Space Information Flow

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1 Space Information Flow Zongpeng Li Dept. of Computer Science, University of Calgary and Institute of Network Coding, CUHK Chuan Wu Department of Computer Science The University of Hong Kong Abstract Network information flow emerged as a fertile research ground over a decade ago, with the advent of network coding that encourages information flows to be encoded when they meet within a data network. In this work, we propose the problem of space information flow the transmission of information flows in a geometric space, instead of in a fixed, existing network topology. In this new model, information flows are free to propagate along any trajectories in a space, and may be encoded wherever they meet. The goal is to minimize a natural network volume that represents the cost of constructing anetwork,whichcansupportend-to-endunicastandmulticast communication demands among terminals at known coordinates. Space information flow models the fundamental problem of information network design, which deserves renewed research attention for taking into account the unique encodability of information flows. We show that designing an information network is indeed different from designing a transportation network. In particular, an optimal multicast network does not necessarily correspond to a Steiner tree, as assumed in past literature. We prove both lower and upper bounds on the cost advantage of network coding (ratio of minimum network cost without coding over that with coding) in geometric multicast network design, and relate the potential benefit of network coding to the Steiner ratio, the cost ratio between a minimum Steiner tree and a minimum spanning tree in geometric spaces. I. INTRODUCTION Network coding [1] is a technique that encourages information flows to be encoded when they meet in the middle of a data network, besides being merely replicated and forwarded. Such a departure from the traditional store-andforward principle in data networking has proven effective in increasing network capacity, especially for one-to-many multicast applications [1], [2]. The field of network information flow studies data transmission with network coding in a given network topology. In this work, we propose to study space information flow, orthetransmissionofinformationflowsin ageometricspace.informationflowsmaypropagatealong any trajectories, may be replicated wherever desired, and may be encoded wherever they meet. Optional relay nodes can be inserted for free at any location. The goal is to minimize a natural bandwidth-distance sum-product ( network volume ), while sustaining end-to-end communication demands among terminals at known coordinates. Space information flow represents a natural and interesting theoretical problem of network coding in space.it constitutes a new research direction lying at the intersection of information theory, networking and geometry. While network information Fig. 1. Multicast information flow in a geometric space: network coding vs routing (Steiner tree). flow blends information theory with graph theory, space information flow further brings geometry into the mix. An important motivation of space information flow is that it models the fundamental problem of information network design, which is indeed different from transportation network design, in terms of both solution optimality and computational complexity. While the latter can nicely model the replicability of information flows, it ignores the encodability of them. Renewed research attention to the min-cost design of information networks is overdue, given that network coding has been proposed for more than a decade. For a given space information flow, viewing each link e as a link to be constructed, with bandwidth c e equal to the link flow rate f e,weobtainablueprintofaninformationnetworkfor satisfying the same end-to-end communication demand. The bandwidth-distance sum-product, or network volume, models the total cost of the network to be constructed, under the natural assumption that the cost of a link is proportional to its length as well as its bandwidth capacity. Fig. 1 [3] depicts an example of the multicast space information flow problem in a 2-D Euclidean space, where the design of an information network is different from that of a transportation network (Steiner tree). Three sources, s 1, s 2 and s 3 are in possess of common information to be multicast to the three receivers t 1, t 2 and t 3 at 1 bps. Thethreesourcesare located at the vertices of an equilateral triangle of unit edge length, and the three receivers lie at the middle points of the three edges of the triangle, respectively. Fig. 1(a) depicts a geometric multicast flow (multicast network blueprint) based on network coding, where each link has flow rate (capacity) of 0.5 bps. ThetotalcostinFig.1(a)is3 3/2. Fig.1(b) shows the optimal solution without network coding, based

2 2 on a minimum Steiner tree, with a cost of 7. The total length 7 is computed based on the facts that (i) at most two Steiner nodes need to be inserted in the optimal Steiner tree solution, and (ii) the three angles around each Steiner node are equal, at 120 [3]. The cost advantage of network coding in this case is 7/(3 3/2) = Weemphasizethatthis 1.8% gap, although small, reveals that space information flow has a fundamentally different structure, and likely a different computational complexity, than the geometric Steiner tree problem, and hence designing a min-cost information network is different from designing a min-cost transportation network. While the example in Fig. 1 involves multiple sources, it is possible to convert it into a single-source multicast problem in 2-D space, where network coding still makes a difference [3]. Space information flow and network information flow, while having subtle and important differences, are also closely related. In particular, connections can be established between the cost advantage in spaces with that in networks [4]. In asubsequentwork[5],ageometricframeworkisdesigned, based on space information flow, for studying the well known multiple unicast conjecture [6], [7] in network coding. Under this geometric framework, unified proofs to existing results on the conjecture, as well as a number of new results, are derived. Our goal in this work is to propose and draw attention to the space information flow problem. We present examples for motivating this new problem, which also lead to lower-bounds on the cost advantage of network coding in a geometric space. We further prove upper-bounds on the cost advantage, in a number of scenarios. Table. I summarizes the best known upper- and lower- bounds on the cost advantage of network coding in different models. Bounds in geometric spaces are proved in this work (multicast) and in a sibling work (multiple unicast, [4]). In particular, our results reveal an interesting connection between the cost advantage of network coding and the Steiner ratio in a geometric space. The potential benefit of network coding is upper-bounded by the cost ratio between aminimumsteinertreeandaminimumspanningtreeinthe same space. TABLE I BENEFITS OF NETWORK CODING:UPPER-&LOWER- BOUNDS directed undirected geometric networks networks spaces multiple conjectured: 1 [4] unicast Ω(n) [6], [7] 1 [6], [7] 2 [8] (2-D) multicast Ω( n) [2] 8/7 [9] The rest of the paper is organized as follows. We review related work in Sec. II, and define models and notations in Sec. III. We study lower-bound and upper-bounds of the cost advantage of network coding in geometric spaces, in Sec. IV, and Sec. V, respectively. In Sec. VI, we discuss the interesting fact that for optimal multicast in a 2-D space, a small finite field suffices for code construction. Sec. VII concludes the paper and proposes future research directions. II. RELATED WORK The coding advantage refers to the ratio of the max achievable throughput with network coding over that without network coding. The cost advantage of network coding refers to the ratio of the optimal cost of satisfying a communication demand without network coding over that with network coding. It is known that these two quantities satisfy a primaldual relation, and their max values are equal in undirected networks [2], [10], [11]. In a sibling work [4], we prove that space information flow represents a more fair paradigm for comparing network coding and routing, than network information flow in either directed or undirected networks. Examples in the existing literature for showing large benefits of network coding are often highly contrived, tailoring a network topology and orientation for network coding, then forcing routing to work in the same. In the space model, network coding and routing can each construct its own network and then select its own network orientation. Li and Li proved that for multicast in undirected networks, the coding advantage is upper-bounded by a small constant of 2 [8]. The result was later proved using two other methods [2], [10]. In this work, we prove a much smaller upper-bound of on the coding advantage in space information flow, for a number of multicast scenarios. In 2004, Li and Li [6] and Harvey et al. [7] conjectured that for multiple unicast sessions in an undirected network, network coding is equivalent to routing. Despite a series of research efforts devoted to the problem [12] [14], rather limited progresses have been made towards settling this fundamental problem in network information flow. In a sibling work [4], we prove the conjecture in its space/geometric version, partially verifying the conjecture. In a subsequent work, Xiahou et al. [5] apply space information flow as a tool to design a framework for analyzing the conjecture. Both Euclidean and non-euclidean spaces are considered, and unified proofs are obtained for both existing and new results on the conjecture. Asketchofapossibleprooftotheoriginalconjecturebased on the framework is also discussed. In another subsequent work, Yin et al. [3] recently studied the space information flow problem as well, focusing on optimal multicast network design in an Euclidean space. They derive a number of properties that an optimal multicast network must satisfy. It is proved that for the most basic case of three multicast terminals in an Euclidean space, network coding does not make a difference. Applying a multicast flow decomposition technique [15], they further show that for n multicast terminals and multicast throughput h, thenumberof relay nodes required for an optimal solution is upper-bounded by n h (n+h 3 (n 1) 2 2) for general h, andisupper-bounded by (2n 3)(2n 2) when h =2. III. MODELS AND NOTATIONS A. Network Information Flow We represent an existing network using a graph G =(V,E). Avectorc Z E + stores capacities of links in E. HereZ + is

3 3 the set of positive integers. Another vector w Q E + represents the distance or cost of links in E, andw e can be interpreted as the cost of routing a unit flow through link e. HereQ + represents the set of positive rational numbers. For min-cost multicast of h unit flows from a source s V to a set of receivers T V,thegoalistominimizeaggregated multicast cost, while sustaining a target multicast throughput r. Without network coding, this translates to the problem of mincost Steiner tree packing in graphs, and is NP-hard. Without loss of generality, we can assume that link capacities in c are scaled, so that the multicast throughput h is an integer. With network coding, the min-cost multicast problem again contains two components, a multicast flow vector f Q E +,and acodeassignmenttothelinkflowsinf. Themin-costflow vector f can be computed through solving a linear program, by exploiting the celebrated multicast rate feasibility condition with network coding [9], [16]. The goal is to support the target multicast rate r while minimizing the total multicast flow cost e w ef e. B. Space Information Flow In the space information flow paradigm, we are given a set of terminal nodes, with (multiple) unicast or multicast communication demands. The space we consider in this work is a h-d Euclidean space, for h 1. Anodeu has coordinate (x 1,u,x 2,u,...,x h,u ). The Euclidean distance between two nodes u and v is ( h uv = i=1 (x i,u x i,v ) 2 ) 1/2 Given an information flow vector f embedded in an Euclidean space, a network G can be induced, over the same nodes and links as in f, byviewingf e as the capacity of e. The distance of e is denoted as e. Thecostoff is then e e f e.thisreflectsthenaturalrulethatthelongerand the wider a communication link is, the more expensive it is. For the sake of cost minimization, apparently, only straight line segments need to be constructed for links in f. Given two vectors p and q, p q = p q cos θ is the inner product of p and q,whereθ is the angle between p and q. IV. LOWER BOUND ON COST ADVANTAGE OF NETWORK CODING Multicast with network coding in a fixed network has been extensively studied. The optimal multicast flow can be computed in polynomial time, based on the following characterization of multicast rate feasibility with network coding: in a directed network, a multicast rate r is feasible with network coding if and only if it is feasible as a unicast rate to each receiver independently [1], [17]. In contrast, the optimal multicast scheme without coding corresponds to the problem of Steiner tree in graphs, and is both NP-hard and APX-hard [18]. The multicast coding advantage is proven to be upperbounded by 2 in undirected networks [2], [8], [10]. sources receivers Fig. 2. Multicast in a 2-D space, with cost advantage of network coding > 1. Terminals(inblack)ontheinnercirclearesources.Terminals on the outer circle are receivers. Solution with network coding shown. For space information flow, a multicast solution without network coding corresponds to the geometric Steiner tree problem, which is still NP-hard, but has a Polynomial-Time Approximation Scheme (PTAS). The complexity of the optimal solution with network coding is unknown. The example in Fig. 1 shows that network coding does make a difference for optimal multicast in spaces. Therefore, it is still possible for the problem of information network design to be efficiently solvable. We now present another example where the cost advantage is larger than in Fig. 1, and obtain the largest value known so far. Theorem 4.1. The max cost advantage for space information flow is lower-bounded by Proof: We provide a constructive proof to the theorem, by designing a (multi-source) multicast problem in a 2-D Euclidean space, for which the cost of a network coding based solution is strictly smaller than the cost of a minimum geometric Steiner tree/forest. As illustrated in Fig. 2, consider identical information available at multiple sources, to be multicast to multiple receivers. In the figure, black nodes on the inner circle are sources, and black nodes on the outer circle are receivers. The solid lines depict a network coding based solution, with flow rate of 0.5 on each link, achieving a multicast throughput of 1. Each relay node (inserted into the 2-D space by the network coding solution) is at the centre of a building block of a equilateral triangle shape. When the total number of such building blocks is odd, such a multicast flow is infeasible without network coding. At least one of the receivers will receive two identical copies of the same flow. Furthermore, without network coding, any routing solution will have a cost strictly higher than that of the coding solution. Note that in a minimum geometric Steiner tree (forest), every relay node must be adjacent to exactly three edges, forming three angles of 120 each [19]. Asolutionthatisvalidwithoutcodingistomodifythe multicast flow in one of the equilateral triangles, replacing the relay-enabled star with two direct links from the source to the two receivers. As a result, the total cost of the multicast flow

4 4 increases from 3 2 k to 3 2 (k 1) + 1. Herek is the number of receivers. The cost advantage in such a pattern is: 3 2 (k 1) k In order for the example to be valid, i.e., forthegeometric structure to be feasible, k =3or k =5is not sufficient. We need k =7at least, and the above cost advantage evaluates to = After it was proved that network coding is equivalent to routing for multiple unicast sessions in a geometric space [4], the example in Fig. 2 became the first example discovered that demonstrates network coding does make a difference for multicast. The smaller example in Fig. 1 came afterwards. The intuition behind the construction of the network pattern in Fig. 2 is explained below. Aplanarnetworkisonethatcanbeembeddedintoaplane without links crossing each other. A multicast flow in 2-D space can always be made planar, by inserting a relay node at where edges cross each other, without affecting the feasibility or the optimality of the solution. We therefore focus on the outer boundary of a planar network. First, due to the optimality of the multicast solution, a relay node has degree 3 with three equal angles at 120.Consequently,theboundaryisunlikely to contain a link between two relay nodes, since that conflicts with planarity. Second, having a link between two terminal nodes on the boundary does not help create a gap between the coding solution and the no-coding solution, since network coding is in general unnecessary for link flows that enter a receiver [20]. Therefore we decided to include alternating relay and receiver nodes on the outer-boundary. Further given the 120 condition at the relays, the shape of the boundary is determined. The final step is to include a number of sources on the inside face. V. UPPER-BOUNDS ON COST ADVANTAGE OF NETWORK CODING A. Multicasting Two Information Flows (h =2) In network information flow, the largest coding advantage observed both in unlimited-size networks [10] and in small networks [9], [11] are based on multicast instances of transmitting two information flows (h = 2) from the sender to all receivers. The two-flow case represents a basic yet fundamental version of multicast. In terms of the necessary base field size, h =2already requires an unbounded field size that grows as the number of receivers does [11]. Recent network coding literature has witnessed a batch of work that focus on the study of two information flows in the unicast setting [21] [23]. We prove an upper-bound of on the cost advantage for multicasting two flows in an Euclidean space, assuming a well accepted conjecture in the field of Steiner trees. In the literature of geometric Steiner trees, the Steiner ratio is the max ratio between the minimum Steiner tree cost and Fig. 3. Decomposing a multicast flow into connected components, based on the code assignment. A planar multicast network example from [24] isshown, with a multicast flow of h =2from the source to 5 receivers. The multicast flow is decomposed into four different components, based on code assignment, such that each component is connected and consists of links transmitting the same information flow only. Each component is shown with a distinct color. the minimum spanning tree cost, for a given set of terminals to be connected. Theorem 5.1. For multicast in a geometric space with h =2, the cost advantage of network coding is upper-bounded by the Steiner ratio. Proof: Let f be the underlying flow vector of a min-cost network coding solution for the two-flow multicast problem in a geometric space. A key step in our proof is to decompose f into a disjoint set of flow components based on flow content, as described below, and illustrated in Fig. 3. Consider a valid code construction over a certain finite field GF (q) for f, achievingmulticastthroughputh =2.We decompose f into a disjoint set of maximally connected flow components, such that link flows within each component are assigned the same information flow for transmission. Such same-content based multicast flow decomposition was first proposed by Fragouli and Soljanin [15], and then utilized in a number of work, e.g., byelrouayhebet al. [25] for studying minimal multicast networks, by Xiahou et al. [24] for studying planar networks, and by Yin et al. [3] for analyzing the number of relay nodes required for optimal multicast in spaces. By the optimality of f, weknowthateachsuchcomponentisa minimum Steiner tree. The root node and leaf nodes of the tree component each receives two linearly independent information flows, since they are either a sender/receiver, or a relay node that can generate a different flow from what it receives. We replace each tree component using a local spanning tree connecting the same root and leaf nodes. By doing so, each tree component has a cost inflation between 1 and the Steiner ratio. After the replacement, every node in the resulting multicast flow can be viewed as a sender or receiver, yielding a broadcast flow f of two unit throughput. The coding advantage is 1 for broadcast [26], and such a broadcast flow can be decomposed into two broadcast trees each with unit throughput, τ 1,τ 2. Let τ denote the total cost of links in a tree τ. Wehave τ 1 + τ 2 = e f e e. Without

5 5 loss of generality, assume τ 1 τ 2. Wecanuseτ 1 for transmitting two unit flows, with total cost at most e f e e. We conclude that the cost advantage of network coding is upper-bounded by the Steiner ratio. Assuming the generally accepted Gilbert-Pollak Conjecture [19], which claims that the Steiner ratio in 2-D Euclidean space is 2/ 3=1.115, weobtainthefollowingtheorem: Theorem 5.2. For multicast in 2-D space with h =2,the cost advantage of network coding is upper-bounded by (conditional on the Gilbert-Pollak Conjecture). B. Bipartite Multicast Flows in Spaces In network information flow, many known example multicast instances where the coding advantage is larger than 1 have atopologyisomorphicorsimilartoacombinationnetwork[2], [11]. A more general type of multicast networks than the 3- layer combination networks are bipartite multicast networks, which are multicast networks without relay-to-relay links. We prove that if the optimal multicast flow with network coding has a bipartite structure, then the cost advantage is upperbounded by the Steiner ratio, and is unconditionally upperbounded by in 2-D. This result is general in that it holds for any multicast throughput h. x x+y s y x+2y x y x+2y Proof: This is due to the fact that in the replacement of a Steiner tree with a spanning tree in the proof of Theorem 5.3, each Steiner tree is a simple star with one relay node only. It is known that in this case, the Steiner ratio is upper-bounded by in a 2-D Euclidean space. We have applied the information flow decomposition technique for proving an upper-bound on the cost advantage, both for multicasting two flows and for bipartite multicast flows. In future work, it will be interesting to further exploit the potential of this approach, for cases where h>2. Itappears that a new technique is needed for such an extension. The challenge lies in the fact that when h > 2, therootofa component resulting from the multicast flow decomposition receives at least 2 but not necessarily h distinct information flows, and hence is not always able to generate an arbitrary combination of the h source flows. VI. THE SUFFICIENCY OF SMALL FINITE FIELDS In network information flow, the best known lower-bound on the sufficient base field size for multicast network coding is the number of receivers 1. As the number of multicast receiver grows, so does the required field size, over which all coding solutions are carried out, in order to guarantee linear independence of information flows received at each receiver [28]. Furthermore, it is also known that multicasting two flows in a network already requires unbounded field size in general. For example, in the 3-layer combination network C n,2 with a single sender at layer 1, n relays at layer 2 and ( n 2) receivers at layer 3, therequiredfieldsizeisatleastn 1 [11], [28]. x+y C A C A Fig. 4. An example of a bipartite multicast network from [24], withunit capacity links and h =2.Thereisnodirectlinkbetweenapairofrelay nodes. The multicast flow decomposition yields four star components. Theorem 5.3. For an embedding of a bipartite multicast flow in a geometric space, the cost advantage of network coding is upper-bounded by the Steiner ratio. Proof: Consider the multicast flow in the optimal network coding solution, along with a valid code assignment. Similar to the proof of Theorem 5.2, we decompose the multicast flow into components, based on the same information flow criterion. Due to the bipartite nature of the multicast flow, each component in the decomposition has a star topology, with exactly one relay node in the middle, as shown in Fig. 4. Replacing such a star with the minimum spanning tree connecting the same terminal nodes, we obtain a broadcast flow. The rest of the proof is the same as that of Theorem 5.2. Theorem 5.4. For an embedding of a bipartite multicast flow in a 2-D Euclidean space, the cost advantage of network coding is (unconditionally) upper-bounded by B D Fig. 5. In a 2-D Euclidean space, if two link flows (A B and C D) intersect in the plane, a new relay node (E) can be introduced at the crossing point, such that the two original crossing links become four links without crossings. Links A E and E B transmit the same information flow as on link A B. SimilarforlinkC D. Originalcodeassignmentremainsvalid, and total flow cost remains unchanged. In contrast, coding over a small field GF (3) is sufficient for multicasting to any number of receivers, for space information flow in 2-D. As shown in Fig. 5, when designing a multicast network in 2-D, we can always make the network planar, by inserting relay nodes to avoid crossing edges. Given a planar network topology, one may combine the multicast flow decomposition technique with four-coloring a planar graph, and prove that coding over GF (3) is sufficient for h =2[24], [25]. Our recent study on network coding in planar networks [24] further reveals that, coding over GF (3) is both necessary and sufficient in both general planar graphs and bipartite planar 1 The latest result is somewhat better, but still equals the number of receivers in the worst case [27]. B E D

6 6 graphs (for necessity, see examples in Fig. 3 and Fig. 4), and such code assignment over GF (3) can be accomplished in linear time. The investigation also leads to a conjecture that coding over a small field is sufficient for any h in planar networks [24]. To summarize, coding over a small field is sufficient for h =2in a 2-D space, and is likely so for h>2 as well, which is being investigated. VII. CONCLUSION AND FUTURE DIRECTIONS A new research direction, space information flow, was proposed in this paper. It studies the optimal transmission of information flows for meeting communication demands in a geometric space. Space information flow models the design of a min-cost information network. We showed that that the optimal multicast network design is different with and without network coding. Although the difference in cost is small, it opens up the possibility for optimal information network design to have efficient, polynomial-time solutions. Upper-bounds and lower bounds were proved on the cost advantage for multicast in a space, which are different from the corresponding bounds in network information flow. This work leaves a number of problems open on space information flow. The complexity of computing an optimal multicast flow in a geometric space is still unknown. A general upper-bound on the multicast cost advantage in a space, for any multicast rate h, is desired. It is also interesting to investigate the possibility of utilizing the multiple unicast result for space information flow to prove the multiple unicast conjecture in network coding [4], [6]. Throughout this paper, we have focused on a version of space information flow that assumes a wireline network is to be constructed. One may also define and study a wireless version of space information flow, for modeling the min-cost design of a wireless network. In wireline information network design, we can place relay nodes anywhere in a space for free, and lay communication cables to connect them. In the wireless version, wireless nodes with omnidirectional antennas may be placed at anywhere in a space, and two nodes may communicate if they are sufficiently close. Total network cost depends on the length of cables in the wireline case, and depends on the number of nodes deployed in the wireless case. The cost advantage in the wireless case could be much larger than that in the wireline case, since the wireless broadcast advantage is particularly beneficial for transmitting encoded flows that may help multiple receivers. REFERENCES [1] R. Ahlswede, N. Cai, S. R. Li, and R. W. Yeung, Network Information Flow, IEEE Transactions on Information Theory, vol.46,no.4,pp , July [2] C. Fragouli and E. Soljanin, Network Coding Fundamentals, Now Publishers, [3] X. Yin, Y. Wang, X. Wang, X. Xue, and Z. 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