A Recursive Partitioning Algorithm for Space Information Flow

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1 A Recursive Partitioning Algorithm for Space Information Flow Jiaqing Huang Department of Electronics and Information Engineering Huazhong University of Science and Technology, China Zongpeng Li Department of Computer Science University of Calgary, Canada Abstract Space Information Flow (SIF) is a new research paradigm that studies network coding in a geometric space, which is different with Network Information Flow (NIF) that studies network coding in a graph. One of the key open problems at the core of SIF is to design an algorithm that computes optimal SIF solutions. A new heuristic SIF algorithm based on nonuniform recursive space partitioning is proposed in this work, for computing SIF for any density distribution of given terminal nodes in 2-D Euclidean space. Simulation results show that the new algorithm has low computational complexity and converges to optimal solutions promptly. I. INTRODUCTION Space Information Flow (SIF) [1][2], proposed in 2011, studies network coding in a geometric space. Departing from Network Information Flow (NIF) [3] that studies network coding in a graph, SIF allows additional relay nodes to be added for helping the communication (See Figure 1). SIF is also different from routing in space, e.g. Euclidean Steiner Minimal Tree () [4]. The first SIF single multicast example Pentagram [5], as illustrated in Figure 2 demonstrates that the performance of network coding in space (SIF) can be strictly better than that of routing in space (e.g. ). In other words, the cost advantage [6] is strictly bigger than 1, where the cost advantage is defined as the ratio of minimum cost necessary for achieving a target throughput by routing over that of network coding. Fig. 1. Graph Space Butterfly Pentagram Graph Space SIF studies network coding in geometric space The Pentagram example [5] is as follows. Figure 2(a) shows six terminal nodes in a 2-D Euclidean space, where five nodes (A to E) form a regular pentagon centered at node F. The radius of the circumscribed circle is 1. The communication demand is for the source F to multicast messages to the sinks (A to E). We will compare the cost between network coding and routing. Here we define cost in terms of the total number of bitmeters of information transmission required for achieving a 1 B C A A A a a+b 1 a 1 5 a+b E B 2 E B a+b E b a b F F 2 b F a b 4 3 b 3 b a a D C D C D (a) (b) (c) Fig. 2. The multicast example of SIF Pentagram (cost advantage ). (a) six terminal nodes in 2-D Euclidean space; (b) using optimal (cost=4.6400/bit); (c) using network coding in space (cost=4.5677/bit). bit end-to-end throughput. With routing (Figure 2(b)), optimal can be computed [7][8] and the cost is /bit. With network coding (Figure 2(c)), we can introduce five additional relay nodes (nodes 1 to 5) each adjacent to three links that form three 120 angles. The total distance is , while every sink receives 2 bits. The normalized cost is /2=4.5677/bit<4.6400/bit (optimal ). The cost advantage is / >1. Despite the seemingly small difference in cost, this example reveals the fact that network coding in space is a fundamentally different problem than routing in space, with a different combinatorial structure and flavor, and likely different computational complexity. As for the applications of SIF, a potential scenario is the planning of wireless sensors in space. For routing in space, Gilbert et al. [4] studied properties of optimal. The computational complexity of is known to be NP-Hard [9]. There exist Polynomial Time Approximation Schemes (PTAS) [10] for that adopt the geometric partitioning approach, similar to the first phase of our proposed SIF algorithm in this work. Along the new direction of SIF, Li and Wu [2] studied multiple-unicast network coding in space. Yin et al. [11] proved a number of properties of multicast network coding in 2-D Euclidean space. Xiahou et al. [12][13] proposed a unified geometric framework in space to investigate the multiple-unicast network coding conjecture in undirected graphs. Huang et al. [5] proposed a heuristic algorithm for SIF, for the case of single multicast. However, algorithm design targeting optimal multicast SIF remains an open problem, which appears particularly challenging we need to compute not only the number of relay nodes, /14/$ IEEE 1460

2 the exact geometric location of each relay node, but also the best way of interconnecting them, and then the best multicast flow over such an interconnection topology. Such optimal SIF algorithm design is currently one of the most important open problems in the paradigm of SIF. Our contribution in this work is to propose a heuristic SIF algorithm based on recursive non-uniform partitioning of the host space, which can adapt to any density distribution of terminal nodes in 2-D Euclidean space. The algorithm has fast convergence and low computational overhead. The main differences between our new SIF algorithm and an existing SIF algorithm [5] include the follows. (1) The previous SIF algorithm rely on uniform partitioning. It suffers from high complexity for clustered terminals (when terminal density in the input is highly non-uniform). An extreme example appears when the distance between two terminal nodes approaches zero, the partitioning parameter q in [5] approaches infinity, making the algorithm infeasible in practice; (2) Our new SIF algorithm adopts a retention mechanism in terms of Linear Programming (LP) input. That is, the balanced relay nodes in the last round are retained and fed as input into the LP computation together with the candidate relay nodes in the current round. The candidate relay nodes are generated from the non-uniform partitioning. And the balanced relay nodes are obtained by means of equilibrium methods of Phase II and represent the local optimization to some extent. Thus, retaining the local optimization can subsequently speedup the convergence; (3) The candidate relay nodes outside the convex hull determined by the given terminal nodes are removed in order to decrease the LP computation. Since the relay nodes in an optimal solution of SIF should be inside the convex hull according to the SIF property [11]. In the rest of the paper, Section II presents the problem formation and definitions. Section III elaborates the detailed steps of the new SIF algorithm. Section IV presents simulation studies. The final section concludes the paper. II. FORMATION AND DEFINITIONS We focus on the multicast SIF problem, which studies mincost multicast network coding in 2-D Euclidean space. The input includes N terminal nodes in a 2-D Euclidean space and a multicast session from one source to a number of sinks. The goal is to compute a min-cost transmission scheme using SIF, which permits the introduction of extra nodes (a set of relay nodes). Cost is defined as e w(e)f(e), where f(e) is the information flow rate of a link e in space, and w(e) is the weight of link e, which equals the Euclidean distance e of e [1][2]. When a SIF scheme is computed, a network G can be induced. A SIF solution is an embedding of a multicast network G into a 2-D Euclidean space. When f(e) required to be always 1, SIF degrades into the problem. A solution to SIF contains two dimensions: the topology of a solution multicast networks/flow and positions of relay nodes inserted into the topology. The former further includes a flow routing scheme that specifies the flow rate on each link. A bounding box of a set of given terminal nodes is the smallest axis-aligned rectangle enclosing them [10]. III. NEW SIF ALGORITHM Our SIF algorithm includes two phases: Phase I for determining the topology of SIF and Phase II for determining the positions of relay nodes. A. Phase I: Computing Optimal SIF Topology Phase I computes the topology, including the number of relay nodes and their topological connection with the terminal nodes, as well as the flow rates on the connection links. Two key techniques are employed in this phase: non-uniform partitioning and Linear Programming (LP). The former helps obtain the candidate relay nodes, which are the centers of a number of rectangular cells from the non-uniform partitioning. The latter solves the partitioning model in Equation (1) to compute the minimum cost as well as the flow rates. Non-uniform partitioning represents the first significant difference from a previous SIF algorithm [5] that resorts to uniform partitioning. Under non-uniform partitioning, through every terminal node, a vertical line and a horizontal line are drawn; a bounding box and a number of sub-rectangles that of different sizes are constructed; every sub-rectangle is recursively partitioned into q q cells. Non-uniform partitioning is able to deal with any density distribution of terminal nodes, especially for non-uniform distributions. Take nine clustering terminals nodes for instance (Figure 3(a)), if uniform partitioning [5] is adopted, it takes at least five rounds to compute the most sufficient candidate relay nodes. As shown in Figure 3(b), the fourth round (i.e. q q=4 4) does not suffice because there is no candidate relay node in the convex hull determined by the three upside terminal nodes. With non-uniform partitioning (Figure 3(c) and 3(d)), the distribution of the candidate relay nodes is in accord with the terminal nodes, which can speedup the convergence of the algorithm. The second key difference from the previous algorithm [5] is to introduce a retention mechanism. The balanced relay nodes after Phase II in the last round are retained and further fed into the LP computation together with the candidate relay nodes in the current round. This retention mechanism can speedup the convergence since the balanced relay nodes by means of equilibrium methods of Phase II represent local optimization to a certain extent. Furthermore, the minimum cost will decrease monotonically. Due to the retention mechanism, there is no need to distinguish the two LP models (partitioning model and OPT model) as in the previous algorithm [5]. We adopt the partitioning LP model here. Minimize cost q = uv A w( uv)f( uv) Subject to : v V (u) f i( vu) = v V (u) f i( uv) i, u f i ( Ti S)=r i f i ( uv) f( uv) i, (1) uv f( uv) 0,f i ( uv) 0 i, uv 1461

3 distance of the complete graph consisting of the terminal nodes and the candidate relay nodes). The detailed steps of Phase I are in Algorithm 1. Fig. 3. Example of non-uniform partitioning vs. uniform partitioning. (a) nine given clustering terminal nodes in 2-D Euclidean space; (b) 4 4 uniform partitioning; (c) non-uniform partitioning: through every terminal node, a vertical line and a horizontal line are drawn to obtain a bounding box and a number of sub-rectangles; (d) non-uniform partitioning: every sub-rectangle is partitioned into q q (e.g. q=2) cells. The centers of the cells inside the convex hull (in red) determined by given terminal nodes are taken as the candidate relay nodes. The solid nodes are terminals, hollow nodes are relays. The partitioning model (Equation (1)) is based on an undirected complete graph generated from non-uniform partitioning. The graph is denoted as G =(V,E), where V is the set of N terminal nodes and the candidate relay nodes as well as the balanced relay nodes, E is the set of undirected links. Due to the bi-directed possibilities of transmission in space, we make links bi-directed and denote a set of directed links as A = { uv, vu uv E}. In the LP objective function, the decision variable f( uv) represents the combined effective flow rate on a link uv. Coefficient w( uv) represents the Euclidean distance uv (= vu = uv ). In the LP constraints, f i ( uv) represents the rate of information flow S T i on a link uv. This kinds of information flows are conceptual in that they share instead of compete for available bandwidth of a link [14]. f( uv) equals to the maximum among all f i ( uv). V (u) and V (u) denote upstream and downstream adjacent set of u in V, respectively. r is a multicast rate from the source S to each sink T i. We assume there is a conceptual link from each sink T i back to the source S with the rate r, for concise representation of flow conservation constraints [14]. The third difference from the previous algorithm [5] is to remove the unnecessary relay nodes outside the convex hull determined by the given terminal nodes, for reduced LP computation overhead (the LP requires computing all-pairs Algorithm 1 SIF Algorithm (Phase I) Require: N terminal nodes, a multicast session Ensure: Output resulting relay nodes 1: Initialization: MINCOST I = MINCOST II =+, partitioning q=2; 2: Compute a convex hull for the N given terminal nodes; 3: Crossing every terminal node, draw a vertical line and a horizontal line to obtain a number of sub-rectangles; 4: Partition every sub-rectangle uniformly into q q cells and obtain centers of all cells; The upperbound number of the centers is (N 1) 2 q 2 ; 5: Choose centers that are inside the convex hull as the candidate relay nodes; 6: Construct a complete graph with N terminal nodes and the candidate relay nodes of the current round as well as the balanced relay nodes of the last round; 7: Solve the partitioning LP model based on the complete graph and output the resulting relay nodes; 8: if cost q <MINCOST I then 9: MINCOST I = cost q 10: end if 11: if Flow rates of all resulting relay nodes == 0 then 12: output MINCOST I and stop. 13: end if B. Phase II: Approach Optimal SIF Positions The aim of Phase II is to refining the locations of relays from Phase I, by moving them to balanced positions that satisfy necessary properties of optimal SIF stable at relay [11]. There are two alternative equilibrium methods: the iterative method [5] and the analytic geometric method. We focus on the latter in this work, to compute the exact positions (i.e. coordinates) of the resulting relay nodes from Phase I. There is a 120 condition: if a relay node has three adjacent links each with equal flow rate, then the balanced position of the relay node should result in three 120 angles among its three adjacent links. The analytic geometry method exploits this fact to compute the exact coordinates of the balanced relay nodes. More specifically, we can apply inner product of two vectors to establish equations. Suppose a relay node (x, y) is connected with three adjacent terminal nodes (x 1,y 1 ),(x 2,y 2 ) and (x 3,y 3 ) (See Figure 4(a)). Two unknown variables (i.e x and y) can be solved by two equations: (x 1 x)(x 2 x)+(y 1 y)(y 2 y) (x1 x) 2 +(y 2 = 1 y) (x 2 x) 2 +(y cos120 2 y) 2 (x 1 x)(x 3 x)+(y 1 y)(y 3 y) (x1 x) 2 +(y 1 y) 2 = (2) (x 3 x) 2 +(y cos120 3 y) 2 The number of equations will vary according to the following different cases. If a relay node (x, y) has one adjacent relay node (x,y ) and two adjacent terminal nodes (x 1,y 1 ) and (x 2,y 2 ), there should be four equations with four unknown 1462

4 (x 1, y 1 ) (x 2, y 2 ) (x, y) (a) (x 3, y 3 ) (x 1, y 1 ) (x 4, y 4 ) (x 2, y 2 ) (x, y) (b) (x', y') (x 3, y 3 ) Fig. 4. The analytic geometry method for equilibrium in Phase II. (a) one relay node with three adjacent terminal nodes; (b) two relay nodes that are connected adjacently and four adjacent terminal nodes. variables, because x and y have another two equations. Suppose the relay node (x,y ) has two adjacent terminal nodes (x 3,y 3 ) and (x 3,y 4 ) (See Figure 4(b)). The four unknown variables (i.e. x, y, x and y ) can be solved by four equations. (x 1 x)(x 2 x)+(y 1 y)(y 2 y) (x1 x) 2 +(y 2 = 1 y) (x 2 x) 2 +(y cos120 2 y) 2 (x 1 x)(x x)+(y 1 y)(y y) (x1 x) 2 +(y 2 = 1 y) (x x) 2 +(y y) cos120 2 (x 1 x)/(y 1 y) =(x 3 x )/(y 3 y ) (x 2 x)/(y 2 y) =(x 4 x )/(y 4 y ) The latter two equations come from the two parallel vectors, for instance, the last equation is due to two parallel vectors (x 2 x, y 2 y) and (x 4 x,y 4 y ). This can reduce the computation overhead. If a relay node has two adjacent relay nodes and one adjacent terminal node, there should be six equations with six unknown variables. For each extra relay node (two unknown coordinates), two extra equations arise. The number of unknown variables and the number of equations are always equal. If a resulting relay node does not have three equal-flow links, the new algorithm adopts the iterative method for equilibrium in our previous algorithm [5], i.e. an iterative process that gradually settles down the SIF topology into a balanced state. In detail, execute repeatedly while the resultant force F i of a resulting relay node R i does not approach zero (i.e. less than a small positive rational number ɛ 1 ), the resulting relay node R i will be moved with a stepsize Δ F i. As the method iterates, the stepsize decreases, for the fine-tuning to 1 gradually converge. An example stepsize sequence is 2 i+3, where i is the iteration number. The detailed steps of Phase II are shown in Algorithm 2. IV. SIMULATIONS RESULTS Our simulations are based on C++ programs and use glpk 4.48 for solving LPs, with the help of MATLAB for solving equations. The optimal is computed by GeoSteiner 3.1 that implements an exact algorithm [7]. A. Case of Clustering Terminal Nodes We compare non-uniform and uniform partitioning with regard to the clustering case mentioned in Figure 3(a). Under (3) Algorithm 2 SIF Algorithm (Phase II) Require: resulting relay nodes of Phase I Ensure: Output a SIF solution 1: if Resulting relay nodes satisfy 120 condition then 2: Apply the analytic geometric method for equilibrium: solve equations to obtain the exact coordinates of the balanced relay nodes according to Section III-B; 3: else 4: Apply the iterative method for equilibrium: 5: while Resultant force >ɛ 1 for some relay nodes do 6: while i :1 Relay node do 7: Compute resultant force F i of relay node R i ; 8: if Resultant force F i >ɛ 1 then 9: Fi = F i +Δ F i 10: end if 11: end while 12: end while 13: end if 14: Construct the second complete graph with N terminal nodes and the balanced relay nodes; 15: Solve partitioning LP model based on the second complete graph; 16: if cost q <MINCOST II then 17: MINCOST II = cost q 18: end if 19: if 0 MINCOST I - MINCOST II <ɛ 2 then 20: Output MINCOST II and stop. 21: else 22: q = q +Δq (e.g. Δq=1); Goto step 3 of Algorithm 1; 23: end if Fig. 5. Uniform partitioning N= Fig. 6. Non-uniform partitioning N=9 uniform partitioning, the relationship between min-cost of LP and q is shown in Figure 5. The overall trend of mincost from the partitioning LP model of Phase I approaches the gradually, with some fluctuations that lead to slow convergence. If the non-uniform partitioning is applied, the relationship between min-cost of LP and q is shown in Figure 6. It shows the new algorithm based on non-uniform partitioning can achieve an optimal solution faster than the previous algorithm based on uniform partitioning. That is because of the adoption of the retention mechanism that the minimum cost decreases monotonically. The SIF results of the clustering case are shown in Figure 7. (1) In the first round, the candidate relay nodes are obtained 1463

5 from non-uniform partitioning (q=2) as shown in Figure 7(a). Nodes outside the convex hull are removed for reducing the LP computation overhead. The resulting relay nodes from the LP computation of Phase I are shown in Figure 7(b). After Phase II, the balanced relay nodes are obtained (See Figure 7(c)). (2) In the second round, the candidate relay nodes from onuniform partitioning (q=3) are shown in Figure 7(d), as well as retained balanced relay nodes of Phase II from the first round. The resulting relay nodes from the LP computation of Phase I are shown in Figure 7(e). After Phase II, the balanced relay nodes are obtained (See Figure 7(f)). After the second round, the min-cost does not change. In this case, SIF degrades to. (a) (b) (c) Fig. 8. Uniform partitioning N= Fig. 9. Non-uniform partitioning N=6 10(a). The resulting relay nodes from the LP computation of Phase I are shown in Figure 10(b). After Phase II, the balanced relay nodes are obtained (See Figure 10(c)). (2) In the second round, the candidate relay nodes from the nonuniform partitioning (q=3) are shown in Figure 10(d), as well as retained balanced relay node of Phase II from the first round. The resulting relay nodes from the LP computation of Phase I are shown in Figure 10(e). After Phase II, the balanced relay nodes are obtained (See Figure 10(f)). After the second round, the min-cost does not change. It is shown from Figure 9 that the SIF algorithm can achieve a smaller value than the optimal. In this case, the SIF can be strictly better than the optimal. (d) (e) (f) Fig. 7. SIF results of Clustering case when N=9. (a) 1 st round: candidate relay nodes from non-uniform partitioning (q=2) of Phase I; (b) 1 st round: resulting relay nodes from the LP computation of Phase I; (c) 1 st round: balanced relay nodes of Phase II; (d) 2 nd round: candidate relay nodes from non-uniform partitioning (q=3) with retained balanced relay nodes; (e) 2 nd round: resulting relay nodes from the LP computation of Phase I; (f) 2 nd round: balanced relay nodes of Phase II. (a) (b) (c) B. The Pentagram Network We compare non-uniform and uniform partitioning with regard to the Pentagram network (Figure 2(a)) where SIF is strictly better than the optimal. Under uniform partitioning, the relationship between min-cost of LP and q is shown in Figure 8. There is a downward fluctuating trend in the min-cost from the partitioning LP model of Phase I. In addition, the min-cost of SIF can be smaller than the after q=6. Under non-uniform partitioning, the relationship between min-cost of LP and q is shown in Figure 9. In the first round, the SIF algorithm can achieve a value that is smaller than. In the second round, the SIF algorithm can achieve a lower value, which does not change in the following rounds. Thus, the new SIF algorithm can achieve an optimal value faster than the previous algorithm. The SIF results of the Pentagram case are shown in Figure 10. (1) In the first round, the candidate relay nodes are obtained from the non-uniform partitioning (q=2) as shown in Figure (d) (e) (f) Fig. 10. SIF results of Pentagram network when N=6. (a) 1 st round: candidate relay nodes from non-uniform partitioning (q=2) of Phase I; (b) 1 st round: resulting relay nodes from the LP computation of Phase I; (c) 1 st round: balanced relay nodes of Phase II; (d) 2 nd round: candidate relay nodes from non-uniform partitioning (q=3) with retained balanced relay nodes; (e) 2 nd round: resulting relay nodes from the LP computation of Phase I; (f) 2 nd round: balanced relay nodes of Phase II. C. The Ladder Network With regard to a Ladder [15] network, the optimal can be computed by GeoSteiner (See Figure 11(a)). Under uniform partitioning, it takes more than ten rounds to approach the ; under non-uniform partitioning, the solution arrives at the optimal in the third round (q=4) (Figure 11(b)). After that, the min-cost does not change. In this case, the SIF 1464

6 degrades to. The new SIF algorithm can achieve an optimal faster than the previous algorithm in [5]. Note, the min-cost value of optimal SIF is unique, while the optimal solutions of SIF may be multiple. (a) (b) Fig. 11. Ladder network when N=10. (a) optimal by GeoSteiner (b) SIF result in the third round (q=4). Furthermore, there are eight resulting relay nodes from the LP computation of Phase I, which are connected together. They satisfy the 120 condition. Thus, we apply the analytic geometry method according to Section III-B to compute the coordinates of the balanced relay nodes by means of establishing 16 equations. D. Random Networks We apply the new SIF algorithm to random networks, which are generated by the Waxman model [16]. In most such random settings, the observed cost advantage equals 1. For instance, the SIF result for N=7 in the third round is shown in Figure 12(b), which equals the optimal (See Figure 12(a)). The new SIF algorithm can achieve the optimal faster than the previous algorithm in [5]. (a) Fig. 12. Random network when N=7. (a) optimal by GeoSteiner (b) SIF result in the third round (q=4). Throughout our simulations, we observed that the resulting relay nodes from LP computation of Phase I satisfy the (b) 120 condition, with no exception. Thus, Phase II of the SIF algorithm can quickly and exactly find the balanced positions of the relay nodes by means of the analytic geometry method. From these observations, Phase II of the proposed algorithm has polynomial complexity; a theoretically proof for such polynomial complexity is left as future work. We conjecture that in all optimal SIF solutions, a relay node always has degree three, with three equal-flow adjacent links, which then have to meet at three 120 angles. A proof of this conjecture will help with a rigorous complexity analysis of our Phase II. V. CONCLUSIONS Multicast SIF is an interesting new problem that studies optimal multicast (with network coding) in space instead of in a graph or network. A key open problem in SIF is to design efficient algorithms that compute optimal SIF solutions. This work combines a recursive non-partitioning method with the linear programming method for a new heuristic SIF algorithm that are empirically shown to always converge to optimal SIF solutions promptly. A natural next step is rigorous theoretical analysis of the algorithm proposed. Our future work include network codes construction and distributed implementation. ACKNOWLEDGMENT This research was supported by National Natural Science Foundation of China(No ). The first author thanks Min Peng, Sijia Gu and Wei Xiong for their helpful discussion. REFERENCES [1] Z. Li. Space information flow. space-information-flow, [2] Z. Li and C. Wu. Space information flow: Multiple unicast. In IEEE ISIT, [3] R. Ahlswede, N. Cai, S.Y.R. Li, and R.W. Yeung. Network information flow. IEEE Trans. on Information Theory, 46(4): , [4] E.N. Gilbert and H.O. Pollak. Steiner minimal trees. SIAM Journal on Applied Mathematics, 16(1):1 29, [5] J. Huang, X. Yin, X. Zhang, X. Du, and Z. Li. On space information flow: Single multicast. In NetCod, [6] S. Maheshwar, Z. Li, and B. Li. Bounding the coding advantage of combination network coding in undirected networks. IEEE Trans. on Information Theory, 58(2): , [7] P. Winter and M. Zachariasen. Euclidean steiner minimum trees: An improved exact algorithm. Networks, 30(3): , [8] J.F. Weng and R.S. Booth. Steiner minimal trees on regular polygons with centre. Discrete Mathematics, 141(1): , [9] J.W. Van Laarhoven. Exact and Heuristic Algorithms for the Euclidean Steiner Tree Problem. PhD thesis, University of Iowa, [10] S. Arora. Polynomial time approximation schemes for euclidean traveling salesman and other geometric problems. Journal of the ACM, 45(5): , [11] X. Yin, Y. Wang, X. Wang, X. Xue, and Z. Li. Min-cost multicast network in euclidean space. In IEEE ISIT, [12] T. Xiahou, C. Wu, J. Huang, and Z. Li. A geometric framework for investigating the multiple unicast network coding conjecture. In NetCod, [13] T. Xiahou, Z. Li, C. Wu, and J. Huang. A geometric perspective to multiple-unicast network coding. IEEE Transactions on Information Theory, 60(5): , [14] Z. Li. Min-cost multicast of selfish information flows. In IEEE INFOCOM, pages , [15] F.R.K. Chung and R.L. Graham. Steiner trees for ladders. Ann. Discrete Math, 2: , [16] B.M. Waxman. Routing of multipoint connections. IEEE J. Select. Areas Commun., 6(9): ,