A Hierarchy of Lower Bounds for Sublinear Additive Spanners
New Upper Bounds on $`(1+\epsilon,\beta)`$-Spanners
Thorup and Zwick gave a very simple randomized construction of an emulator with size $`O(kn^{1-\frac{1}{2^{k+1}-1}})`$ and stretch function $`f(d) = d + O(kd^{1-1/k} + 3^k)`$. Alternatively, one can view this as a $`(1+\epsilon, O(k/\epsilon)^{k-1})`$-emulator for every $`\epsilon>0`$, where the optimal choice of $`\epsilon`$, as a function of $`d`$, is $`\epsilon = \Theta(k/d^{1/k})`$.
The Thorup-Zwick Emulator
The Thorup-Zwick emulator is parameterized by an integer $`k\ge 2`$. Let $`G=(V,E)`$ be the input graph. One samples vertex sets $`V = V_0 \supset V_1 \supset V_2 \cdots \supset V_{k}`$ where vertices in $`V_i`$ are promoted to $`V_{i+1}`$ with probability $`q_{i+1}/q_i`$, so $`\E[|V_i|] = q_i n`$. Define $`p_i(u)`$ to be the closest $`V_i`$-vertex to $`u`$, breaking ties in a consistent manner. Define $`\Ball(u,r) = \{v \;|\; \dist(u,v) \le r\}`$ to be the set of vertices inside the radius-$`r`$ ball centered at $`u`$ and let $`\Ball_i(u)`$ be short for $`\Ball(u, \dist(u,p_i(u))-1)`$. For $`i\geq k+1`$, $`p_i(u)`$ does not exist and $`\Ball_i(u)`$ is the entire graph, by definition. The emulator edge set consists of $`E_0 \cup E_1 \cup \cdots \cup E_{k}`$, where $`E_i`$ is defined as follows.
E_i = \left\{(u,v) \;|\; u,v\in V_i \mbox{ and } v\in \Ball_{i+1}^{\ }(u)\right\} \;\cup\; \left\{(u,p_{i+1}^{\ }(u)) \;|\; u\in V\right\}.
The length of all emulator edges is precisely the distance between their endpoints in $`G`$. Since $`|\Ball_{i+1}(u)|`$ is $`q_{i+1}^{-1}`$ in expectation, the expected number of edges contributed by $`E_i`$ is $`n + nq_i^2/q_{i+1}`$, for $`i < k`$, and is $`(nq_{k})^2`$ when $`i=k`$. Setting $`q_i = n^{-\f{2^i-1}{2^{k+1}-1}}`$ makes the size of the emulator $`O(kn^{1+\f{1}{2^{k+1}-1}})`$ in expectation. In order to obtain a $`d + O(kd^{1-\f{1}{k}})`$-type stretch bound for all distances $`d \le D`$, it actually suffices to restrict $`E_i`$ to pairs at distance at most $`(r+2)^i`$, where $`r = D^{1/k}`$. Letting $`P(u,v)`$ be any shortest path from $`u`$ to $`v`$, the subgraph $`S_{TZ}(k,r) = (V, E_0'\cup E_1'\cup\cdots\cup E_{k}')`$ is a spanner, where
E_i' = \bigcup_{\substack{(u,v)\in E_i \; :\\ v\in \Ball(u,(r+2)^i)}} P(u,v).
As we show in Sections 4.2 and 4.3, the spanner $`S_{TZ}(k,r)`$ behaves exactly like the emulator for all distances up to $`D`$, i.e., it has stretch function $`d+O(kd^{1-\f{1}{k}})`$ for all sufficiently large $`d\le D`$.1 However, choosing the optimum sampling probabilities as a function of $`r,k,n`$ is no longer trivial. Since each path in $`E_i`$ contributes $`(r+2)^i`$ edges, the spanner size is on the order of $`kn`$ (for paths of the form $`P(u,p_i(u))`$) plus
\frac{n}{q_1} + \frac{nq_1^2r}{q_2} + \frac{nq_2^2r^2}{q_3} + \cdots \frac{nq_{k-1}^2r^{k-1}}{q_{k}} + (nq_{k})^2r^{k}.
Assuming this sum is minimized when $`E_0',E_1',\ldots,E_{k}'`$ contribute equally, we have the following equalities:
\begin{align*}
q_2 &= rq_1^3 & \mbox{(balancing $E_0'$ and $E_1'$)}\\
q_3 &= r^2 q_2^2 q_1 & \mbox{(balancing $E_0'$ and $E_2'$)}\\
&\vdots\\
q_{k} &= r^{k-1} q_{k-1}^2q_1 & \mbox{(balancing $E_0'$ and $E_{k-1}'$)}\\
%
\intertext{If $q_i$ is constrained to be of the form $n^{-g(i)} r^{-h(i)}$, these equalities are satisfied when}
g(i) &= 2g(i-1) + g(1) & \mbox{(for $i\ge 2$)}\\
&= (2^i-1)g(1) & \mbox{(by induction)}\\
\mbox{and } \; h(i) &= 2h(i-1) + h(1) - (i-1) & \mbox{(for $i\ge 2$)}\\
&= (2^i-1)h(1) - 2^i + (i+1) & \mbox{(by induction)}\\
%\end{align*}
\intertext{So $q_{k} = n^{-(2^{k}-1)g(1)} r^{-(2^{k}-1)h(1) + 2^{k} - (k+1)}$. Plugging this equality
into the expression for $|E_{k}'|$ and balancing with $|E_0'|$, we have,
\[
|E_{k}'| \,=\, (nq_{k})^2 r^{k} \,=\, n^{2-2(2^{k}-1)g(1)} r^{-2[(2^{k}-1)h(1) - 2^{k} + (k+1)] + k} \,=\, n^{1+g(1)} r^{h(1)} \,=\, |E_0'|,
\]
which is minimized when
}
%\begin{align*}
g(1) &= \f{1}{2^{k+1}-1}\\
h(1) &= \f{2^{k+1} - (k+2)}{2^{k+1}-1}.
\end{align*}
For example, when $`k=2`$ and $`h(1)=4/7`$ this leads to a $`d+O(\sqrt{d})`$-spanner for distances $`d \le D = r^2`$ having size $`O(r^{4/7}n^{8/7}) = O(D^{2/7}n^{8/7})`$. Since $`h(1)`$ is strictly less than 1 for any fixed $`k`$, the spanner size is always $`o(rkn^{1+\f{1}{2^{k+1}-1}}) = o(D^{1/k}kn^{1+\f{1}{2^{k+1}-1}})`$.
Even Sparser $`(1+\epsilon,\beta)`$-Spanners
In order to form an even sparser spanner we substitute for $`E_1'`$ a subgraph whose size has no dependence on $`r`$ but preserves the relevant distances well enough, up to an additive +2 error. The following theorem is proved using the same path-buying algorithm for constructing additive 6-spanners . The algorithm begins with the subgraph $`E_0'`$ and supplements it with an $`\tilde{E}_1`$ to guarantee +2 stretch for each $`u,v\in V_1`$ that were connected by an edge in $`E_1`$.
(see and ) Suppose $`V_1,V_2`$ are sampled with probability $`q_1`$ and $`q_2`$, with $`q_2 < q_1`$. Then there is an edge-set $`\tilde{E}_1`$ with expected size $`O(nq_1^2/q_2)`$ such that if $`u,v\in V_1`$ and $`v \in \Ball_2(u)`$, then
\dist_{E_0' \cup \tilde{E}_1}(u,v) \le \dist(u,v) + 2.
Proof. (sketch) We assume the reader is familiar with the path-buying algorithm and its analysis . Let $`\mathcal{P} \subset {V_1 \choose 2}`$ be the pairs for which we are guaranteeing good stretch, i.e., $`\{u,v\} \in \mathcal{P}`$ if $`u\in \Ball_2(v)`$ or $`v\in \Ball_2(u)`$. Since $`|\Ball_2(u)|`$ is $`1/q_2`$ in expectation, $`|\mathcal{P}|`$ is $`O(nq_1^2/q_2)`$ in expectation. For each $`\{u,v\} \in \mathcal{P}`$ we evaluate $`P(u,v)`$ and buy it (set $`\tilde{E}_1 \leftarrow \tilde{E}_1 \cup P(u,v)`$) if its current value exceeds its cost. The value is the number of pairs $`\{x,y\} \in \mathcal{P}`$ with $`x,y`$ adjacent to $`P(u,v)`$ for which $`\dist_{E_0'\cup \tilde{E}_1 \cup P(u,v)}(x,y) < \dist_{E_0'\cup \tilde{E}_1}(u,v)`$. It is argued by the pigeonhole principle that any path $`P(u,v)`$ not bought has $`\dist_{E_0' \cup \tilde{E}_1}(u,v) \le \dist(u,v)+2`$, and that each pair in $`\mathcal{P}`$ is charged for $`O(1)`$ edges in $`\tilde{E}_1`$. ◻
We sample vertex sets $`V = V_0 \supset V_1 \supset \cdots \supset V_{k}`$ as before and construct the spanner $`S(k,r)`$ with edge set $`E_0' \cup \tilde{E}_1 \cup E_2' \cup \cdots \cup E_{k}'`$, where $`\tilde{E}_1`$ is the edge set from Theorem [thm:path-buying]. The expected size of the entire spanner is therefore
\frac{n}{q_1} + \frac{nq_1^2}{q_2} + \frac{nq_2^2r^2}{q_3} + \cdots + \frac{nq_{k-1}^2r^{k-1}}{q_{k}} + (nq_{k})^2r^{k}.
Letting $`q_i = n^{-\f{2^i-1}{2^{k+1}-1}} r^{-h(i)}`$, we balance the contribution of $`E_0',\tilde{E}_1,E_2',\ldots,E_{k}'`$ by having $`h`$ satisfy the following.
\begin{align*}
h(2) &= 3h(1) & \mbox{(balancing $\tilde{E}_1$ and $E_0'$)}\\
\mbox{and for $i\ge 3$, }\, h(i) &= 2h(i-1) + h(1) - (i-1) & \mbox{(balancing $E_{i-1}'$ and $E_0'$)}\\
&= (2^i-1)h(1) - 3\cdot 2^{i-2} + (i+1) & \mbox{(by induction)}
\end{align*}
Following similar calculations, it follows that the spanner size is minimized when
h(1)= \f{3\cdot 2^{k-1} - (k+2)}{2^{k+1}-1}.
We shall prove shortly that this spanner is, indeed, a $`d + O(kd^{1-\f{1}{k}} + 3^k)`$-spanner. For example, when $`k=1`$ we have $`h(1)=2/7`$, so it is a $`d+O(\sqrt{d})`$-spanner for all $`d\le D \le r^2`$ with size $`O(r^{2/7}n^{8/7}) = O(D^{1/7}n^{8/7})`$. For any fixed $`k`$, $`h(1) < 3/4`$, so the spanner has size $`o(D^{\frac{3}{4k}} kn^{1+\f{1}{2^{k+1}-1}})`$.
We were able to substitute $`\tilde{E}_1`$ for $`E_1'`$ without disturbing the exponent $`1+\f{1}{2^{k+1}-1}`$ of the spanner, but only because the path-buying algorithm buys $`O(nq_1^2/q_2)`$ additional edges when initialized with the edge set $`E_0'`$. In general we can use to substitute an $`\tilde{E}_i`$ for $`E_i'`$, but its size is $`O(n\sqrt{q_i/q_{i+1}})`$. This improves the exponent attached to $`r`$ but worsens the exponent attached to $`n`$. For example, balancing $`E_2'`$ and $`E_0'`$ lets us put $`q_3 = (q_1)^7r^{O(1)}`$, whereas balancing $`\tilde{E}_2`$ and $`E_0'`$ forces $`q_3 = (q_1)^5`$.
Stretch Analysis
We analyze the stretch of the spanner $`S=S(k,r)`$ with edge set $`E_0' \cup \tilde{E}_1 \cup E_2' \cup \cdots \cup E_{k}'`$. We will first consider two vertices $`u,v`$ at distance at most $`\ell^i`$, for some integers $`\ell\ge 2, i\ge 0`$. We will assume for the time being that $`r=\infty`$ and calculate specific quantities related to the spanner distance $`\dist_S(u,v)`$ without considering the constraints imposed by a finite $`r`$. Once these quantities are calculated, it will be clear that the analysis goes through, so long as $`\ell \le r`$. The pair $`u,v`$ can be either complete or incomplete (or both), as explained in the following definition.
Define $`\{\Succ{\ell}{i}, \, \Fail{\ell}{i}\}_{\ell \in [2,r], i\ge 0}`$ to be integers such that for all $`u,v`$ with $`\dist(u,v) \le \ell^i`$, at least one of the following inequalities holds. Here $`S=S(k,r)`$ is the spanner.
\begin{align*}
\dist_{S}(u,v) &\le \dist(u,v) + \Succ{\ell}{i} & \mbox{(``$u\cdots v$ is {\em complete}'')}\\
\dist_{S}(u, p_{i+1}(u)) &\le \Fail{\ell}{i} & \mbox{(``$u\cdots v$ is {\em incomplete}'')}
\end{align*}
The following values for $`\{\Succ{\ell}{i}, \, \Fail{\ell}{i}\}_{\ell\in [2,r], i\ge 0}`$ satisfy Definition [def:succfail].
\begin{align*}
\Succ{\ell}{0} &= 0 & \mbox{ for all $\ell$}\\
\Fail{\ell}{0} &= 1 & \mbox{ for all $\ell$}\\
\Succ{\ell}{1} &= 6 & \mbox{ for all $\ell$}\\
\Fail{\ell}{1} &= \ell+3 & \mbox{ for all $\ell$}\\
\Succ{\ell}{i} &= \min\left\{\begin{array}{l}
\ell\cdot \Succ{\ell}{i-1}\\
(\ell-1)\cdot \Succ{\ell}{i-1} + 4\cdot \Fail{\ell}{i-1}
\end{array}\right. & \mbox{ for all $\ell$ and $i\geq 2$}\\
\Fail{\ell}{i} &= \ell^i + 3\cdot \Fail{\ell}{i-1} & \mbox{ for all $\ell$ and $i\ge 2$}
\end{align*}
Proof. In the base case $`(i=0)`$, we have $`\ell^0 = 1`$, so $`u`$ and $`v`$ are adjacent in the input graph. If $`(u,v)\in E_0'`$ then $`\dist_{S}(u,v)=1`$ and if $`(u,v)\not\in E_0'`$ then it must be that $`\dist(u,p_1(u))=1`$, so $`\Succ{\ell}{0}=0, \Fail{\ell}{0}=1`$ satisfy Definition [def:succfail] for all $`\ell`$.
When $`i>0`$, partition the shortest path from $`u`$ to $`v`$ into at most $`\ell`$ segments with length $`\ell^{i-1}`$, and let $`u_j`$ be the vertex on the path at distance $`j\ell^{i-1}`$ from $`u`$. For the sake of simplicity, assume $`\dist(u,v)=\ell^i`$, so $`v= u_{\ell}`$. Each segment from $`u_j`$ to $`u_{j+1}`$ is classified as either complete or incomplete. If all segments are complete then $`\dist_{S}(u,v) \le \dist_{S}(u,v) \le \dist(u,v) + \ell\cdot \Succ{\ell}{i-1}`$. If there is at least one incomplete segment, let there be $`s`$ complete segments on a prefix of the path and $`s'`$ complete segments on a suffix of the path, where $`s+s' \le \ell-1`$. It follows that
\begin{align*}
\dist_{S}(u,p_{i}(u_s)) &\le \dist_{S}(u,u_s) + \dist_{S}(u_s,p_i(u_s))\\
&\le \dist(u,u_s) + s\cdot \Succ{\ell}{i-1} + \Fail{\ell}{i-1}\\
\dist_{S}(v,p_{i}(u_{\ell-s'})) &\le \dist_{S}(v,u_{\ell-s'}) + \dist_{S}(u_{\ell-s'},p_i(u_{\ell-s'})) \\
&\le \dist(v,u_{\ell-s'}) + s'\cdot \Succ{\ell}{i-1} + \Fail{\ell}{i-1}.
%\end{align*}
\intertext{If $p_i(u_{\ell-s'}) \not\in \Ball_{i+1}(p_i(u_s))$ then}
%\begin{align*}
\dist_{S}(u,p_{i+1}(u)) &\le \dist(u,p_i(u_{s})) + \dist(p_i(u_s), p_i(u_{\ell-s'})) \\
&\le (\ell-s')\ell^{i-1} + 3\Fail{\ell}{i}\\
&\le \ell^i + 3\Fail{\ell}{i} & \mbox{worst case when $s'=0$.}
\end{align*}
and the path from $`u`$ to $`v`$ is incomplete. On the other hand, if $`p_i(u_{\ell-s'}) \in \Ball_{i+1}(p_i(u_s))`$ then $`S`$ contains a shortest (or nearly shortest, if $`i=1`$) path from $`p_i(u_s)`$ to $`p_i(u_{\ell-s'})`$, so
\begin{align*}
\dist_{S}(u,v) &\le \dist_{S}(u,p_i(u_s)) + \dist_{S}(p_i(u_s), p_i(u_{\ell-s'})) \zero{+ \dist_{S}(p_i(u_{\ell-s'}), v)} \\
&\le [s(\ell^{i-1} + \Succ{\ell}{i-1}) + \Fail{\ell}{i-1}] & \mbox{from $u$ to $p_i(u_s)$}\\
&\hcm[.5] + [(\ell-s-s')\ell^{i-1} + 2\Fail{\ell}{i-1} \;\; \{+2\}] & \mbox{from $p_i(u_s)$ to $p_i(u_{\ell-s'})$}\\
&\hcm[.5] + [s'(\ell^{i-1} + \Succ{\ell}{i-1}) + \Fail{\ell}{i-1}] & \mbox{from $p_i(u_{\ell-s'})$ to $v$}\\
&\le \dist(u,v) + (\ell-1)\Succ{\ell}{i-1} + 4\Fail{\ell}{i-1} \;\; \{+2\} &\mbox{worst case when $s+s'=\ell-1$}
\end{align*}
where the $`\{+2\}`$ is only present if $`i=1`$. We satisfy Definition [def:succfail] by setting $`\Succ{\ell}{1}=6, \Fail{\ell}{1}=\ell+3`$, and, for $`i\ge 2`$, $`\Fail{\ell}{i} = \ell^i + 3\Fail{\ell}{i-1}`$ and $`\Succ{\ell}{i}`$ to be the maximum of $`\ell\cdot \Succ{\ell}{i-1}`$ and $`(\ell-1)\Succ{\ell}{i-1}+4\Fail{\ell}{i-1}`$. ◻
We now find closed form bounds for $`\Succ{\ell}{i}`$ and $`\Fail{\ell}{i}`$.
The values defined inductively in Lemma [lem:succfail-recursive] satisfy the following (in)equalities.
\begin{align*}
\Fail{2}{i} &= 3^{i+1} - 2^{i+1}\\
\Succ{2}{i} &\le 3^{i+1}\\
%
\Fail{3}{i} &= (i+1)3^i\\
\Succ{3}{i} &\le 4i3^i\\
%
\intertext{Define $c_{\ell} = \ell/(\ell-3)$. For all $\ell\ge 4$ and $i\ge 1$,}
\Fail{\ell}{i} &\le c_{\ell} \ell^i \\
\Succ{\ell}{i} &\le \min\left\{\begin{array}{l}
4c_{\ell} \ell^i\\
(4c_{\ell}i+2)\ell^{i-1}
\end{array}\right.
\end{align*}
Proof. All bounds are established by induction on $`i`$. The cases when $`\ell \in \{2,3\}`$ are left as an exercise. When $`\ell \ge 4`$ the base cases $`i\in\{0,1\}`$ clearly hold. For incomplete paths and $`i\ge 2`$ we have
\begin{align*}
\Fail{\ell}{i} &= \ell^i + 3\cdot \Fail{\ell}{i-1} & \mbox{(by definition)}\\
&\le \ell^i(1 + 3c_{\ell}/\ell) \;\le\; c_\ell \ell^i & \mbox{(induction hypothesis, $c_{\ell} = \f{\ell}{\ell-3}$)}
\intertext{and for complete paths we have two cases,}
\Succ{\ell}{i} &= (\ell-1)\Succ{\ell}{i-1} + 4\Fail{\ell}{i-1} & \mbox{(by definition)}\\
&\le (\ell-1)4c_{\ell}\ell^{i-1} + 4c_{\ell}\ell^{i-1} & \mbox{(1st induction hypothesis)}\\
&= 4c_{\ell}\ell^i\\
\mbox{and }\; &\le (\ell-1)(4c_{\ell}(i-1)+2)\ell^{i-2} + 4c_{\ell}\ell^{i-1} & \mbox{(2nd induction hypothesis)}\\
&\le (4c_{\ell} i+2)\ell^{i-1}.
\end{align*}
◻
Observe that when we check whether $`p_i(u_{\ell-s'}) \in \Ball_{i+1}(p_i(u_s))`$, $`i\ge 2`$, the distance between $`p_i(u_{\ell-s'})`$ and $`p_i(u_{s})`$ is maximized when $`s=s'=0`$; it is at most
\ell^i + 2\Fail{\ell}{i-1} = \ell^i + 2c_{\ell}\ell^{i-1} \; < (\ell+2)^i.
Thus, as long as $`\ell \le r`$, the criterion $`p_i(u_{\ell-s'}) \in \Ball(p_i(u_s), (r+2)^i)`$ will also hold. This retroactively justifies the constraint $`\ell \le r`$ in Lemma [lem:succfail-recursive].
The spanner $`S(k,r)`$ has size $`O(r^hkn^{1+\f{1}{2^{k+1}-1}})`$, where $`h = \frac{3\cdot 2^{k-1} - (k+2)}{2^{k+1}-1} < 3/4`$. Its stretch changes as a function of the distance $`d`$ being approximated.
-
For $`d\ge 2^{k}`$ it is a multiplicative $`O((3/2)^k)`$-spanner.
-
For $`d\ge 3^{k}`$ it is a multiplicative $`O(k)`$-spanner.
-
For $`d\ge \ell^{k}`$, $`\ell\in[4,k)`$, it is a multiplicative $`(5+O(1/\ell))`$-spanner, and when $`\ell \in [k,r]`$ it is a multiplicative $`(1 + (4k+O(1))/\ell)`$-spanner.
$`S(k,r)`$ is a $`(1+\epsilon, ((4k+O(1))/\epsilon)^{k-1})`$-spanner for every $`\epsilon`$ such that $`(4k+O(1))/\epsilon < r`$. Its stretch function can also be expressed as $`f(d) = d + (4+o(1))kd^{1-\f{1}{k}} + 3^k`$ for all $`d \le r^k`$, and $`f(d) = d + (4+o(1))kd/r`$ for larger $`d`$, where the $`o(1)`$s go to zero as $`d`$ increases.
Proof. Let $`\dist(u,v) \ge 2^{k}`$ be the distance to be approximated and $`\ell = \floor{d^{\f{1}{k}}}`$, where $`\ell \le r`$. Partition the shortest path $`P(u,v)`$ into intervals of length precisely $`\ell^{k-1}`$, with at most one shorter interval. Since $`d < (\ell+1)^{k}`$, there are between $`\ell`$ and $`\floor{(\ell+1)(1+1/\ell)^{k-1}}`$ intervals. If all intervals are complete then $`\dist_S(u,v) \le d + \ceil{\frac{d}{\ell^{k-1}}}\Succ{\ell}{k-1}`$. If at least one is incomplete then $`\dist_S(u,v) \le d + (\ceil{\frac{d}{\ell^{k-1}}}-1)\Succ{\ell}{k-1} + 4\Fail{\ell}{k-1}`$. If $`\ell \in [3,k-1]`$ then according to Lemma [lem:succfail-closedform], $`\Succ{\ell}{k-1}=4\Fail{\ell}{k-1}`$ and we are indifferent between these two possibilities. If $`\ell \ge k`$ or $`\ell=2`$ then $`\Succ{\ell}{k-1} < 4\Fail{\ell}{k-1}`$, so the second case is worse. When $`\ell=2`$ we have
\begin{align*}
\lefteqn{d + \paren{\ceil{\frac{d}{2^{k-1}}}-1}\Succ{2}{k-1} + 4\Fail{2}{k-1}}\\
&< d + \paren{\ceil{\frac{d}{2^{k-1}}}-1}3^{k} + 4\cdot 3^{k}\\
&\le d(1 + 3(3/2)^{k-1}) + 4\cdot 3^{k}\\
%
\intertext{So $S(k,r)$ is a multiplicative $O((3/2)^{k-1})$-spanner for $d\ge 2^k$. This is a non-trivial multiplicative stretch. Traditional multiplicative stretch spanners with size $n^{1+\f{1}{2^{k+1}-1}}$ size stretch some pairs by a factor of $2^{k+2}-3$.
When $\ell=3$ we have}
\lefteqn{d + \paren{\ceil{\frac{d}{3^{k-1}}}-1}\Succ{3}{k-1} + 4\Fail{3}{k-1}}\\
&< d + \paren{\ceil{\frac{d}{3^{k-1}}}-1}4(k-1)3^{k-1} + 4k3^{k-1}\\
&\le d(1 + 4(k-1)/3 + 4k/3) & \mbox{(since $3^{k-1} \le d/3$)}
\intertext{Thus $S(k,r)$ functions as a multiplicative $O(k)$-spanner when $d\ge 3^{k}$. When $\ell \in [4,k)$,}
\lefteqn{d + \paren{\ceil{\frac{d}{\ell^{k-1}}}-1}\Succ{\ell}{k-1} + 4\Fail{\ell}{k-1}}\\
&< d + \paren{\ceil{\frac{d}{\ell^{k-1}}}-1}4c_{\ell} \ell^{k-1} + 4c_{\ell}\ell^{k-1}\\
&< d(1 + 4c_{\ell} + 4c_{\ell}/\ell) \\
&= d\paren{1+\f{4(\ell+1)}{\ell-3}} = (5+O(\fr{1}{\ell}))d & \mbox{(since $\ell^{k-1} \le d/\ell$, $c_\ell = \ell/(\ell-3)$)}
\intertext{The multiplicative stretch of $S(k,r)$ tends to $5$ as $d$ increases from $3^{k}$ to $(k-1)^{k}$.
When $\ell \ge k$ we have}
\lefteqn{d + \paren{\ceil{\frac{d}{\ell^{k-1}}}-1}\Succ{\ell}{k-1} + 4\Fail{\ell}{k-1}}\\
&\le d + \paren{\ceil{\frac{d}{\ell^{k-1}}}-1}(4c_{\ell}(k-1)+2)\ell^{k-2} \zero{\:+\: 4c_{\ell}\ell^{k-1}}\\
&\le d\paren{1+\frac{4c_\ell k + 2}{\ell}} & \mbox{$\ell^{k-1} \le d/\ell$}
\end{align*}
When $`d \ge \ell^k \ge k^k`$ the multiplicative stretch is $`1 + (1+o(1))4k/\ell`$, where the $`o(1) = O(1/\ell)`$ tends to zero as $`\ell`$ increases. When $`\ell \ge (4c_{\ell}k + 2)/\epsilon`$ the multiplicative stretch becomes $`1+\epsilon`$.
One may confirm that by setting $`\ell = \floor{d^{\f{1}{k}}}`$, in all the cases above the stretch function of $`S(k,r)`$ can be expressed as $`f(d) = d + O(kd^{1-\f{1}{k}} + 3^k)`$, for $`\ell^k \le d \le r^k`$, and $`f(d) = d + O(kd/r)`$ for $`d \ge r^k`$. The leading constants in the terms $`O(kd^{1-\f{1}{k}})`$ and $`O(kd/r)`$ tend to $`4`$ as $`d`$ increases. ◻
Setting $`r = (4k + O(1))/\epsilon`$, we obtain a $`(1+\epsilon, O(k/\epsilon)^{k-1})`$-spanner with size $`O((k/\epsilon)^h kn^{1+\f{1}{2^{k+1}-1}})`$. This spanner is sparsest when $`\epsilon>0`$ is a fixed constant and $`k = \log_2\log_2 n - O(1)`$: it is then a $`(1+\epsilon, ((4+o(1))\log\log n)^{\log\log n-O(1)})`$-spanner with size $`O(n(\log\log n)^{7/4})`$. When $`k=\log\log n`$ it is possible to reduce the size of this spanner to $`O(kn + nr^{3/4}) = O(n(\log\log n + (\epsilon^{-1}\log\log n)^{3/4}))`$. The $`kn`$ term reflects the cost of the paths $`\{P(u,p_i(u))\}_{u\in V, i\in [1,k]}`$. Rather than equalize the remaining contribution of $`E_0',\ldots,E_{k}'`$, one chooses the sampling probabilities such that $`|\tilde{E}_1|`$ and $`|E_0'|`$ are balanced and $`|E_2'|,|E_3'|,\ldots,|E_{k}'|`$ decay geometrically.
Even sparser $`(1+\epsilon,\beta)`$-spanners are known, but they have slightly worse tradeoffs. Pettie constructed a $`(1+\epsilon, O(\epsilon^{-1}\log\log n)^{\log \log n})`$-spanner with size $`O(n\log\log(\epsilon^{-1}\log\log n))`$.
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Thorup and Zwick also noted that their emulator can be converted to a spanner, but their sketch of how to do this was incorrect. ↩︎