Low-congestion shortcut and graph parameters

Abstract

Distributed graph algorithms in the standard CONGEST model often exhibit the time-complexity lower bound of ${\tilde{\Omega }}(\sqrt{n} + D)$ rounds for several global problems, where n denotes the number of nodes and D the diameter of the input graph. Because such a lower bound is derived from special “hard-core” instances, it does not necessarily apply to specific popular graph classes such as planar graphs. The concept of low-congestion shortcuts was initiated by Ghaffari and Haeupler [SODA2016] for addressing the design of CONGEST algorithms running fast in restricted network topologies. In particular, given a graph class ${\mathcal {C}}$, an f-round algorithm for constructing shortcuts of quality q for any instance in ${\mathcal {C}}$ results in ${\tilde{O}}(q + f)$-round algorithms for solving several fundamental graph problems such as minimum spanning tree and minimum cut, for ${\mathcal {C}}$. The main interest on this line is to identify the graph classes allowing the shortcuts that are efficient in the sense of breaking ${\tilde{O}}(\sqrt{n}+D)$-round general lower bounds. In this study, we consider the relationship between the quality of low-congestion shortcuts and the following four major graph parameters: doubling dimension, chordality, diameter, and clique-width. The key ingredient of the upper-bound side is a novel shortcut construction technique known as short-hop extension, which might be of independent interest.

1 Introduction

1.1 Background

The CONGEST is one of the standard message-passing models when considering distributed graph algorithms. It is defined as a round-based synchronous system with limited bandwidth, where each link can transfer $O(\log n)$-bit information per round (n is the number of nodes in the system). Because global distributed tasks such as minimum spanning tree (MST) inherently require each node to access the information far apart from itself, the $\Omega (D)$-round complexity becomes an universal lower bound applied to any network topology, where D represents the diameter of the input topology. Although D-round computation is sufficiently long to make some information reach all the nodes in the network, the constraint of limited bandwidth precludes the centralized solution that one node collects the information of the entire network topology. Thus, the round complexity of CONGEST algorithms for global tasks is typically represented in the form of ${\tilde{O}}(n^c + D)$ or ${\tilde{O}}(n^cD)$ for some constant $0 \le c \le 2$¹. The main complexity-theoretic question is the extent to which we can make c small (ideally $c =0$, which matches the universal lower bound). Unfortunately, achieving such a universal bound is impossible for several fundamental problems including MST, which typically exhibits the lower bound of ${\tilde{\Omega }}(\sqrt{n} + D)$ rounds for general graphs.

Most of the ${\tilde{\Omega }}(n^c + D)$-round lower bounds for some $c > 0$ are derived from special “hard-core” instances, and do not necessarily apply to popular graph classes such as planar graphs, which evokes the interest of developing efficient distributed graph algorithms for specific graph classes. In the last few years, the study along this line has rapidly made progress, where the concepts of partwise aggregation and low-congestion shortcuts play an important role. In the partwise aggregation problem, all nodes in the network are initially partitioned into a number of disjoint-connected subgraphs known as a part. The goal of this problem is to perform a certain type of distributed task independently within all the parts in parallel. The executable tasks cover several standard operations, such as broadcast, convergecast, leader election, and finding minimum. The low-congestion shortcut is a framework for solving the partwise aggregation problem, which is initiated by Ghaffari and Haeupler [15]. The key difficulty of the partwise aggregation problem appears when the diameter of a part is much larger than the diameter D of the original graph. Because the diameter of a part can become $\Omega (n)$ in the worst case scenario, the naive solution that performs the aggregation task only by in-part communication causes the expensive $\Omega (n)$-round running time. A low-congestion shortcut is defined as the sets of links augmented to each part to accelerate the aggregation task there. Its efficiency is characterized by the following two quality parameters: The dilation is the maximum diameter of all the parts after the augmentation, whereas the congestion is the maximum edge congestion of all edges e, where the edge congestion of e is defined as the number of parts augmenting e. In the application of low-congestion shortcuts, the performance of an algorithm typically relies on the sum of the dilation and congestion. Hence, we simply refer to the value of dilation plus congestion as the quality of the shortcut. It is known that any low-congestion shortcut with quality q and O(f)-round construction time yields an ${\tilde{O}}(f + q)$-round solution for the partwise aggregation problem, and ${\tilde{O}}(f + q)$-round partwise aggregation yields the efficient solutions for several fundamental graph problems. Precisely, the following meta-theorem holds:

Theorem 1

(Ghaffari and Haeupler [15], Haeupler and Li [23]) Let ${\mathcal {G}}$ be a graph class allowing the low-congestion shortcut with quality O(q) that can be constructed in O(f) rounds in the CONGEST model. Then, there exist three algorithms solving (1) the MST problem in ${\tilde{O}}(f + q)$ rounds, (2) the $(1 + \epsilon )$-approximate minimum cut problem in ${\tilde{O}}(f + q)$ rounds for any $\epsilon = \Omega (1)$, and (3) $O(n^{O(\log \log n) / \log \beta })$-approximate weighted single-source shortest path problem in ${\tilde{\Omega }}((f + q)\beta )$ rounds for any $\beta = \Omega (\mathrm {polylog}(n))$².

Conversely, if we obtain a time-complexity lower bound for any problem stated above, then it also applies to the partwise aggregation and low-congestion shortcuts (with respect to quality plus construction time). In fact, the ${\tilde{O}}(\sqrt{n} + D)$-round lower bound of shortcuts for general graphs is deduced from the lower bound of MST. Meanwhile, the existence of efficient (in the sense of breaking the general lower bound) low-congestion shortcuts is known for several major graph classes as well as its construction algorithms [15, 17, 18, 21, 22, 24].

1.2 Our result

Herein, we study the relationship between several major graph parameters and the quality of low-congestion shortcuts. In particular, we focus on the following four parameters, that is: (1) doubling dimension, (2) chordality, (3) diameter, and (4) clique width. The precise statement of our results is as follows:

• There is an O(1)-round algorithm that constructs a low congestion shortcut with quality ${\tilde{O}}(D^{x})$ for any doubling dimension-x graph.

• There is an O(1)-round algorithm that constructs a low-congestion shortcut with quality O(kD) for any k-chordal graph. When $k=O(1)$, its quality matches the $\Omega (D)$-universal lower bound.

• For $k\le D$ and $kD\le \sqrt{n}$, there exists a k-chordal graph where the construction of MST requires ${\tilde{\Omega }}(kD)$ rounds, implying that the quality plus construction time of our algorithm is nearly optimal up to polylogarithmic factors.

• There exists an algorithm for constructing a low-congestion shortcut with quality ${\tilde{O}}(n^{1/4})$ in ${\tilde{O}}(n^{1/4})$ rounds for any graph of diameter three. In addition, there exists an algorithm for constructing a low-congestion shortcut with quality ${\tilde{O}}(n^{1/3})$ in ${\tilde{O}}(n^{1/3})$ rounds for any graph of diameter four. These results are similar to the long-standing complexity gap of the MST construction in graphs with small diameters, which was originally proposed by Lotker et al. [32].

• We present a negative instance certifying that bounded clique-width does not help in the construction of good-quality shortcuts. More precisely, we provide an instance of clique-width six, where the construction of MST is as expensive as the general case, i.e., ${\tilde{\Omega }}(\sqrt{n}+D)$ rounds.

Table 1 summarizes the state-of-the-art upper and lower bounds for low-congestion shortcuts. Notably, all the parameters considered in this study are independent of the other (known) parameters admitting good shortcuts (e.g., treewidth and genus) because the graphs of bounded doubling dimension, chordality, or diameter can contain the clique of an arbitrary size and thus, are not a subclass of any minor-excluded graphs. Therefore, any result presented in this paper is not a corollary of past results.

For proving our upper bounds, we propose a novel scheme for shortcut construction, known as short-hop extension. The simplest form of this scheme is 1-hop extension, where each node in a part assumes all the incident edges to be the shortcut of its own part. Surprisingly, this very simple construction admits nearly optimal shortcuts for graphs of bounded chordality or doubling dimension. For graphs of diameters of three or four, the 2-hop extension (that is, each node in a part takes all the two-length paths starting from itself as the shortcut) clearly yield O(1) dilation, but the second edge in each path suffers from high congestion. Our algorithm circumvents this matter through random choice of the second edges based on hash functions, which is simple though far from triviality to bound the quality of constructed shortcuts. The analytic part includes several new ideas and may be of independent interest.

Table 1 Quality bounds of low-congestion shortcuts for specific graph classes

1.3 Related work

The MST problem is one of the most fundamental problems in distributed graph algorithms. As well as its own importance, MST has several applications for solving other graph problems in distributed settings (e.g., detecting connected components and minimum cut). Several studies have addressed the design of efficient MST algorithms in the CONGEST model [11, 12, 16, 19, 20, 25, 29, 35, 36], and the round-complexity of MST construction is a central topic in distributed complexity theory [8, 9, 32, 33, 37, 38]. The inherent difficulty of MST construction is solving the partwise aggregation (minimum) problem efficiently. This viewpoint was first identified by Ghaffari and Haeupler [15] explicitly, as well as an efficient algorithm for solving MST construction in planar graphs. The concept of low-congestion shortcuts is newly invented herein for encapsulating the difficulty of partwise aggregation. Recently, several follow-up papers have been published to extend the applicability of low-congestion shortcuts, which break the known general lower bounds of MST and its applications in specific graph classes: This line covers bounded-genus graphs [15, 22], bounded-treewidth graphs [22], graphs with excluded minors [24], and expander graphs [17, 18] (see Table 1). All the shortcuts stated here belong to the class of tree-restricted shortcuts, where the shortcut edges augmented to each part are a subgraph of a precomputed spanning tree (typically a breadth first search (BFS) tree). It is shown that there exists a universal algorithm for computing tree-restricted shortcuts [21]. To the best of our knowledge, the upper bounds presented in this paper are the first to exhibit non-trivial shortcuts not belonging to the tree-restricted class. The application of low-congestion shortcuts is not limited to MST. As stated in Theorem 1, low-congestion shortcuts also admits efficient solutions for an approximate minimum cut, and a single-source shortest path. In addition, a few algorithms utilize low-congestion shortcuts as an important building block, e.g., the depth first search in planar graphs [23], approximate tree decomposition [30], along with diameter and distance labeling scheme in planar graphs [31]. Haeupler et al. [20] show a message-reduction scheme of shortcut-based algorithms, which drops the total number of messages exchanged by the algorithm into ${\tilde{O}}(m)$, where m denotes the number of links in the network. On the negative side, it is known that the hardness of (approximate) a diameter cannot be encapsulated by low-congestion shortcuts. Abboud et al. [1] showed a hard-core family of unweighted graphs with $O(\log n)$ treewidth, where any diameter computation in the CONGEST model requires ${\tilde{\Omega }}(n)$ rounds. Because any graph with $O(\log n)$ treewidth admits a low-congestion shortcut of quality ${\tilde{O}}(D)$, this result implies that it is not possible to compute the diameter of graphs efficiently by using only the property of low-congestion shortcuts.

Although our results exhibit a tight upper bound for graphs of diameter three or four, a more generalized lower bound is known for small-diameter graphs. [38]. For any $\log n \ge D \ge 3$, it is proved that there exists a network topology that incurs the ${\tilde{\Omega }}\left( n^{(D-2)/(2D-2)}\right)$-round time complexity for any MST algorithm. In more restricted cases of $D = 1$ and $D= 2$, Jurdzinski et al. [25] and Lotker et al. [32] showed O(1)- and $O(\log n)$-round MST algorithms, respectively.

1.4 Outline of the paper

The paper is organized as follows. In Sect. 2, we introduce the formal definitions of the CONGEST model, partwise aggregation, and low-congestion shortcuts, and other miscellaneous terminologies and notations. In Sect. 3, we present a shortcut construction for graphs of bounded doubling dimensions. In Sect. 4, we show the upper and lower bounds for shortcuts and MST in k-chordal graphs. Section 5 provides the shortcut algorithms for graphs of diameters of three or four. In Sect. 6, we prove the hardness result for bounded clique-width graphs. The paper is concluded in Sect. 7.

2 Preliminaries

2.1 CONGEST model

Throughout this paper, we denote [a, b] as the set of integers at least a and at most b. A distributed system is represented by a simple undirected connected graph $G=(V,E)$, where V denotes the set of nodes and E the set of edges. Let n and m be the numbers of nodes and edges, respectively, and D be the diameter of G. Each node has an ID from ${\mathbb {N}}$ (which is represented by $O(\log n)$ bits. In the CONGEST model, the computation follows a round-based synchrony. In one round, each node sends messages to its neighbors, receives messages from its neighbors, and executes the local computation. It is guaranteed that every message sent in a round is delivered to the destination within the same round. Each link can transfer $O(\log n)$-bit information (bidirectionally) per round, and each node can inject different messages to its incident links. Each node has no prior knowledge of the network topology. except for its neighbor IDs. Given a graph H for which the node and link sets are not explicitly specified, we denote them by $V_H$ and $E_H$, respectively. Let N(v) be the set of nodes that are adjacent to v in G, and let $N^{+}(v) = N(v)\cup \{v\}$. We define $N(S) = \cup _{s \in S} N(s)$, and $N^{+}(S) = \cup _{s \in S} N^+(s)$, for any $S \subseteq V$. For two node subsets $X, Y \subseteq V$, we also define $E(X, Y) = \{(u, v) \in E \mid u \in X, v \in Y\}$. If X (resp. Y) is a singleton $X = \{w\}$, (resp. $Y=\{w\}$), we describe E(X, Y) as E(w, Y) (resp. E(X, w)). The distance (that is, the number of edges in the shortest path) between two nodes u and v in G is denoted by $\mathrm {dist}_{G}(u,v)$, and we let U be a path in G. With a small abuse of notations, we treat U as the sequence of nodes or edges representing the path, as the set of nodes or edges in the path, or the subgraph of G forming the path. A path $U=\{s_0,s_1,\dots ,s_{\ell }\}$ is referred to as chordless if and only if for any two nodes $s_{i},s_{j}\in U$ and $|i-j|\ge 2$, it holds that $(s_{i},s_{j}) \notin E_{G}$

2.2 Partwise aggregation

The partwise aggregation is a communication abstraction defined over a set ${\mathcal {P}} = \{P_1, P_2, \dots , P_N\}$ of mutually disjoint and connected subgraphs known as parts, and provides simultaneous fast group communication among the nodes in each $P_i$. It is formally defined as follows:

Definition 1

(Partwise Aggregation (PA)) Let ${\mathcal {P}} = \{P_1, P_2, \dots , P_N\}$ be the set of connected mutually-disjoint subgraphs of G, and each node $v \in V_{P_i}$ maintains variable $b^i_v$ storing an input value $x^i_v \in X$. The output of the partwise aggregation problem is to assign $\oplus _{w \in P_i} x^i_w$ with $b^i_v$ for any $v \in V_{P_i}$, where $\oplus$ is an arbitrary associative and commutative binary operation over X.

The straightforward solution of the partwise aggregation problem in the CONGEST model is to perform the convergecast and broadcast in each part $P_{i}$ independently. In particular, we construct a BFS tree for each part $P_i$ (after the selection of the root by any leader election algorithm). The time complexity is proportional to the diameter of each part $P_i$, which can be large ($\Omega (n)$ in the worst case) independently of the diameter of G.

2.3 (d, c)-shortcut

As we stated in the introduction, the notion of low-congestion shortcuts is introduced for quickly solving the partwise aggregation problem (for some specific graph classes). Formal definition of (d, c)-shortcuts is provided as follows:

Definition 2

(Ghaffari and Haeupler [15]) Given a graph $G=(V,E)$ and partition ${\mathcal {P}} = \{P_1, P_2, \dots , P_N\}$, of G into node-disjoint and connected subgraphs, we define a (d, c)-shortcut of G and ${\mathcal {P}}$ as a set of subgraphs ${\mathcal {H}} = \{H_1, H_2, \dots , H_N\}$ of G such that;

• 1. For each i, the diameter of $P_{i}+H_i$ is at most d (d-dilation).

• 2. For each edge $e\in E$, the number of subgraphs $P_{i}+H_{i}$ containing e is at most c (c-congestion).

The values of d and c for a (d, c)-shortcut ${\mathcal {H}}$ are known as the dilation and congestion of ${\mathcal {H}}$. As a general statement, a (d, c)-shortcut that is constructed in f rounds admits the solution of the partwise aggregation problem in ${\tilde{O}}(d+c+f)$ rounds [14, 15]. Because the parameter $d + c$ asymptotically affects the performance of the application, we refer to the value of $d + c$ as the quality of (d, c)-shortcuts. A low-congestion shortcut with quality q is simply known as a q-shortcut.

2.4 Lower-bound framework

Fig. 1

Example of ${\mathcal {G}}(O(lb),b,l,O(\log n))$

To prove the lower bound of MST, we introduce a simplified version of the framework by Das Sarma et al. [38]. In this framework, we consider the graph class ${\mathcal {G}}(n,b,\ell ,c)$ that is defined below. A vertex set $X\subseteq V$ is known as connected if the subgraph induced by X is connected.

Definition 3

For $n,b,c\ge 0$ and $\ell \ge 3$, the graph class ${\mathcal {G}}(n,b,\ell ,c)$ is defined as the set of n-vertex graphs $G = (V, E)$ satisfying the following conditions:

• (C1) The vertex set V is partitioned into $\ell$ disjoint vertex sets ${\mathcal {X}} = \{X_1, X_2, \dots , X_{\ell }\}$, such that $X_1$ and $X_{\ell }$, are singletons (let $X_1 = \{s\}$ and $X_{\ell } = \{r\}$).

• (C2) The vertex set $V \backslash \{s, r\}$ is partitioned into b disjoint connected sets ${\mathcal {Q}} = \{Q_1,\dots ,Q_{b}\}$ such that $|E(X_1, Q_i)| \ge 1$ and $|E(X_{l}, Q_i)| \ge 1$ hold for any $1\le i \le b$.

• (C3) Let $R_{i}=\bigcup _{i+1 \le j \le \ell }X_{j}$ and $L_{i}=\bigcup _{0 \le j \le \ell -i}X_{j}$. For $2 \le i \le \ell /2-1$, $|E(R_{i}, N(R_{i})\setminus R_{i-1})| \le c$, and $|E(L_{i}, N(L_{i})\setminus L_{i-1})| \le c$.

Figure 1 shows the vertex partition ${\mathcal {X}}$ and ${\mathcal {Q}}$ for the hard-core instances presented in the original proof by Das Sarma et al. [38]. This graph belongs to ${\mathcal {G}}(O(\ell b),b,\ell ,O(\log n))$. For class ${\mathcal {G}}(n,b,\ell ,c)$, the following theorem holds, which is just a corollary of the result by Das Sarma et al. [38]:

Theorem 2

(Das Sarma et al. [38]) For any graph $G \in {\mathcal {G}}(n,b,\ell ,c)$ and any MST algorithm A, there exists an edge-weight function $w_{A,G}:E \rightarrow {\mathbb {N}}$ such that the execution of A in G requires ${\tilde{\Omega }}(\min \{b/c,\ell /2-1\})$ rounds. This bound holds with high probability even if A is a randomized algorithm.

2.5 1-hop extension scheme

Throughout this study we utilize the 1-hop extension scheme for shortcut construction, which is stated as follows:

For any $V_{P_i} \subseteq V$, node $v \in V_{P_{i}}$ adds each incident edge (v, u) to $H_i$ and informs u of $(v, u) \in H_i$.

It is trivial to implement this scheme using only one round in the CONGEST model. Because each node belongs to one part, the congestion of each edge is at most two. Hence, the technical challenge of this scheme is bound dilation. For the proof, we introduce the concept of (a, b)-path dominating set, which characterizes the graphs allowing good shortcuts through 1-hop extension.

Definition 4

Given a path $U \subseteq G$, a (a, b)-path dominating set $S \subseteq V_G$ of U is a node subset satisfying the following two conditions:

• For any $u \in V_U$, there exists $s \in N^+(S)$ such that $\mathrm {dist}_U(u, s) \le a$ holds.

• $|S| \le b$.

It is easy to check that if S is a (a, b)-path dominating set of U, $S \cap N^+(U)$ is also a (a, b)-path dominating set of U. Thus, in the following argument, we assume that any (a, b)-path dominating set for U is a subset of $N^+(U)$ without loss of generality. We say that G is (a, b)-path dominating if and only if any chordless path $U \subseteq G$ has a (a, b)-path dominating set. By definition, any graph having a dominating set of size b is (0, b)-path dominating.

Lemma 1

The 1-hop extension constructs an $O((a+1)b)$-shortcut for any (a, b)-path dominating graph.

Proof

Because the congestion bound is trivial, we focus on bounding dilation. Let G be any (a, b)-path dominating graph, $P_i$ be any part of G, and $H_i$ is the shortcut through 1-hop extension for part $P_i$. Let $U = (s_0, s_1, \dots , s_\ell )$ be any shortest path in $P_i$. Because U is the shortest, it is chordless, and thus it has a (a, b)-path dominating set $S_U$ of size $b' \le b$. Let $Z_{U} =(V_{Z_{U}},E_{Z_{U}})$ be the subgraph of G such that $V_{Z_{U}}=U \cup S_{U}$ and $E_{Z_{U}} = E(U,U) \cup E(U,S_{U})$ holds. Because $S_U \subseteq N^+(U) \subseteq N^+(P_i)$, every edge in $E(U, S_{U})$ is a shortcut for $H_i$. Thus, to prove the lemma, it suffices to show that $\mathrm {dist}_{Z_{U}}(s_0, s_{\ell }) = O((a+1)b')$, for any U. The proof is by the induction on $b'$, that is, we show that every chordless path in $P_i$ having $(a, b')$-path dominating set $S_U$ of size $b' \le b$ satisfying $\mathrm {dist}_{Z_{U}}(s_i, s_j)$ is at most $(2a+3)b'$ for all $b' \le b$. (Basis) The case of $b'=1$: Let w be the unique node in $S_{U}$, i be the minimum index such that $s_{i}\in N^{+}(w)$ holds, and j be the maximum index such that $s_{j}\in N^{+}(w)$ holds. Because $S_{U}=\{w\}$ is a (a, 1)-path dominating set, we obtain $\mathrm {dist}_{Z_{U}}(s_0,s_{\ell }) \le \mathrm {dist}_{Z_{U}}(s_0,s_{i})+\mathrm {dist}_{Z_{U}}(s_{j},s_{\ell }) + 2 \le 2a+2$. (Inductive Step) Suppose as the induction hypothesis that any chordless path $U' = (s'_0, s'_1, \dots s'_{\ell '})$ in $P_i$ having $(a,b'')$-path dominating set of size $b'' < b'$ satisfies $\mathrm {dist}_{Z_{U'}}(s'_{0},s'_{\ell '})\le (2a+3)b''$. Let i be the minimum index such that $s_{i}\in N^{+}(S_{U})$ holds, w is any node in $S_{U} \cap N^{+}(s_i)$, and j be the maximum index such that $s_{j}\in N^{+}(w)$ (see Fig. 2). If $\ell - j \le a$, we obtain $\mathrm {dist}_{Z_{U}}(s_0,s_{\ell }) \le \mathrm {dist}_{Z_{U}}(s_0,s_{i})+\mathrm {dist}_{Z_{U}}(s_{j},s_{\ell }) + 2 \le 2a+2$. In the case of $\ell -j > a$, any node $s_{h}$ for $j+a+1\le h \le \ell$ has no node $s_{h'}$ such that $s_{h'}\in N^{+}(w)$ and $|h'-h|\le a$ hold. Hence, the vertex set $S_{U} \backslash \{w\}$ is a $(a,b^{*})$-path dominating set of (chordless) subpath $U^{*} = (s_{j+a+1}, \dots , s_{\ell })$. Because $b^{*} < b'$ holds, by the induction hypothesis, we obtain $\mathrm {dist}_{Z_{U}}(s_{j+a+1},s_{\ell }) \le \mathrm {dist}_{Z_{U^{*}}}(s_{j+a+1},s_{\ell }) \le (2a+3)b^{*}$. It follows that $\mathrm {dist}_{Z_{U}}(s_0, s_{\ell }) \le \mathrm {dist}_{Z_{U}}(u,s_{i})+\mathrm {dist}_{Z_{U}}(s_{j},s_{j+a+1}) + \mathrm {dist}_{Z_{U}}(s_{j+a+1},s_{\ell }) + 2 \le a + (a+1) +(2a+3)b^{*} + 2 \le (2a+3)b'$. The lemma holds. $\square$

Fig. 2

Proof of Lemma 1

3 Low-congestion shortcut for constant doubling dimension graphs

A pair of a set V and the associated function $\mathrm {dist}: V \times V \rightarrow {\mathbb {R}}$ is known as a metric space if and only if the following three conditions hold: (1) $\mathrm {dist}(u,v)=0$ if and only if $v=u$, (2) $\mathrm {dist}(u,v)=\mathrm {dist}(v, u)$ for all $u,v\in V$, and (3) $\mathrm {dist}(u,v)\le \mathrm {dist}(u,w)+\mathrm {dist}(w,v)$ for all $u,v,w \in V$. The doubling dimension of a metric space V is the smallest positive integer x such that it is possible to cover the ball $B(v,r)= \{u\mid \mathrm {dist}(v,u)<r\}$ of radius r with the union of at most $2^{x}$ balls of radius r/2 for any $v \in V$ and $r>0$. A graph $G = (V, E)$ has a doubling dimension x if $(V, \mathrm {dist}_G)$ is a metric space of the doubling dimension x. The graphs of bounded doubling dimensions can be assumed to be a generalization of unit disk graphs, and are often considered in the context of distributed computing [7, 10, 27, 28]. The main results of this section are that the graphs of bounded doubling dimensions allow a good shortcut. We have the following theorem:

Theorem 3

Let x be the doubling dimension of the graph G. Then, there is an O(1)-round algorithm that constructs low-congestion shortcuts with quality ${\tilde{O}}(D^{x})$.

The theorem is obtained by combining the following lemma with Lemma 1. Recall that any graph having a dominating set of size ${\tilde{O}}(D^{x})$ is $(0, {\tilde{O}}(D^{x}))$-path dominating.

Lemma 2

Let G be any graph of the doubling dimension x. There is a dominating set of size ${\tilde{O}}(D^{x})$ in graph G.

Proof

We show that G is covered by at most $2^{ix}$ balls with radius $(D/2^{i})$ for any $1\le i \le \log D$. The lemma is obtained by setting $i=\log D$. The proof follows the induction on i. (Basis) The case of $i = 1$ is obtained from the definition of the doubling dimension. (Inductive step) Suppose as the induction hypothesis that there exists at most $2^{ix}$ balls with radius $D/2^{i}$ that cover the graph G. By the definition of the doubling dimension, each ball with radius $D/2^{i}$ can be covered at most by $2^{x}$ balls with radius $D/2^{i+1}$. Therefore, there exist at most $2^{(i+1)x}$ balls with radius $D/2^{i+1}$ that cover the graph G. The lemma is proved. $\square$

4 Low-congestion shortcut for k-Chordal graphs

4.1 k-Chordal graph

A graph G is k-chordal if and only if every cycle of length larger than k has a chord (equivalently, G contains no induced cycle of length larger than k). In particular, 3-chordal graphs are simply known as chordal graphs, which are known to be related to various intersection graph families such as interval graphs [13, 34]. The main results of this section are the following two theorems:

Theorem 4

There is an O(1)-round algorithm that constructs an O(kD)-shortcut for any k-chordal graph.

Theorem 5

For $k\le D$ and $kD\le \sqrt{n}$, there exists an unweighted k-chordal graph $G = (V, E)$ where for any MST algorithm A, there exists an edge-weight function $w_A : E \rightarrow {\mathbb {N}}$ such that the running time of A becomes ${\tilde{\Omega }}(kD)$ rounds.

4.2 Proof of Theorem 4

The Theorem 4 is deduced from the following lemma and Lemma 1.

Lemma 3

Any k-chordal graph is $(k,D+1)$-path dominating.

Proof

Let $U=\{u=s'_{0},s'_{1},\dots ,s'_{\ell '}=v\}$ be any chordless path, and let $S_{U}=\{u=s'_{0},s'_{1},\dots ,s'_{\ell '}=v\}$ be the shortest path from u to v. We show that $S_{U}$ is a (k, D)-path dominating set for U. Because the diameter of graph G is D, $|S_{U}|\le D$ holds. Thus, it suffices to show that for any $v \in V_{U}$, there exists $v' \in N^{+}(U)$ such that $\mathrm {dist}_{U}(v, v')\le k$ holds. Suppose that there exists a vertex $s_i$ that satisfies $|i-j| > k$ for any $s_{j}\in N^{+}(S_{U}) \cap U$. Let $l_{i}$ be the maximum index satisfying $s_{l_{i}} \in N^{+}(S_{U}) \cap U$ and $l_{i}\le i$, and $r_{i}$ be the minimum index that satisfies $s_{r_{i}} \in N^{+}(S_{U}) \cap U$ and $i \le r_{i}$. Because $s_{0}$ and $s_{\ell }$ are included in $N^{+}(S_{U})$, $s_{l_{i}}$ and $s_{r_{i}}$ always exist.

First, we consider the case in which there exists $w \in N^{+}(s_{l_{i}}) \cap N^{+}(s_{r_{i}}) \cap S_{U}$. Let C be the cycle consisting of $\{s_{l_i},\dots ,s_{r_i}, w\}$. Because the length of C is at least k, C has a chord, but the subpath $\{s_{l_i},\dots ,s_{r_i}\}$ is chordless; thus, there are no edges $(s_{j},s_{h}) \in E_{G}$ for any $l_{i}\le j < h \le r_{j}$ and $h-j\ge 2$. By the definition of $l_{i}$ and $r_{i}$, for any $l_{i}< j < r_{i}$, the node $s_{j}$ is not included in $N^{+}(w)$; that is, C has no chord, but this is a contradiction.

Next, we consider the case in which there are no $w\in N^{+}(s_{l_{i}})\cap N^{+}(s_{r_{i}})\cap S_{U}$. We choose y and $y'$ satisfying the following conditions (see Fig. 3):

• $s'_{y}\in N^{+}(s_{l_{i}})$ and $s'_{y'}\in N^{+}(s_{r_{i}})$.

• Letting $y_{max}= \max (y, y')$ and $y_{min}=\min (y,y')$, for any $y_{min}< j< y_{max}$, there is no $s'_{j}$ that satisfies $s'_{j}\in N^{+}(s_{l_{i}}) \cup N^{+}(s_{r_{i}})$.

We consider the cycle C consisting of $\{s'_{y' _{min}},\dots , s'_{y'_{max}}\}$, $(s'_{y'},s_{r_{i}})$, $\{s_{l_{i}},\dots ,s_{r_{i}}\}$, and $(s_{l_{i}},s'_{y'})$. Because the length of C is at least k, C has a chord. Notably, any shortest paths are chordless paths. Thus, there are no edges $(s'_{j},s'_{h}) \in E_{G}$ for any $y_{min}\le j < h \le y_{max}$ and $h-j\ge 2$. Because the subpath $\{s_{l_i},\dots ,s_{r_i}\}$ is chordless, there are no edges $(s_{j},s_{h}) \in E_{G}$ for any $l_{i}\le j < h \le r_{j}$ and $h-j\ge 2$. By the definition of $l_{i}$, $r_{i}$, y, and $y'$, we obtain $E(\{s_{l_i},\dots ,s_{r_i}\},\{s'_{y'_{min}},\dots , s'_{y' _{max}}\}) =\{(s_{l_{i}},s'_{y'}),(s'_{y'},s_{r_{i}})\}$. Consequently, we can conclude that C has no chord. It is a contradiction; thus, the lemma is proved. $\square$

Fig. 3

Proof of Lemma 3

4.3 Proof of Theorem 5

We first introduce the instance mentioned in Theorem 5. Because it has two additional parameters $x\ge 0$ and $N\ge 2$ as well as k, we refer to this instance as $G(k,x,N) = (V(k, x, N), E(k,x,N))$ in the following argument: The parameters x and N are adjusted later to obtain the claimed lower bound. Let $K = k/2 - 1$ be short. The vertex set and edge set of G(k, x, N) is defined as follows:

• $V(k, x, N) = \{v_{1,j} \mid 0 \le j \le x\} \cup \{v_{i,j} | 2 \le i \le N , 0 \le j \le xK \}$.

• $E(k, x, N) = E_1 \cup E_2 \cup E_3 \cup E_4$ such that $E_1 = \{\{v_{1,j},v_{1,j+1}\} \mid 0 \le j \le x-1\}$, $E_2 = \{\{v_{i,j},v_{i,j+1}\} \mid 2 \le i \le N , 0 \le j \le xK - 1\}$, $E_3 = \{\{v_{1,j},v_{i,h}\} \mid 2 \le i \le N , 0 \le j \le x, h = jK\}$, and $E_4 = \{\{v_{i,h},v_{j,h}\} \mid 2 \le i, j \le N, i \ne j, h\bmod {K} = 0\}$.

Figure 4 illustrates the graph G(k, x, N).

Fig. 4

Example of k-chordal graph G(k, x, N)

It is cumbersome to check whether this graph is k-chordal, but straightforward. One can show the following lemma.

Lemma 4

For $x\ge 0$ and $N\ge 2$, G(k, x, N) is the k-chordal.

Proof

For simplicity, we give some of the vertices a name $v'_{xy}$ as follows:

• $v'_{1,j} = v_{1,j} (0 \le j \le x)$

• $v'_{i,j} = v_{i,h} (2 \le i \le N , 0 \le j \le x , h = jK)$.

We define a subset of vertices known as row and column. The i-th row $R_i$ is defined as $R_i=\{v'_{i,j} | 0 \le j \le x\}$, and the i-th column $C_i$ is defined as $C_i=\{v'_{j,i} | 1 \le j \le N\}$. First, we consider the diameter of G(k, x, N). For $2\le i \le N$ and $0\le j \le xK$, we have $\min _{0\le k \le x} \mathrm {dist}(v'_{1,k} , v_{i,j}) = \min _{0\le k \le x} \mathrm {dist}(v'_{i,k} , v_{i,j}) + 1 \le K/2 + 1$. For $0\le i\le x$ and $0\le j\le x$, $\mathrm {dist}(v'_{1,i},v_{1,j})\le x-1$, holds; thus, the diameter of G(k, x, N) is at most $K+1+x$. We consider a cycle X in G(k, x, N). Let l and r be the minimum/maximum indices of the rows X intersect, Similarly, let t and b be the minimum/maximum indices of the columns X intersect. Let m be the index such that $|C_m \cap X|$ maximizes, and let $a_m = |C_m \cap X|$ for short. Any cycle X applies to one of the following four cases:

• 1. $r-l \ge 2$ holds.

• 2. $a_m \ge 3$ and $r-l\ne 0$ hold.

• 3. $r-l = 0$ holds.

• 14. $r-l = 1$ and $a_m = 2$ hold.

We show that Lemma 4 holds for all the cases (Fig. 5 almost states the proof).

• 1. The case of $r-l \ge 2$: Through the construction of G(k, x, N), l-r path intersects $(l+1)$-column at least twice. Let u and v be the intersection of the X and $(l+1)$-column. Because $C_{l+1}$ is clique, u and v are adjacent. Thus the edge (u, v) is a chord of X.

• 2. The case of $a_m \ge 3$ and $r-l\ne 0$: There exists two vertices in $C_{m}$, which are not adjacent in X. Because $C_m$ is a clique, there exists an edge between them, and this edge is a chord of X.

• 3. The case of $r-l = 0$: The cycle X is a clique in graph G and the lemma holds clearly.

• 4. The case of $r-l = 1$ and $a_m=2$: The cycle consists of four vertices $v'_{t, l}$,$v'_{t, r}$,$v'_{b, l}$, $v'_{b, r}$ and two paths, that is; the paths connecting $v'_{t, l}$ with $v'_{t, r}$, and $v'_{b, l}$ with $v'_{b, r}$. It follows $\mathrm {dist}(v'_{t, l},v'_{t, r}) \le K = k/2 - 1$, $\mathrm {dist}(v'_{b ,l},v'_{b, r}) = K = k/2 - 1$, and $\mathrm {dist}(v'_{t, l},v'_{b, l}) = \mathrm {dist}(v'_{t, r},v'_{b, r}) = 1$. Thus, the length of X is at most k.

The lemma is proved. $\square$

Fig. 5

Proof of Lemma 4

The proof of Theorem 5 follows the framework of Das Sarma et al. [38]. It suffices to prove the following lemma. Theorem 5 is obtained by combining this lemma with Theorem 2.

Lemma 5

For any $D>2K$ and $N\ge 2kD$, $G(k,D-K,N) \in {\mathcal {G}}(n,N,(D-K)K+3,1)$, holds.

Proof

We define ${\mathcal {X}} = \{X_1,X_2,\dots ,X_{(D-K)K+3}\}$ for $G(k,D-K,N)$ as follows:

• $X_{i} = \{v_{1,0}\}$ ($i = 1$).

• $X_{i} = \{v_{j,0}\mid 2\le j \le N\}$ ($i = 2$).

• $X_{i} = \{v_{j,i-2}\mid 2\le j \le N\}\cup \{v_{(i-2)/K,1} \}$ ($3 \le i \le (D-K) K,i \bmod {K} = 2$).

• $X_{i} = \{v_{j,i-2} \mid 2\le j \le N \}$ ($3\le i \le (D-K)K, i \bmod {K} \ne 2$).

• $X_{i} = \{v_{j,(D-K)K-1}\mid 2\le j \le N\}$ ($i=(D-K)K+2$).

• $X_{i} = \{v_{1,(D-K)}\}$ ($i=(D-K)K+3$)

We define ${\mathcal {Q}} = \{Q_1,Q_2,\dots ,Q_{N}\}$ for $G(k,D-K,N)$ as follows:

• $Q_{i} = \{v_{1,j}\mid 1\le j \le (D-K)-1 \}$ ($i=1$).

• $Q_{i} = \{v_{i,j}\mid 0\le j \le (D-K)K\}$ ($2\le i\le N$)

It is easy to check that (C1) and (C2) are satisfied. Thus, we only show that (C3) is satisfied. We have $E(R_{i},N(R_{i})\backslash R_{i-1})$ and $E(L_{i},N(L_i)\backslash L_{i})$ as follows:

• $E(R_{i},N(R_{i})\backslash R_{i-1}) = \{v_{1,0}\}$ ($i=2$).

• $E(R_{i},N(R_{i})\backslash R_{i-1}) = \{v_{1,\lfloor (i-1)/K\rfloor }$ ($3\le i\le ((D-K)K)/2, i\bmod {K}\ne 2$).

• $E(R_{i},N(R_{i})\backslash R_{i-1}) = \emptyset$ ($3\le i\le ((D-K)K)/2, i\bmod {K}= 2$).

• $E(L_{i},N(L_{i})\backslash L_{i-1}) = \{v_{1,D-K}\}$ ($i=2$).

• $E(L_{i},N(L_{i})\backslash L_{i-1}) = v_{1, D-K-\lfloor (i-2)/K\rfloor }\}$ ($3\le i\le ((D-K)K)/2, i\bmod {K}\ne 2$).

• $E(L_{i},N(L_{i})\backslash L_{i-1}) = \emptyset$ ($3\le i\le ((D-K)K)/2, i\bmod {K}= 2$).

Thus, we have $| E(R_{i} , N(R_{i})\backslash R_{i-1})| \le 1$, and $|E( L_{i} , N ( L_{i} ) \backslash L_{i-1} ) | \le 1$. Therefore, we can prove that the graph $G(k,D-K,N)$ is included in ${\mathcal {G}}(n,N,(D-K)K+3,1)$. $\square$

5 Low-congestion shortcut for small diameter graphs

Let $\kappa _D = n^{(D - 2)/(2D - 2)}$. Note that $\kappa _3 = n^{1/4}$, and $\kappa _4 = n^{1/3}$ hold. The main result in this section is the following theorem:

Theorem 6

For any graph of diameter $D \in \{3, 4\}$, there exists an algorithm for constructing low-congestion shortcuts with quality ${\tilde{O}}(\kappa _{D})$ in ${\tilde{O}}(\kappa _{D})$ rounds.

5.1 Centralized construction

In the following argument, we use terminology “whp.” (with high probability) to mean that the event considered occurs with probability $1 - n^{-\omega (1)}$ (or equivalently $1 - e^{-\omega (\log n)}$). For simplicity of the proof, we treat any whp. event as if it necessarily occurs. (i.e., with P=1). Because the analysis below handles only a polynomially bounded number of whp. events, the standard union-bound argument guarantees that everything simultaneously occurs whp; that is, any consequence yielded by the analysis also occurs whp. Because the proof is constructive, we first present the algorithms for $D = 3$ and 4. They are described as a (unified) centralized algorithm, and the distributed implementation is explained later. Let $N'$ be the number of parts whose diameter is greater than $12\kappa _D\log ^3 n$ (say large part). Assume that $P_1, P_2, \dots , P_{N'}$ are large without loss of generality. Because each part $P_i$ ($1 \le i \le N'$) contains at least $\kappa _D$ nodes, $N' \le n / \kappa _D$ holds clearly. Our technical challenge is to reduce the dilation of the large part. To this end, we separate the large part into the subparts whose diameters are ${\tilde{O}}(\kappa _D)$, and shows that the shortcut edges establish at least one length-D path between any two subparts. Note that this separation scheme is introduced only for the analysis, and the algorithm does not actually construct it. The detailed explanation of the scheme is explained later. First, each large part computes 1-hop extension. As shown in the previous section, the 1-hop extension only increases the congestion by O(1). Therefore, it suffices to show that at least one shortcut path of length $D-2$ is established between any two extended subparts. For the case of $D = 3$, the independent sampling of each edge with probability $1/n^{1/2}$ guarantees the construction of such paths (of length $D - 2 = 1$). For the case of $D = 4$, we introduce a new edge sampling scheme based on hash functions of limited independence, which positively correlates two edges incident to a common vertex, and thus amplifies the probability of establishing length-2 shortcut paths without too much increase of congestion. The precise description of the algorithms is stated below. It is applied to each large part $P_i$ for the construction of $H_i$.

• 1. Each node $v \in V_{P_i}$ adds its incident edges to $H_i$ (i.e., compute the 1-hop extension).

• 2. This step adopts two different strategies according to the value of D. ($D = 3$) Each node $u \in N^{+}(V_{P_i})$ adds each incident edge (u, v) to $H_i$ with probability $1 / n^{1/2}$. ($D = 4$). Let ${\mathcal {Y}} = [1, n^{1/3}/\log n]$. We first prepare an $(n^{1/3}\log ^3 n)$-wise independent hash function $h: [0, N - 1] \times V \rightarrow {\mathcal {Y}}$,³. At node $u \in V$, each incident edge (u, v) satisfying $v \in N^{+}(V_{P_i})$ is independently sampled with probability 1/h(u, i). All the sampled edges are added to $H_i.$

We show that this algorithm provides a low-congestion shortcut of quality ${\tilde{O}}(\kappa _D)$. First, we observe the bound for congestion. Let $H^1_i$ be the set of the edges added to $H_i$ in the first step, and $H^2_i$ be those added in the second step. Because the congestion of the 1-hop extension is negligibly small, it suffices to consider the congestion incurred by step 2. Intuitively, we can believe the congestion of ${\tilde{O}}(\kappa _D)$ as the expected congestion of each edge is ${\tilde{O}}(\kappa _D)$: Because the total number of large parts is at most $n / \kappa _D$, the expected congestion of each edge incurred in step 2 is $n / \kappa _D \cdot (1 / n^{1/2}) = O(n^{1/4})$ for $D = 3$, and $(n/\kappa _D)\sum _{y \in {\mathcal {Y}}} (1/y) \cdot (1/|{\mathcal {Y}}|) \le (n/\kappa _D) \cdot (\log n / |{\mathcal {Y}}|) = {\tilde{O}}(n^{1/3})$ for $D = 4$.

Lemma 6

The congestion of the constructed shortcut is ${\tilde{O}}(\kappa _D)$ whp.

Proof

It suffices to show that the congestion of any edge $e = (u, v)\in E$ is ${\tilde{O}}(\kappa _D)$, whp. For simplicity of proof, we observe an undirected edge $e = (u, v)$ as two (directed) edges (u, v) and (v, u), and distinguish the events by adding (u, v) to shortcuts by u and that by v; that is, the former is recognized as adding (u, v), whereas the latter is recognized by adding (v, u). Clearly, the asymptotic bound holding for directed edge (u, v) also holds for the corresponding undirected edge (u, v) actually existing in G (which is at most twice of the directed bound). Because the first step of the algorithm increases the congestion of each directed edge at most by one, it suffices to show that the congestion incurred by the second step is at most ${\tilde{O}}(\kappa _D)$.

Let $X_i$ be the indicator random variable for event $(u, v) \in H^2_i$, and $X = \sum _{i} X_i$. The goal of the proof is to show that $X = {\tilde{O}} (\kappa _D)$ holds whp. The cases of $D = 3$ and $D = 4$ are proved separately. ($D = 3$) Because at most $n/ \kappa _3$ large parts exist, we have that ${\mathbb {E}}[X] \le (n/\kappa _3) \cdot (1/n^{1/2}) = n^{1/4} = \kappa _3.$ The straightforward application of the Chernoff bound to X allows us to bound the congestion of e by at most $2\kappa _3$ with probability $1 - e^{-\Omega (n^{1/4})}$. ($D = 4$) Let ${\mathcal {P}}'$ be the subset of all large parts $P_j$ such that $u \in N^+(P_j)$ holds. Consider an arbitrary partition of $\mathcal {P'}$ into several groups with a size of at least $(n^{1/3}\log ^3 n)/2$ and at most $n^{1/3}\log ^3 n$. Let q be the number of groups. Each group was identified by a number $\ell \in [1, q]$. We refer to the $\ell$-th group as ${\mathcal {P}}^{\ell }$. Fixing $\ell$, we bound the number of parts in ${\mathcal {P}}^{\ell }$ using $e = (u, v)$ as the shortcut edge. Let $Y_{i}$ be the value of h(u, i). For $P_i \in {\mathcal {P}}^{\ell }$, the probability that $X_{i} = 1$ is

$$\begin{aligned} \Pr [X_{i} = 1]&=\sum _{y \in {\mathcal {Y}}} \Pr [Y_{i} = y] \frac{1}{y}\\&= \frac{ Har \left( |{\mathcal {Y}}|\right) }{|{\mathcal {Y}}|}, \end{aligned}$$

where $Har (x)$ is the harmonic number of x, i.e., $\sum _{1 \le i \le x} i^{-1}$. Letting $X^{\ell } = \sum _{j \in P^{\ell }} X_{j}$, we have ${\mathbb {E}}[X^{\ell }] = (|P^{\ell }| Har (|{\mathcal {Y}}|))/|{\mathcal {Y}}|$. Because $Har (x) \ge 1$, we have ${\mathbb {E}}[X^{\ell }] \ge |P^{\ell }|/|{\mathcal {Y}}| = (\log ^4 n)/2$. As the hash function h is $(n^{1/3}\log ^3 n)$-wise independent, it is easy to check that $X_{ 1}, X_{ 2}, \dots , X_{ p^{\ell }}$ are independent. We apply Chernoff bound to $X^{\ell }$ and obtain $\Pr [X^{\ell } \le 2{\mathbb {E}}[X^{\ell }]] \ge 1 - e^{-\Omega ({\mathbb {E}}[X^{\ell }])} = 1 - e^{-\Omega (\log ^4 n)}.$ It implies that for any $\ell$, at most $2{\mathbb {E}}[X^{\ell }]$ groups use (u, v) as their shortcut edges. The total congestion of (u, v) is obtained by summing up $2{\mathbb {E}}[X^{\ell }]$ for all $\ell \in [1, q]$, which results in the following:

$$\begin{aligned} \sum _{\ell }2 {\mathbb {E}}[X^{\ell }]&\le \sum _{\ell } \frac{2|P^{\ell }| \log n }{ |{\mathcal {Y}}|}\\&= \frac{2|{\mathcal {P}}'|\log n}{|{\mathcal {Y}}|} \\&= {\tilde{O}}(n^{1/3}). \end{aligned}$$

The lemma is proved. $\square$

For bounding dilation, we first introduce several preliminary notions and terminologies. Given a graph $G = (V, E)$, a subset $S \subset V$ is known as an $(\alpha , \beta )$-ruling set if it satisfies the following: (1) for any $u, v \in S$, $\mathrm {dist}_G(u, v) \ge \alpha$ holds, and (2) for any node $v \in V$, there exists $u \in S$ such that $\mathrm {dist}_G(v, u) \le \beta$ holds. It is known that there exists an $(\alpha , \alpha +1)$-ruling set for any graph G [2]. Let ${\hat{P}}_i = P_i + H^1_i$ for short. For the analysis of $P_i$’s dilation, it suffices to consider the case where the diameter of ${\hat{P}}_{i}$ is greater than $12\kappa _D \log ^{3}n$. We first consider an $(\alpha , \alpha + 1)$-ruling set of ${\hat{P}}_i$ for $\alpha = 12\kappa _D \log ^3 n$, which is denoted by $S = \{s_0, s_1, \dots , s_z\}$. Note that this ruling set is introduced only for the analysis, and the algorithm does not actually construct it. The key observation of the proof is that for any $s_j$ ($1 \le j \le z$) $H_i$ contains a path of length ${\tilde{O}}(\kappa _D)$ from $s_0$ to $s_j$ whp. It follows that any two nodes $u, v \in V_{{\hat{P}}_i}$ are connected by a path of length ${\tilde{O}}(\kappa _D)$ in $P_i + H_i$ because any node in $V_{{\hat{P}}_i}$ has at least one ruling set node within distance $\alpha + 1$ in $P_i + H^1_i$.

To prove the above-mentioned claim, we further introduce the notion of terminal sets. A terminal set $T_j \subseteq V_{P_i}$ associated with $s_j \in S$ ($0 \le j \le z$) is the subset of $V_{P_i}$ satisfying (1) $|T_j| \ge \kappa _D \log ^3 n$, (2) $\mathrm {dist}_{P_i + H_i}(s_j, x) \le 6 \kappa _D \log ^3 n$ for any $x \in T_j$, and (3) $N^+(x) \cap N^+(y) = \emptyset$ for any $x, y \in T_j$ (note that $N^+(\cdot )$ is the set of neighbors in G, not in $P_i + H^1_i$). We can show that such a set always exists.

Lemma 7

Let $S = \{s_0, s_1, \dots , s_z\}$ be any $(\alpha , \alpha + 1)$-ruling set of ${\hat{P}}_i$ for $\alpha = 14\kappa _D \log ^3 n$, there always exists a family of the terminal sets ${\mathcal {T}} = \{T_0, T_1, \dots , T_z\}$ associated with S.

Proof

The proof is constructive. Let $c = 6 \kappa _D \log ^3 n$. We take an arbitrary shortest path $Q = (s_j = u_0, u_1, u_2, \dots , u_{c})$ of length c in $P_i + H^1_i$, starting from $s_j \in S$. Because no two nodes in $N^+(V_{P_i}) \setminus V_{P_i}$ are adjacent in $P_i + H^1_i$, Q contains no two consecutive nodes, which are both in $N^+(V_{P_i}) \setminus V_{P_i}$; implying that at least half of the nodes in Q belong to $V_{P_i}$. Let $q' = (u'_{0}, u'_{1}, \dots u'_{c'})$ be the subsequence of Q consisting of the nodes in $V_{P_i}$. Then, we define $T_j = \{u'_{0}, u'_{3}, \dots , u'_{3\lfloor c'/3 \rfloor }\}$, which satisfies the three properties of terminal sets: It is easy to check that the first and second properties hold. In addition, one can show that $\mathrm {dist}_G(u'_x, u'_{x+a}) \ge 3$ (which is equivalent to $N^+(u'_x) \cap N^+(u'_{x + a}) = \emptyset$) holds for any $a \ge 3$, and $x \in [1, c' - a]$: Suppose that $\mathrm {dist}_G(u'_x, u'_{x+a}) \le 2$ holds for some $a \ge 3$ and $x \in [1, c' - a]$, the distance between $u'_x$ and $u'_{x+a}$ implies $N^+(u'_x) \cap N^+(u'_{x+a}) \ne \emptyset$, and thus $\mathrm {dist}_{{\hat{P}}_i}(u'_x, u'_{x+a}) \le 2$ holds. Then, bypassing the subpath from $u'_{x}$ to $u'_{x+a}$ in Q through a distance-two path, we obtain a path from $s_j$ to $u_c$ shorter than Q. This contradicts the fact that Q is the shortest path. $\square$

The second property of terminal sets and the following lemma deduces that $\mathrm {dist}_{P_i + H_i}(s_0, s_j) = {\tilde{O}}(\kappa _D)$ holds for any $j \in [0, z]$.

Lemma 8

Let $S = \{s_0, s_1, \dots , s_z\}$ be any $(\alpha , \alpha + 1)$-ruling set of ${\hat{P}}_i$ for $\alpha = 14\kappa _D \log ^3 n$, and ${\mathcal {T}} = \{T_0, T_1, \dots , T_z\}$ be a family of terminal sets associated with S. For any $j \in [0, z]$, there exist $u \in T_0$ and $v \in T_j$ such that $\mathrm {dist}_{P_i + H_i}(u, v) = O(1)$ holds.

Proof

Because the distances $s_0$ and $s_j$ are at least $14\kappa _D\log ^3 n$, we have $N^+(T_0) \cap N^+(T_j) = \emptyset$. The proof is divided into the cases of $D = 3$ and $D = 4$. ($D = 3$) Under the conditions of $N^+(T_0) \cap N^+(T_j) = \emptyset$ and $D = 3$, there exists a path of length exactly three from any node $a \in T_0$ to any node $b \in T_j$. Letting $e_{a, b}$ be the second edge in that path, we define $F = \{e_{a, b} \mid a \in T_0, b \in T_j\}$. By the third property of the terminal sets and because $N^+(T_0) \cap N^+(T_j) = \emptyset$, for any two edges $(x_1, y_1), (x_2, y_2) \in F$, either $x_1 \ne x_2$, or $y_1 \ne y_2$ holds; that is, $e_{a_1, b_1} \ne e_{a_2, b_2}$ holds for any $a_1, a_2 \in T_0$, $b_1, b_2 \in T_j$ and $(a_{1},b_{1})\ne (a_{2},b_{2})$. The first property of terminal sets implies that $|F| = |T_0||T_j| \ge (\kappa _D \log ^3 n)^2$. Because each edge in F is added to $H^2_i$, with probability $1/n^{1/2} = 1/ \kappa _D^2$, the probability that no edge in F is added to $H^2_i$ is at most $(1 - 1 / \kappa _D^2)^{(\kappa _D \log ^3 n)^2} \le e^{-\Omega (\log ^6 n)}$; that is, an edge $e_{a, b}$ is added to $H_i$ whp. and then $\mathrm {dist}_{P_i + H_i}(a, b) \le 3$ holds. ($D = 4$) For any node $u \in T_0$ and $v \in T_j$, there exists a path from u to v of length three or four in G. That path necessarily contains a length-two sub-path $P_2(u, v) = (a_{uv}, b_{uv}, c_{uv})$ such that $a_{uv} \in N^+(u)$ and $c_{uv} \in N^+(v)$ hold (if $P_2(u, v)$ is not uniquely determined, an arbitrary length-two sub-path is chosen). We refer to $(a_{uv}, b_{uv})$ and $(b_{uv}, c_{uv})$, the first and second edges of $P_2(u, v)$, respectively. (See Fig. 6.) Let ${\mathcal {P}}_2 = \{P_2(u, v) \mid u \in T_0, v \in T_j\}$, $G'$ is the union of $P_2(u, v)$ for all $u \in T_0$ and $v \in T_j$, and ${\mathcal {P}}^e_2 = \{P_2(u, v) \in {\mathcal {P}}_2, \mid e \in P_2(u, v)\}$ for any $e \in E_{G'}$. We first bound the size of ${\mathcal {P}}^e_2$. Assume that e is the first edge of a path in ${\mathcal {P}}^e_2$. Let $e = (a, b)$ and $u \in T_0$ be the (unique) node such that $a \in N^+(u)$ holds. Because at most $|T_j|$ paths in ${\mathcal {P}}_2$ can start from a node in $N^+(u)$, the number of paths in ${\mathcal {P}}_2$ using e as their first edges is at most $|T_j|$. Similarly, if e is the second edge of some path in ${\mathcal {P}}^e_2$, at most $|T_0|$ paths in ${\mathcal {P}}_2$ can contain e as their second edges. Although some edges may be used as both the first and second edges, the total number of paths using e is bounded by $|T_0| + |T_j| = 2 \kappa _D \log ^3 n$, implying that any path $P_2(u, v)$ can share edges with at most $4 \kappa _D \log ^3 n$ edges, and thus, ${\mathcal {P}}_2$ contains at least $|T_0||T_j| / (4\kappa _D \log ^3 n + 1) \ge \kappa _D \log ^3 n/ 5$ edge-disjoint paths. Let ${\mathcal {P}}'_2 \subseteq {\mathcal {P}}_2$ be the maximum-cardinality subset of ${\mathcal {P}}_2$ such that any $P_2(u_1, v_1), P_2(u_2, v_2) \in {\mathcal {P}}'_2$ is edge-disjoint. We define $B = \{ b \mid (a, b, c) \in {\mathcal {P}}'_2\}$. Let $\Delta (b)$ be the number of paths in ${\mathcal {P}}'_2$, containing $b \in B$ as the center. Owing to the edge disjointness of ${\mathcal {P}}'_2$, we have $|E_{G}(N^+(T_0), b)| \ge \Delta (b)$, and $|E_{G}(N^+(T_j), b)| \ge \Delta (b)$ for any $b \in B$. Let $Y_b$ be the value of h(b, i), and $X_b$ be the indicator random variable that takes one if a path in ${\mathcal {P}}_2$, which contains b as the center, is added to $H_i$, and zero otherwise. Let X and Y be the indicator random variables corresponding to the events of $\bigvee _{b \in B} (X_b = 1)$ and $\bigvee _{b \in B} (Y_b \le \Delta (b)/\log ^2 n)$ respectively. By the definition of $\Delta (b)$, the probability of adding no first edge with b and the node in $T_{0}$ as the endpoints to $H_{i}$ is at most $\left( 1 - 1/h(b,i)\right) ^{\Delta (b)}$. Similarly, the probability of adding no second edge with b and the node in $T_{j}$ as the endpoints to $H_{i}$ is at most $\left( 1 - 1/h(b,i)\right) ^{\Delta (b)}$. Then, we obtain $\Pr [X_b = 1 \mid Y_b = y] \ge 1 - \left( 1 - 1/y\right) ^{\Delta (b)} - \left( 1 - 1/y\right) ^{\Delta (b)} \ge 1- 2e^{-\Delta (b)/y},$ and thus, $\Pr [X_b = 1 \mid Y_b \le \max \{1, \Delta (b)/\log ^2 n\}] \ge 1 - e^{-\Omega (\log ^2 n)}$ holds. Note that, if $Y_{b}=1$, then all incident edges of b included in ${\mathcal {P}}'_2$ must be added to $H_{i}$, so $X_{b}=1$ always holds. Therefore, $\Pr [X = 1 \mid Y = 1] \ge 1 - e^{-\Omega (\log ^2 n)}$ holds. Because h is $(n^{1/3}\log ^3 n)$-wise independent, $Y_b$ for all $b \in B$ are independent. Thus, we obtain the following:

$$\begin{aligned} \Pr [Y = 1]&= 1 - \Pr [Y = 0] \\&=1 - \Pr \left[ \bigwedge _{b \in B} Y_b> \max \left\{ 1,\frac{\Delta (b)}{\log ^2 n}\right\} \right] \\&=1 - \prod _{b \in B , \Delta (b)> 2\log ^2 n} \Pr \left[ Y_b> \frac{\Delta (b)}{\log ^2 n}\right] \\&\quad \times \prod _{b \in B , \Delta (b) \le 2\log ^2 n} \Pr \left[ Y_b> 1\right] ,\\&=1 - \prod _{b \in B, \Delta (b)>2\log ^2 n} \left( 1 - \left\lfloor \frac{\Delta (b)}{n^{\frac{1}{3}} \log n}\right\rfloor \right) \\&\quad \times \prod _{b \in B , \Delta (b)\le 2\log ^2 n} \left( 1 - \frac{\log n}{n^{\frac{1}{3}}}\right) , \\&=1 - \prod _{b \in B, \Delta (b)>2\log ^2 n} \left( 1 - \frac{\Delta (b)}{2n^{\frac{1}{3}} \log n}\right) \\&\quad \times \prod _{b \in B , \Delta (b)\le 2\log ^2 n} \left( 1 - \frac{2\log ^2 n}{2n^{\frac{1}{3}}\log n}\right) , \\&\ge 1 - exp\left( -\sum _{b \in B, \Delta (b)> 2\log ^2 n} \frac{\Delta (b)}{2n^{\frac{1}{3}} \log n}\right) \\&\quad \times exp\left( -\sum _{b \in B, \Delta (b) \le 2\log ^2 n}\frac{2\log ^2 n}{2n^{\frac{1}{3}}\log n}\right) , \\&\ge 1 - exp\left( -\sum _{b \in B, \Delta (b) > 2\log ^2 n} \frac{\Delta (b)}{2n^{\frac{1}{3}} \log n}\right) \\&\quad \times exp\left( -\sum _{b \in B, \Delta (b) \le 2\log ^2 n}\frac{\Delta (b)}{2n^{\frac{1}{3}}\log n}\right) , \\&=1 - exp\left( -\sum _{b \in B} \frac{\Delta (b)}{2n^{\frac{1}{3}} \log n}\right) \\&\ge 1 - exp\left( -\frac{|{\mathcal {P}}'_2|}{2n^{\frac{1}{3}} \log n}\right) ,\\&\ge 1 - exp(-\Omega (\log ^2 n)). \end{aligned}$$

Consequently, we have that $\Pr [X = 1] \ge \Pr [X = 1 \wedge Y = 1] \Pr [Y = 1] \ge \left( 1 - e^{-\Omega (\log n)}\right) ^2.$ The lemma is proved. $\square$

Fig. 6

Proof of Lemma 8

5.2 Distributed implementation

We explain the above-mentioned implementation details of the algorithm in the CONGEST model as follows:

• (Preprocessing) In the algorithm stated above, the shortcut construction is performed only for large parts, which is crucial to bound the congestion of each edge. Thus, as a preprocessing task, each node has to know if its own part is large (i.e., having a diameter larger than $\kappa _D$) or not. The exact identification of the diameter is usually a hard task; only an asymptotic identification is sufficient for achieving the shortcut quality stated above, where the parts of diameter $\omega (\kappa _D)$ and diameter $o(\kappa _D)$ must be identified as large and small ones, but those of diameter $\Theta (\kappa _D)$ are arbitrarily identified. This loose identification is easily implemented through simple distance-bounded aggregation. The algorithm for part $P_i$ is as follows: (1) In the first round, each node in $P_i$ sends its ID to all the neighbors, and (2) in the subsequent rounds, each node forwards the minimum ID received thus far. The algorithm executes this message propagation during $\kappa _D$ rounds. If the diameter is substantially larger than $\kappa _D$, the minimum ID in $P_i$ does not reach all the nodes in $P_i$. Then, there exists an edge whose endpoints identify different minimum IDs. The one-more-round propagation allows those endpoints to know the part is large. Thereafter, they start to broadcast the signal “large” using the following $\kappa _D$ rounds. If $\kappa _D$ is large, the signal “large” is invoked at several nodes in $P_i$, and $\kappa _D$-round propagation guarantees that every node receives the signal. That is, any node in $P_i$ identifies that $P_i$ is large. The running time of this task is $O(\kappa _D)$ rounds.

• (Step 1) As we stated, the 1-hop extension is implemented in one round. In this step, each node $v \in V_{P_i}$ tells all the neighbors if $P_i$ is large or not. Consequently, if part $P_i$ is identified as a large one, all the nodes in $N^+(P_i)$ know it after this step.

• (Step 2) The algorithm for $D=3$ is trivial. For $D = 4$, there are two non-trivial matters. The first matter is the preparation of hash function h. We realize it by sharing a random seed of $O(n^{1/3}\log ^3 n\log |{\mathcal {Y}}|)$-bit length in advance. A standard construction by Wegman and Carter [39] allows each node to construct the desired h in common. Sharing the random seed is implemented by the broadcast of one $O(n^{1/3}\log ^3 n \log |{\mathcal {Y}}|)$-bit message, i.e., taking ${\tilde{O}}(\kappa _D)$ rounds. The second matter is to address that u does not know whether $P_i$ is large or not, and/or if v belongs to $N^+(P_i)$. It makes u difficult to determine if (u, v) should be added to $H_i$. Instead, our algorithm simulates the task of u by the nodes in N(u). More precisely, each node $v \in N^+(V_{P_i})$ adds each incident edge (u, v) to $H_i$ with probability 1/h(u, i). Because $v \in N^+(P_i)$, v knows if $P_i$ is large or not (informed in step 1), and v can also compute h(u, i) locally. Thus, the choice of (u, v) is locally decidable at v. Because this simulation is completely equivalent to the centralized version, the analysis of the quality also applies.

It is easy to check that the construction time of the distributed implementation above is ${\tilde{O}}(\kappa _D)$ in total.

6 Low-congestion shortcut for bounded clique-width graphs

Let $G = (V, E)$ be a graph. A k-graph ($k \ge 1$) is a graph whose vertices are labeled by integers in [1, k]. A k-graph is naturally defined as a triple (V, E, f), where f is the labeling function $f : V \rightarrow [1, k]$. The clique width $G= (V, E)$ is the minimum k such that there exists a k-graph $G = (V, E, f)$, which is constructed through repeated application of the following four operations: (1) introduce: create a graph of a single node v with label $i \in [1,k]$, (2) disjoint union: take the union $G \cup H$ of two k-graphs G and H, (3) relabel: given $i, j \in [1, k]$, change all the labels i in the graph to j, and (4) join: given $i, j \in [1, k]$, connect all vertices labeled by i, with all vertices labeled by j by edges.

The clique-width is invented first as a parameter to capture the tractability for an easy subclass of high treewidth graphs [3, 6]; that is, the class of bounded clique-width can contain several graphs with high treewidths. In centralized settings, one can often obtain polynomial-time algorithms for several non-deterministic-polynomial complete (NP-complete) problems under the assumption of bounded clique-width. Courcelle et al. [5] showed that for some fixed k, any problem that can be expressed in Monadic second-order logic with quantification over vertices ($\hbox {MSO}_{{1}}$) can be solved in linear time on any class of graphs of clique-width at most k if we obtain k-expressions that define the input graph. Coudert et al. [4] showed several polynomial dependencies in the fixed parameter algorithm (P-FPT) in a bounded clique-width graph. The following negative result, however, states that bounding clique-width does not admit any good solution for the MST problem (and thus, for the low-congestion shortcut).

Theorem 7

There exists an unweighted n-vertex graph $G = (V, E)$ of clique-width six, where for any MST algorithm A, there exists an edge-weight function $w_A : E \rightarrow {\mathbb {N}}$ such that the running time of A becomes ${\tilde{\Omega }}(\sqrt{n}+D)$ rounds.

We introduce the instance stated in this theorem, which is denoted by $G(\Gamma , p)$ ($\Gamma$ and p are the parameters fixed later), using the operations specified in the definition of clique width; that is, this introduction itself becomes the proof of clique-width six. Let ${\mathcal {G}}(\Gamma )$ be the set of 6-graphs that contains one node with label 1, $\Gamma$ nodes with label 2, and $\Gamma$ nodes with label 3, and all other nodes are labeled by 4. Then, we define the binary operation $\oplus$ over ${\mathcal {G}}(\Gamma )$. For any $G, H \in {\mathcal {G}}(\Gamma )$, the graph $G \oplus H$ is defined as the graph obtained through the following operations: (1) Relabel 2 in G with 5 and relabel 3 in H with 6, (2) take the disjoint union $G \cup H$, (3) join with labels 5 and 6, (4) relabel 5 and 6 with 4, and then 1 with 5, (5) add a node with label 1 through operation introduce, (6) join with 1 and 5, and (7) relabel 5 with 4. This process is illustrated in Fig. 7.

Fig. 7

Graph $G \oplus H$

Now we are ready to define $G(\Gamma , p)$. The construction is recursive. First, we define $G(\Gamma , 1)$ as follows: (1) Prepare a $(2\Gamma )$-biclique $K_{\Gamma , \Gamma }$, where one side has label 2, and the other side has label 3. Notably, two labels suffice to construct $K_{\Gamma , \Gamma }$. (2) Add three nodes with labels 1, 5, and 6 through operation introduce. (3) Join with labels 2 and 5, and with 3 and 6. (4) Join with label 1 and 5, and with 1 and 6. (5) Relabel 5 and 6 with 4. Thereafter, we define $G(\Gamma , p) = G(\Gamma , p-1) \oplus G(\Gamma , p- 1)$. The instance claimed in Theorem 7 is $G(\sqrt{n}, \log n /2)$, which is illustrated in Fig. 8. This instance is very close to the standard hard-core instance used in prior work (for example, [37, 38]. See Fig. 1). Thus it is not difficult to observe that ${\tilde{\Omega }}(\sqrt{n})$-round lower bound for MST construction also applies to $G(\sqrt{n}, \log n / 2)$. It suffices to show the subsequent lemma. Combined with Theorem 2, we obtain Theorem 7.

Lemma 9

$G(\Gamma ,p) \in {\mathcal {G}}(O(\Gamma (2^{p}+2)),\Gamma ,2^{p}+2,3p)$.

Proof

First, let us formally specify the graph $G(\Gamma , p)$, which is defined as follows (vertex IDs introduced below are described in Fig. 8):

• $V(\Gamma ,p)=T\cup \bigcup _{1\le l \le \Gamma } V_{l}$ such that $T=\{u^{j}_{i} \mid 0 \le i \le 2^{j}-1, 0 \le j \le p\}$, $V_{l}=\{v^{l}_{i} \mid 0 \le i \le 2^{p}-1\}$.

• $E(\Gamma ,p)=E_{1}\cup E_{2}\cup E_{3}$ such that $E_{1}=\{(u^{j}_{i},u^{j-1}_{\lfloor \frac{i}{2} \rfloor }) \mid 0 \le i \le 2^{j}-1,1\le j \le p\}$. $E_{2}=\{(u^{p}_{i},v^{j}_{i})\mid 0 \le i \le 2^{p}-1, 1 \le j \le \Gamma \}$, $E_{3}=\{(v^{j}_{i},v^{k}_{i+1})\mid 0 \le i \le 2^{p}-2, 1\le j \le \Gamma ,1\le k \le \Gamma \}$.

We define ${\mathcal {X}} = \{X_1,X_2,\dots ,X_{2^{p}+2}\}$ for graph $G(\Gamma ,p)$ as follows:

• $X_{i} = \{u^{p}_{0}\}$ ($i=1$).

• $X_{i} = \{v^{j}_{0}\mid 1\le j \le \Gamma \}$ ($i=2$).

• $X_{i} = \{v^{j}_{i-2}\mid 2\le j \le N\} \cup \{u^{p-j}_{(i-1)/2^{j}-1}\mid 0\le j \le p, i-1\bmod {2^{j}}=0 \}$ ($3\le i \le 2^{p}-1$).

• $X_{i} = \{v^{j}_{2^{p}-2}\mid 1\le j \le \Gamma \} \cup \{u^{p}_{2^{p}-2}\}\cup \{u^{j}_{2^{j}-1} \mid 0\le j \le p-1 \}$ ($i=2^{p}$).

• $X_{i} = \{v^{j}_{2^{p}-1}\mid 1\le j \le \Gamma \}$ ($i=2^{p}+1$)

• $X_{i} =\{u^{p}_{2^{p}-1}\}$ ($i=2^{p}+2$)

We define ${\mathcal {Q}}= \{Q_1 , Q_2 , \dots , Q_{\Gamma }\}$ for graph $G(\Gamma ,p)$ as follows:

• $Q_{i} = V_{1}\cup \left( T\backslash (s \cup r)\right)$ ($\left( i=1\right)$).

• $Q_{i} = V_{i}$ ($\left( 2\le i\le \Gamma \right)$).

It is easy to check that (C1) and (C2) are satisfied. Thus, we only show that (C3) is satisfied. Let $V_{R_{i}}=R_{i} \cap \bigcup _{j=1}^{\Gamma }V_j$. For $2\le i\le (2^{p}+2)/2$, we have $(N(V_{R_{i}})\backslash R_{i-1}) = \emptyset$. For any $\ell$ and $1\le i \le 2^{p-2}$, if $u^{p}_{i}$ is included in $R_{\ell }$, then the neighbors of $u^{p}_{i}$ are included in $R_{\ell }$. For any $\ell$, $1\le i \le p$ and $0\le j \le 2^{i}-2$, if $u^{i}_{j}$ is included in $R_{\ell }$, then $u^{i}_{j+1}$ is included in $R_{\ell }$. Let $u^{i}(R_{\ell })$ be the leftmost vertex whose level is i of T and included in $R_{\ell }$. For any $\ell$, $1\le i \le p$ and $0\le j \le 2^{i}-1$, if $u^{i}_{j}\ne u^{i}(R_{\ell })$ and $u^{i}_{j}$ are included in $R_{\ell }$, then the parent of $u^{i}_{j}$ is included in $R_{\ell }$. Thus, $|(N(R_\ell )\backslash R_{\ell -1})|$ only includes neighbors of $u^{i}(R_{\ell })$ for $1\le i \le p$ and $2\le \ell \le (2^{p}+2)/2$. Because the tree T is a binary tree, $u^{i}(R_{\ell })$ has at most 3 neighbors in T. Therefore we have $|E\left( (N(R_i)\backslash R_{i-1}) \right) |\le 3p$. Similarly, we have $|E\left( (N(L_i)\backslash L_{i-1}) \right) |\le 3p$. Therefore, we can prove that the graph $G(\Gamma ,p)$ is included in ${\mathcal {G}}(O(\Gamma (2^{p}+2)),\Gamma ,2^{p}+2,3p)$. By Theorem 2, the lower bound for constructing the MST in ${\mathcal {G}}(O(\Gamma (2^{p}+2)),\Gamma ,2^{p}+2,3p)$ is ${\tilde{\Omega }}((\min \{\Gamma /3p,\left( (2^{p}+2\right) /2-1\})$. When $\Gamma =\Theta (\sqrt{n})$ and $2^{p}=\Theta (\sqrt{n})$, we obtain the ${\tilde{\Omega }}(\sqrt{n})$ lower bound. $\square$

Fig. 8

Example of clique-width 6 graph $G(\Gamma ,p)$

7 Conclusion

In this study, we have shown the upper and lower bounds for the round complexity of shortcut construction and MST in k-chordal graphs, diameter-three or four graphs, and bounded clique-width graphs. We presented an O(1)-round algorithm, constructing an optimal O(kD)-quality shortcut for any k-chordal graphs. We also presented the algorithms for constructing optimal low-congestion shortcuts with quality ${\tilde{O}}(\kappa _D)$ in ${\tilde{O}}(\kappa _D)$ rounds for $D=3$ and 4, which yields the optimal algorithms for MST matching the known lower bounds by Lotker et al. [32]. On the negative side, O(1)- clique width does not allow us to have good shortcuts. We conclude this paper by posing three related open problems as follows: (1) Can we have good shortcuts for $D \ge 5$? (2) Can we have good shortcuts for the k-clique width where $k\le 5$? (3) While the bounded clique width does not contribute to solving MST efficiently, it seems to provide several edge-disjoint paths (not necessarily short). Can we find any problem that can use the benefit of bounded clique width?