can be considered a reduced al phabet, say A : = { a 1 , . . . , a l } . • Coding: Let A 0 denote the subset of th e alphabet A for which the functio n ev aluates only to 0, irrespective of t he value of X 2 , and let A 1 denote the subset of A which always ev aluates to 1. Clearly , from th e equiv alence class structure, we have | A 0 | ≤ 1 and | A 1 | ≤ 1. Using the Kraft inequality as in Theorems 6 and 7, we obtain a scheme S ∗ N with compl exity log 2 ( 2 l − | A 0 | − | A 1 | ) . B. Computing symmetri c Bo olean fu nctions on tr ee networks Consider a t ree graph T = ( V , E ) , wi th node set V = { 0 , 1 , . . . , n } and edge set E . Each nod e i has a Boolean variable X i ∈ { 0 , 1 } , and ev ery node wants to com pute a given symmetri c Bool ean function f ( X 1 , X 2 , . . . , X n ) . Again, we allow for block compu tation and con sider all strategies where nod es can transm it in any sequence with possi ble repetitions, subject to: • On any edge e = ( i , j ) , either node i transmits or node j transmits, or neither , and this is determined from the previous transmission s. • Nod e i ’ s transmissio n can depend on t he previous transm issions and th e m easurement block X N i . For sum-threshol d functions , we hav e a com putation and communication st rategy that is optimal for each link . Theor em 8: Consider a tree network where we want t o compute th e function Π θ ( X 1 , . . . , X n ) . Let us focus on a single edge e ≡ ( i , j ) whose remova l d isconnects t he graph i nto comp onents A e and V \ A e , with | A e | ≤ | V \ A e | . For any s trategy S N ∈ S N , the number of bits exchanged along edge e ≡ ( i , j ) , denoted by C e ( Π θ ( X 1 , . . . , X n ) , S N , N ) , is lower bounded b y C e ( Π θ ( X 1 , . . . , X n ) , S N , N ) ≥ N log 2 { min ( 2 θ + 1 , 2 | A e | + 2 , 2 ( n − θ + 1 ) + 1 ) } . Further , there exists a strategy S ∗ N such that for any edge e , C e ( Π θ ( X 1 , . . . , X n ) , S ∗ N , N ) ≤ ⌈ N log 2 { min ( 2 θ + 1 , 2 | A e | + 2 , 2 ( n − θ + 1 ) + 1 ) }⌉ . The complexity of com puting Π θ ( X 1 , . . . , X n ) is given by C e ( Π θ ( X 1 , . . . , X n )) = log 2 { min ( 2 θ + 1 , 2 | A e | + 2 , 2 ( n − θ + 1 ) + 1 ) } . Pr oof: Given a tree network T , ev ery edge e is a cut edge. Consider an edge e whose remov al creates components A e and V \ A e , with | A e | ≤ | V \ A e | . Now let us aggregate the nodes in A e and also those in V \ A e , and view this as a problem with t wo nodes connected by edge e . Clearly the complexity of computi ng the function Π θ ( X A e , X V \ A e ) is a lower bound on th e worst-case total number of bits that must be exchanged on edge e un der any strategy S N . Hence we obtain C e ( Π θ ( X 1 , . . . , X n ) , S N , N ) ≥ N log 2 { min ( 2 θ + 1 , 2 | A e | + 2 , 2 ( n − θ + 1 ) + 1 ) } . The achiev abl e s trategy S ∗ N is derived from the achiev able strategy for the two node case in Theorem 6. While t he transmis sions back and forth along any edge wil l be exactly t he same, we need to orchestrate these transm issions so that condi tions of causali ty are maintained. Pick any node, say r , to be th e root . This i nduces a partial order on the tree network. W e start with each leaf in the net work transmi tting its codew ord to the parent. Once a parent no de obtains a codeword from each of its chi ldren, it has suffi cient k nowledge to disambi guate the letters of the effec tive alphabet of the sub tree, and subsequently it transmits a code word to i ts parent. Thus cod e words are t ransmitted from child nodes to parent no des unt il t he root is reached. The root can then compute the value of the function and n ow sends t he appropriate replies to its children. The children then compute the function and send appropriate replies, and so on. Th is sequential strategy depends critically on the fact that, in t he two node problem, we derived optimal st rategies starting from either nod e. For any edge e , the worst-case total number of bits exchanged is give n by C e ( Π θ ( X 1 , . . . , X n ) , S ∗ N , N ) ≤ ⌈ N log 2 { min ( 2 θ + 1 , 2 | A e | + 2 , 2 ( n − θ + 1 ) + 1 ) }⌉ . ✷ One can similarly deriv e an approximately optimal strategy for sum-interval functi ons, which we s tate here wit hout proof. Theor em 9: Consider a tree network where we want to comput e the function Π [ a , b ] ( X 1 , . . . , X n ) , with b ≤ n 2 . Let us focus on a single edge e ≡ ( i , j ) whose remov al di sconnects th e graph into components A e and V \ A e , wi th | A e | ≤ | V \ A e | . For any strategy S N ∈ S N , the n umber of bits exchanged along edge e ≡ ( i , j ) , denoted by C e ( f , S N , N ) is lower bounded by C e ( Π [ a , b ] ( X 1 , . . . , X n ) , S N , N ) ≥ N log 2 { min ( 2 b − a + 3 , | A e | + 1 ) } . Further th ere exists a strategy S ∗ N such that for any edge e , C e ( Π [ a , b ] ( X 1 , . . . , X n ) , S ∗ N , N ) ≤ ⌈ N log 2 { min ( 2 ( b + 1 ) + 1 , 2 | A e | + 2 ) }⌉ . C. Extens ion t o non-bi nary alp habets The extension to the case w here each n ode draws measurements from a n on-binary alph abet is imm ediate. Cons ider a t ree network with n nodes where node i has a m easurement X i ∈ { 0 , 1 , . . . , l i − 1 } . Suppose all nodes want to com pute a gi ven functi on which on ly depends on the value of X 1 + X 2 + . . . + X n . W e can define sum-threshold functions in analogo us fashion and deriv e an optim al strategy for computatio n. Theor em 10: Consider a tree network where we want to compute a sum -threshold function , Π θ ( X 1 , . . . , X n ) , of non-binary measurements. Let us focus on a single edge e whose remov al disconnects the graph into components A e and V \ A e . L et us define l A e : = ∑ i ∈ A e l i . T hen the complexity of computin g Π θ ( X 1 , . . . , X n ) is given by C e ( Π θ ( X 1 , . . . , X n )) = log 2 { min ( 2 θ + 1 , 2 m in ( l A e , l V \ A e ) + 2 , 2 ( l V − θ + 1 ) + 1 ) } . Theorem 9 also extends to th e case of non-binary alp habets. D. Computing sum-th r eshol d funct ions in general graphs W e now consider the computati on of sum -threshold functions in general graphs where the alphabet i s not restricted to be binary . A cut is defined t o be a s et of edges F ⊆ E wh ich disconnect t he network into two component s A F and V \ A F . Lemma 5 (Cut-set bound): Consider a general network G = ( V , E ) , where node i has mea- surement X i ∈ { 0 , 1 , . . . , l i − 1 } and all nodes want t o compute t he fun ction Π θ ( X 1 , . . . , X n ) . Given a cut F which separates A F from V \ A F , t he cut-set lower bound specifies that: For any st rategy S N , the number of bit s exchanged on the edges in F is lower bo unded by C F ( Π θ ( X 1 , . . . , X n ) , S N , N ) ≥ N log 2 ( min { 2 θ + 1 , 2 m F + 2 , 2 ( l V − θ + 1 ) + 1 ) } . where l A F = ∑ i ∈ A F l i and m F = min ( l A F , l V \ A F ) . A natural achiev able strategy i s to pick a spanni ng subtree of edges and use the optim al strategy on this su btree. The con vex hull of t he rate vectors of the subt ree aggregation schemes, is an achiev able region. W e wish to compare t his with the cut-set region. T o simplify matters, consider a complete graph G wh ere each node i has a measurement X i ∈ { 0 , . . . , l − 1 } . Let R ach be t he maximum symmetri c ratepoint achiev able by aggregating along trees, and R cu t be t he minimum sym metric ratepoi nt that satisfies the cut-set constraints. Theor em 11: For the computati on of sum-threshol d fun ctions on compl ete graphs, R ach ≤ 2 ( 1 − 1 n )) R cu t . In fact, this approximatio n ratio is t ight. Pr oof: Let us assume without loss of generality that θ ≤ n . l 2 . Consid er all cuts of the type ( { i } , V \ { i } ) . This y ields R cu t ≥ max i ∈ V  min ( log 2 ( 2 θ + 1 ) , log 2 ( 2 l i + 2 )) n − 1  . Now consider the achie vable scheme which emp loys each of the n star g raphs for equal s ized sub-blocks of measurements . The rate on edge ( i , j ) is give n by 1 n  min ( log 2 ( 2 θ + 1 ) , log 2 ( 2 l i + 2 )) + min ( log 2 ( 2 θ + 1 ) , log 2 ( 2 l j + 2 ))  Hence we have R ach ≤ 2 n ( min ( log 2 ( 2 θ + 1 ) , max i ∈ V { log 2 ( 2 l i + 2 ) } )) ≤ 2  1 − 1 n  R cu t . T ight Example: Suppose l 1 = l 2 = . . . = l n = l and θ > l , then R cu t = 1 n − 1 min ( log 2 ( 2 θ + 1 ) , log 2 ( 2 l + 2 )) Further , from the symm etry of the prob lem, i t i s clear t hat th e opti mal schem e is to employ th e n star graphs for equal su b-blocks of measurement s. This gives a sym metric achie vable poi nt of R ach = 2 n min ( log 2 ( 2 θ + 1 ) , log 2 ( 2 l + 2 ) ) = 2  1 − 1 n  R cu t . E. Linear Pr ogramming F o rmulation The above approach of restricting attention to aggregation along st ar graphs , gives in to a con venient Linear Programm ing (LP) formulati on. Consid er a compl ete g raph G . Let us define the rate region achie vable by star graphs in the following way ˜ R ach = { A λ : || λ || 1 = 1 } where A is a n × n ( n − 1 ) 2 matrix where a ie th entry is th e m inimum num ber of bi ts th at must be s ent along edge e und er tree aggregation scheme T i . The vector λ is t he relativ e weights assigned to the different trees. W e want to compare t he rate vectors achieved by t his schem e with the rate vectors that sati sfy the cut const raints. Let r ∈ R cu t be a given rate vector which s atisfies the cut constraint s of L emma 1. Now , we seek to find an achiev able rate vector th at is within a θ factor of r , and further , we want t o find the minimu m value of such a θ . Th is can formulated as a li near prog ram Min. θ s.t. A λ ≤ θ r || λ || 1 ≥ 1 λ ≥ 0, θ ≥ 0 Thus we can obtain the opti mal assi gnment λ ∗ and the opt imal factor θ ∗ . Note th at this assignment d epends o n the g iv en rate vector r ∈ R cu t . W e can also write simi lar such LPs for ot her cl asses of trees. V . C O N C L U D I N G R E M A R K S In t his paper , we have addressed some problems that arise i n the context of inform ation aggre- gation in sensor networks. Whi le t he general problem of devising optimal strategies for fun ction computation in wireless networks appears formidable, we have simplified it by abst racting out the medium access cont rol probl em and analyzing the problem o f funct ion com putation in graphs. W e hav e started with the problem of zero error fun ction com putation in directed graphs, and analyzed b oth worst case and av erage case metrics. For directed tree graphs , we have constructed optimal encoding schemes on each edge. This matches th e cut-set lower bound s. For general D A Gs, we have p rovided an out er bo und on the rate region, and an achiev able region based on aggregating along subtrees. Wh ile we hav e presented some examples where tree aggregation schemes are optimal, it remains to quant ify the sub-optimality of t ree aggregation schemes in general. W e hav e also addressed the com putation of sym metric Boolean funct ions i n und irected graphs, where all nodes want to compute the function . F or the case of com puting sum-threshold functions in undirected trees, we have derived th e opti mal strategy for each edge. The ac hiev able scheme for block computatio n in volves a layering o f transmiss ions that is reminiscent of message passing . Our framew ork can be generalized to handle functions of integer measurements which onl y depend o n the sum of the measurements. The extension t o general graphs is very i nteresting and appears significantly h arder . Howe ver , a cut-set lower bo und can be im mediately derived, and in some special cases one can show that subtree aggregation schemes provide a 2-OPT soluti on. Once again, it remains to stu dy the suboptimali ty of tree agg regation schemes in general graphs. R E F E R E N C E S [1] R . Ahlswede, N. Cai, S. R . Li, and R. W . Y eung. Network information flow . IEEE T ransa ctions on Information Theory , 46(4):1204– 1216, July 2000. [2] R . Appuswam y , M. Franceschetti, N. Karamchandani, and K. Zeger . 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