Abstract
Space Information Flow (SIF) is a new promising research area that studies network coding in geometric space, such as Euclidean space. The design of algorithms that compute the optimal SIF solutions remains one of the key open problems in SIF. This work proposes the first exact SIF algorithm and a heuristic SIF algorithm that compute min-cost multicast network coding for N (N ≥ 3) given terminal nodes in 2-D Euclidean space. Furthermore, we find that the Butterfly network in Euclidean space is the second example besides the Pentagram network where SIF is strictly better than Euclidean Steiner minimal tree. The exact algorithm design is based on two key techniques: Delaunay triangulation and linear programming. Delaunay triangulation technique helps to find practically good candidate relay nodes, after which a min-cost multicast linear programming model is solved over the terminal nodes and the candidate relay nodes, to compute the optimal multicast network topology, including the optimal relay nodes selected by linear programming from all the candidate relay nodes and the flow rates on the connection links. The heuristic algorithm design is also based on Delaunay triangulation and linear programming techniques. The exact algorithm can achieve the optimal SIF solution with an exponential computational complexity, while the heuristic algorithm can achieve the sub-optimal SIF solution with a polynomial computational complexity. We prove the correctness of the exact SIF algorithm. The simulation results show the effectiveness of the heuristic SIF algorithm.
Highlights
Network Information Flow (NIF) [1], proposed in 2000, studies network coding in graphs
Space Information Flow (SIF) is different from Minimum Spanning Tree (MST) in a way that MST interconnects all the terminal nodes of a given set by a network of direct terminal-to-terminal links with the smallest possible total length, without any additional relay nodes [7], while additional relay nodes are required in SIF [2, 3]
In a recent subsequent work, Huang and Li [16] proposed a polynomial-time heuristic algorithm based on non-uniform partitioning and Delaunay triangulation techniques for solving the problem of min-cost multicast network coding in 2-D Euclidean space
Summary
Network Information Flow (NIF) [1], proposed in 2000, studies network coding in graphs. SIF aims to minimize the total bandwidth-distance sum-product (‘network volume’) in geometric space with a certain throughput requirement, allowing network coding to be used and additional relay nodes to be inserted to connect a given set of terminal nodes, while sustaining end-to-end communication demands among terminals at known coordinates [2]. We propose the first exact algorithm with an exponential computational complexity that computes the optimal SIF solution, i.e., the optimal positions of the relay nodes (optimal topology of the network), as well as the flow rate assignments on the connection links in single multicast SIF.
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