# Structural Characteristics of Two-Sender Index Coding

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## Abstract

**:**

## 1. Introduction

- Macro-cell networks with caching helpers [14]—cellular networks deploying dedicated nodes, called helpers, with large storage capacity instead of femto-cell access points to reduce backhaul loads,
- cooperative data exchange [15]—peer-to-peer networks with data exchange within a group of closely-located wireless nodes, and
- distributed storage—storage networks where data are distributed over multiple storage devices/locations.

#### 1.1. Prior Works

#### 1.2. Our Work and Contributions

**Proposing a new coloring concept for confusion graphs in TSUIC, called two-sender graph coloring (Definition 8, Section 4.3)**: For SSUIC, the chromatic number of its confusion graph gives the optimal broadcast rate and the corresponding index code (for a specific message size). However, for TSUIC, as the two senders (encoders) contain some messages in common, the standard method of graph coloring of the confusion graph may not lead us to an index code. In this regard, we need a different kind of coloring function in TSUIC, and thus, in this paper, we propose a novel coloring technique to color the confusion graphs in TSUIC, and its optimization gives the optimal broadcast rate and optimal index code.**Presenting a way of grouping the vertices of confusion graphs in TSUIC (Appendix B)**: By exploiting the symmetry of the confusion graph, we propose a way of grouping its vertices for analysis purposes mainly in its two-sender graph coloring. In particular, this grouping helps us to analyze the number of colors used in two-sender graph coloring of a confusion graph.**Deriving the optimal broadcast rates of TSUIC problems as a function of the optimal broadcast rates of its sub-problems (Theorems 4–8)**: We divide a TSUIC problem into three independent sub-problems based on the requested messages by receivers, specifically whether the messages are present in only one of the senders or in both senders. Now in TSUIC, considering the interactions (defined by side-information available at the receivers) between these three independent sub-problems, we derive the optimal broadcast rate (in both asymptotic and non-asymptotic regimes in the message size) of the problem as a function of the optimal broadcast rates of its sub-problems. Moreover, we bound the optimal broadcast rate, and show that the bounds are tight for several classes of TSUIC instances (sometimes with conditions). Furthermore, we find a class of TSUIC instances where a TSUIC scheme can achieve the same optimal broadcast rate as the same instances when the two senders form a single sender having all messages.**Characterizing a class of TSUIC instances where a certain type of side-information is not critical (Corollary 1)**: For a class of TSUIC instances, we prove that certain interactions between the three independent sub-problems can be removed without affecting the optimal broadcast rate (in the asymptotic regime). This means that those interactions are not critical.**Generalizing the results of some classes of TSUIC problems to multiple senders (Section 6)**: For some classes of TSUIC problems, we generalize the two-sender graph coloring of confusion graphs and the proposed grouping of their vertices. Then, we compute the optimal broadcast rates of those problems as a function of the optimal broadcast rates of their sub-problems.

## 2. Problem Definitions and Graphical Representation

#### 2.1. Problem Setup

**Definition**

**1**(Two-sender index code)

**.**

- (i)
- an encoding function for each sender ${S}_{s}$, ${\mathcal{F}}_{s}:{\{0,1\}}^{|{\mathcal{M}}_{s}|\times t}\to {\{0,1\}}^{{p}_{s}}$ such that ${\mathcal{C}}_{s}={\mathcal{F}}_{s}({\mathcal{M}}_{s})$, and
- (ii)
- a decoding function for every receiver r, ${\mathcal{G}}_{r}:{\{0,1\}}^{({\Sigma}_{s=1}^{2}{p}_{s}+|{\mathcal{H}}_{r}|\times t)}\to {\{0,1\}}^{t}$ such that ${x}_{r}=\phantom{\rule{3.33333pt}{0ex}}{\mathcal{G}}_{r}({\mathcal{C}}_{1},{\mathcal{C}}_{2},{\mathcal{H}}_{r})$.

**Definition**

**2**(Broadcast rate or aggregate normalized codelength)

**.**

**Definition**

**3**(Optimal broadcast rate)

**.**

**Remark**

**1.**

#### 2.2. Representation of the Receivers’ Side-Information and the Senders’ Message Setting in TSUIC Problems

**Definition**

**4**(Directed graphs and undirected graphs)

**.**

**Definition**

**5**(Constraint due to the two senders)

**.**

## 3. A New Way of Classifying TSUIC Problems and Main Results

#### 3.1. Interactions between ${D}_{1}$, ${D}_{2}$ and ${D}_{3}$

#### 3.2. A Compact Representation of Interactions

#### 3.3. A Classification of the Interactions

#### 3.4. Main Results

## 4. Confusion Graphs and Their Coloring

#### 4.1. Confusion Graphs

**Definition**

**6**(Confusion graph)

**.**

- (i)
- $V({\Gamma}_{t}(D))=\{{u}^{N}:{u}^{N}\in {\{0,1\}}^{t\times N}\}$, and
- (ii)
- $E({\Gamma}_{t}(D))=\{({u}^{N},{v}^{N}):{u}^{N},{v}^{N}\in V({\Gamma}_{t}(D)),\phantom{\rule{4pt}{0ex}}\mathit{and}\phantom{\rule{4pt}{0ex}}{u}^{N}\phantom{\rule{4pt}{0ex}}\mathit{and}\phantom{\rule{4pt}{0ex}}{v}^{N}\phantom{\rule{4pt}{0ex}}\mathit{are}\phantom{\rule{4.pt}{0ex}}\mathit{confusable}\}$.

#### 4.2. A Review of Confusion Graph Coloring for SSUIC

**Definition**

**7**(Graph coloring and Chromatic number)

**.**

**Theorem**

**1.**

#### 4.3. Proposed Confusion Graph Coloring for TSUIC

- Without loss of generality, we assume ${x}_{1},{x}_{2},\cdots ,{x}_{{n}_{1}}$ to be the messages requested by vertices in $V({D}_{1})$, ${x}_{{n}_{1}+1},{x}_{{n}_{1}+2},\cdots ,{x}_{{n}_{1}+{n}_{2}}$ the messages requested by vertices in $V({D}_{2})$, and ${x}_{{n}_{1}+{n}_{2}+1},{x}_{{n}_{1}+{n}_{2}+2},\cdots ,{x}_{{n}_{1}+{n}_{2}+{n}_{3}}$ the messages requested by vertices in $V({D}_{3})$ with $N=\phantom{\rule{3.33333pt}{0ex}}{n}_{1}+\phantom{\rule{3.33333pt}{0ex}}{n}_{2}+\phantom{\rule{3.33333pt}{0ex}}{n}_{3}$.
- Indices $i,{i}_{1},{i}_{2}\in \{1,2,\cdots ,{2}^{t{n}_{1}}\}$, $j,{j}_{1},{j}_{2}\in \{1,2,\cdots ,{2}^{t{n}_{2}}\}$ and $k,{k}_{1},{k}_{2}\in \{1,2,\cdots ,{2}^{t{n}_{3}}\}$ are used in the representation of possible realizations of words of $t{n}_{1}$, $t{n}_{2}$ and $t{n}_{3}$ bits, respectively. For convenience, we use three indices (e.g., $i,{i}_{1},{i}_{2}$) for the same set of numbers, where the first index (e.g., i) is used for a general case, and the remaining two indices (e.g., ${i}_{1}$ and ${i}_{2}$) are used to indicate any two words within the group of words.
- We group the bits associated with the messages requested by vertices of ${D}_{{i}^{\prime}}$, ${i}^{\prime}\in \{1,2,3\}$. Within each group, each realization of the bits, i.e., each member in ${\{0,1\}}^{t{n}_{{i}^{\prime}}}$ is represented by a unique label ${\mathbf{b}}_{{D}_{{i}^{\prime}}}^{{j}^{\prime}}$, ${j}^{\prime}\in \{1,2,\cdots ,{2}^{t{n}_{{i}^{\prime}}}\}$. Figure A1a in Appendix A outlines each tuple ${\mathbf{b}}_{{D}_{{i}^{\prime}}}^{{j}^{\prime}}$ for $t=1$. Each message tuple $({x}_{1},\cdots ,{x}_{N})$ realization can then be uniquely written as $({\mathbf{b}}_{{D}_{1}}^{i},{\mathbf{b}}_{{D}_{2}}^{j},{\mathbf{b}}_{{D}_{3}}^{k})$ for some $i,j,k$.

**Definition**

**8**

**.**Let two onto functions ${J}_{1}:{\{0,1\}}^{t{n}_{1}}\times {\{0,1\}}^{t{n}_{3}}\to {\mathcal{J}}_{1}$, and ${J}_{2}:{\{0,1\}}^{t{n}_{2}}\times {\{0,1\}}^{t{n}_{3}}\to {\mathcal{J}}_{2}$ be the coloring functions carried out by senders ${S}_{1}$ and ${S}_{2}$, respectively. A proper two-sender graph coloring of ${\Gamma}_{t}(D)$ is an onto function ${J}_{o}:{\{0,1\}}^{t{n}_{1}}\times {\{0,1\}}^{t{n}_{2}}\times {\{0,1\}}^{t{n}_{3}}\to {\mathcal{J}}_{1}\times {\mathcal{J}}_{2}$ where ${J}_{o}(({\mathit{b}}_{{D}_{1}}^{i},{\mathit{b}}_{{D}_{2}}^{j},{\mathit{b}}_{{D}_{3}}^{k}))=({J}_{1}({\mathit{b}}_{{D}_{1}}^{i},{\mathit{b}}_{{D}_{3}}^{k}),{J}_{2}({\mathit{b}}_{{D}_{2}}^{j},{\mathit{b}}_{{D}_{3}}^{k}))$ such that if $({\mathit{b}}_{{D}_{1}}^{{i}_{1}},{\mathit{b}}_{{D}_{2}}^{{j}_{1}},{\mathit{b}}_{{D}_{3}}^{{k}_{1}})$ and $({\mathit{b}}_{{D}_{1}}^{{i}_{2}},{\mathit{b}}_{{D}_{2}}^{{j}_{2}},{\mathit{b}}_{{D}_{3}}^{{k}_{2}})$ are adjacent vertices of ${\Gamma}_{t}(D)$, then ${J}_{o}(({\mathit{b}}_{{D}_{1}}^{{i}_{1}},{\mathit{b}}_{{D}_{2}}^{{j}_{1}},{\mathit{b}}_{{D}_{3}}^{{k}_{1}}))\ne {J}_{o}(({\mathit{b}}_{{D}_{1}}^{{i}_{2}},{\mathit{b}}_{{D}_{2}}^{{j}_{2}},{\mathit{b}}_{{D}_{3}}^{{k}_{2}}))$.

**Remark**

**2.**

#### 4.4. A Few Lemmas for the TSUIC Confusion Graph Coloring

**Lemma**

**1.**

**Proof.**

**Lemma**

**2.**

**Lemma**

**3.**

**Lemma**

**4.**

## 5. The Optimal Broadcast Rate for TSUIC

**Theorem**

**2.**

**Proof.**

**Example**

**1.**

#### 5.1. Lower Bounds

**Lemma**

**5**(A simple lower bound)

**.**

**Lemma**

**6**(A lower bound)

**.**

**Proof.**

**Theorem**

**3.**

- (i)
- $\beta (D)=\beta ({D}_{a})+\beta ({D}_{b})$ if there is (i) no interaction between ${D}_{a}$ and ${D}_{b}$ (i.e., no ${D}_{a}\phantom{\rule{3.33333pt}{0ex}}\to \phantom{\rule{3.33333pt}{0ex}}{D}_{b}$ and ${D}_{b}\phantom{\rule{3.33333pt}{0ex}}\to \phantom{\rule{3.33333pt}{0ex}}{D}_{a}$), or (ii) a one-way interaction (either partially or fully participated) between ${D}_{a}$ and ${D}_{b}$, i.e., either ${D}_{a}\phantom{\rule{3.33333pt}{0ex}}\to \phantom{\rule{3.33333pt}{0ex}}{D}_{b}$ or ${D}_{b}\phantom{\rule{3.33333pt}{0ex}}\to \phantom{\rule{3.33333pt}{0ex}}{D}_{a}$, but not both and
- (ii)
- $\beta (D)=\mathit{max}\{\beta ({D}_{a}),\beta ({D}_{b})\}$ if there is a fully participated both way interaction between ${D}_{a}$ and ${D}_{b}$ (i.e., fully participated ${D}_{a}\rightleftarrows {D}_{b}$).

#### 5.2. Optimal Broadcast Rates for CASE I and CASE II-A: The Arcs between ${D}_{1}$, ${D}_{2}$ and ${D}_{3}$ Are Not Critical in Asymptotic Regime in the Message Size

**Theorem**

**4**(CASE I)

**.**

**Proof.**

**Example**

**2.**

**Proposition**

**1.**

**Proof.**

**Definition**

**9**(Vertex-transitive graphs)

**.**

**Remark**

**3.**

**Theorem**

**5**(CASE I and CASE II-A)

**.**

**Proof.**

**Remark**

**4.**

**Corollary**

**1.**

**Proof.**

#### 5.3. Optimal Broadcast Rates for CASE II-B

**Theorem**

**6**(CASE II-B)

**.**

**Proof.**

**Corollary**

**2.**

**Proof.**

**Remark**

**5.**

**Example**

**3.**

#### 5.4. Optimal Broadcast Rates for CASE II-C: An Upper Bound, and Special Cases Where the Upper Bound Is Tight

**Theorem**

**7**(CASE II-C)

**.**

- (i)
- ${\beta}_{t}(D,{G}_{o})\le {\beta}_{t}({D}_{2})+\mathit{max}\{{\beta}_{t}({D}_{1}),{\beta}_{t}({D}_{3})\}$,
- (ii)
- ${\beta}_{t}(D,{G}_{o})={\beta}_{t}({D}_{1})+{\beta}_{t}({D}_{2})$ if ${\beta}_{t}({D}_{1})\ge {\beta}_{t}({D}_{3})$,
- (iii)
- $\beta (D,{G}_{o})\le \beta ({D}_{2})+\mathit{max}\{\beta ({D}_{1}),\beta ({D}_{3})\}$, and
- (iv)
- $\beta (D,{G}_{o})=\beta ({D}_{1})+\beta ({D}_{2})$ if $\beta ({D}_{1})\ge \beta ({D}_{3})$.

**Proof.**

**Proposition**

**2**(CASE II-C)

**.**

**Proof.**

**Corollary**

**3.**

**Proof.**

**Corollary**

**4.**

- (i)
- if $\beta ({D}_{1})\le \beta ({D}_{3})$, then $\beta (D,{G}_{o})=\beta ({D}_{2})+\beta ({D}_{3})=\beta (D)$,
- (ii)
- if $\beta ({D}_{1})>\beta ({D}_{3})$, then
- (a)
- if $\beta ({D}_{1})\ge \beta ({D}_{2})+\beta ({D}_{3})$, then $\beta (D,{G}_{o})=\beta ({D}_{1})+\beta ({D}_{2})\ge \beta ({D}_{1})=\beta (D)$, with a strict inequality if ${D}_{2}$ is non-empty, and
- (b)
- if $\beta ({D}_{1})\le \beta ({D}_{2})+\beta ({D}_{3})$, then $\beta (D,{G}_{o})=\beta ({D}_{1})+\beta ({D}_{2})>\beta ({D}_{2})+\beta ({D}_{3})=\beta (D)$ for a non-empty ${D}_{2}$.

**Proof.**

**Remark**

**6.**

- (i)
- $\{\beta ({D}^{32},{G}_{o})=\beta ({D}^{32})=\beta ({D}_{2})+\beta ({D}_{3})\}\le \beta ({D}^{28})\le \beta ({D}^{28},{G}_{o})\le \{\beta ({D}^{26})=\beta ({D}^{26},{G}_{o})=\beta ({D}_{2})+\beta ({D}_{3})\}$ from Proposition 2, Theorem 3, Corollarys 3 and 4, and Lemma 5. This implies $\beta ({D}^{28})=\beta ({D}^{28},{G}_{o})=\beta ({D}_{2})+\beta ({D}_{3})$.
- (ii)
- $\{\beta ({D}^{31},{G}_{o})=\beta ({D}^{31})=\beta ({D}_{2})+\beta ({D}_{3})\}\le \beta ({D}^{29})\le \beta ({D}^{29},{G}_{o})\le \{\beta ({D}^{27})=\beta ({D}^{27},{G}_{o})=\beta ({D}_{2})+\beta ({D}_{3})\}$ from Proposition 2, Theorem 3, Corollarys 3 and 4, and Lemma 5. This implies $\beta ({D}^{29})=\beta ({D}^{29},{G}_{o})=\beta ({D}_{2})+\beta ({D}_{3})$.

**Example**

**4.**

#### 5.5. Optimal Broadcast Rates for CASE II-D

**Theorem**

**8**(CASE II-D)

**.**

- (i)
- ${\beta}_{t}(D,{G}_{o})\le \mathit{max}\{{\beta}_{t}({D}_{1}),{\beta}_{t}({D}_{3})\}+\mathit{max}\{{\beta}_{t}({D}_{2}),{\beta}_{t}({D}_{3})\}$,
- (ii)
- ${\beta}_{t}(D,{G}_{o})={\beta}_{t}({D}_{1})+{\beta}_{t}({D}_{2})$ if ${\beta}_{t}({D}_{3})\le \mathit{min}\{{\beta}_{t}({D}_{1}),{\beta}_{t}({D}_{2})\}$,
- (iii)
- $\beta (D,{G}_{o})\le \mathit{max}\{\beta ({D}_{1}),\beta ({D}_{3})\}+\mathit{max}\{\beta ({D}_{2}),\beta ({D}_{3})\}$ , and
- (iv)
- $\beta (D,{G}_{o})=\beta ({D}_{1})+\beta ({D}_{2})$ if $\beta ({D}_{3})\le \mathit{min}\{\beta ({D}_{1}),\beta ({D}_{2})\}$.

**Proof.**

**Proposition**

**3.**

**Proof.**

## 6. Generalizing the Results of Some Classes of TSUIC Problems to Multiple Senders

- Vertices labeled by all $({\mathbf{b}}_{{D}_{1}}^{{i}_{1}},{\mathbf{b}}_{{D}_{2}}^{{i}_{2}},{\mathbf{b}}_{{D}_{3}}^{{i}_{3}},\cdots ,{\mathbf{b}}_{{D}_{{N}^{\prime}}}^{{i}_{{N}^{\prime}}},{\mathbf{b}}_{{D}_{{N}^{\prime}+1}}^{k})$, ${i}_{j}\in \{1,2,\cdots ,{2}^{t{n}_{j}}\}$, $j\in \{1,2,\cdots ,{N}^{\prime}\}$ and $k\in \{1,2,\cdots ,{2}^{t{n}_{{N}^{\prime}+1}}\}$, with the same ${\mathbf{b}}_{{D}_{{N}^{\prime}+1}}^{k}$ sub-label,
- any row sub-block consists of vertices labeled by all $({\mathbf{b}}_{{D}_{1}}^{{i}_{1}},{\mathbf{b}}_{{D}_{2}}^{{i}_{2}},{\mathbf{b}}_{{D}_{3}}^{{i}_{3}},\cdots ,{\mathbf{b}}_{{D}_{{N}^{\prime}}}^{{i}_{{N}^{\prime}}},{\mathbf{b}}_{{D}_{{N}^{\prime}+1}}^{k})$, ${i}_{j}\in \{1,2,\cdots ,{2}^{t{n}_{j}}\}$, $j\in \{1,2,\cdots ,{N}^{\prime}\}$, with the same ${\mathbf{b}}_{{D}_{2}}^{{i}_{2}},{\mathbf{b}}_{{D}_{3}}^{{i}_{3}},\cdots ,{\mathbf{b}}_{{D}_{{N}^{\prime}}}^{{i}_{{N}^{\prime}}},{\mathbf{b}}_{{D}_{{N}^{\prime}+1}}^{k}$ sub-labels, and
- any ${i}_{1}$-th column sub-block consists of vertices labeled by all $({\mathbf{b}}_{{D}_{1}}^{{i}_{1}},{\mathbf{b}}_{{D}_{2}}^{{i}_{2}},{\mathbf{b}}_{{D}_{3}}^{{i}_{3}},\cdots ,{\mathbf{b}}_{{D}_{{N}^{\prime}}}^{{i}_{{N}^{\prime}}},{\mathbf{b}}_{{D}_{{N}^{\prime}+1}}^{k})$, ${i}_{j}\in \{1,2,\cdots ,{2}^{t{n}_{j}}\}$, $j\in \{1,2,\cdots ,{N}^{\prime}\}$, with the same ${\mathbf{b}}_{{D}_{1}}^{{i}_{1}}$ and ${\mathbf{b}}_{{D}_{{N}^{\prime}+1}}^{k}$ sub-labels. Moreover, in contrast to SSUIC, there are multiple sub-labels other than ${\mathbf{b}}_{{D}_{1}}^{{i}_{1}}$ and ${\mathbf{b}}_{{D}_{{N}^{\prime}+1}}^{k}$ in MSUIC, so we arrange the vertices of any ${i}_{1}$-th column sub-block as dictated by Figure A2 in Appendix A. Clearly, a block has ${2}^{t{n}_{1}}$ column sub-blocks and ${2}^{t({\sum}_{i=2}^{{N}^{\prime}}{n}_{i})}$ row sub-blocks.

**Proposition**

**4.**

**Proof.**

**Theorem**

**9.**

**Remark**

**7.**

## 7. Discussion

- The role of side-information of the vertices in $V({D}_{3})$ (vertices requesting the common messages) about the messages requested by vertices in $V({D}_{1})\cup V({D}_{2})$ (vertices requesting the private messages) in TSUIC: It is proved in SSUIC that, if the interaction between ${D}_{1}$, ${D}_{2}$ and ${D}_{3}$ is acyclic, i.e., $f(D)$ belongs to one of the digraphs in CASE I, then $\beta (D)=\beta ({D}_{1})+\beta ({D}_{2})+\beta ({D}_{3})$ (by using Theorem 3). This means that the arcs contributing acyclic interactions between the sub-digraphs of D can be removed without affecting the optimal broadcast rate of D; in other words, those are non-critical arcs. In this paper, we have proved that this result is also true in TSUIC (by Theorem 5). Moreover, in TSUIC, we have proved that, for D, if the vertices in $V({D}_{3})$ have no side-information about the messages requested by vertices in $V({D}_{1})\cup V({D}_{2})$, i.e., ${d}_{f(D)}^{+}(3)=\varnothing $, then by Theorem 5, we have $\beta (D,{G}_{o})=\beta ({D}_{1})+\beta ({D}_{2})+\beta ({D}_{3})$ (behaves like having acyclic interactions between ${D}_{1}$, ${D}_{2}$ and ${D}_{3}$). Under this condition, any arc that is contributing any interaction between ${D}_{1}$, ${D}_{2}$ and ${D}_{3}$ is non-critical.
- Non-critical arcs in SSUIC are not necessarily non-critical in TSUIC: We illustrate this with an example. Consider the TSUIC problem stated in Example 1 (whose $f(D)={\mathrm{H}}_{33}$). In SSUIC, we know that the optimal broadcast rate $\beta (D)=2$. This problem has an arc $(3,1)$ that is non-critical in SSUIC (its removal does not change the optimal broadcast rate), but it is critical in TSUIC. This can be understood from the following: In SSUIC, we can remove the arc $(3,1)\in A(D)$, and still form a valid index code $\{{x}_{1}\oplus {x}_{2},\phantom{\rule{4pt}{0ex}}{x}_{3}\}$ that achieves $\beta (D)$. This infers that removing the arc $(3,1)$ does not affect the optimal broadcast rate in SSUIC. However, in TSUIC, if we remove the arc $(3,1)\in A(D)$, then the new problem, say ${D}^{\prime}$, has $\beta ({D}^{\prime},{G}_{o})=3$ (applying Theorem 5), whereas we get a valid two-sender index code $\{{x}_{1}\oplus {x}_{3},\phantom{\rule{4pt}{0ex}}{x}_{2}\oplus {x}_{3}\}$ of codelength two if we consider $(3,1)\in A(D)$. Now, it is evident that there exist cases in TSUIC where some side-information (e.g., $(1,2)$ and $(2,1)$) cannot be exploited directly during encoding by senders because of the constraint due to the two senders. However, that side-information can be utilized during decoding process at receivers’ end due to the presence of other helping side-information (e.g., $(3,1)$). Thus, this helping side-information can be critical in TSUIC. This observation was also made by Sadeghi et al. [20] for MSUIC under a different performance metric (rate region with fixed capacity links).

## 8. Concluding Remarks and Open Problems

**Study of the critical edges in the TSUIC problems**: It is observed that the non-critical arcs in SSUIC can be critical arcs in TSUIC. This requires further study.**Study of a general distributed index coding**: As our study is a step towards understanding multi-sender index coding, it is left as a future work to extend the approaches implemented and the results obtained in this paper to more general setups.**Finding the optimal broadcast rates of TSUIC problems with cyclic-partially-participated interactions:**The analysis of D with partially-participated interactions between its sub-digraphs ${D}_{1}$, ${D}_{2}$ and ${D}_{3}$ is left as a future work.

## Author Contributions

## Funding

## Conflicts of Interest

## Abbreviations

MSUIC | Multi-sender unicast index coding |

SMSUIC | Special multi-sender unicast index coding |

SSUIC | Single-sender unicast index coding |

TSUIC | Two-sender unicast index coding |

UIC | Unicast index coding |

## Appendix A

**Figure A1.**(

**a**) Each ${n}_{{i}^{\prime}}$-bit tuple ${\mathbf{b}}_{{D}_{{i}^{\prime}}}^{{j}^{\prime}}$ with its respective bits for $t=1$, where ${i}^{\prime}\in \{1,2,3\}$ and ${j}^{\prime}\in \{1,2,\cdots ,{2}^{{n}_{{i}^{\prime}}}\}$, and (

**b**) representation of the vertices (e.g., $({\mathbf{b}}_{{D}_{1}}^{{i}_{1}},{\mathbf{b}}_{{D}_{2}}^{{j}_{1}},{\mathbf{b}}_{{D}_{3}}^{{k}_{1}})$) and sets of vertices (e.g., ${\mathcal{B}}_{{\mathbf{b}}_{{D}_{2}}^{{j}_{1}},{\mathbf{b}}_{{D}_{3}}^{{k}_{1}}}$), each represented by a dotted line, in a confusion graph.

**Figure A2.**Arrangement of vertices in ${i}_{1}$-th column sub-block of a k-th block of a ${\Gamma}_{t}(D)$, where, for example, the first, second and the last vertices are labeled $({\mathbf{b}}_{{D}_{1}}^{{i}_{1}},{\mathbf{b}}_{{D}_{2}}^{1},{\mathbf{b}}_{{D}_{3}}^{1},\cdots ,{\mathbf{b}}_{{D}_{{N}^{\prime}-1}}^{1},{\mathbf{b}}_{{D}_{{N}^{\prime}}}^{1},{\mathbf{b}}_{{D}_{{N}^{\prime}+1}}^{k})$, $({\mathbf{b}}_{{D}_{1}}^{{i}_{1}},{\mathbf{b}}_{{D}_{2}}^{1},{\mathbf{b}}_{{D}_{3}}^{1},\cdots ,{\mathbf{b}}_{{D}_{{N}^{\prime}-1}}^{1},{\mathbf{b}}_{{D}_{{N}^{\prime}}}^{2},{\mathbf{b}}_{{D}_{{N}^{\prime}+1}}^{k})$, and $({\mathbf{b}}_{{D}_{1}}^{{i}_{1}},{\mathbf{b}}_{{D}_{2}}^{{2}^{t{n}_{2}}},{\mathbf{b}}_{{D}_{3}}^{{2}^{t{n}_{3}}},\cdots ,{\mathbf{b}}_{{D}_{{N}^{\prime}-1}}^{{2}^{t{n}_{{N}^{\prime}-1}}},{\mathbf{b}}_{{D}_{{N}^{\prime}}}^{{2}^{t{N}^{\prime}}},{\mathbf{b}}_{{D}_{{N}^{\prime}+1}}^{k})$, respectively. Observe that this sub-block has ${2}^{t({\sum}_{i=2}^{{N}^{\prime}}{n}_{i})}$ vertices in total.

## Appendix B. Proposed Grouping of the Vertices of ${\mathbf{\Gamma}}_{\mathit{t}}\mathbf{(}\mathit{D}\mathbf{)}$ and Its Characteristics

**Definition**

**A1**(Block)

**.**

**Definition**

**A2**(Column sub-block)

**.**

**Definition**

**A3**(Row sub-block)

**.**

**Figure A3.**Functional block digraph of grouping the vertices of a confusion graph. The vertices of a confusion graph are all the possible realizations of words of $tN$ bits.

**Example**

**A1.**

#### Some Lemmas

**Lemma**

**A1.**

**Proof.**

**Lemma**

**A2.**

**Lemma**

**A3.**

**Proof.**

**Lemma**

**A4.**

**Lemma**

**A5.**

**Proof.**

## Appendix C. Proof of Proposition 1

#### Appendix C.1. An Example

**Example**

**A2.**

**Definition**

**A4**(Inter-block edge and Intra-block edge)

**.**

**Figure A4.**(

**a**) A given side-information digraph D such that $f(D)={\mathrm{H}}_{15}$, and a source-constraint graph ${G}_{o}$; (

**b**) the confusion graph ${\Gamma}_{1}(D)$, and its two-sender graph coloring (where each vertex is assigned with an ordered pair of colors such that the first color is always associated with ${S}_{1}$ and the second color is always associated with ${S}_{2}$). The edges are shown in color only for an illustration purpose; it is not an edge coloring.

#### Appendix C.1.1. Intra-Block Coloring

#### Appendix C.1.2. Inter-Block Coloring

#### Appendix C.2. Ingredients for the Proof

**Lemma**

**A6.**

**Proof.**

**Lemma**

**A7.**

**Proof.**

**Lemma**

**A8.**

**Proof.**

**Lemma**

**A9.**

**Proof.**

**Lemma**

**A10.**

**Proof.**

#### Appendix C.3. Proof of Proposition 1

#### Appendix C.3.1. Construction and Coloring of Γ_{t} (D^{16})

- (i)
- Edges in $E({\Gamma}_{t}({D}^{16}))$ due to the confusion at some vertices in $V({D}_{1})$: The confusion at any vertex in $V({D}_{1})$ contributes to only intra-edges due to Lemma A8.
- (ii)
- Edges in $E({\Gamma}_{t}({D}^{16}))$ due to the confusion at some vertices in $V({D}_{2})$: The confusion at any vertex in $V({D}_{2})$ contributes to only intra-edges due to Lemma A8.
- (iii)
- Edges in $E({\Gamma}_{t}({D}^{16}))$ due to the confusion at some vertices in $V({D}_{3})$: If there exists an edge due to the confusion at some vertices in $V({D}_{3})$ between any vertex pair $(({\mathbf{b}}_{{D}_{1}}^{{i}_{1}},{\mathbf{b}}_{{D}_{2}}^{{j}_{1}},{\mathbf{b}}_{{D}_{3}}^{{k}_{1}}),({\mathbf{b}}_{{D}_{1}}^{{i}_{2}},{\mathbf{b}}_{{D}_{2}}^{{j}_{2}},{\mathbf{b}}_{{D}_{3}}^{{k}_{2}}))$, then each of the vertices in the ${k}_{1}$-th block has edges with all the vertices in the ${k}_{2}$-th block. This is because any vertex in $V({D}_{3})$ has no message requested by any vertex in $V({D}_{1})\cup V({D}_{2})$ as its side-information. This results in no effect due to a change in bits of ${\mathbf{b}}_{{D}_{1}}^{i}$ or ${\mathbf{b}}_{{D}_{2}}^{j}$ sub-label once we have an edge due to confusion at some receivers in $V({D}_{3})$, which corresponds to the change in bits of the ${\mathbf{b}}_{{D}_{3}}^{k}$ sub-label.

#### Appendix C.3.2. Construction and Coloring of Γ_{t} (D^{1})

## Appendix D. Proof of Theorem 6

- Let $\mathcal{C}({D}_{{i}^{\prime}})$ be an index code (linear or nonlinear,) having a codeword length of $|\mathcal{C}({D}_{{i}^{\prime}})|$ bits, for a given t (message bits) that achieves ${\beta}_{t}({D}_{{i}^{\prime}})$. For convenience, we represent $|\mathcal{C}({D}_{{i}^{\prime}})|$ by ${\ell}^{*}(\mathcal{C}({D}_{{i}^{\prime}}))$ such that ${\ell}^{*}(\mathcal{C}({D}_{{i}^{\prime}}))={\beta}_{t}({D}_{{i}^{\prime}})$.
- Let the sequence of bits in $\mathcal{C}({D}_{{i}^{\prime}})$ be $({w}_{1}^{{i}^{\prime}},{w}_{2}^{{i}^{\prime}},\cdots ,{w}_{{\ell}^{*}(\mathcal{C}({D}_{{i}^{\prime}}))}^{{i}^{\prime}})$, where ${w}_{m}^{{i}^{\prime}}\in \{0,1\}$, $m\in \{1,2,\cdots ,{\ell}^{*}(\mathcal{C}({D}_{{i}^{\prime}}))\}$.
- Let ${\mathcal{C}}^{1}({D}_{{i}^{\prime}})=({w}_{1}^{{i}^{\prime}},{w}_{2}^{{i}^{\prime}},\cdots ,{w}_{{\ell}_{1}^{*}(\mathcal{C}({D}_{{i}^{\prime}}))}^{{i}^{\prime}})$ and ${\mathcal{C}}^{2}({D}_{{i}^{\prime}})=({w}_{{\ell}_{1}^{*}(\mathcal{C}({D}_{{i}^{\prime}}))+1}^{{i}^{\prime}},{w}_{{\ell}_{1}^{*}(\mathcal{C}({D}_{{i}^{\prime}}))+2}^{{i}^{\prime}},\cdots ,{w}_{{\ell}^{*}(\mathcal{C}({D}_{{i}^{\prime}}))}^{{i}^{\prime}})$ with $|{\mathcal{C}}^{2}({D}_{{i}^{\prime}})|={\ell}_{2}^{*}(\mathcal{C}({D}_{{i}^{\prime}}))$ be two parts of the sequence of bits of a codeword of $\mathcal{C}({D}_{{i}^{\prime}})$ such that $\mathcal{C}({D}_{{i}^{\prime}})=({\mathcal{C}}^{1}({D}_{{i}^{\prime}}),{\mathcal{C}}^{2}({D}_{{i}^{\prime}}))$ with ${\ell}^{*}(\mathcal{C}({D}_{{i}^{\prime}}))={\ell}_{1}^{*}(\mathcal{C}({D}_{{i}^{\prime}}))+{\ell}_{2}^{*}(\mathcal{C}({D}_{{i}^{\prime}}))$.
- For any two codes $\mathcal{C}({D}_{{i}^{\prime}})$ and $\mathcal{C}({D}_{{j}^{\prime}})$ with codeword lengths of ${\ell}^{*}(\mathcal{C}({D}_{{i}^{\prime}}))$ and ${\ell}^{*}(\mathcal{C}({D}_{{j}^{\prime}}))$ bits, respectively, $\mathcal{C}({D}_{{i}^{\prime}})\oplus \mathcal{C}({D}_{{j}^{\prime}})$ refers to the bit-wise XOR of bits of $\mathcal{C}({D}_{{i}^{\prime}})$ and $\mathcal{C}({D}_{{j}^{\prime}})$ with zero padding if ${\ell}^{*}(\mathcal{C}({D}_{{i}^{\prime}}))\ne {\ell}^{*}(\mathcal{C}({D}_{{j}^{\prime}}))$. This means $\mathcal{C}({D}_{{i}^{\prime}})\oplus \mathcal{C}({D}_{{j}^{\prime}})$ contains $\mathrm{max}\{{\ell}^{*}(\mathcal{C}({D}_{{i}^{\prime}})),{\ell}^{*}(\mathcal{C}({D}_{{j}^{\prime}}))\}$ bits. For example, if $\mathcal{C}({D}_{{i}^{\prime}})=(101)$ and $\mathcal{C}({D}_{{j}^{\prime}})=(001101)$, then $\mathcal{C}({D}_{{i}^{\prime}})\oplus \mathcal{C}({D}_{{j}^{\prime}})=(101000)\oplus (001101)=(100101)$.

## Appendix E. Proof of Theorem 7

## Appendix F. Proof of Theorem 8

**Proof.**

## Appendix G. Proof of Proposition 4

**Lemma**

**A11.**

**Proof.**

**Lemma**

**A12.**

**Proof.**

**Lemma**

**A13.**

**Proof.**

**Lemma**

**A14.**