Near-Additive Spanners In Low Polynomial Deterministic CONGEST Time

TL;DR

This paper presents a low polynomial deterministic algorithm in the CONGEST model for constructing sparse near-additive graph spanners, matching the efficiency of existing randomized algorithms and improving over previous deterministic methods.

Contribution

It introduces the first efficient deterministic CONGEST algorithm for near-additive spanners with arbitrary sparsity, achieving near state-of-the-art performance.

Findings

Deterministic CONGEST algorithm runs in low polynomial time

Constructs arbitrarily sparse near-additive spanners

Matches parameters of randomized algorithms in efficiency and sparsity

Abstract

Given parameters $\alpha\geq 1,\beta\geq 0$, a subgraph $G'=(V,H)$ of an $n$-vertex unweighted undirected graph $G=(V,E)$ is called an $(\alpha,\beta)$-spanner if for every pair $u,v\in V$ of vertices, $d_{G'}(u,v)\leq \alpha d_{G}(u,v)+\beta$. If $\beta=0$ the spanner is called a multiplicative $\alpha$-spanner, and if $\alpha = 1+\epsilon$, for an arbitrarily small $\epsilon>0$, the spanner is said to be a near-additive one. Graph spanners are a fundamental and extremely well-studied combinatorial construct, with a multitude of applications in distributed computing and in other areas. Near-additive spanners, introduced in [EP01], preserve large distances much more faithfully than multiplicative spanners. Also, recent lower bounds [AB15] ruled out the existence of arbitrarily sparse purely additive spanners (i.e., spanners with $\alpha=1$), and therefore near-additive spanners provide the best approximation of distances that one can hope for. Numerous distributed algorithms for constructing sparse near-additive spanners exist. In particular, there are now known efficient randomized algorithms in the CONGEST model that construct such spanners [EN17], and also there are efficient deterministic algorithms in the LOCAL model [DGPV09]. The only known deterministic CONGEST-model algorithm for the problem [Elk01] requires superlinear time in $n$. We remedy the situation and devise an efficient deterministic CONGEST-model algorithm for constructing arbitrarily sparse near-additive spanners. The running time of our algorithm is low polynomial, i.e., roughly $O(\beta \cdot n^\rho)$, where $\rho > 0$ is an arbitrarily small positive constant that affects the additive term $\beta$. In general, the parameters of our algorithm and of the resulting spanner are at the same ballpark as the respective parameters of the state-of-the-art randomized algorithm for the problem due to [EN17].

Equations

$$\beta ={\left(\frac{O\left(\log \kappa \rho +{\rho }^{-1}\right)}{\rho \cdot ϵ}\right)}^{\log \kappa \rho +{\rho }^{-1}+O(1)}.$$

$$R_0=0\text{ and }{R}_{i+1}=2{ϵ}^{-i}/\rho +(5/\rho )R_i.$$

$${\delta }_i={ϵ}^{-i}+2R_i.$$

$$V=V{P}_0\supseteq V{P}_1\supseteq \cdots \supseteq V{P}_{\ell }\supseteq V{P}_{\ell +1}=\varnothing .$$

$$V{U}^{(i-1)}\cap V{P}_i=\varnothing .$$

$$\begin{array}[]{clllll}R_{i+1}&=&({2}/{\rho})\cdot\epsilon^{-i}+\left({5}/{\rho}\right)\sum_{j=0}^{i-1}\frac{2}{\rho}\left(\frac{1}{\epsilon}\right)^{j}\left({5}/{\rho}\right)^{i-1-j}&=&\sum_{j=0}^{i}\frac{2}{\rho}\left(\frac{1}{\epsilon}\right)^{j}\left({5}/{\rho}\right)^{i-j}.\end{array}$$

$$\begin{array}[]{cllllllllllll}R_{i}&\leq&\frac{4}{\rho}\cdot\epsilon^{-{(i-1)}}.\end{array}$$

$$\begin{array}[]{cllllllll}Rad(P_{i})&=&Rad(\widehat{P}_{i-1})&\leq&R_{i}&\leq&\frac{4}{\rho}\cdot\epsilon^{-{(i-1)}}.\end{array}$$

$${\delta }_i\le O\left((1/ϵ{)}^i\right).$$

$$\begin{array}[]{ccccccc}\sum_{i=0}^{\ell}O\left(\rho^{-1}\cdot\delta_{i}\cdot n^{\rho}\right)&\leq&n^{\rho}\cdot\rho^{-1}\sum_{i=0}^{\ell}O\left(\epsilon^{-i}\right)&\leq&O\left(n^{\rho}\cdot\rho^{-1}\cdot\epsilon^{-{\ell}}\right).\end{array}$$

$$|P_{i_0+1}|\le n^{1-\frac{2^{i_0+1}-1}{\kappa }}.$$

$$|P_{i_0+1}|\le n^{1-\frac{2^{i_0+1}-1}{\kappa }}=n^{1-\frac{2^{\lfloor \log \kappa \rho \rfloor +1}-1}{\kappa }}\le n^{1+\frac{1}{\kappa }-\rho }=n^{1+\frac{1}{\kappa }-(i_0+1-i_0)\rho }.$$

$$|{\hat{P}}_i|\le |P_i|/n^{\rho }\le n^{1+\frac{1}{\kappa }-(i-i_0)\rho -\rho }=n^{1+\frac{1}{\kappa }-(i+1-i_0)\rho }.$$

$$O(n+n^{1+1/\kappa }\cdot {\delta }_i).$$

$$\begin{array}[]{ccccccc}\sum_{i=0}^{\ell}O\left(n^{1+{1}/{\kappa}}\cdot\delta_{i}\right)&\leq&\sum_{i=0}^{\ell}O\left(n^{1+{1}/{\kappa}}\cdot\epsilon^{-i}\right)&\leq&O\left(n^{1+{1}/{\kappa}}\cdot\epsilon^{-{\ell}}\right)\end{array}$$

$$\begin{array}[]{cllll}d_{H}(w,r_{C^{\prime}})&\leq&3R_{j}+1+R_{i}&\leq&2R_{i}+1.\end{array}$$

$$d_H(u,v)=\left(1+(30\cdot ϵ\cdot i){\rho }^{-1}\right)d_G(u,v)+6\sum _{j=0}^i{R}_j\cdot 2^{i-j}$$

$$\begin{array}[]{lllllllllll}d_{G}(r_{C_{l}},r_{C_{r}})&\leq d_{G}(r_{C_{l}},z_{l})+d_{G}(z_{l},z_{r})+d_{G}(z_{r},r_{C_{r}})&\leq d_{G}(z_{l},z_{r})+2R_{i}&\leq\epsilon^{-i}+2R_{i}&=\delta_{i}.\end{array}$$

$$\begin{array}[]{lllllllllll}d_{H}(x,w_{l})&\leq&\left(1+{(30\epsilon(i-1))\cdot\rho^{-1}}\right)\cdot d_{G}(x,w_{l})+6\sum_{j=1}^{i-1}R_{j}\cdot 2^{i-1-j}\\ d_{H}(w_{r},y)&\leq&\left(1+{(30\epsilon(i-1))\cdot\rho^{-1}}\right)\cdot d_{G}(w_{r},y)+6\sum_{j=1}^{i-1}R_{j}\cdot 2^{i-1-j}.\end{array}$$

$$\begin{array}[]{lllll}d_{H}(x,y)&\leq&d_{H}(x,w_{l})+d_{H}(w_{l},r_{C_{l}})+d_{H}(r_{C_{l}},r_{C_{r}})+d_{H}(r_{C_{r}},w_{r})+d_{H}(w_{r},y)\\ &\leq&d_{H}(x,w_{l})+2R_{i}+1+d_{G}(z_{l},z_{r})+2R_{i}+2R_{i}+1+d_{H}(w_{r},y)\\ &\leq&\left(1+{(30\epsilon(i-1))}{\cdot\rho^{-1}}\right)d_{H}(x,y)+6R_{i}+2\cdot 6\sum_{j=1}^{i-1}R_{j}\cdot 2^{i-1-j}\\ &=&\left(1+{(30\epsilon(i-1))}{\cdot\rho^{-1}}\right)d_{H}(x,y)+6\sum_{j=1}^{i}R_{j}\cdot 2^{i-j}.\end{array}$$

$$\begin{array}[]{lllll}d_{H}(u,v)&\leq&\left(1+\frac{30\epsilon(i-1)}{\rho}\right)d_{G}(u,v)+\left\lfloor\frac{d_{G}(u,v)}{\epsilon^{-i}}\right\rfloor 6\sum_{j=1}^{i}R_{j}\cdot 2^{i-j}+\par 6\sum_{j=1}^{i}R_{j}\cdot 2^{i-j}\\ &\leq&d_{G}(u,v)\left[\left(1+\frac{30\epsilon(i-1)}{\rho}\right)+6\epsilon^{i}\cdot\sum_{j=1}^{i}R_{j}\cdot 2^{i-j}\right]+6\sum_{j=1}^{i}R_{j}\cdot 2^{i-j}.\\ \end{array}$$

$$\begin{array}[]{cllllllllllll}6\sum_{j=1}^{i}R_{j}\cdot 2^{i-j}\par&\leq&\frac{24}{\rho\cdot\epsilon^{i-1}\cdot(1-2\epsilon)}\par&\leq&\frac{30}{\rho\cdot\epsilon^{i-1}}.\\ \end{array}$$

$$\begin{array}[]{lllll}d_{H}(u,v)&\leq d_{G}(u,v)\left[1+\frac{30\cdot\epsilon\cdot(i-1)}{\rho}+\par\frac{30\epsilon}{\rho}\right]\par+6\sum\limits_{j=1}^{i}R_{j}2^{i-j}\par&\leq d_{G}(u,v)\left(1+\frac{30\cdot\epsilon\cdot i}{\rho}\right)\par+6\sum\limits_{j=1}^{i}R_{j}2^{i-j}.\end{array}$$

$$\begin{array}[]{lcccccccc}\beta&=&\left(({30\cdot\ell})/({\rho\cdot\epsilon^{\prime}})\right)^{\ell}&=&\epsilon^{-{\ell}}.\end{array}$$

$$\beta ={\left(\frac{30\cdot \ell }{\rho \cdot {ϵ}^{\prime }}\right)}^{\ell }={\left(\frac{30\cdot (\log \kappa \rho +{\rho }^{-1}+O(1))}{\rho \cdot {ϵ}^{\prime }}\right)}^{\log \kappa \rho +{\rho }^{-1}+O(1)}.$$

$$\begin{array}[]{lclclclc}|H|&=O(n^{1+{1}/{\kappa}}\cdot\epsilon^{-{\ell}})&=&O(\beta\cdot n^{1+{1}/{\kappa}}).\end{array}$$

$$\begin{array}[]{lclclclc}O\left(n^{\rho}\cdot\rho^{-1}\epsilon^{-{\ell}}\right)=O\left(\beta\cdot n^{\rho}\cdot\rho^{-1}\right).\par\end{array}$$

$$\beta ={\left(\frac{O\left(\log \kappa \rho +{\rho }^{-1}\right)}{\rho \cdot ϵ}\right)}^{\log \kappa \rho +{\rho }^{-1}+O(1)}.$$

Full text

Graph spanners [Awe85, PS89] are a fundamental and extremely well-studied combinatorial construct, with a multitude of applications in distributed computing and in other areas. Near-additive spanners, introduced in [EP01], preserve large distances much more faithfully than the more traditional multiplicative spanners. Also, recent lower bounds [AB15] ruled out the existence of arbitrarily sparse purely additive spanners (i.e., spanners with $\alpha=1$), and therefore showed that essentially near-additive spanners provide the best approximation of distances that one can hope for.

Numerous distributed algorithms, for constructing sparse near-additive spanners, were devised in [Elk01, EZ06, DGPV09, Pet10, EN17]. In particular, there are now known efficient randomized algorithms in the CONGEST model that construct such spanners [EN17], and also there are efficient deterministic algorithms in the LOCAL model [DGPV09]. However, the only known deterministic CONGEST-model algorithm for the problem [Elk01] requires super-linear time in $n$. In this paper we remedy the situation and devise an efficient deterministic CONGEST-model algorithm for constructing arbitrarily sparse near-additive spanners.

The running time of our algorithm is low polynomial, i.e., roughly $O(\beta\cdot n^{\rho})$, where $\rho>0$ is an arbitrarily small positive constant that affects the additive term $\beta$. In general, the parameters of our new algorithm and of the resulting spanner are at the same ballpark as the respective parameters of the state-of-the-art randomized algorithm for the problem due to [EN17].

1 Introduction

Graph spanners were introduced in the context of distributed algorithms by Awerbuch, Peleg and their co-authors [Awe85, PU87, PS89, PU89, AP90] in the end of the eighties, and were extensively studied since then. See, e.g., [DGPV09, GK18, GP17, ABP17, AB15, EN17, EP01, BS07, BKMP10], and the references therein. Numerous applications of spanners in Distributed Algorithms and other related areas were discovered in [Awe85, PU87, ACIM99, SS99, DHZ00, EP01, Elk05, EZ06, HKN16, TZ01, RTZ05]. Also, spanners are a fundamental combinatorial construct, and their existential properties are being extensively explored [ADD*+*93, PS89, EP01, BKMP10, TZ06, Pet09, AB15, ENS15, ABP17].

In this paper, we focus on unweighted, undirected graphs. Given a graph $G=(V,E)$, a subgraph $G^{\prime}=(V,H)$, $H\subseteq E$, is said to be a (multiplicative) $t$-spanner of $G$, if for every pair $u,v\in V$ of vertices, we have $d_{G^{\prime}}(u,v)\leq t\cdot d_{G}(u,v)$, for a parameter $t\geq 1$, where $d_{G}$ (respectively, $d_{G^{\prime}}$) stands for the distance in the graph $G$ (respectively, in the subgraph $G^{\prime}$).

A fundamental tradeoff for multiplicative spanners is that for any $\kappa=1,2,\dots$, and any $n$-vertex graph $G=(V,E)$, there exists a $(2\kappa-1)$-spanner with $O(n^{1+{1}/{\kappa}})$ edges [ADD*+*93, PS89]. Assuming Erdös-Simonovits girth conjecture (see e.g., [ES82]), this tradeoff is optimal. Efficient distributed algorithms for constructing multiplicative spanners that (nearly) realize this tradeoff were given in [ABCP93, Coh98, BS07, Elk07, DGPV08, EN17, DMZ10, GP17, GK18].

A different variety of spanners, called near-additive spanners, was presented by Elkin and Peleg in [EP01]. They showed that for every $\epsilon>0$ and $\kappa=1,2,\dots$, there exists $\beta=\beta(\kappa,\epsilon)$, such that for every $n$-vertex graph $G=(V,E)$ there exists a $(1+\epsilon,\beta)$-spanner $G^{\prime}=(V,E)$, $H\subseteq E$, with $O_{\epsilon,\kappa}(n^{1+{1}/{\kappa}})$ edges. (The subgraph $G^{\prime}$ is called a $(1+\epsilon,\beta)$-spanner of $G$ if for every pair $u,v\in V$ of vertices, it holds that $d_{G^{\prime}}(u,v)\leq(1+\epsilon)d_{G}(u,v)+\beta$.)

In [EP01], the additive error $\beta$ is given by $\beta_{EP}=\left(\frac{{\log\kappa}}{\epsilon}\right)^{{\log\kappa}-1}$, and it remains the state-of-the-art. These spanners approximate large distances much more faithfully than multiplicative spanners. Recent lower bounds of Abboud and Bodwin [AB15], based on previous results of Coppersmith and Elkin [CE05], showed that this type of spanners is essentially the best one can hope for, as in general arbitrarily sparse purely additive spanners (i.e., with $\epsilon=0$) do not exist. Abboud et al. [ABP17] showed that the dependence on $\epsilon$ in [EP01] is nearly tight. Specifically, they showed a lower bound of $\beta=\Omega\left(\frac{1}{\epsilon{\log k}}\right)^{{\log\kappa}-1}$.

Near-additive spanners were extensively studied in the past two decades [EP01, Elk05, EZ06, TZ06, DGP07, DGPV09, Pet09, Pet10, AB15, ABP17, EN17], both in the context of distributed algorithms and in other settings. (In particular, they were used in the streaming setting [EZ06, EN17], and for dynamic algorithms [BR11, HKN16].) Efficient distributed deterministic constructions of such spanners using large messages (the so-called LOCAL model; see Section 1.3.1 for relevant definitions) were given by Derbel et al. [DGPV09]. They devised two constructions. In the first one, the running time is $O(\beta)$, and the additive term $\beta$ is is roughly $\epsilon^{-{(\kappa-2)}}$, i.e., exponentially larger than $\beta_{EP}$. In the second construction of [DGPV09], the running time is $O(\beta\cdot 2^{O(\sqrt{{\log n}})})$, and $\beta$ is roughly the same as in [EP01], i.e., $\beta\approx\beta_{EP}$.

Derbel et al. [DGPV09] explicitly raised the question of bulding $(1+\epsilon,\beta)$-spanners in the distributed CONGEST model (i.e., using short messages; see Section 1.3.1 for the definition), and this question remained open since their work. Efficient distributed randomized algorithms (in the CONGEST model) for constructing near-additive spanners were given in [EZ06, Pet10, EN17]. The state-of-the-art algorithm is due to [EN17]. They devised a randomized algorithm that constructs $(1+\epsilon,\beta)$-spanners with $O_{\epsilon,\rho,\kappa}(n^{1+{1}/{\kappa}})$ edges, with a low polynomial running time, that is, running time ${O}_{\epsilon,\kappa,\rho}(n^{\rho})$, for arbitrarily small $\epsilon,\rho,{1}/{\kappa}>0$, where $\beta_{EN}=\beta(\epsilon,\rho,\kappa)=O\left(\frac{{\log\kappa\rho}+\rho^{-1}}{\epsilon}\right)^{{\log\kappa\rho}+\rho^{-1}+O(1)}$. Note that the additive term $\beta_{EN}$ of [EN17] is at the same ballpark as the existential bound $\beta_{EP}$ of [EP01].

The importance of devising deterministic distributed algorithms that work in the CONGEST model for constructing spanners was recently articulated by Barenboim et al. [BEG15], Grossman and Parter [GP17] and Ghaffari and Kuhn [GK18]. All these authors devised algorithms for constructing multiplicative spanners. The only deterministic CONGEST-model algorithm for building $(1+\epsilon,\beta)$-spanners with $\tilde{O}_{\epsilon,\kappa,\rho}(n^{1+{1}/{\kappa}})$ edges is due to [Elk05]. That algorithm suffers however from a superlinear in $n$ running time, $O(n^{1+\frac{1}{2\kappa}})$, and from additive term $\beta_{E}=\left(\frac{\kappa}{\epsilon}\right)^{{\log\kappa}}\cdot(\rho^{-1})^{\rho^{-1}}$, which is significantly larger than $\beta_{EN}$ (the additive term of [EN17]). Also, the number of edges in the resulting spanner is by a $polylog(n)$ factor larger than the desired bound of $O_{\epsilon,\kappa,\rho}(n^{1+{1}/{\kappa}})$ (see [EP01, Pet10, EN17, ABP17]).

In the current paper we devise a distributed deterministic algorithm that works in the CONGEST model, and for any $\epsilon>0,\rho>0$ and $\kappa=1,2,\dots,$ and for any $n$-vertex graph, it constructs a $(1+\epsilon,\beta)$-spanner with $O_{\epsilon,\kappa,\rho}(n^{1+{1}/{\kappa}})$ edges in low polynomial, specifically, $O_{\epsilon,\kappa,\rho}(n^{\rho})$ time, and its additive term $\beta$ is at the same ballpark as $\beta_{EN}$. Specifically, our additive term satisfies:

[TABLE]

See Table 1 for a concise comparison of our algorithm with the only previously-existing distributed deterministic CONGEST-model algorithm for constructing near-additive spanners [Elk01]. See also Table 2 in Appendix B for a concise overview of distributed algorithms, both deterministic and randomized, in both LOCAL and CONGEST models, for computing near-additive spanners.

1.1   Technical Overview and Related Work

Our algorithm builds upon Elkin-Peleg’s superclustering-and-interconnection approach (see [EP01, EN16, EN17] for an elaborate discussion of this technique). Its randomized distributed implementation by Elkin and Neiman [EN17] relies on random sampling of clusters. Those sampled clusters join nearby unsampled clusters to create superclusters, and this is iteratively repeated.

Our basic idea is to replace the random sampling by constructing ruling sets (see Section 1.3.2 for definition) using a deterministic procedure by Schneider et al. [SEW13]. Recently, Ghaffari and Kuhn [GK18] used network decompositions for derandomizing algorithms that construct multiplicative spanners, hitting and dominating sets. All existing deterministic constructions of network decompositions [AGLP89, PS96, BEG15] employ ruling sets. On the other hand, our approach avoids constructing network decomposition, but rather directly uses ruling sets for derandomization.

1.2   Outline

Section 1.3 provides necessary definitions for understanding this paper. Section 2 contains our spanner construction along with an analysis of the running time, size and stretch of the spanner. Appendix A contains a variant of the Bellman-Ford algorithm that is used by the construction in Section 2. Finally, Appendix B contains a table that summarizes known results concerning near-additive spanners.

1.3   Preliminaries

1.3.1   The Computational Model

In the distributed model [Pel00] we have processors residing in vertices of the graph. The processors communicate with their graph neighbors in synchronous rounds. In the CONGEST model, messages are limited to $O(1)$ words, i.e., $O(1)$ edge weights or ID numbers. In the LOCAL model, the length of messages is unbounded. The running time of an algorithm in the distributed model is the worst case number of communication rounds that the algorithm requires. Throughout this paper, we assume that all vertices have unique IDs such that for all $v,\ v.ID\in[n]$, and all vertices know their ID. Moreover, we assume that all vertices know the number of vertices $n$. In fact, our results apply even if vertices know an estimate $\tilde{n}$ for $n$, where $n\leq\tilde{n}\leq poly(n)$.

1.3.2   Ruling Sets

Given a graph $G=(V,E)$, a set of vertices $W\subseteq V$ and parameters $\zeta,\eta\geq 0$, a set of vertices $A\subseteq W$ is said to be a $(\zeta,\eta)$-ruling set for $A$ if for every pair of vertices $u,v\in A$, the distance between them in $G$ is at least $\zeta$, and for every $u\in W$ there exists a representative $v\in A$ such that the distance between $u,v$ is at most $\eta$.

2 A Deterministic Distributed Construction of Near-Additive Spanners

2.1   Overview

Let $G=(V,E)$ be an unweighted, undirected graph on $n$ vertices, and let $\epsilon>0$, $\kappa=1,2,\ldots$ and ${1}/{\kappa}\leq\rho<{1}/{2}$. The algorithm constructs a sparse $(1+\epsilon,\beta)-spanner$ $H=(V,E_{H})$ of $G$ where $\beta(\kappa,\epsilon,\rho)$ is a function of the parameters $\kappa,\epsilon,\rho$, given by eq. 1, and $|H|=O(\beta n^{1+{1}/{\kappa}})$, in deterministic time ${O}_{\epsilon,\kappa,\rho}(n^{\rho})$ in the CONGEST model.

The overall structure of our algorithm is reminiscent of that in [EN17]. However, unlike the algorithm of [EN17], the current algorithm does not use any randomization. We first provide a high-level overview of the algorithm.

The algorithm begins by initializing $E_{H}$ as an empty set, and proceeds in phases. It begins by partitioning $V$ into singletons clusters $P_{0}=\{\{v\}\ |\ v\in V\}$. Each phase $i$, for $i=0,\ldots,\ell$, receives as input a collection of clusters $P_{i}$, and distance and degree threshold parameters $\delta_{i},\deg_{i}$. The parameters $\delta_{i}$ and $deg_{i}$ will be specified later. We set the maximum index of a phase $\ell$ by $\ell=\left\lfloor{\log\kappa\rho}\right\rfloor+\lceil\frac{\kappa+1}{\kappa\rho}\rceil-1$, as in [EN17].

Each of the clusters constructed by our algorithm will have a designated center vertex. Throughout the algorithm, we denote by $r_{C}$ the center of the cluster $C$ and say that $C$ is centered around $r_{C}$. For a cluster $C$, define $Rad(C)=max\{d_{H}(r_{C},v)\ |\ v\in C\}$, and for a set of clusters $P_{i}$, define $Rad(P_{i})=max\{Rad(C)\ |\ C\in P_{i}\}$. For a collection $P_{i}$, we denote by $S_{i}$ the set of centers of $P_{i}$, i.e., $S_{i}=\{r_{C}\ |\ C\in P_{i}\}$.

Intuitively, in each phase we wish to add paths between pairs of cluster centers that are close to one another. However, if a center has many centers close to it, i.e., it is popular, this can add too many edges to the spanner. Two cluster centers $r_{C},r_{C^{\prime}}$ are said to be close, if $d_{G}(r_{C},r_{C^{\prime}})\leq\delta_{i}$. A cluster $C$ and its center $r_{C}$ are said to be popular if $r_{C}$ has at least $deg_{i}$ cluster centers that are close to it. In order to avoid adding too many edges to the spanner, each phase consists of two steps, the superclustering step and the interconnection step.

The superclustering step of phase $i$ detects popular clusters, and builds larger clusters around them (see Figure 1 for an illustration). For each new cluster $C$, a BFS tree of the cluster $C$ is added to the spanner $H$ (see Figure 2). The collection of new clusters is the input for phase $i+1$. This allows us to defer the work on highly dense areas in the graph to later phases of the algorithm.

In the interconnection step of phase $i$, clusters that have not been superclustered in this phase are connected to one another. For each center $r_{C}$ that is not superclustered in this phase, paths are added to all centers of clusters that are close to it. As the center $r_{C}$ is not superclustered, it is not popular, and so we will not add too many paths to the resulting spanner $H$.

In the last phase $\ell$, the superclustering step is skipped and we move directly to the interconnection step. We will show that the number of clusters in the final collection of clusters $P_{\ell}$ is small. Specifically, we show that even if every pair of clusters in $P_{\ell}$ is interconnected by a path, the number of added edges will still be relatively small. Thus the superclustering step of phase $\ell$ can be safely skipped. This concludes the high-level overview of the algorithm.

The distance parameters are defined as follows. Define

[TABLE]

We will show (see Lemma 2.3 below) that $R_{i}$ is an upper bound on the radius of clusters in $P_{i}$, that is, $Rad(P_{i})\leq R_{i}$ for all $0\leq i\leq\ell$. Also, the distance threshold parameter $\delta_{i}$ is given by:

[TABLE]

The degree threshold $deg_{i}$ affects the number of clusters in each phase, and thus the number of phases of the algorithm. It also affects the number of edges added to the spanner by the interconnection step. Ideally, we would like to set $deg_{i}=n^{\frac{2^{i}}{\kappa}}$ for all $i$. However, the algorithm requires $\Omega(deg_{i})$ time to execute phase $i$. Thus, we must keep $deg_{i}\leq n^{\rho}$ for all phases, as we aim at running time of roughly $O(n^{\rho})$.

For this reason, we partition phases $0,1,\dots,\ell-1$ into two stages, the exponential growth stage and the fixed growth stage. In the exponential growth stage, that consists of phases $0,\ldots,i_{0}=\left\lfloor{\log(\kappa\rho)}\right\rfloor$, we set $deg_{i}=n^{\frac{2^{i}}{\kappa}}$. In the fixed growth stage, which consists of phases $i_{0}+1,\ldots,i_{1}=i_{0}+\lceil\frac{\kappa+1}{\kappa\rho}\rceil-2=\ell-1$, we set $deg_{i}=n^{\rho}$. Observe that for every index $i$, we have $deg_{i}\leq n^{\rho}$. The concluding phase $\ell$ is not a part of either these stages, as the number of clusters in $P_{\ell}$ is at most $n^{\rho}$. For notational purposes, we define $deg_{\ell}=n^{\rho}$. However, we note that as there are at most $n^{\rho}$ clusters in $P_{\ell}$, there are no popular clusters in this phase.

We will now discuss the differences between the current algorithm and the algorithm of [EN17]. In [EN17], the superclustering step uses a randomized selection of vertices to cover the popular cluster centers. The current algorithm replaces this selection with a deterministic procedure that computes a ruling set that covers these centers. For this aim, we use the algorithm of [SEW13, KMW18] that computes ruling sets efficiently in the CONGEST model. As a result of the deterministic procedure, the distance parameter (and as a result, the radii of clusters) for each phase in the current algorithm is larger than in [EN17]. For this reason, the additive term that the current algorithm provides is slightly inferior to the additive term of [EN17]. The number of phases and the degree parameter sequence remain as in [EN17].

The deterministic detection of popular clusters’ centers requires vertices to acquire information regarding their $\delta_{i}$-neighborhood. This information is later utilized by our algorithm in the interconnection step as well. Thus, while in the algorithm of [EN17] the interconnection step executes Bellman-Ford explorations, the current algorithm relies on already acquired information. Therefore, the current execution of the interconnection step is simpler than the execution of the interconnection step in [EN17].

2.2   Superclustering

This section provides details of the execution of the superclustering steps for all phases $i\in[0,i_{1}=\ell-1]$, i.e., all phases other than the (concluding) phase $\ell$ on which there is no superclustering step. Recall that for the exponential growth stage, we have $deg_{i}=n^{\frac{2^{i}}{\kappa}}$. For the fixed growth stage, we have $deg_{i}=n^{\rho}$. The input to phase $i$ is a set of clusters $P_{i}$. The phase begins by detecting popular clusters. To do so, we employ Algorithm 1, described in Appendix A, with the input $(G,P_{i},\deg_{i},\delta_{i})$. The following theorem, which is proven in Appendix A, summarizes the properties of the returned vertex set $W_{i}$.

Theorem 2.1**.**

Given a graph $G=(V,E)$, a collection of clusters $P_{i}$ centered around cluster centers $S_{i}$ and parameters $\delta_{i},\ \deg_{i}$, Algorithm 1 returns a set $W_{i}$ in $O(deg_{i}\cdot\delta_{i})$ time such that:

$W_{i}$* is the set of all centers of popular clusters from $P_{i}$.* 2. 2.

Every cluster center $r_{C}\in S_{i}$ that did not join $W_{i}$ knows the identities of all the centers $r_{C^{\prime}}\in S_{i}$ such that $d_{G}(r_{C},r_{C^{\prime}})\leq\delta_{i}$. Furthermore, for each pair of such centers $r_{C},r_{C^{\prime}}$, there is a shortest path $\pi$ between them such that all vertices on $\pi$ know their distance from $r_{C^{\prime}}$.

Next, we wish to select a subset of vertices from $W_{i}$ to grow large clusters around them. On the one hand, the number of clusters in $P_{i+1}$ must be significantly smaller than the number of clusters in $P_{i}$. On the other hand, all cluster centers in $W_{i}$ must be superclustered. In the centralized version of this algorithm, Elkin and Peleg [EP01] run multiple consecutive scans to detect popular clusters and build superclusters around them. In the distributed model of computation, this requires too much time. In the distributed randomized version of this algorithm, Elkin and Neiman [EN17] randomly select centers to grow superclusters around. In this deterministic distributed version, we select vertices by constructing a ruling set for the set $W_{i}$. We use the algorithm given in [SEW13, KMW18], on the set $W_{i}$ with parameters $q=2\delta_{i},c=\rho^{-1}$. The following theorem summarizes the properties of the returned ruling set $RS_{i}$.

Theorem 2.2**.**

[SEW13, KMW18]* Given a graph $G=(V,E)$, a set of vertices $W_{i}\subseteq V$ and parameters $q\in\{1,2,\ldots\},c>1$, one can compute a $(q+1,cq)$-ruling subset for $W_{i}$ in $O(q\cdot c\cdot n^{\frac{1}{c}})$ deterministic time, in the CONGEST model.*

By Theorem 2.2, the returned subset $RS_{i}$ is a $(2\delta_{i}+1,({2}/{\rho})\cdot\delta_{i})$-ruling set for the set $W_{i}$.

The ruling set $RS_{i}$ is $(2\delta_{i}+1)$-separated. This is done in order to guarantee that the sets of vertices in radius $\delta_{i}$ around each vertex in $RS_{i}$ are pairwise disjoint (for an illustration, see Figure 3). This allows us to bound the size of the ruling set $RS_{i}$, and eventually the size of the collection $P_{i+1}$.

We are now ready to create large superclusters. A BFS exploration rooted at the set $RS_{i}$ is executed to depth $({2}/{\rho})\cdot\delta_{i}$ in $G$. As a result a forest $F_{i}$ is constructed, rooted at vertices of $RS_{i}$.

For a cluster center $r_{C^{\prime}}\in S_{i}\backslash RS_{i}$ that is spanned by $F_{i}$, let $r_{C}$ be the root of the forest tree of $F_{i}$ to which $r_{C^{\prime}}$ belongs. The cluster $C^{\prime}$ now becomes superclustered in the cluster $\widehat{C}$ centered around $C$.

The center $r_{C}$ of $C$ becomes the new cluster center of $\widehat{C}$, i.e., $r_{\widehat{C}}\leftarrow r_{C}$. The vertex set of the new supercluster $\widehat{C}$ is the union of the vertex set of the original cluster $C$, with the vertex sets of all clusters $C^{\prime}$ which are superclustered into $\widehat{C}$. We denote by $V({C})$ the vertex set of a cluster ${C}$.

For every cluster center $r_{C^{\prime}}$ that is spanned by the tree in $F_{i}$ rooted at $r_{C}$, the path in $F_{i}$ from $r_{C}$ to $r_{C^{\prime}}$ is added to the spanner $H$ (see Figure 4). Note that the path itself is not added to the cluster $\widehat{C}$. Recall that $H$ is initialized as an empty set.

We define $\widehat{P_{i}}$ as the set of new superclusters, $\widehat{C}$, that were built by the superclustering step of phase $i$. We set $P_{i+1}=\widehat{P_{i}}$. Note that the set $S_{i+1}$ of cluster centers of $P_{i+1}$ is given by $S_{i+1}=RS_{i}$.

Next we show that $R_{i}$ is an upper bound on $Rad(P_{i})$, the radius of clusters in phase $i$.

Lemma 2.3**.**

For all integer $0\leq i\leq\ell$, we have $Rad(P_{i})\leq R_{i}$.

Proof.

We will prove the lemma by induction on the index of the phase $i$. For $i=0$, observe that $Rad(P_{0})=0$, and $R_{0}=0$ by definition. Thus the induction base case holds. For the inductive step, recall that by definition $R_{i+1}={2\epsilon^{-i}}/{\rho}+\left({5}/{\rho}\right)R_{i}$.

For analyzing $Rad(\widehat{P_{i}})$, consider a vertex $u\in\widehat{C}$. If $u\in C$, by the induction hypothesis, we have $d_{H}(r_{C},u)\leq R_{i}$, and since $\rho<1$ we have that $R_{i}\leq R_{i+1}$. Otherwise, $u\in C^{\prime}$ for some $C^{\prime}\in P_{i}$ such that $C^{\prime}$ is clustered into $\widehat{C}$ in the superclustering step of phase $i$. By the induction hypothesis, we have $d_{H}(r_{C^{\prime}},u)\leq R_{i}$. Since $C^{\prime}$ is clustered into $\widehat{C}$, we have $d_{G}(r_{C},r_{C^{\prime}})\leq\frac{2}{\rho}\delta_{i}$. Moreover, a shortest $r_{C}-r_{C^{\prime}}$ path was added to $H$, and by eq. 3, we have $d_{H}(r_{C},r_{C^{\prime}})\leq\frac{2}{\rho}\delta_{i}=\rho^{-1}\left[2\epsilon^{-i}+4R_{i}\right]$. Since $\rho<1$, we have $d_{H}(r_{C},u)\ \leq\ R_{i+1}$. Therefore, $Rad(P_{i+1})=Rad(\widehat{P_{i}})\leq R_{i+1}$. ∎

Lemma 2.4**.**

All popular clusters in phase $i$ are superclustered into clusters of $\widehat{P}_{i}$.

Proof.

Let $C^{\prime}$ be a popular cluster. Then, $r_{C^{\prime}}$ belongs to the set $W_{i}$ returned by Algorithm 1. There is a vertex $r_{C}$ in the ruling subset $RS_{i}$ computed for the set $W_{i}$, such that $d_{G}(r_{C},r_{C^{\prime}})\leq\frac{2}{\rho}\delta_{i}$. Hence the BFS exploration that is executed from the set $RS_{i}$ to depth $\frac{2}{\rho}\delta_{i}$ reaches $r_{C^{\prime}}$. Thus $r_{C^{\prime}}$ is spanned by the forest $F_{i}$, and it is superclustered into some supercluster in $\widehat{P}_{i}$. ∎

This concludes the analysis of the superclustering step of phase $i$.

2.3   Interconnection

Next we provide the details of the execution of the interconnection step. Let $i\in[0,\ell]$. Denote by $U_{i}$ the set of clusters of $P_{i}$ which were not superclustered into clusters of $\widehat{P}_{i}$. For phase $\ell$, the superclustering step is skipped. Therefore, we set $U_{\ell}=P_{\ell}$.

In the interconnection step for $i\geq 0$, we wish to connect clusters $C\in U_{i}$ to all the clusters $C^{\prime}\in P_{i}$ that are close to them, i.e., such that $d_{G}(r_{C},r_{C^{\prime}})\leq\delta_{i}$. Note that we will connect a cluster $C\in U_{i}$ to all clusters $C^{\prime}\in P_{i}$ that are close to it. This is regardless of whether $C^{\prime}$ has been superclustered in this phase or not (see Figure 5). However, by Lemma 2.4, every cluster in $U_{i}$ is not popular. As $C\in U_{i}$, it has at most $deg_{i}$ clusters close to it. Thus we can connect it to these other nearby clusters without adding too many edges to the spanner. By Theorem 2.1, the center $r_{C}$ of each such cluster $C$ knows all other centers $r_{C^{\prime}}$ such that $C^{\prime}\in P_{i}$ and $d_{G}(r_{C},r_{C^{\prime}})\leq\delta_{i}$, and the distances to them. Thus, each center $r_{C}$ of a cluster $C\in U_{i}$ already knows which are the clusters it needs to connect to. For each $C^{\prime}\in P_{i}$ such that $d_{G}(r_{C},r_{C^{\prime}})\leq\delta_{i}$, the center $r_{C}$ traces back the message that informed $r_{C}$ regarding $r_{C^{\prime}}$ (see Theorem 2.1), and a shortest path between $r_{C}$ and $r_{C^{\prime}}$ is added to the spanner $H$.

The interconnection of the concluding phase $\ell$ is slightly different. As the superclustering step of this phase is skipped, we employ Algorithm 1, described in Appendix A, with the input $(G,P_{\ell},deg_{\ell},\delta_{\ell})$. The output set $W_{i}$ is an empty set111 Algorithm 1 does not explicitly return the set $W_{i}$. Rather each vertex $v\in V$ knows whether it belongs to the output $W_{i}$ or not. In the execution of Algorithm 1 in phase $\ell$, all vertices know that they do not belong to the output $W_{i}$., but the execution of the algorithm allows vertices to acquire the required information for the interconnection step. This completes the description of the interconnection step.

Denote by $U^{(i)}$ the union of all sets $U_{0},U_{1},\dots,U_{i}$, i.e., $U^{(i)}=\bigcup_{j=0}^{i}U_{j}$. The following corollary shows that the set $U^{(\ell)}$ is a partition of $V$.

Corollary 2.5**.**

The set $U^{(\ell)}$ defined by $U^{(\ell)}=\bigcup_{i=0}^{\ell}U_{i}$ is a partition of $V$.

The following definitions will be used in the proof of Lemma 2.6 below. Denote by $VU_{i}$ the set of vertices $v$ such that there is a cluster $C\in U_{i}$ that contains $v$, i.e., $VU_{i}=\{v\ |\ \exists C\in U_{i}\textit{ such that }v\in C\}$. Denote by $VP_{i}$ the set of vertices $v$ such that there is a cluster $C\in P_{i}$ that contains $v$, i.e., $VP_{i}=\{v\ |\ \exists C\in P_{i}\textit{ such that }v\in C\}$. Denote $VU^{(i)}$ the union of all sets $VU_{0},VU_{1},\dots,VU_{i}$,. i.e., $VU^{(i)}=\{v\ |\ \exists C\in U^{(i)},\textit{ such that }v\in C\}$. For notational convenience, we will also define $P_{\ell+1}$, $VP_{\ell+1}$ and $VU_{\ell+1}$ to be empty sets.

Lemma 2.6**.**

For every index $i\in[0,\ell]$, the set $U^{(i)}$ is a partition of $VU^{(i)}$.

Proof.

The proof is by induction on the index of the phase $i$.

For $i=0$, the set $U_{0}$ is the set of all clusters that were not superclustered in phase [math]. As all these clusters are singletons, we have that $U_{0}$ is a partition of $VU^{(0)}$ to singletons. Note that every vertex $v\in V$ that does not belong to $VU^{(0)}$, belongs to some cluster $C\in P_{1}$.

Assume that $U^{(i-1)}$ is a partition of $VU^{(i-1)}$, for some index $i$ in the range $[1,\ell]$. We will prove that $U^{(i)}$ is a partition of $VU^{(i)}$.

For the induction step, first observe that for every vertex $v$ that belongs to a cluster $\widehat{C}\in P_{i}$, there is a cluster $C\in P_{i-1}$ such that $v$ belongs to $C$. It follows that

[TABLE]

Moreover, each vertex can belong to at most one cluster in $P_{i-1}$. For each cluster $C\in P_{i-1}$, either $C$ has been superclustered into a cluster of $P_{i}$, or it has joined $U_{i-1}$ (note that this condition also applies for $i=\ell+1$, as $P_{\ell+1}=\emptyset$, and $U_{\ell}=P_{\ell}$). These two cases are mutually exclusive. Thus $VU_{i-1}\cup VP_{i}=VP_{i-1}$, and $VU_{i-1}\cap VP_{i}=\emptyset$. By eq. 4, it follows that

[TABLE]

Let $v$ be a vertex in $VU^{(i)}$. We will consider now two complementary cases.

Case 1: $v$ does not belong to $VP_{i}$. Then, $v$ does not belong to $VU_{i}$. Thus it belongs to $VU^{(i-1)}$, and by the induction hypothesis there is exactly one cluster $C\in U^{(i-1)}$ such that $v\in C$. Moreover, since $v$ does not belong to $VP_{i}$, there is no cluster $C^{\prime}\in P_{i}$ such that $v\in C$. Therefore $v\notin VU_{i}$. It follows that $v$ belongs to exactly one cluster $C\in U^{(i)}$.

Case 2: $v$ belongs to $VP_{i}$. Then by eq. 5, $v$ does not belong to $VU^{(i-1)}$. By the induction hypothesis, since $U^{(i-1)}$ is a partition of $VU^{(i-1)}$, there is no cluster $C\in U^{(i-1)}$ such that $v$ belongs to $C$. Since $v$ does belong to $VU^{(i)}$, it follows that there is exactly one cluster $C^{\prime}\in U_{i}$ such that $v\in C^{\prime}$. Therefore, $v$ belongs to exactly one cluster $C\in U^{(i)}$.

Hence $U^{(i)}$ us a partition of $VU^{(i)}$.

∎

Consider the set $VU^{(\ell)}$. Note that $VP_{0}=V$, and that for every index $i\in[0,\ell]$, each vertex $v$ that belongs to $VP_{i}$ either belongs to $VU_{i}$ or to $VP_{i+1}$. Since $VP_{\ell+1}=\emptyset$ it follows that $VU^{(\ell)}=V$. As a corollary, we conclude

Corollary 2.5 The set $U^{(\ell)}$ defined by $U^{(\ell)}=\bigcup_{i=0}^{\ell}U_{i}$ is a partition of $V$.

2.4   Analysis of the Construction

In this section, we analyze the running time, the size and the stretch of our construction. We begin by analyzing the radii of clusters’ collections $P_{i}$ for $i\in[\ell]$. By Lemma 2.3, these radii are upper bounded by $R_{i}$’s. In the next lemma, we derive an explicit expression for these upper bounds. This upper bound is later used to bound $\delta_{i}$, which is used in Sections 2.4.1, 2.4.2 and 2.4.3 to analyze the running time, size and stretch of the spanner, respectively. Finally, in Section 2.4.4, we rescale $\epsilon$ to derive our ultimate result.

Lemma 2.7**.**

For $i\in[\ell]$, we have $R_{i}=\sum_{j=0}^{i-1}{2}/{\rho}\cdot\epsilon^{-j}\cdot\left({5}/{\rho}\right)^{i-1-j}.$

Proof.

The proof is by induction on the index $i$.

Recall that $R_{0}=0$ and $R_{i}$ is given by (see eq. 2) $R_{i+1}=\frac{2}{\rho}\epsilon^{-i}+\left({5}/{\rho}\right)R_{i}$.

For $i=1$, it is easy to verify that $\sum_{j=0}^{i-1}{2}/{\rho}\cdot\epsilon^{-j}\cdot\left({5}/{\rho}\right)^{i-1-j}=2/\rho$ and also $\frac{2}{\rho}\epsilon^{-{0}}+\left({5}/{\rho}\right)R_{0}=2/\rho$. So the base case holds. For the induction step, by the induction hypothesis we have:

[TABLE]

∎

Assume that $\rho\geq 10\epsilon$. (This assumption will not affect our ultimate result. See Section 2.4.4.) Observe that Lemma 2.7 implies that $R_{i}\leq\frac{2}{\rho-5\epsilon}\cdot\epsilon^{-{(i-1)}}$.

In particular, we have $\frac{2}{\rho-5\epsilon}\cdot\epsilon^{-{(i-1)}}\leq\frac{4}{\rho}\cdot\epsilon^{-{(i-1)}}.$ It follows that:

[TABLE]

Recall that $P_{i}=\widehat{P}_{i-1}$. Hence $Rad(P_{i})=Rad(\widehat{P}_{i-1})$. By Lemma 2.3, we have for all $i\in[0,\ell]$, (assuming $\rho\geq 10\epsilon$),

[TABLE]

Recall that $\delta_{i}=\epsilon^{-i}+2R_{i}$, and that $R_{0}=0$, and $\delta_{0}=1$. For $i\in[\ell]$, by eq. 6, we obtain: $\delta_{i}\leq\epsilon^{-i}+\frac{8}{\rho}\cdot\epsilon^{-{(i-1)}}$. Assume that $\rho\geq 10\epsilon$. Then $\frac{8}{\rho}\epsilon^{-{(i-1)}}\leq\epsilon^{-i}$. It follows that for all $i\in[0,\ell]$:

[TABLE]

2.4.1   Analysis of the Running Time

In this section, we analyze the running time of the entire algorithm. We begin with the following lemma which analyses the running time of a single phase of the algorithm.

Lemma 2.8**.**

For all $i\in[0,\ell]$, the running time of phase $i$ is $O\left(\rho^{-1}\cdot\delta_{i}\cdot n^{\rho}\right)$.

Proof.

The superclustering step consists of running Algorithm 1, which requires $O(deg_{i}\cdot\delta_{i})$ time (see Theorem 2.1). Constructing a $(2\delta_{i}+1,\frac{2}{\rho}\delta_{i})$-ruling set requires $O(\rho^{-1}\cdot\delta_{i}\cdot n^{\rho})$ time (see Theorem 2.2). The BFS exploration to depth $\frac{2}{\rho}\delta_{i}$ that builds superclusters requires $O(\rho^{-1}\delta_{i})$ time. (Note that no congestion occurs on this step.) In total, the superclustering step of phase $i$ requires $O(deg_{i}\cdot\delta_{i}+\rho^{-1}\delta_{i}\cdot n^{\rho}+\rho^{-1}\delta_{i})$ time. Since for all $i$, $deg_{i}\leq n^{\rho}$, it follows that the superclustering step of phase $i$ requires $O(\rho^{-1}\cdot\delta_{i}\cdot n^{\rho})$ time.

As for the interconnection step, note that each cluster that needs to add a path to the spanner $H$ in the interconnection step of phase $i$ knows all the clusters it needs to add a path to. Moreover, by Theorem 2.1, each cluster center $r_{C}$, $C\in U_{i}$, can trace back the shortest path that a message regarding a nearby cluster center $r_{C^{\prime}}$ took to reach $r_{C}$, and add the edges along this route to $H$. By Theorem 2.1, this requires $O(deg_{i}\cdot\delta_{i})$ time. Thus the running time of the interconnection step of a given phase is dominated by the running time of the superclustering step of this phase. An exception to that is the concluding phase, in which there is no superclustering step. In this phase, we execute Algorithm 1 with parameters $n^{\rho},\delta_{\ell}$. By Theorem 2.1 this requires $O(n^{\rho}\cdot\delta_{\ell})$ time. We then trace back the shortest paths in $O(n^{\rho}\cdot\delta_{\ell})$ time. Thus the running time of the interconnection step of each phase $i\geq 0$ is dominated by $O(\rho^{-1}\cdot\delta_{i}\cdot n^{\rho})$, and so the running time of a single phase of the algorithm is $O(\rho^{-1}\cdot\delta_{i}\cdot n^{\rho})$.

∎

By Lemma 2.8 and eq. 8, the running time of the entire algorithm is bounded by:

[TABLE]

Corollary 2.9**.**

The running time of the algorithm is bounded by $O\left({n^{\rho}}{\rho^{-1}}\epsilon^{-{\ell}}\right)$.

2.4.2   Analysis of the Number of Edges

In this section, we analyze the number of edges added to the spanner $H$. First, observe that in the superclustering step of phase $i$, the edges that are added to $H$ are a subset of the BFS forest $F_{i}$. Thus in each phase $O(n)$ edges are added to $H$ by the superclustering step.

We will now analyze the number of edges added by the interconnection step. In the interconnection step of phase $i$, a path is added to the spanner from each cluster $C$ in $U_{i}$ to clusters in $P_{i}$ that are close to it. As $C$ belongs to $U_{i}$, it is not popular. Thus it has at most $deg_{i}$ other clusters in $P_{i}$ that are close to it. To bound the size of $U_{i}$, we bound the size of $P_{i}$, as $U_{i}\subseteq P_{i}$. The following lemma provides an upper bound on the size of $P_{i}$ in the exponential growth stage.

The next two lemmas provide an upper bound on the size of the cluster collection $P_{i}$ in the exponential growth and the fixed growth stages.

Lemma 2.10**.**

For $i\in[0,i_{0}+1=\left\lfloor\log(\kappa\rho)\right\rfloor+1]$, we have $|{P}_{i}|\leq n^{1-\frac{2^{i}-1}{\kappa}}.$

Proof.

We will prove the lemma by induction on the index of the phase $i$.

For $i=0$, the right-hand side is $n^{1-\frac{2^{0}-1}{\kappa}}=n$. Thus the claim is trivial.

Recall that for each index $i$, the set $RS_{i}$ is the ruling set computed in phase $i$, and that $S_{i}$ is the set of centers of clusters in $P_{i}$. Since for each phase $i\geq 1$, the clusters of the collection $P_{i}$ are centered around vertices of $RS_{i-1}$, for $i\geq 1$, we have $S_{i}=RS_{i-1}$. The set $RS_{i}$ is a $(2\delta_{i}+1,\frac{2}{\rho}\delta_{i})$-ruling set for $W_{i}$. By Theorem 2.1, all vertices in $W_{i}$ are popular cluster centers. Thus, for every $r_{C}\in W_{i}$, it holds that $|\Gamma^{(\delta_{i})}(r_{C})\cap S_{i}|\geq deg_{i}$.

By Theorem 2.2, the set $RS_{i}$ is $(2\delta_{i}+1)$-separated, i.e., for every pair of distinct cluster centers $r_{C},r_{C^{\prime}}\in RS_{i}$ we have $d_{G}(r_{C},r_{C^{\prime}})\geq 2\delta_{i}+1$. Thus, for every pair of distinct centers $r_{C},r_{C^{\prime}}\in RS_{i}$, their $\delta_{i}$-neighborhoods are disjoint, i.e., $\Gamma^{(\delta_{i})}(r_{C})\cap\Gamma^{(\delta_{i})}(r_{C^{\prime}})=\emptyset$.

Together with the induction hypothesis, and since for i, $0\leq i\leq i_{0}$, $deg_{i}=n^{\frac{2^{i}}{\kappa}}$, this implies that $|\widehat{P}_{i}|\leq\frac{|{P}_{i}|}{deg_{i}}\leq{n^{1-\frac{2^{i}-1}{\kappa}}}/{n^{\frac{2^{i}}{\kappa}}}=n^{1-\frac{2^{i+1}-1}{\kappa}}$.

As $P_{i+1}=\widehat{P}_{i}$, we conclude that $|{P}_{i+1}|\leq n^{1-\frac{2^{i+1}-1}{\kappa}}$. ∎

Observe that by Lemma 2.10, for the phase ${i_{0}+1}$, we have that:

[TABLE]

Lemma 2.11**.**

For $i_{0}+1\leq i\leq\ell$, it holds that $|{P}_{i}|\quad\leq\quad n^{1+\frac{1}{\kappa}-(i-i_{0})\rho}.$

Proof.

The proof is by induction on the index of the phase $i$.

For the base case, by eq. 10,

[TABLE]

For $i>i_{0}+1$, we have $S_{i}=RS_{i-1}$. The set $RS_{i}$ is a $(2\delta_{i}+1,\frac{2}{\rho}\delta_{i})$-ruling set for $W_{i}$. By Theorem 2.1, all vertices in $W_{i}$ are popular, i.e., for every cluster center $r_{C}\in W_{i}$, it holds that $|\Gamma^{(\delta_{i})}(r_{C})\cap S_{i}|\geq n^{\rho}.$

By Theorem 2.2, the set $RS_{i}$ is $(\delta_{i}+1)$-separated, i.e., for every pair of distinct cluster centers $r_{C},r_{C^{\prime}}\in RS_{i}$, we have $d_{G}(r_{C},r_{C^{\prime}})\geq 2\delta_{i}+1.$

Thus, for every pair of centers $r_{C},r_{C^{\prime}}\in RS_{i}$, $\Gamma^{(\delta_{i})}(r_{C})\cap\Gamma^{(\delta_{i})}(r_{C^{\prime}})=\emptyset$. Together with the induction hypothesis, this implies that

[TABLE]

Since $P_{i+1}=\widehat{P}_{i}$, it follows that $|{P}_{i+1}|\leq n^{1+\frac{1}{\kappa}-(i+1-i_{0})\rho}$. ∎

The next lemma provides an upper bound on the number of edges added to the spanner by each phase $i\in[0,\ell]$.

Lemma 2.12**.**

For all $i\in[0,\ell]$, in each phase $i$, $O(n^{1+{1}/{\kappa}}\cdot\delta_{i})$ edges are added to the spanner $H$.

Proof.

In the interconnection step of phase [math], we add all edges adjacent to unpopular clusters. Note that all vertices $v\in U_{0}$, have at most $n^{1/\kappa}$ neighbors, i.e., $|\Gamma(v)|<n^{1/\kappa}$. Hence the number of edges added by the interconnection step of phase [math] is $O(n^{1+\frac{1}{\kappa}})$.

Let $i\in[0,i_{1}]$ be an index (recall that $i_{1}=i_{0}+\lceil\frac{\kappa+1}{\kappa\rho}\rceil-2=\ell-1$). Each cluster center $r_{C}$, such that $C\in U_{i}$, initiates a BFS exploration to depth $\delta_{i}$. For each cluster center $r_{C^{\prime}}$, $C^{\prime}\in P_{i}$, that is discovered by this exploration, a shortest $r_{C},r_{C^{\prime}}$ path is added to the spanner $H$. Since $C\in U_{i}$, the cluster center $r_{C}$ has at most $deg_{i}$ other cluster centers $r_{C^{\prime}}\in S_{i}$ within distance $\delta_{i}$ from it. So, the number of paths that are added to the spanner in the interconnection step of phase $i$ is at most: $|U_{i}|\cdot deg_{i}\leq|P_{i}|\cdot deg_{i}$.

For $i\in[0,i_{0}=\left\lfloor\log(\kappa\rho)\right\rfloor]$, by Lemma 2.10, we have $|P_{i}|\cdot deg_{i}\leq n^{1-\frac{2^{i}-1}{\kappa}}\cdot n^{\frac{2^{i}}{\kappa}}=n^{1+\frac{1}{\kappa}}$.

For $i\in[i_{0}+1,i_{1}]$, by Lemma 2.11, we have $|P_{i}|\cdot deg_{i}\leq n^{1+\frac{1}{\kappa}-(i-i_{0})\rho}\cdot n^{\rho}\leq n^{1+\frac{1}{\kappa}}$.

So in each phase $i\in[0,i_{1}]$, the number of edges added to the spanner $H$ by the interconnection step is $O(n^{1+\frac{1}{\kappa}}\cdot\delta_{i})$, and the total number of edges added to $H$ by the superclustering and the interconnection steps of phase $i$ is:

[TABLE]

We will now discuss the number of edges added to $H$ by phase $\ell$, i.e., by the concluding phase. In this phase, we skip the superclustering step and execute the interconnection step. We set $U_{\ell}=P_{\ell}$. Note that by Lemma 2.11, for the last ($\ell=i_{0}+\lceil\frac{\kappa+1}{\kappa\rho}\rceil-1$) phase, the size of the input collection $P_{\ell}$ is bounded by $|{P}_{\ell}|\leq n^{1+\frac{1}{\kappa}-(\ell-i_{0})\rho}=n^{1+\frac{1}{\kappa}-(i_{0}+\lceil\frac{\kappa+1}{\kappa\rho}\rceil-1-i_{0})\rho}\leq n^{1+\frac{1}{\kappa}-(\frac{\kappa+1}{\kappa}-\rho)}\leq n^{\rho}$.

Therefore, even if every pair of clusters in $P_{\ell}$ is connected by the interconnection step of phase $\ell$, at most $O(n^{2\rho}\cdot\delta_{\ell})$ edges are added to $H$ by the concluding phase. As we set $\rho\leq\frac{1}{2}$, the concluding phase adds at most $O(n\cdot\delta_{\ell})$ edges to the spanner $H$.

Hence, the total number of edges added to the spanner $H$ by a phase $i\in[0,\ell]$ is $O(n^{1+{1}/{\kappa}}\cdot\delta_{i})$.

∎

By Lemma 2.12 and eq. 8, the number of edges added to the spanner $H$ by all phases of the algorithm is bounded by:

[TABLE]

Corollary 2.13**.**

The size of the spanner $H$ is bounded by $|E_{H}|=O\left(n^{1+{1}/{\kappa}}\cdot\epsilon^{-{\ell}}\right)$.

2.4.3   Analysis of the Stretch

In this section, we analyze the stretch of the spanner $H$. The following two lemmas provide the necessary tools for Lemma 2.16, in which we analyze the stretch of the spanner $H$.

Lemma 2.14**.**

For all $i\in[0,\ell]$, for every pair of clusters $C\in{U}_{i}$, $C^{\prime}\in{P}_{i}$ at distance at most $\left(\frac{1}{\epsilon}\right)^{i}$ from one another, a shortest path between the clusters centers $r_{C},r_{C^{\prime}}$ was added to the spanner $H$.

Proof.

Let $i\in[0,\ell]$, and let $C\in{U}_{i}$, $C^{\prime}\in{P}_{i}$ a pair of clusters at distance at most $\left(\frac{1}{\epsilon}\right)^{i}$ from one another. Since $R_{i}$ is an upper bound on $Rad(P_{i})$ (see Lemma 2.3), we have that $d_{G}(r_{C},r_{C^{\prime}})\leq 2R_{i}+\epsilon^{-i}=\delta_{i}$. Recall that by Lemma 2.4, all popular clusters in phase $i$ are superclustered into clusters of $P_{i+1}$. Thus they do not belong to $U_{i}$. It follows that centers of clusters of $U_{i}$ do not belong to $W_{i}$. Thus, since $C\in{U}_{i}$, by Theorem 2.1, in the interconnection step of phase $i$ the center $r_{C}$ of $C$ knows the distance to $r_{C^{\prime}}$. So $r_{C}$ adds a shortest $r_{C}-r_{C^{\prime}}$ path to the spanner $H$. ∎

We now provide an upper bound on the distance in $H$ between neighboring clusters in $G$.

Lemma 2.15**.**

Consider a pair of indices $0\leq j<i\leq\ell$, and a pair of neighboring clusters $C\in U_{j}$ and $C^{\prime}\in U_{i}$. Let $w$ be some vertex in $C$, and let $r_{C^{\prime}}$ be the center of the cluster $C^{\prime}$. Let $\rho<1$. Then, there is a path in $H$ between $w$ and $r_{C^{\prime}}$ of length at most $3R_{j}+1+R_{i}$.

Proof.

Let $(z^{\prime},z)\in E$ be an edge such that $z\in C$ and $z^{\prime}\in C^{\prime}$ (see Figure 6 for an illustration). We will first consider phase $j$. Let $C^{\prime\prime}\in P_{j}$ be the cluster such that $z^{\prime}\in C^{\prime\prime}$, that is, $C^{\prime\prime}$ is the cluster of $z^{\prime}$ in the $j$th phase. Observe that $C\in U_{j}$, and $d_{G}(C,C^{\prime\prime})=1$. Then by Lemma 2.3, we have $d_{G}(r_{C},r_{C^{\prime\prime}})\leq 2R_{j}+1$. Note that $2R_{j}+1\leq\epsilon^{-{j}}+2R_{j}=\delta_{j}$, for all $j\in[0,\ell]$. Thus, a shortest $r_{C}-r_{C^{\prime\prime}}$ path was added to $H$ in the interconnection step of phase $j$. By Lemma 2.3, there is a path from $w$ to $r_{C}$ in $H$ of length at most $R_{j}$, and there is a path from $r_{C^{\prime\prime}}$ to $r_{C^{\prime}}$ of length at most $R_{i}$. Note also that $\epsilon^{-{j}}\leq R_{j}$. Thus, $d_{H}(w,r_{C^{\prime}})\leq d_{H}(w,r_{C})+d_{H}(r_{C},r_{C^{\prime\prime}})+d_{H}(r_{C^{\prime\prime}},r_{C^{\prime}})\leq 3R_{j}+1+R_{i}$.

∎

Since $R_{i+1}=\frac{2}{\rho}\epsilon^{-i}+\left({5}/{\rho}\right)R_{i}$, and since $\rho<1$ and $j<i$, we have $3R_{j}\leq R_{i}$. It follows that:

[TABLE]

We are now ready to analyze the stretch of our spanner. Recall that for every index $i$, the set $U^{(i)}$ is the union of all sets $U_{0},U_{1},\dots,U_{i}$.

Lemma 2.16**.**

Assume $\epsilon\leq\frac{1}{10}$ and $\rho\geq 10\epsilon$. Consider a pair of vertices $u,v\in V$. Fix a shortest path $\pi(u,v)$ between them in $G$, and suppose that for some $i\leq\ell$ all the vertices of $\pi(u,v)$ are all clustered in the set $U^{(i)}$. Then,

[TABLE]

Proof.

The proof is by induction on $i$. For the base case, $i=0$, all the vertices on $\pi(u,v)$ are $U_{0}$-clustered, and so all the edges of the path are in the spanner, and $d_{G}(u,v)=d_{H}(u,v)$.

For the induction step, we first consider a pair of vertices $x,y$ such that there is a path $\pi(x,y)$ in $G$ of length at most $\epsilon^{-i}$, and such that all the vertices on the path are clustered in the set $U^{(i)}$.

To simplify presentation, we will imagine that the vertices of $\pi(x,y)$ appear from left to right (see Figure 7 for an illustration), where $x$ (respectively, $y$) is the leftmost (respectively, rightmost) vertex of the path. Let $z_{l},z_{r}$ be the leftmost and rightmost $U_{i}$-clustered vertices on $\pi(x,y)$, respectively, and let $C_{l},C_{r}$ ($r_{C_{l}},r_{C_{r}}$) be their respective clusters (cluster centers).

We note that it is possible that $z_{l},z_{r}$ belong to the same cluster, i.e, $C_{l}=C_{r}$. However, this case is simpler, as in the analysis the path between the cluster centers $r_{C_{l}},r_{C_{r}}$ will simply be an empty path. We also note that it is possible that there is no $U_{i}$-clustered vertex on the path $\pi(x,y)$. In this case, we can take $i^{\prime}$ to be the maximal index such that there is a vertex on $\pi(x,y)$ that is $U_{i^{\prime}}$-clustered. Replacing the index $i$ with an index $i^{\prime}$ such that $i^{\prime}<i$ will only decrease the bound on the distance between $x,y$ in $H$. Therefore, without loss of generality, we assume that there exists a vertex on $\pi(x,y)$ that is $U_{i}$ clustered. Let $w_{l}$ (respectively, $w_{r}$) be the neighbor of $z_{l}$ (respectively, $z_{r}$) on the sub-path $\pi(x,z_{l})$ (respectively, $\pi(z_{r},y)$). Observe that $w_{l},w_{r}$ are $U^{(i-1)}$ clustered. By eq. 12, $d_{H}(w_{l},r_{C_{l}})\leq 2R_{i}+1\ \text{ and }\ d_{H}(r_{C_{r}},w_{r})\leq 2R_{i}+1$.

Note that as $|\pi(x,y)|\leq\epsilon^{-i}$, we have:

[TABLE]

From this and from the fact that $C_{l},C_{r}\in U_{i}$, it follows that a shortest path between $r_{C_{l}},r_{C_{r}}$ was added to the spanner $H$ in the interconnection step of phase $i$. Thus $d_{G}(r_{C_{l}},r_{C_{r}})=d_{H}(r_{C_{l}},r_{C_{r}})$, and $d_{H}(r_{C_{l}},r_{C_{r}})\leq d_{G}(z_{l},z_{r})+2R_{i}$.

As all the vertices in the sub-paths $\pi(x,w_{l}),\pi(w_{r},y)$ are $U^{(i-1)}$ clustered, the induction hypothesis applies to them. Hence

[TABLE]

Therefore $d_{H}(x,y)$ is bounded by:

[TABLE]

Now, consider a pair $u,v\in V$ such that all vertices of $\pi(u,v)$ are $U^{(i)}$ clustered, but $\pi(u,v)$ may be of an arbitrarily large length. We divide the path into segments of length exactly $\epsilon^{-i}$, except for maybe one segment that can be shorter (see Figure 8). By eq. 14:

[TABLE]

It is left to show that $6{\epsilon^{i}}\cdot\sum_{j=1}^{i}R_{j}\cdot 2^{i-j}\leq\frac{30\epsilon}{\rho}$.

First, we will provide an upper bound on $6\sum_{j=1}^{i}R_{j}\cdot 2^{i-j}$. Recall that by eq. 7, $R_{i}\leq({4}/{\rho})\cdot\epsilon^{-{(i-1)}}$. It follows that $6\sum_{j=1}^{i}R_{j}\cdot 2^{i-j}\leq 6\sum_{j=1}^{i}({4}/{\rho})\cdot\epsilon^{-{(j-1)}}\cdot 2^{i-j}={24}/({\rho\cdot\epsilon^{i-1}\cdot(1-2\epsilon)})$. Recall that $\epsilon<\frac{1}{10}$. Thus

[TABLE]

This yields: $6{\epsilon^{i}}\cdot\sum_{j=1}^{i}R_{j}\cdot 2^{i-j}\leq{\epsilon^{i}}\cdot{30}/({\rho\cdot\epsilon^{i-1}})=({30\epsilon})/{\rho}$. To conclude,

[TABLE]

∎

Recall that by Corollary 2.5, all vertices in $V$ are $U^{(\ell)}$-clustered, thus for any pair of vertices, the condition of Lemma 2.16 holds for $i=\ell$ and for every vertex $v\in V$ there is an index $i$ such that $v$ is $U_{i}$-clustered. By Lemma 2.16 and eq. 16,

Corollary 2.17**.**

For every pair $u,v\in V$, for $\rho\geq 10\epsilon$, $d_{H}(u,v)\leq\left(1+\frac{30\cdot\epsilon\cdot\ell}{\rho}\right)\cdot d_{G}(u,v)+\frac{30}{\rho\cdot\epsilon^{\ell-1}}$.

2.4.4   Rescaling

In this section, we rescale $\epsilon$ to obtain a $(1+\epsilon,\beta)$-spanner. Set $\epsilon^{\prime}=({30\cdot\epsilon\cdot\ell})/{\rho}$.

The condition $\epsilon<\frac{1}{10}$ now translates to $({\epsilon^{\prime}\cdot\rho})/({30\ell})<{1}/{10}$, i.e, $\epsilon^{\prime}<\frac{3\ell}{\rho}$. We replace it with a much stronger condition, $\epsilon^{\prime}<1$. The condition $\rho\geq 10\epsilon$ translates to $\rho=({30\cdot\epsilon\cdot\ell})/{\epsilon^{\prime}}>10\epsilon$, i.e., $\epsilon^{\prime}<3\ell$, which holds trivially for $\epsilon^{\prime}<1$ and $\ell\geq 1$.

It follows that the additive term of the stretch now translates to $\frac{30}{\rho\cdot\epsilon^{\ell-1}}\leq\left(\frac{30\cdot\ell}{\rho\cdot\epsilon^{\prime}}\right)^{\ell}.$

Denote $\beta=\left(({30\cdot\ell})/({\rho\cdot\epsilon^{\prime}})\right)^{\ell}.$ Observe that

[TABLE]

Thus we obtain stretch $\left(1+\epsilon^{\prime},\left(({30\cdot\ell})({\rho\cdot\epsilon^{\prime}})\right)^{\ell}\right).$

Recall that $\ell=\left\lfloor{\log\kappa\rho}\right\rfloor+\lceil\frac{\kappa+1}{\kappa\rho}\rceil-1\leq\log\kappa\rho+\rho^{-1}+O(1)$. It follows that:

[TABLE]

By Corollary 2.13 and eq. 17, the number of edges in the spanner $H$ is:

[TABLE]

By Corollary 2.9 and eq. 17, the time required to construct the spanner $H$ is:

[TABLE]

Denote now $\epsilon=\epsilon^{\prime}$.

Corollary 2.18**.**

For any parameters $0<\epsilon\leq 1$, $\kappa\geq 2$, and ${1}/{\kappa}\leq\rho<{1}/{2}$, and any $n$-vertex graph $G=(V,E)$, our algorithm constructs a $(1+\epsilon,\beta)-spanner$ with $O(\beta\cdot n^{1+{1}/{\kappa}})$ edges, in $O\left(\beta\cdot n^{\rho}\cdot\rho^{-1}\right)$ deterministic time in the CONGEST model, where

[TABLE]

Appendix A Detecting Popular Clusters

Given a graph $G=(V,E)$, a set of clusters $P_{i}$ centered around the vertices in $S_{i}$, parameters $\delta_{i},deg_{i}$, we say that the cluster $C$ and its center $r_{C}$ are popular if $|\Gamma^{(\delta_{i})}(r_{C})\cap S_{i}|\geq deg_{i}$, that is, if the center $r_{C}$ has at least $deg_{i}$ cluster centers within distance $\delta_{i}$ from it. This section provides a procedure that allows each cluster center $r_{C}$ for $C\in P_{i}$ to know if it is popular.

The procedure runs a modified BFS exploration from each vertex $r_{C}\in S_{i}$. In this exploration, each vertex $u\in V$ maintains a list of all the cluster centers it learned about and the shortest known distance to them. The algorithm runs for $\delta_{i}$ phases. We note that the phases of this algorithm are different from the phases of the main algorithm described in Section 2. Phase [math] consists of a single round. Each phase $i>0$ consists of $deg_{i}$ rounds. Intuitively, the phases can be thought of as single rounds, in which vertices can send $deg_{i}$ messages. In round [math] (also called phase [math]), each vertex $r_{C}\in S_{i}$ sends the message $\langle r_{C},0\rangle$ to all its neighbors. The first part of the message is the ID of the original sender. The second part is the distance that the message has traversed so far. In each phase $i>0$, each vertex $u\in V$ increments the distance on each message it received in the previous phase, and forwards these messages. Note that in each phase $j>0$, i.e., rounds $deg_{i}\cdot(j-1)+1$ to $deg_{i}\cdot j$ for $1\leq j\leq\delta_{i}$, each vertex sends messages that traversed exactly $j-1$ edges before reaching it. If a vertex $v\in V$ received messages of the form $\langle r_{C},j\rangle$ regarding more than $deg_{i}$ centers, it will arbitrarily choose $deg_{i}$ of these messages and forward them. All other messages are discarded, i.e., they will never be sent.

If a center $r_{C}\in S_{i}$ receives messages regarding at least $deg_{i}$ other clusters, it joins $A$. Each vertex $u$ maintains a list of all the first $deg_{i}$ vertices it has learned about. The vertex $u$ maintains the shortest known distance to them, and its neighbor from which the message regarding each such center arrived. This is in order to trace back a shortest path from $u$ to vertices it has learned about. The pseudo-code of the algorithm (Algorithm 1) is provided below.

Next, we show that when the algorithm terminates, each vertex $u\in V$ maintains information regarding some of the centers in $S_{i}$ that are close to it. That is, $u$ knows the IDs of these vertices, its distance to them, and its neighbor that informed it of them.

Lemma A.1**.**

When Algorithm 1 terminates, each vertex $u\in V$ knows at least $\min\{deg_{i},|\Gamma^{(\delta_{i})}(u)\cap S_{i}|\}$ centers from $S_{i}$ with distance at most $\delta_{i}$ from $u$.

Proof.

We will prove by induction on the index of the phase $j$, that by the end of phase $j$, for all $j\geq 0$, each vertex $u$ knows at least $\min\{deg_{i},|\Gamma^{(j)}(u)\cap S_{i}|\}$ centers from $S_{i}$ with distance at most $j$ from $u$.

For $j=0$ the claim is trivial since $\Gamma^{(0)}(u)=\{u\}$.

For $j>0$, assume inductively that by the end of phase $j-1$, each vertex $v\in V$ knows $\min\{deg_{i},|\Gamma^{(j-1)}(v)\cap S_{i}|\}$ centers from $S_{i}$ with distance at most $j-1$ from it.

Let $j>0$ and let $u$ be a vertex. The immediate neighbors of $u$ in $G$ have exactly $|\Gamma^{(j)}(u)\cap S_{i}|$ distinct vertices from $S_{i}$ within distance at most $j-1$ from them. By the induction hypothesis, each vertex $v$ which is a neighbor of $u$ knows at least $\min\{deg_{i},|\Gamma^{(j-1)}(v)\cap S_{i}|\}$ centers from $S_{i}$ within distance at most $j-1$ from it.

If one of these neighbors has at least $deg_{i}$ vertices from $S_{i}$ within distance exactly $j-1$ from it, then it has sent $deg_{i}$ messages to $u$ in phase $j$. Thus by the end of phase $j$, $u$ knows at least $deg_{i}$ vertices from $S_{i}$ within distance at most $j$ from it and the claim holds. Otherwise, in phase $j$, each vertex $v$ which is a neighbor of $u$, sent to $u$ messages regarding all vertices in $S_{i}$ within distance exactly $j-1$ from $v$. Moreover, by the induction hypothesis, at the end of phase $j-1$, the vertex $u$ already knows $\min\{deg_{i},|\Gamma^{(j-1)}(u)\cap S_{i}|\}$ centers from $S_{i}$ with distance at most $j-1$ from $u$. It follows that by the end of phase $j$, the vertex $u$ knows $\min\{deg_{i},|\Gamma^{(j)}(u)\cap S_{i}|\}$ centers from $S_{i}$ with distance at most $j$ from it. ∎

We are now ready to prove Theorem 2.1. For convenience, it is stated here again:

Theorem 2.1 Given a graph $G=(V,E)$, a collection of clusters $P_{i}$ centered around vertices $S_{i}$ and parameters $\delta_{i},\ \deg_{i}$, Algorithm 1 returns a set $A$ in $O(deg_{i}\cdot\delta_{i})$ time such that:

$A$* is the set of all centers of popular clusters from $P_{i}$.* 2. 2.

Every cluster center $r_{C}\in S_{i}$ that did not join $A$ knows the identities of all the centers $r_{C^{\prime}}\in S_{i}$ such that $d_{G}(r_{C},r_{C^{\prime}})\leq\delta_{i}$. Furthermore, for each pair of such centers $r_{C},r_{C^{\prime}}$, there is a shortest path $\pi$ between them such that all vertices on $\pi$ know their distance from $r_{C^{\prime}}$.

Proof.

(1) Let $C\in P_{i}$ be a popular cluster, i.e., $r_{C}$ has at least $deg_{i}$ other centers in $S_{i}$ within distance $\delta_{i}$ from it. By Lemma A.1, when the algorithm terminates, $r_{C}$ knows at least $deg_{i}$ other cluster centers from $S_{i}$. Thus it joins $A$.

Let $r_{C}\in A$. Observe that $r_{C}$ joined $A$ because it learned about at least $deg_{i}$ clusters within distance $\delta_{i}$ from it. Thus it is a popular cluster.

(2) Let $r_{C}\in S_{i}\backslash A$. Since $r_{C}$ did not join $A$, it has learned about less than $deg_{i}$ other cluster centers from $S_{i}$. Therefore, by Lemma A.1, the vertex $u$ has learned about $\min\{deg_{i},|\Gamma^{(\delta_{i})}(r_{C})\cap S_{i}|\}=|\Gamma^{(\delta_{i})}(r_{C})\cap S_{i}|$ vertices in $S_{i}$ with distance at most $\delta_{i}$ from it. Hence it knows the identities of all the centers $r_{C^{\prime}}\in S_{i}$ such that $d_{G}(r_{C},r_{C^{\prime}})\leq\delta_{i}$.

Let $r_{C}$ be a center that did not join $A$, and let $r_{C^{\prime}}$ be a center $d_{G}(r_{C^{\prime}},r_{C})\leq\delta_{i}$. The messages regarding $r_{C^{\prime}}$ reached $r_{C}$ along some shortest $(r_{C^{\prime}},r_{C})$ path $\pi$. Since each vertex $v\in V$ maintains a list with all (up to $deg_{i}$) the centers it has learned about and the distances to them, all the vertices on the shortest path $\pi$ know their distance to $r_{C^{\prime}}$.

The running time of Algorithm 1 is trivially $O(deg_{i}\cdot\delta_{i})$, as it is executed for $\delta_{i}$ phases, each of which lasts up to $deg_{i}$ rounds. ∎

Appendix B Previous Results

In this section, we review relevant previous results for near-additive spanners.