Thorup-Zwick Emulators are Universally Optimal Hopsets

TL;DR

This paper shows that Thorup-Zwick emulators are also universal hopsets with near-optimal size, improving bounds on hopset construction and related graph sparsification structures.

Contribution

It demonstrates that Thorup-Zwick's sublinear additive emulators serve as universal hopsets and improves their size bounds with minor modifications.

Findings

Thorup-Zwick emulators are also $(O(k/eta)^k,eta)$-hopsets.

Modified construction reduces hopset size to $O(n^{1+1/(2^{k+1}-1)})$.

Results improve bounds on sublinear additive emulators and graph spanners.

Abstract

A $(\beta,\epsilon)$-$\textit{hopset}$ is, informally, a weighted edge set that, when added to a graph, allows one to get from point $a$ to point $b$ using a path with at most $\beta$ edges ("hops") and length $(1+\epsilon)\mathrm{dist}(a,b)$. In this paper we observe that Thorup and Zwick's $\textit{sublinear additive}$ emulators are also actually $(O(k/\epsilon)^k,\epsilon)$-hopsets for every $\epsilon>0$, and that with a small change to the Thorup-Zwick construction, the size of the hopset can be made $O(n^{1+\frac{1}{2^{k+1}-1}})$. As corollaries, we also shave "$k$" factors off the size of Thorup and Zwick's sublinear additive emulators and the sparsest known $(1+\epsilon,O(k/\epsilon)^{k-1})$-spanners, due to Abboud, Bodwin, and Pettie.

Figures

Tables

Table 1. Table 1: Tradeoffs between size and hopbound of previous hopsets. Fix the parameter κ = 2 k + 1 − 1 𝜅 superscript 2 𝑘 1 1 \kappa=2^{k+1}-1 to compare  [ 7 , 9 ] against the lower bound  [ 1 ] and the new result.

Authors	Size	Hopbound	Stretch
Klein and Subramanian [14]	$O\left(n\right)$	$O\left(\sqrt{n}\log n\right)$	1
Thorup and Zwick [19]	$O\left(\kappa n^{1+1/\kappa }\right)$	2	$2\kappa -1$
Cohen [7]	$O\left(n^{1+\frac{1}{\kappa }}\cdot \log n\right)$	${\left(\left(\log n\right)/ϵ\right)}^{O\left(\log \kappa \right)}$	$1+ϵ$
Elkin and Neiman [9]	$O\left(n^{1+\frac{1}{\kappa }}\log n\log \kappa \right)$	$O{\left(\left(\log \kappa \right)/ϵ\right)}^{\log \kappa }$	$1+ϵ$
Abboud, Bodwin, and Pettie [1]	$n^{1+\frac{1}{2^{k+1}-1}-\delta }$    $⟶$	$\Omega \left(c_k/{ϵ}^{k+1}\right)$	$1+ϵ$
New	$O\left(n^{1+\frac{1}{2^{k+1}-1}}\right)$	$O{\left(k/ϵ\right)}^k$	$1+ϵ$

Equations

$${\mathrm{dist}}_G(u,v)\le {\mathrm{dist}}_{G\cup H}^{(\beta )}(u,v)\le (1+ϵ){\mathrm{dist}}_G(u,v).$$

$$\mathcal{B}(v)=\{u\in V_i|{\mathrm{dist}}_G(v,u)<{\mathrm{dist}}_G(v,p_{i+1}(v))\}$$

$$E_i=\bigcup _{v\in V_i\setminus V_{i+1}}\left\{(v,u)|u\in \mathcal{B}(v)\cup \{p_{i+1}(v)\}\right\}.$$

$$n{q}_i^2/q_{i+1}$$

$$=n^{1-\frac{2^{i+1}-2}{2^{k+1}-1}+\frac{2^{i+1}-1}{2^{k+1}-1}}\cdot 2^{-2^{i+1}-2i+2-(-2^{i+1}-i)}$$

$$=n^{1+\frac{1}{2^{k+1}-1}}\cdot 2^{-i+2}.$$

$$(n{q}_k{)}^2$$

$$=n^{1+\frac{1}{2^{k+1}-1}}{2}^{-2^{k+1}-2k+2}\ll n^{1+\frac{1}{2^{k+1}-1}}\cdot 2^{-k+2},$$

$$\sum _{i=0}^{k}E[|E_i|]\le n^{1+\frac{1}{2^{k+1}-1}}\left(\sum _{i=0}^k{2}^{-i+2}\right)=O(n^{1+\frac{1}{2^{k+1}-1}}).$$

$$h_0$$

$$h_i$$

$${\mathrm{dist}}_{G\cup H}^{(h_i)}(u,v)$$

$$\le {\mathrm{dist}}_G(u,v)+((r+4{)}^i-r^i)\mu ,$$

$${\mathrm{dist}}_{G\cup H}^{(h_{i-1})}(u_{j_1},u^{\prime })$$

$${\mathrm{dist}}_{G\cup H}^{(h_{i-1})}(u_{j_2},v^{\prime })$$

$${\mathrm{dist}}_{G\cup H}^{(2h_{i-1}+1)}(u_{j_1},u_{j_2})-{\mathrm{dist}}_G(u_{j_1},u_{j_2})$$

$$\le 2{\mathrm{dist}}_{G\cup H}^{(h_{i-1})}(u_{j_1},u^{\prime })+2{\mathrm{dist}}_{G\cup H}^{(h_{i-1})}(v^{\prime },u_{j_2})$$

$$\le 4(r+4{)}^{i-1}\mu .$$

$${\mathrm{dist}}_{G\cup H}^{(h_i)}(u,v)-{\mathrm{dist}}_G(u,v)$$

$$\le \left[(r-1)((r+4{)}^{i-1}-r^{i-1})+4(r+4{)}^{i-1}\right]\mu$$

$$=\left[(r+3)(r+4{)}^{i-1}-r^i+r^{i-1}\right]\mu$$

$$\le ((r+4{)}^i-r^i)\mu$$

$${\mathrm{dist}}_{G\cup H}^{(h_{i-1}+1)}(u_{j_1},u^{\prime \prime })$$

$$\le {\mathrm{dist}}_{G\cup H}^{(h_{i-1})}(u_{j_1},u^{\prime })+{\mathrm{dist}}_G(u^{\prime },v^{\prime })$$

$$\le 2{\mathrm{dist}}_{G\cup H}^{(h_{i-1})}(u_{j_1},u^{\prime })+{\mathrm{dist}}_G(u_{j_1},u_{j_2})+{\mathrm{dist}}_{G\cup H}^{(h_{i-1})}(u_{j_2},v^{\prime })$$

$$\le 3(r+4{)}^{i-1}\mu +{\mathrm{dist}}_G(u_{j_1},u_{j_2}).$$

$${\mathrm{dist}}_{G\cup H}^{((r-1)h_{i-1}+r-1)}(u,u_{j_1})$$

$${\mathrm{dist}}_{G\cup H}^{(h_i)}(u,u^{\prime \prime })$$

$$\le \left[(r-1)((r+4{)}^{i-1}-r^{i-1})+3(r+4{)}^{i-1}\right]\mu +{\mathrm{dist}}_G(u,u_{j_2})$$

$$\le \left[(r+2)(r+4{)}^{i-1}-r^i+r^{i-1}\right]\mu +{\mathrm{dist}}_G(u,u_{j_2})$$

$$\le \left[(r+4{)}^i-r^i\right]\mu +{\mathrm{dist}}_G(u,u_{j_2})$$

$$\le (r+4{)}^i\mu$$

$$d_{G\cup H}^{(h_k)}(u,v)$$

$$=d+\left(\frac{4k}{r}+\frac{4^2\left(\genfrac{}{}{0ex}{}{k}{2}\right)}{r^2}+\frac{4^3\left(\genfrac{}{}{0ex}{}{k}{3}\right)}{r^3}+\cdots \right)d$$

$$\le \left(1+{ϵ}^{\prime }+\frac{{ϵ}^{\prime 2}}{2!}+\frac{{ϵ}^{\prime 3}}{3!}+\cdots \right)d$$

$$\le e^{{ϵ}^{\prime }}d=(1+ϵ)d.$$

Full text

1 Introduction

Let $G=(V,E,w)$ be a weighted undirected graph. Define $\operatorname{\mathrm{dist}}_{G}^{(\beta)}(u,v)$ to be the length of the shortest path from $u$ to $v$ in $G$ that uses at most $\beta$ edges, or “hops.” Whereas $\operatorname{\mathrm{dist}}_{G}=\operatorname{\mathrm{dist}}_{G}^{(\infty)}$ is a metric, $\operatorname{\mathrm{dist}}_{G}^{(\beta)}$ is not in general. A set $H\subset{V\choose 2}$ of weighted edges is called a $(\beta,\epsilon)$-hopset if for every $u,v\in V$,

[TABLE]

Background.

Cohen [7] formally defined the notion of a hopset, but the idea was latent in earlier work [21, 14, 6, 18]. Cohen’s $(\beta,\epsilon)$-hopset had size $O(n^{1+1/\kappa}\log n)$ and $\beta=(\epsilon^{-1}\log n)^{O(\log\kappa)}$. Elkin and Neiman [9] showed that a constant hopbound $\beta$ suffices (when $\kappa,\epsilon$ are constants). In particular, their hopset has size $O(n^{1+1/\kappa}\log n\log\kappa)$ and $\beta=O(\epsilon^{-1}\log\kappa)^{\log\kappa}$. Abboud, Bodwin, and Pettie [1] recently proved that the tradeoffs of [9] are essentially optimal: for any integer $k$, any hopset of size $n^{1+\frac{1}{2^{k+1}-1}-\delta}$ must have $\beta=\Omega(c_{k}/\epsilon^{k+1})$, where $c_{k}$ is a constant depending only on $k$.111Note that setting $\kappa=2^{k+1}-1$ in the Elkin-Neiman construction gives $\beta=O(k/\epsilon)^{k}$, where $\log\kappa=\lfloor\log\kappa\rfloor=k$. Thus, saving any $\delta$ in the exponent of the hopset increases $\beta$ significantly. In general, the statement of [9] obscures the nature of the tradeoff: there are not distinct tradeoffs for each $\kappa\in\{1,2,3,\ldots\}$, but only for $\kappa\in\{1,3,7,\ldots,2^{k+1}-1,\ldots\}$. There are other constructions of hopsets [5, 11, 12, 16] that are designed for parallel or dynamic environments; their tradeoffs (between hopset size and hopbound) are worse than [7, 9] and the ones presented here. See Table 1.

Hopsets, Emulators, and Spanners.

Recall that $G$ is an undirected graph, possibly weighted. A spanner is a subgraph of $G$ such that $\operatorname{\mathrm{dist}}_{H}(u,v)\leq f(\operatorname{\mathrm{dist}}_{G}(u,v))$ for some nondecreasing stretch function $f$. An emulator of an unweighted graph $G$ is a weighted edge set $H$ such that $\operatorname{\mathrm{dist}}_{H}(u,v)\in[\operatorname{\mathrm{dist}}_{G}(u,v),f(\operatorname{\mathrm{dist}}_{G}(u,v))]$. Syntactically, the definition of hopsets is closely related to emulators. The difference is that hopsets have a hopbound constraint but are allowed to use original edges in $G$ whereas emulators must use only $H$. The purpose of emulators is to compress the graph metric $\operatorname{\mathrm{dist}}_{G}$: ideally $|H|\ll|E(G)|$. Historically, the literature on hopset constructions [7, 9] has been noticeably more complex than those of spanners and emulators, many of which [3, 2, 8, 20, 4, 15, 1] are quite elegant. Our goal in this work is to demonstrate that there is nothing intrinsically complex about hopsets, and that a very simple construction improves on all prior constructions and matches the Abboud-Bodwin-Pettie lower bound.

New Results.

Thorup and Zwick [20] designed their emulator for unweighted graphs, and proved that it has size $O(kn^{1+\frac{1}{2^{k+1}-1}})$ and a sublinear additive stretch function $f(d)=d+O(kd^{1-1/k})$. In this paper we show that the Thorup-Zwick emulator, when applied to a weighted graph, produces a $(\beta,\epsilon)$-hopset that achieves every point on the Abboud-Bodwin-Pettie [1] lower bound tradeoff curve. Moreover, with two subtle modifications to the construction, we can reduce the size to $O(n^{1+\frac{1}{2^{k+1}-1}})$, shaving off a factor $k$. Our technique also applies to other constructions, and as corollaries we improve the size of Thorup and Zwick’s emulator [20] and Abboud, Bodwin, and Pettie’s $(1+\epsilon,\beta)$-spanners.222A $(1+\epsilon,\beta)$-spanner of an unweighted graph is one with stretch function $f(d)=(1+\epsilon)d+\beta$.

Theorem 1**.**

Fix any weighted graph $G$ and integer $k\geq 1$. There is a $(\beta,\epsilon)$-hopset for $G$ with size $O(n^{1+\frac{1}{2^{k+1}-1}})$ and $\beta=2\left(\frac{(4+o(1))k}{\epsilon}\right)^{k}$.

Theorem 2**.**

(cf. [20]) Fix any unweighted graph $G$ and integer $k\geq 1$. There is a sublinear additive emulator $H$ for $G$ with size $O(n^{1+\frac{1}{2^{k+1}-1}})$ and stretch function $f(d)=d+(4+o(1))kd^{1-1/k}$.

Theorem 3**.**

(cf. [1]) Fix any unweighted graph $G$, integer $k\geq 1$, and real $\epsilon>0$. There is a $(1+\epsilon,((4+o(1))k/\epsilon)^{k-1})$-spanner $H$ for $G$ with size $O((k/\epsilon)^{h}n^{1+\frac{1}{2^{k+1}-1}})$, where $h=\frac{3\cdot 2^{k-1}-(k+2)}{2^{k+1}-1}<3/4$.

Remark 1**.**

In recent and independent technical report, Elkin and Neiman [10] also observed that Thorup and Zwick’s emulator yields an essentially optimal hopset. They proposed a modification to Thorup and Zwick’s construction that reduces the size to $O(n^{1+\frac{1}{2^{k+1}-1}})$ (eliminating a factor $k$), but increases the hopbound $\beta$ from $O(k/\epsilon)^{k}$ to $O((k+1)/\epsilon)^{k+1}$. For example, their technique does not imply any of the improvements found in Theorems 1, 2, or 3.

2 The Hopset Construction

In this section, we present the construction of the hopset based on Thorup and Zwick’s emulator [20], then analyze its size, stretch, and hopbound.

The construction is parameterized by an integer $k\geq 1$ and a set $\{q_{i}\}$ of sampling probabilities. Let $V=V_{0}\supseteq V_{1}\supseteq V_{2}\supseteq\cdots\supseteq V_{k}\supseteq V_{k+1}=\emptyset$ be the vertex sets in each layer. For each $i\in[0,k)$, each vertex in $V_{i}$ is independently promoted to $V_{i+1}$ with probability $q_{i+1}/q_{i}$. Thus $\operatorname{\mathrm{E}}[|V_{i}|]=nq_{i}$. For each vertex $v\in V$ and $i\in[1,k]$, define $p_{i}(v)$ to be any vertex in $V_{i}$ such that $\operatorname{\mathrm{dist}}_{G}(v,p_{i}(v))=\operatorname{\mathrm{dist}}_{G}(v,V_{i})$. For any vertex $v\in V_{i}\setminus V_{i+1}$, define $\mathcal{B}(v)$ to be:

[TABLE]

Note that $p_{k+1}(v)$ does not exist; by convention $\operatorname{\mathrm{dist}}_{G}(v,p_{k+1}(v))=\infty$. The hopset is defined to be $H=E_{0}\cup E_{1}\cup\cdots\cup E_{k}$, where

[TABLE]

The length of an edge in $H$ is always the distance between its endpoints. This concludes the description of the construction.

2.1 Size Analysis

The expected size of $E_{i}$ is at most $\operatorname{\mathrm{E}}[|V_{i}|](q_{i}/q_{i+1})=nq_{i}^{2}/q_{i+1}$, for each $i\in[0,k)$, and is $(nq_{k})^{2}$ if $i=k$. Following Pettie [17], we choose $\{q_{i}\}$ such that the layers of the hopset have geometrically decaying sizes. Setting $q_{i}=n^{-\frac{2^{i}-1}{2^{k+1}-1}}\cdot 2^{-2^{i}-i+1}$, the expected size of $E_{i}$, for $i\in[0,k)$, is

[TABLE]

The expected size of $E_{k}$ is

[TABLE]

so the expected size of $H$ is at most

[TABLE]

2.2 Stretch and Hopbound Analysis

Let us first give an informal sketch of the analysis. Let $a,b$ be vertices. Choose an integer $r\geq 2$, and imagine dividing up the shortest $a$–$b$ path into $r^{k}$ intervals of length $\mu=\operatorname{\mathrm{dist}}_{G}(a,b)/r^{k}$, where $\mu$ defines one “unit” of length. Once $r$ and $\mu$ are fixed we prove that given any two vertices $u,v$ at distance at most $r^{i}\mu$, there is either an $h_{i}$-hop path from $u$ to $v$ with additive stretch $O(ir^{i-1})\cdot\mu$, or there is an $h_{i}$-hop path from $u$ to a $V_{i+1}$-vertex with length $(r^{i}+O(ir^{i-1}))\cdot\mu$. Of course, when $i=k$ the set $V_{k+1}=\emptyset$ is empty, so we cannot be in the second case. Since, by definition of $\mu$, $\operatorname{\mathrm{dist}}_{G}(a,b)\leq r^{k}\mu$, there must be an $h_{k}$-hop path with additive stretch $O(kr^{k-1})\cdot\mu$. In order for this stretch to be $\epsilon\operatorname{\mathrm{dist}}_{G}(a,b)$ we must set $r=\Theta(k/\epsilon)$.

So, to recap, the integer parameter $r=\Theta(k/\epsilon)$ depends on the desired stretch $\epsilon$, and $r$ determines the hopcount sequence $(h_{i})$, which is defined inductively as follows.

[TABLE]

The parameter $\beta$ of the hopset is exactly $h_{k}$. It is straightforward to show that $h_{k}<2(r+1)^{k}$. Once $r$ and $(h_{i})$ are fixed, Theorem 4 is proved by induction.

Theorem 4**.**

For any fixed real $\mu$ (the “unit”), for all $i\in[0,k]$ and any pair $u,v\in V$ such that $\operatorname{\mathrm{dist}}_{G}(u,v)\leq r^{i}\mu$, at least one of the following statements holds.

1. (i)

$\operatorname{\mathrm{dist}}_{G\cup H}^{(h_{i})}(u,v)\leq\operatorname{\mathrm{dist}}_{G}(u,v)+((r+4)^{i}-r^{i})\mu$, 2. (ii)

There exists $u_{i+1}\in V_{i+1}$ such that $\operatorname{\mathrm{dist}}_{G\cup H}^{(h_{i})}(u,u_{i+1})\leq(r+4)^{i}\mu$.

Proof.

The proof is by induction on $i$. In the base case $i=0$ and $h_{0}=1$. Let $u,v\in V$ with $\operatorname{\mathrm{dist}}_{G}(u,v)\leq r^{0}\mu=\mu$. If $(u,v)\in H$ then $\operatorname{\mathrm{dist}}_{G\cup H}^{(1)}(u,v)=\operatorname{\mathrm{dist}}_{G}(u,v)$ so (i) holds. Otherwise, $(u,v)\notin H$, meaning $v\notin\mathcal{B}(u)$. If $u\in V_{0}\setminus V_{1}$ then $\operatorname{\mathrm{dist}}_{G\cup H}^{(1)}(u,p_{1}(u))\leq\operatorname{\mathrm{dist}}_{G}(u,v)\leq\mu$, and if $u\in V_{1}$ then $p_{1}(u)=u$, so $\operatorname{\mathrm{dist}}_{G\cup H}^{(1)}(u,p_{1}(u))=0$. In either case, (ii) holds.

Now assume $i>0$. Consider vertices $u,v\in V$ with $\operatorname{\mathrm{dist}}_{G}(u,v)\leq r^{i}\mu$ and let $P$ be a shortest $u$–$v$ path in $G$. Then, as shown in Figure 1, we partition $P$ into at most $2r-1$ segments $\langle u_{0}=u,u_{1}\rangle$, $\langle u_{1},u_{2}\rangle$, $\ldots$, $\langle u_{\ell-1},u_{\ell}=v\rangle$ as follows. Starting at $u_{0}=u$, we pick $u_{1}$ to be the farthest vertex on $P$ such that $\operatorname{\mathrm{dist}}_{G}(u_{0},u_{1})\leq r^{i-1}\mu$, and let $(u_{1},u_{2})$ be the next edge on the path.333Note that if the first edge has length more than $r^{i-1}\mu$, then $u_{1}=u_{0}$. Repeat the process until we reach $u_{\ell}=v$, oscillating between selecting segments that have length at most $r^{i-1}\mu$ and single edges.

• •

Multi-hop segment: the shortest path from $u_{s}$ to $u_{s+1}$ satisfies $\operatorname{\mathrm{dist}}_{G}(u_{s},u_{s+1})\leq r^{i-1}\mu$.

• •

Single-hop segment: the segment is actually an edge $(u_{s},u_{s+1})\in E$.

By the induction hypothesis, each multi-hop segment satisfies (i) or (ii) within $h_{i-1}$ hops. Moreover, in each greedy iteration the sum of the lengths from picked multi-hop segment and immediately followed single-hop segment is strictly greater than $r^{i}\mu$ except the last one. Therefore, by the pigeonhole principle, there are at most $r$ multi-hop segments on $P$ and at most $r-1$ single-hop segments on $P$.

If condition (i) holds for all multi-hop segments, then in at most $rh_{i-1}+r-1\leq h_{i}$ hops,

[TABLE]

and condition (i) holds for $P$.

Otherwise, condition (i) does not hold for at least one multi-hop segment. Consider the first multi-hop segment $\langle u_{j_{1}},u_{j_{1}+1}\rangle$ and the last multi-hop segment $\langle u_{j_{2}-1},u_{j_{2}}\rangle$ that do not satisfy condition (i). By condition (ii), there exist $u^{\prime}$ and $v^{\prime}\in V_{i}$ satisfying

[TABLE]

Now we have two cases depending on whether $(u^{\prime},v^{\prime})\in H$ or not. If $(u^{\prime},v^{\prime})\in H$, then by the triangle inequality, we can get from $u_{j_{1}}$ to $u_{j_{2}}$ with $2h_{i-1}+1$ hops and additive stretch

[TABLE]

We know there are a total of at most $r-1$ multi-hop segments satisfying condition (i). Hence, within at most $(r-1)h_{i-1}+r-1+2h_{i-1}+1\leq h_{i}$ hops, we can get from $u$ to $v$ with additive stretch

[TABLE]

and condition (i) holds for $P$ in this case.

On the other hand, suppose that $(u^{\prime},v^{\prime})\notin H$. Since both $u^{\prime},v^{\prime}\in V_{i}$ but $(u^{\prime},v^{\prime})\notin H$, we know that $u^{\prime\prime}=p_{i+1}(u^{\prime})\in V_{i+1}$ must exist with $\operatorname{\mathrm{dist}}_{H}^{(1)}(u^{\prime},u^{\prime\prime})\leq\operatorname{\mathrm{dist}}_{G}(u^{\prime},v^{\prime})$. Hence, we can get from $u_{j_{1}}$ to $u^{\prime\prime}$ via an $(h_{i-1}+1)$-hop path with length

[TABLE]

Similar to the previous case, there are at most $r-1$ multi-hop segments appeared before $u_{j_{1}}$, and all of them are satisfying condition (i). Hence, the surplus

[TABLE]

Therefore, in at most $(r-1)h_{i-1}+r-1+h_{i-1}+1\leq h_{i}$ hops,

[TABLE]

∎

Proof of Theorem 1.

Fix $u,v\in V$ and $d=\operatorname{\mathrm{dist}}_{G}(u,v)$. Define $\epsilon^{\prime}=\ln(1+\epsilon)$. Notice that $1/\epsilon^{\prime}=(1+o(1))(1/\epsilon)$. Set $r=\lceil 4k/\epsilon^{\prime}\rceil=\Theta(k/\epsilon)$ and $\mu=d/r^{k}$. By Theorem 4, since $V_{k+1}=\emptyset$, condition (i) must hold: within $h_{k}<2(r+1)^{k}$ hops we have

[TABLE]

∎

Observe that if we set $k=\log\log n-O(1)$ the size becomes linear.

Corollary 1**.**

Every $n$-vertex graph has an $O(n)$-size $(\beta,\epsilon)$-hopset with $\beta=2(\frac{(4+o(1))k}{\epsilon})^{k}$ and $k=\log\log n-O(1)$.

3 Conclusion

In this paper our goal was to demonstrate that hopset constructions need not be complex, and that optimal hopsets can be constructed with a simple and elegant algorithm, namely a small modification to Thorup and Zwick’s emulator construction [20]. From a purely quantitative perspective our hopsets also improve on the sparseness and/or hopbound of other constructions [7, 9, 10]. As a happy byproduct of our construction, we also shave small factors off the best sublinear additive emulators [20] and $(1+\epsilon,\beta)$-spanners [1].

We now have a good understanding of the tradeoffs available between $\beta$ and the hopset size when the stretch is fixed at $1+\epsilon$, $\epsilon>0$ being a small real. However, when $\epsilon=0$ or $\epsilon$ is large, there are still gaps between the best upper and lower bounds. For example, when $\epsilon=0$ a trivial hopset444Let $H$ be a clique on a set of $\sqrt{n}$ vertices chosen uniformly at random. has size $O(n)$ with $\beta=O(\sqrt{n}\log n)$. A construction of Hesse [13] (see also [1, §6]) implies that $\beta$ must be at least $n^{\delta}$ for some $\delta$, but it is open whether $O(n)$-size hopsets exist with $\beta\ll\sqrt{n}$. At the other extreme, Thorup and Zwick’s distance oracles imply that $O(\kappa n^{1+1/\kappa})$-size hopsets exist with $\beta=2$ and stretch $2\kappa-1$. Is this tradeoff optimal? Are there other tradeoffs available when $\beta$ is a fixed constant (say 3 or 4), independent of $\kappa$?

Acknowledgement.

Thanks to Richard Peng for help with the references for zero-stretch hopsets.