Fig 1.
Network coding in space (SIF).
Fig 2.
The Pentagram example to illustrate a single multicast Space Information Flow problem in 2-D Euclidean space.
(A) Six terminal nodes. (B) Euclidean Steiner Minimal Tree (cost = 4.6400/bit). (C) Minimum multicast flow with network coding in space (cost = 4.5677/bit).
Fig 3.
Delaunay triangulation (solid lines) and Voronoi diagram (dashed lines) for 20 points.
Fig 4.
Replacing unbalanced relay node with a balanced relay node.
(A) Unbalanced relay node R with three adjacent terminal nodes A, B, C and one non-adjacent terminal node D. (B) Balanced relay node R′ with three adjacent terminal nodes A, B, C and one non-adjacent terminal node D.
Fig 5.
Delaunay triangles generated by Delaunay triangulation from Pentagram.
Fig 6.
All possible candidate Steiner nodes for 1 DT to 4 DT concatenations.
(A) Possible Steiner nodes in 1 DT. (B) Possible Steiner nodes in 2 DT. (C) Possible Steiner nodes in 3 DT. (D) Possible Steiner nodes in 4 DT.
Fig 7.
The optimal SIF topology of Pentagram.
Fig 8.
Finding the candidate relay node in a triangle.
Fig 9.
Finding the candidate relay nodes in a quadrilateral.
Fig 10.
The MST cost, the cost of SIF solutions after concatenating m adjacent Delaunay triangles and ESMT cost.
Fig 11.
The relative error percentage: .
Table 1.
The cost advantage: .
Fig 12.
The cost of optimal SIF solutions for special Cases.
Fig 13.
The network topologies of Case 9 after concatenating 1 to 3 adjacent Delaunay triangles.
(A) Concatenation of 1 Delaunay triangle. (B) Concatenation of 2 Delaunay triangles. (C) Concatenation of 3 Delaunay triangles.
Fig 14.
The ESMT topology for Case 9 by GeoSteiner.
Fig 15.
The MST topology for Case 9 by Matlab.
Fig 16.
The network topologies of Case 14 after concatenating 1 to 4 adjacent Delaunay triangles.
(A) Concatenation of 1 Delaunay triangle. (B) Concatenation of 2 Delaunay triangles. (C) Concatenation of 3 Delaunay triangles. (D) Concatenation of 4 Delaunay triangles.
Fig 17.
The ESMT topology for Case 14 by GeoSteiner.
Fig 18.
The MST topology for Case 14 by Matlab.
Fig 19.
The network topologies of Case 15 after concatenating 1 to 3 adjacent Delaunay triangles.
(A) Concatenation of 1 Delaunay triangle. (B) Concatenation of 2 Delaunay triangles. (C) Concatenation of 3 Delaunay triangles.
Fig 20.
The ESMT topology for Case 15 by GeoSteiner.
Fig 21.
The MST topology for Case 15 by Matlab.
Fig 22.
MST topology by Matlab for Pentagram network.
Fig 23.
The network topologies of Butterfly network after concatenating 1 and 2 adjacent Delaunay triangles.
(A) Concatenation of 1 Delaunay triangle. (B) Concatenation of 2 Delaunay triangles.
Fig 24.
MST topology by Matlab for Butterfly network.
Fig 25.
ESMT topology by GeoSteiner for Butterfly network.
Fig 26.
SIF result for random network after concatenating 1 and 2 adjacent Delaunay triangles, when N = 9.
(A) Concatenation of 1 Delaunay triangle. (B) Concatenation of 2 Delaunay triangles.
Fig 27.
MST result by Matlab for random network.
Fig 28.
The optimal ESMT by GeoSteiner for random network.