Order optimal information spreading using algebraic gossip

Abstract

In this paper we study gossip based information spreading with bounded message sizes. We use algebraic gossip to disseminate $k$ distinct messages to all $n$ nodes in a network. For arbitrary networks we provide a new upper bound for uniform algebraic gossip of $O((k+\log n + D)\varDelta )$ rounds with high probability, where $D$ and $\varDelta$ are the diameter and the maximum degree in the network, respectively. For many topologies and selections of $k$ this bound improves previous results, in particular, for graphs with a constant maximum degree it implies that uniform gossip is order optimal and the stopping time is $\varTheta (k + D)$. To eliminate the factor of $\varDelta$ from the upper bound we propose a non-uniform gossip protocol, TAG, which is based on algebraic gossip and an arbitrary spanning tree protocol $\mathcal{S }$. The stopping time of TAG is $O(k+\log n +d(\mathcal{S })+t(\mathcal{S }))$, where $t(\mathcal{S })$ is the stopping time of the spanning tree protocol, and $d(\mathcal{S })$ is the diameter of the spanning tree. We provide two general cases in which this bound leads to an order optimal protocol. The first is for $k=\varOmega (n)$, where, using a simple gossip broadcast protocol that creates a spanning tree in at most linear time, we show that TAG finishes after $\varTheta (n)$ rounds for any graph. The second uses a sophisticated, recent gossip protocol to build a fast spanning tree on graphs with large weak conductance. In turn, this leads to the optimally of TAG on these graphs for $k=\varOmega (\text{ polylog }(n))$. The technique used in our proofs relies on queuing theory, which is an interesting approach that can be useful in future gossip analysis.

Appendix

1.1 Proof of Lemma 10

Lemma 10

(restated). For any connected graph $G_n$ with $n$ nodes, the sum of the degrees of the nodes along any shortest path between any two nodes $v$ and $u$ is at most $3n$.

Proof

Without loss of generality, consider a BFS spanning tree of $G$ rooted at some node $v$, and some arbitrary leaf $u$. We will find the maximum degree of the node located on the path $(v\rightarrow u)$ at distance $i$ from the root $v$. Clearly, such a node can be connected only to the following nodes:

• Nodes that are located at distance $i-1$ from the root. (It can not be connected to the nodes that are closer to the root (than $i-1$) since then, its distance from the root would be $i-1$ which contradicts the given BFS execution).

• Nodes that are at the same distance $i$ from the root.

• Nodes that are located at distance $i+1$ from the root. (It can not be connected to the nodes that are farther from the root (than $i+1$) since then, their distance from the root would be $i+1$ which contradicts the given BFS execution).

Let us define $m_i$ as the number of nodes at distance $i$ from the root. Clearly, $\sum _{i=0}^{n-1}m_i=n$. (The node at distance $0$ is the root $v$). The degree of a node (at distance $i$ from the root) can be at most: $d_i\le (m_{i-1}+m_{i}+m_{i+1})$. Thus, the sum of degrees on a path of length $l$ from the root to a leaf is at most: $d=\sum _{i=0}^{l}d_i$. Since $l\le n-1,\, d=\sum _{i=0}^{l}d_i\le \sum _{i=0}^{n-1}d_i = \sum _{i=0}^{n-1}(m_{i-1}+m_{i}+m_{i+1}) \le 3n$. $\square$

1.2 Proof of Lemma 11

Lemma 11

(restated). Let $X$ be a sum of $m$ independent and identically distributed geometric random variables (each one with parameter $p>0$) and $\text{ E }\left[ X\right] =\tfrac{m}{p}$. Then, for $\alpha >1$:

$$\begin{aligned} \Pr \left( X \le \alpha \text{ E }\left[ X\right] \right) > 1-\left( \alpha e^{1-\alpha }\right) ^m. \end{aligned}$$

(26)

Proof

First, we will define $Y$ as the sum of $k$ independent Bernoulli random variables, i.e., $Y=\sum _{i=1}^{k}Y_i$, where $Y_i \sim Bernoulli(p)$. Let us notice that:

$$\begin{aligned} \Pr \left( X \le k\right) = \Pr \left( Y \ge m\right) \end{aligned}$$

(27)

The last is true since the event of observing at least $m$ successes in a sequence of $k$ Bernoulli trials implies that the sum of $m$ independent geometric random variables is no more than $k$. On the other hand, if the sum of $m$ independent geometric random variables is no more than $k$ it implies that $m$ successes occurred no later than the $k$-th trial and thus $Y\ge m$.

Now we will use a Chernoff bound for the sum of independent Bernoulli random variables presented in [26]: For any $0<\delta < 1$ and $\mu = \text{ E }\left[ Y\right]$:

$$\begin{aligned} \Pr \left( Y \le (1-\delta )\mu \right) \le \left( \frac{e^{-\delta }}{(1-\delta )^{1-\delta }}\right) ^\mu . \end{aligned}$$

(28)

Since $\mu =\text{ E }\left[ Y\right] =kp$, and by letting $\delta =\frac{kp-m}{kp}$ we obtain:

$$\begin{aligned} \Pr \left( Y \le (1-\delta )\mu \right)&= \Pr \left( Y \le m \right) \le \left( \frac{m}{e^{\frac{m-kp}{m}}kp}\right) ^{-m}.\quad \end{aligned}$$

(29)

$$\begin{aligned} \Pr \left( Y \ge m\right)&> 1-\left( \frac{m}{e^{\frac{m-kp}{m}}kp}\right) ^{-m}\quad \end{aligned}$$

(30)

$$\begin{aligned} \Pr \left( X \le k\right)&> 1- \left( \frac{m}{e^{\frac{m-kp}{m}}kp}\right) ^{-m} \end{aligned}$$

(31)

By substituting $k=\alpha \tfrac{m}{p}=\alpha \text{ E }\left[ X\right]$ (where $\alpha >1$) we obtain:

$$\begin{aligned} \Pr \left( X \le \alpha \text{ E }\left[ X\right] \right) > 1-\left( \frac{e^{\alpha }}{e\alpha }\right) ^{-m} \end{aligned}$$

(32)

$\square$