On efficient distributed construction of near optimal routing schemes

Abstract

Given a distributed network represented by a weighted undirected graph $G=(V,E)$ on n vertices, and a parameter k, we devise a randomized distributed algorithm that whp computes a routing scheme in $O(n^{1/2+1/k}+D)\cdot n^{o(1)}$ rounds, where D is the hop-diameter of the network. Moreover, for odd k, the running time of our algorithm is $O(n^{1/2 + 1/(2k)} + D) \cdot n^{o(1)}$. Our running time nearly matches the lower bound of $\tilde{\Omega }(n^{1/2}+D)$ rounds (which holds for any scheme with polynomial stretch). The routing tables are of size $\tilde{O}(n^{1/k})$, the labels are of size $O(k\log ^2n)$, and every packet is routed on a path suffering stretch at most $4k-5+o(1)$. Our construction nearly matches the state-of-the-art for routing schemes built in a centralized sequential manner. The previous best algorithms for building routing tables in a distributed small messages model were by Lenzen and Patt-Shamir (In: Symposium on theory of computing conference, STOC’13, Palo Alto, CA, USA, 2013) and Lenzen and Patt-Shamir (In: Proceedings of the 2015 ACM symposium on principles of distributed computing, PODC 2015, Donostia-San Sebastián, Spain, 2015). The former has similar properties but suffers from substantially larger routing tables of size $O(n^{1/2+1/k})$, while the latter has sub-optimal running time of $\tilde{O}(\min \{(nD)^{1/2}\cdot n^{1/k},n^{2/3+2/(3k)}+D\})$.

Appendix: Proof of Theorem 3

Let $X\subseteq V$ be a set of vertices so that each $v\in V$ is sampled to X independently with probability $1/\sqrt{n}$. Define $V'=A\cup X$, and note that with high probability $B=4\sqrt{n}\ln n\ge |V'|$ (since it is given that $|A|\le 2\sqrt{n}\ln n$). Apply the same preprocessing steps as in Sect. 3.3.1 with $V'$ as defined here, to obtain a graph $G''$ on $V'$ satisfying (13).

Computing approximate SPT for $V'$ The first step is to compute the values $(\hat{d}(v),\hat{z}(v))$ for vertices $v\in V'$. Every vertex in $v\in A$ initializes its values as (0, v), while $v\notin A$ sets $(\infty ,\bot )$. Conduct $\beta =\min \{2^{\tilde{O}(\sqrt{\log n})},(\log n)^{O(k)}\}$ iterations of Bellman–Ford rooted at A: at every iteration, every vertex $v\in V'$ broadcasts its pair $(\hat{d}(v),\hat{z}(v))$ to the entire graph, and if $u\in V'$ has $w''(u,v)+\hat{d}(v)<\hat{d}(u)$, then u updates its pair to be $(w''(u,v)+\hat{d}(v),\hat{z}(v))$. (Recall that $w''$ is the edge weight function of $G''$, where the latter is the virtual graph given by Theorem 1 augmented with the hopset edges of Theorem 2.)

The number of rounds required to construct $G''$ is $(n^{1/2+1/(2k)}+D)\cdot \min \{2^{\tilde{O}(\sqrt{\log n})},(\log n)^{O(k)}\}$, and by Lemma 1 this term also bounds the number of rounds it takes to broadcast the $O(|V'|\cdot \beta )$ messages for the Bellman–Ford iterations.

Extending the SPT to V At the end of the $\beta$ iterations of Bellman–Ford, every vertex $u\in V$ knows $(\hat{d}(v),\hat{z}(v))$ for every $v\in V'$. Every vertex $u\in V$ computes

$$\begin{aligned} \hat{d}(u)=\min _{v\in V'}\{d_{uv}+\hat{d}(v)\}, \end{aligned}$$

(40)

and sets $\hat{z}(u)=\hat{z}(v)$, where $v\in V'$ is the minimizer of (40). (Recall that $d_{uv}$ is the value computed in Theorem 1.)

Analysis We assume all the events of Claim 3 hold (which happens with high probability). For $u\in V$ let $z_u\in A$ be a vertex satisfying $d_G(u,z_u)=d_G(u,A)$. Since we performed $\beta$ iterations of Bellman–Ford, using (13) with $v\in V'$ and $z_v\in A\subseteq V'$ we have that $v'$ satisfies (5).

Consider now some $u\in V$, and let $v\in V'$ be the minimizer in (40). The left hand side of (5) holds, as the fact that $v\in V'$ satisfies (5) implies

$$\begin{aligned} d_{uv}+\hat{d}(v)&\mathop {\ge }\limits ^{(2)} d_G^{(B)}(u,v) +d_G(v,A)\\&\ge d_G(u,v)+d_G(v,A)\ge d(u,A). \end{aligned}$$

For the right hand side of (5): in the case that $h(u,z_u)\le B$, by (2) we get that

$$\begin{aligned} \hat{d}(u)\le & {} d_{uz_u}+\hat{d}(z_u)\le (1+\epsilon )d_G^{(B)}(u,z_u)+0 \\= & {} (1+\epsilon )d_G(u,z_u). \end{aligned}$$

Otherwise $h(u,z_u)> B$, and by Claim 3 there exists $v\in X\subseteq V'$ on the shortest path in G from u to $z_u$ with $h(u,v)\le B$. Since (5) holds for v,

$$\begin{aligned} \hat{d}(u)&\le d_{uv}+\hat{d}(v)\\&\mathop {\le }\limits ^{(2)} (1+\epsilon )d_G^{(B)}(u,v)+ (1+\epsilon )d_G(v,A)\\&\le (1+\epsilon )d_G(u,v)+ (1+\epsilon )d_G(v,z_u)\\&=(1+\epsilon )d_G(u,z_u). \end{aligned}$$