Efficient Network Coding in Planar Multicast Networks Tang Xiahou - - PowerPoint PPT Presentation

Network Coding Problem Statement and Motivation Contributions Efficient Network Coding in General Planar Networks Efficient Network Coding in Relay-coface Networks Summary

Efficient Network Coding in Planar Multicast Networks

Tang Xiahou

Department of Computer Science University of Calgary

August 16, 2012

Tang Xiahou Efficient Network Coding in Planar Multicast Networks

Network Coding Problem Statement and Motivation Contributions Efficient Network Coding in General Planar Networks Efficient Network Coding in Relay-coface Networks Summary

Outline

1

Network Coding

2

Problem Statement and Motivation

3

Contributions

4

Efficient Network Coding in General Planar Networks

5

Efficient Network Coding in Relay-coface Networks

Tang Xiahou Efficient Network Coding in Planar Multicast Networks

Network Coding Problem Statement and Motivation Contributions Efficient Network Coding in General Planar Networks Efficient Network Coding in Relay-coface Networks Summary

Butterfly Network

S T1 T2

v x y y v1 v2 v3

S T1 T2

v x y x v1 v2 v3 (a) (b) x y y x x y

S → T1: Sv1T1, Sv2vv3T1 S → T2: Sv1vv3T2, Sv2T2

What to be transmitted on vv3?

Tang Xiahou Efficient Network Coding in Planar Multicast Networks

Network Coding Problem Statement and Motivation Contributions Efficient Network Coding in General Planar Networks Efficient Network Coding in Relay-coface Networks Summary

Butterfly Network

S T1 T2

v x y y v1 v2 v3

S T1 T2

v x y x v1 v2 v3 (a) (b) x y y x x y

S → T1: Sv1T1, Sv2vv3T1 S → T2: Sv1vv3T2, Sv2T2

What to be transmitted on vv3? =

⇒ f(vv3) = x + y

Tang Xiahou Efficient Network Coding in Planar Multicast Networks

Network Coding Problem Statement and Motivation Contributions Efficient Network Coding in General Planar Networks Efficient Network Coding in Relay-coface Networks Summary

Problem Statement

Goal: “efficient” network coding in the planar or pseudo planar network of study.

Small field sizes (≤ 4) such that encoding and decoding computations are most efficient. Linear code assignment algorithms to construct a feasible network code solution

Single-source planar multicast networks with optimum throughput 2.

Tang Xiahou Efficient Network Coding in Planar Multicast Networks

Network Coding Problem Statement and Motivation Contributions Efficient Network Coding in General Planar Networks Efficient Network Coding in Relay-coface Networks Summary

Motivation

Some real-world networks exhibit planar topologies. For general networks, the two-flow multicast case requires unbounded field sizes and leads to the largest known throughput benefit of network coding.

v1 S v2 v3 v4 v5 Tang Xiahou Efficient Network Coding in Planar Multicast Networks

Network Coding Problem Statement and Motivation Contributions Efficient Network Coding in General Planar Networks Efficient Network Coding in Relay-coface Networks Summary

Contributions

Category Illustration Field Size Code Assignment Complexity Outer-planar Networks tree packing suffices O(|V|) Relay-coface Networks q = 2 O(|V|) over GF(2) Terminal-coface Networks q = 2 O(|V|) over GF(2)† Planar Networks q = 3 O(|V|2) over GF(3) O(|V|) over GF(4) Apex Networks q = 4 O(|V|2) over GF(4) O(|V|) over GF(5)

†: Partially proven.

Tang Xiahou Efficient Network Coding in Planar Multicast Networks

Network Coding Problem Statement and Motivation Contributions Efficient Network Coding in General Planar Networks Efficient Network Coding in Relay-coface Networks Summary

“Necessity” of GF(3) in General Planar Networks

An example planar multicast network requiring field GF(3)

S

• v1

S v2 v3 v4 v1 v2 v3 v4

Tang Xiahou Efficient Network Coding in Planar Multicast Networks

Network Coding Problem Statement and Motivation Contributions Efficient Network Coding in General Planar Networks Efficient Network Coding in Relay-coface Networks Summary

“Sufficiency” of GF(3) in General Planar Networks

Equivalence between a network code solution and a proper face coloring

Proof Ideas: Step 1: Construct a subtree graph G′ Step 2: Expand some nodes in G′ Reason: To build an equivalence between each network code over G and a proper face coloring over G′ Step 3: Color the subtree graph G′

Tang Xiahou Efficient Network Coding in Planar Multicast Networks

Network Coding Problem Statement and Motivation Contributions Efficient Network Coding in General Planar Networks Efficient Network Coding in Relay-coface Networks Summary

Step 1: Construct a Subtree Graph G′

Sufficiency” of GF(3) in General Planar Networks

Starting from S, along each outgoing edge, search maximal subtrees until meeting nodes with in-degree 2. Repeat this process for each node with in-degree 2. The subtree root must be S or a node with in-degree 2. Leaves must have in-degree 2.

S T1 T2 v e1 e2 e3 e4 e6 e7 e5 e8 e9

(a) (b) Tang Xiahou Efficient Network Coding in Planar Multicast Networks

Network Coding Problem Statement and Motivation Contributions Efficient Network Coding in General Planar Networks Efficient Network Coding in Relay-coface Networks Summary

Step 2: Expand some nodes in G′

Sufficiency” of GF(3) in General Planar Networks

A node may be a common leaf of two subtrees which are not neighboring in G′. To make the two subtree faces neighboring, expand the node to their common boundary.

Tang Xiahou Efficient Network Coding in Planar Multicast Networks

Network Coding Problem Statement and Motivation Contributions Efficient Network Coding in General Planar Networks Efficient Network Coding in Relay-coface Networks Summary

Step 3: Color the subtree graph G′

Sufficiency” of GF(3) in General Planar Networks

Color G′ in four colors or five colors. Four-coloring can be finished in quadratic time (Robertson et al. 1996) while five-coloring can be finished in linear time (Frederickson 1984). For each node with in-degree 2 in G, it is a common leaf of two subtrees, which corresponds to two neighboring faces in G′. For the other nodes with in-degree 1 in G, they must be nodes of subtrees and can obtain flows from the root of subtrees.

Tang Xiahou Efficient Network Coding in Planar Multicast Networks

Network Coding Problem Statement and Motivation Contributions Efficient Network Coding in General Planar Networks Efficient Network Coding in Relay-coface Networks Summary

“Sufficiency” of GF(2) in Relay-coface Networks

Equivalence between a network code solution and a proper vertex coloring

Proof Ideas: Step 1: Construct a bipartite multicast network G1 Step 2: Map a network code solution in G′ to a vertex coloring of a series parallel graph K4 minor free 3-colorable

Tang Xiahou Efficient Network Coding in Planar Multicast Networks

Network Coding Problem Statement and Motivation Contributions Efficient Network Coding in General Planar Networks Efficient Network Coding in Relay-coface Networks Summary

Step 1: Construct a bipartite multicast network G1

“Sufficiency” of GF(2) in Relay-coface Networks

For each node with in-degree 2 connected to a node with in-degree 2, remove it. For each node with in-degree 1, remove its incoming edge and connect it to the source S.

S T1 T2 v S T1 T2 v

Tang Xiahou Efficient Network Coding in Planar Multicast Networks

Network Coding Problem Statement and Motivation Contributions Efficient Network Coding in General Planar Networks Efficient Network Coding in Relay-coface Networks Summary

Step 2: Map a network code in G1 to a vertex coloring

• f a series parallel graph

“Sufficiency” of GF(2) in Relay-coface Networks

Remove S and its adjacent edges. Connect two relay nodes if they are neighboring to a common node with in-degree 2. The graph we obtain is planar and thus K4 minor free. It is 3-colorable (Seymour 1990).

S T1 T2 v

Tang Xiahou Efficient Network Coding in Planar Multicast Networks

Network Coding Problem Statement and Motivation Contributions Efficient Network Coding in General Planar Networks Efficient Network Coding in Relay-coface Networks Summary

Summary

Network topology affects the lower-bound on field size which admits a linear network code solution. Planarity may constrain the necessity of large field in multicast networks. Small field size (≤ 3) is sufficient in planar networks to achieve optimum throughput 2. Linear code assignment algorithms are proposed for planar networks.

Tang Xiahou Efficient Network Coding in Planar Multicast Networks

Network Coding Problem Statement and Motivation Contributions Efficient Network Coding in General Planar Networks Efficient Network Coding in Relay-coface Networks Summary

Open Problems

Is GF(2) sufficient for general terminal-coface planar networks? What about the case with general through h ≥ 3? What is the coding advantage of multicasting in planar networks?

directed: 3/2 ? undirected: 10/9 ?

Tang Xiahou Efficient Network Coding in Planar Multicast Networks