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Network encoding complexity: Exact values, bounds, and inequalities

Results of this paper have been partially presented in the 49th Allerton Conference on Communication, Control and Computing [15], and the 2012 International Symposium on Information Theory and its Applications [14]

The last author would like to thank the support from the University Grants Committee of the Hong Kong Special Administrative Region, China under grant No. AoE/E-02/08 and the support from Research Grants Council of the Hong Kong Special Administrative Region, China under grant No. 17301814.
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  • For an acyclic directed network with multiple pairs of sources and sinks and a set of Menger's paths connecting each pair of source and sink, it is known that the number of mergings among these Menger's paths is closely related to network encoding complexity. In this paper, we focus on networks with two pairs of sources and sinks and we derive bounds on and exact values of two functions relevant to encoding complexity for such networks.

    Mathematics Subject Classification: Primary: 94A29; Secondary: 05C40.


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  • Figure 1.  Paths $\beta_1, \beta_2$ merge at edge $A \to B$ and at merged subpath (or merging) $A \to B \to C \to D$, and paths $\beta_1, \beta_2, \beta_3$ merge at edge $B \to C$ and at merged subpath (or merging) $B\to C$ (Here, arrows in the figure represents edges, and the terminal points of arrows should be naturally interpreted as vertices; the same convention applies to other figures in this paper)

    Figure 2.  (a) Network coding in the butterfly network (b) Network coding in a two-way channel (c) Network coding in two sessions of unicast

    Figure 3.  An example of a reroutable graph

    Figure 4.  Two examples of merging sequences (as in Remark 3.3, all the mergings in this figure are represented by solid dots instead)

    Figure 5.  Two examples of alternating sequences

    Figure 6.  (a) A non-reroutable (2, 3)-graph with 8 mergings (b) An example of a (2, 5)-graph

    Figure 7.  Graph $\mathcal{E}(4, 4)$ with $9$ mergings

    Figure 8.  (a) Graph $\mathcal{F}(3, 3)$ with $11$ mergings (b) Splitting of $R_1$ in $\mathcal{F}(3, 3)$

    Figure 9.  Concatenation of $\mathcal{F}(2, 2)$ and a non-reroutable $(2, 2)$-graph

    Figure 10.  Partition a $(3, c_2)$-graph into blocks

    Figure 11.  Case 1

    Figure 12.  Case 2

    Figure 13.  Concatenation of two (3, 3)-graphs

    Figure 14.  Concatenation of a (2, 2)-graph and a (4, 4)-graph

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