A face cover perspective to $\ell_1$ embeddings of planar graphs

TL;DR

This paper improves bounds on embedding terminal sets of planar graphs into  space with low distortion, leading to better approximations for the sparsest cut problem when terminals are coverable by few faces.

Contribution

It introduces an improved upper bound of O((\u221a{\u03b3})) for embedding terminals in planar graphs into , advancing previous bounds and impacting sparsest cut approximations.

Findings

Enhanced embedding bounds to O(((b3)))

Provides a polynomial time approximation for the sparsest cut problem

Connects embedding distortion with flow-cut gap in planar graphs

Abstract

It was conjectured by Gupta et al. [Combinatorica04] that every planar graph can be embedded into $\ell_1$ with constant distortion. However, given an $n$-vertex weighted planar graph, the best upper bound on the distortion is only $O(\sqrt{\log n})$, by Rao [SoCG99]. In this paper we study the case where there is a set $K$ of terminals, and the goal is to embed only the terminals into $\ell_1$ with low distortion. In a seminal paper, Okamura and Seymour [J.Comb.Theory81] showed that if all the terminals lie on a single face, they can be embedded isometrically into $\ell_1$. The more general case, where the set of terminals can be covered by $\gamma$ faces, was studied by Lee and Sidiropoulos [STOC09] and Chekuri et al. [J.Comb.Theory13]. The state of the art is an upper bound of $O(\log \gamma)$ by Krauthgamer, Lee and Rika [SODA19]. Our contribution is a further improvement on the upper bound to $O(\sqrt{\log\gamma})$. Since every planar graph has at most $O(n)$ faces, any further improvement on this result, will be a major breakthrough, directly improving upon Rao's long standing upper bound. Moreover, it is well known that the flow-cut gap equals to the distortion of the best embedding into $\ell_1$. Therefore, our result provides a polynomial time $O(\sqrt{\log \gamma})$-approximation to the sparsest cut problem on planar graphs, for the case where all the demand pairs can be covered by $\gamma$ faces.

Figures

Equations

$$\frac{1}{\tau }\cdot d_G(u,v)\le \Vert f(u)-f(v){\Vert }_1\le \rho \cdot d_G(u,v).$$

$${\mathbb{E}}_{(H,f_H)\sim \mathcal{D}}\left[d_H\left(f_H(u),f_H(v)\right)\right]\le t\cdot d_G(u,v).$$

$${\mathbb{E}}_{T^{\prime }\sim \mathcal{D}}\left[d_T(v,u)\right]$$

$$=\sum _{G^{\prime }\in \mathrm{supp}({\mathcal{D}}_{OP})}\underset{{\mathcal{D}}_{OP}}{\mathrm{Pr}}\left[G^{\prime }\right]\cdot O\left(d_{G^{\prime }}(v,u)\right)=O\left(d_G(v,u)\right).$$

$${\Vert f(v)-f(u)\Vert }_1$$

$$\le 2\cdot \Vert f_{\mathrm{tw}}(v)-f_{\mathrm{tw}}(u)\Vert \le 4\cdot d_T(v,u).$$

$${\Vert f(v)-f(u)\Vert }_1$$

$$\le \sum _T\mathrm{Pr}\left[T\right]\cdot 4\cdot d_T(v,u)=4\cdot {\mathbb{E}}_{T\sim \mathcal{D}}\left[d_T(v,u)\right]=O(d_G(v,u)).$$

$${\tilde{f}}_t(v)\doteq \{\begin{array}{ll}{\lambda }_v\cdot f_t(v)& \text{if }t=t_v+1,\\ (1-{\lambda }_v)\cdot f_t(v)& \text{if }t=t_v,\\ \vec{0}& \text{otherwise}.\end{array}$$

$${\Vert f(v)\Vert }_1$$

$$=(1-{\lambda }_v)\cdot 2^{t_v}+{\lambda }_v\cdot 2^{t_v+1}$$

$$=\frac{2^{t_v+1}-d_G(v,\mathcal{B})}{2^{t_v}}\cdot 2^{t_v}+\frac{d_G(v,\mathcal{B})-2^{t_v}}{2^{t_v}}\cdot 2^{t_v+1}=d_G(v,\mathcal{B}).$$

$${\Vert f(v)-f(u)\Vert }_1\le {\Vert f(v)\Vert }_1+{\Vert f(u)\Vert }_1\stackrel{(\text{ref}eq:boundedNorm)}{=}{d}_G(v,\mathcal{B})+d_G(u,\mathcal{B})\le d_G(v,u).$$

$${\Vert {\tilde{f}}_t(v)-{\tilde{f}}_t(u)\Vert }_1$$

$$\le \min \left\{p_t,q_t\right\}\cdot {\Vert f_t(v)-f_t(u)\Vert }_1+\left|p_t-q_t\right|\cdot \max \left\{{\Vert f_t(v)\Vert }_1,{\Vert f_t(u)\Vert }_1\right\}$$

$$\le O(d_G(u,v))+\left|p_t-q_t\right|\cdot 2^t.$$

$${\lambda }_v-{\lambda }_u=\frac{d_G\left(v,\mathcal{B}\right)-2^{t_v}}{2^{t_v}}-\frac{d_G\left(u,\mathcal{B}\right)-2^{t_v}}{2^{t_v}}=\frac{d_G\left(v,\mathcal{B}\right)-d_G\left(u,\mathcal{B}\right)}{2^{t_v}}\le \frac{d_G(u,v)}{2^{t_v}}.$$

$${\lambda }_v+(1-{\lambda }_u)$$

$$\le \frac{\left(d_G\left(v,\mathcal{B}\right)-2^{t_v}\right)+\left(2^{t_u+1}-d_G\left(u,\mathcal{B}\right)\right)}{2^{t_u}}\le \frac{d_G(u,v)}{2^{t_u}}.$$

$$\begin{array}[]{ccccc}\left|p_{t_{v}+1}-q_{t_{v}+1}\right|&=&\lambda_{v}&=&O(d_{G}(u,v)/2^{t_{v}})\\ \left|p_{t_{v}}-q_{t_{v}}\right|&=&\left|1-\lambda_{v}-\lambda_{u}\right|&=&O(d_{G}(u,v)/2^{t_{v}})\\ \left|p_{t_{u}}-q_{t_{u}}\right|&=&1-\lambda_{u}&=&O(d_{G}(u,v)/2^{t_{v}})\end{array}~{}~{}.$$

$${\Vert f_t(v)-f_t(u)\Vert }_1\ge \min \left\{d_{G[\mathcal{I}]}(v,u),2^t\right\}\ge \frac{d_G(v,u)}{2c},$$

$${\Vert f(v)-f(u)\Vert }_1$$

$$\ge \left|{\Vert p_t\cdot f_t(v)\Vert }_1-{\Vert q_t\cdot f_t(u)\Vert }_1\right|$$

$$=\left|p_t-q_t\right|\cdot 2^t>\frac{d_G(v,u)}{S}.$$

$${\Vert f(v)-f(u)\Vert }_1$$

$$\ge q_t\cdot {\Vert f_t(v)-f_t(u)\Vert }_1-\left|p_t-q_t\right|\cdot \Vert f_t(v)\Vert$$

$$\stackrel{(\text{ref}eq:NoScaleBOund)}{\ge }\frac{1}{3}\cdot \frac{d_G(v,u)}{2c}-\frac{d_G(v,u)}{S}=\frac{d_G(v,u)}{S}.$$

Full text

1. 1 Introduction
2. 2 Preliminaries
3. 3 Partial Shortest Path Decomposition
4. 4 Uniformly Truncated Embedding
5. 5 Non-Uniformly Truncated Embedding
6. 6 Construction of a PSPD for Planar Graphs
7. 7 Embedding Parametrized by Face Cover: Proof of Theorem 1

1 Introduction

Metric embeddings is a widely used algorithmic technique that have numerous applications, notably in approximation, online and distributed algorithms. In particular, embeddings into $\ell_{1}$ have implications to graph partitioning problems. Specifically, the ratio between the Sparsest Cut and the maximum multicommodity flow (also called flow cut gap) is upper bounded by the distortion of the optimal embedding into $\ell_{1}$ (see [LLR95, GNRS04]).

Given a weighted graph $G=(V,E,w)$ with the shortest path metric $d_{G}$, and embedding $f:V\to\ell_{1}$, the contraction and expansion of $f$ are the smallest $\tau,\rho$, respectively, such that for every pair $u,v\in V$,

[TABLE]

The distortion of the embedding is $\tau\cdot\rho$. If $\tau=1$ (resp. $\rho=1$) we say that the embedding is non-contractive (expansive). If $\rho=O(1)$, we say that the embedding is Lipschitz.

In this paper we focus on embeddings of planar graphs into $\ell_{1}$. Rao [Rao99] showed that every $n$-vertex planar graph can be embedded into $\ell_{1}$ with distortion $O(\sqrt{\log n})$. The best known lower bound is $2$ by Lee and Raghavendra [LR10]. A long standing conjecture by Gupta et al. [GNRS04] states that every graph family excluding a fixed minor, and in particular planar graphs, can be embedded into $\ell_{1}$ with constant distortion.

Consider the case where there is a set $K\subseteq V$ of terminals, and we are only interested in embedding the terminals into $\ell_{1}$. This version is sufficient for the flow cut-gap equivalence, where the terminals are the vertices with demands. Better embeddings might be constructed when $K$ has a special structure. A face cover of $G$ is a set of faces such that every terminal belongs to some face from the set (see Figure 1 for an illustration). Given a drawing of $G$ in the plane, denote by $\gamma(G,K)$ the minimal size of a face cover. It was shown by Hurkens, Schrijver and Tardos [HST86], that the result of Okamura and Seymour [OS81] implies that if $\gamma(G,K)=1$, that is, if all the terminals lie on a single face, then $K$ embeds isometrically into $\ell_{1}$ (a special case is when $G$ is outerplanar). For the general case, where $\gamma(G,K)=\gamma\geq 1$, the methods of Lee and Sidiropoulos [LS09] imply distortion of $2^{O(\gamma)}$. Chekuri, Shepherd and Weibel [CSW13] constructed an embedding with distortion of $3\gamma$. Recently, Krauthgamer, Lee and Rika [KLR19] managed to construct an embedding into $\ell_{1}$ with $O(\log\gamma)$ distortion by first applying a stochastic embedding into trees. This method has benefits, since trees are very simple to work with. Additionally, the result of [KLR19] is tight w.r.t stochastic embedding into trees. We improve upon [KLR19] by embedding directly into $\ell_{1}$.

Theorem 1**.**

Let $G=(V,E,w)$ be a weighted planar graph with a given drawing in the plane and $K\subseteq V$ a set of terminals. There is an embedding of $K$ into $\ell_{1}$ with distortion $O(\sqrt{\log\gamma(G,K)})$. Moreover, this embedding can be constructed in polynomial time.

Since every $n$-vertex graph has $O(n)$ faces, by setting $K=V$, Theorem 1 re-proves the celebrated result of Rao [Rao99]. Moreover, any improvement upon Theorem 1 will be a major breakthrough.

In addition, Theorem 1 has implication on the sparsest cut problem. Let $c:E\rightarrow\mathbb{R}_{+}$ be an assignment of capacities to the edges, and $d:{K\choose 2}\rightarrow\mathbb{R}_{+}$ assignment of demands to terminal pairs. The sparsity of a cut $S$ is the ratio between the capacity of the edges crossing the cut to the demands crossing the cut. The sparsest cut is the cut with minimal sparsity. Theorem 1 implies:

Corollary 1**.**

Let $G=(V,E)$ be a weighted planar graph with a given drawing in the plane, $K\subseteq V$ a set of terminals, capacities $c:E\rightarrow\mathbb{R}_{+}$ and demands $d:{K\choose 2}\rightarrow\mathbb{R}_{+}$. Let $\gamma(G,K)=\gamma$. Then there is a polynomial time $O(\sqrt{\log\gamma})$-approximation algorithm for the sparsest-cut problem.

See [LLR95, GNRS04, KLR19] for further details.

1.1 Technical Ideas

Abraham et al. [AFGN22], among other results, constructed an $O(\sqrt{\log n})$-distortion embedding of planar graphs into $\ell_{1}$. This embedding is based on shortest path decompositions (SPD). Even though the distortion is similar, the new embedding is very different from the classic embedding of Rao [Rao99]. An SPD is a hierarchical decomposition of a graph using shortest paths. The first level of the partition is simply $V$. In level $i$, all the clusters are connected. To construct level $i+1$, we remove a single shortest path from every cluster of level $i$. Level $i+1$ consists of the remaining connected components. This process is repeated until all the vertices are removed. The SPDdepth is the depth of the hierarchy. Using cycle separators [Mil86] it is possible to create an SPD of depth $O(\log n)$ for every planar graph. [AFGN22] showed that every graph which admits an SPD of depth $k$, can be embedded into $\ell_{1}$ with distortion $O(\sqrt{k})$. In particular $O(\sqrt{\log n})$ for planar graphs.

In this paper we generalize the notion of SPD by defining partial SPD (PSPD). The difference is that in PSPD we do not need all the vertices to be removed. That is, in PSPD the last level of the hierarchy is allowed to be non-empty. Given a planar graph $G$ with a terminal set $K$ and a face cover of size $\gamma(G,K)=\gamma$, using cycle separators [Mil86] we create a PSPD of depth $O(\log\gamma)$, such that for every cluster $C$ in the lower level of the hierarchy, all the remaining terminals $K\cap C$ lie on a single face. In other words, each such cluster is an Okamura-Seymour (O-S) graph.

We invoke the embedding of [AFGN22] on our PSPD, as a result we get an embedding with expansion $O(\sqrt{\log\gamma})$, where every pair of terminals $v,u$ that either was separated by the PSPD, or lie close enough to some removed shortest path, has constant contraction. The remaining work is to take care of terminal pairs that remained in the same cluster, and lie far from the cluster boundary. As each such cluster is O-S graph, it embeds isometrically to $\ell_{1}$. However, we cannot simply embed each cluster independently of the entire graph. Such an oblivious embedding will create an unbounded expansion, as close-by pairs might belong to different clusters.

Our solution, and the main technical part of the paper, is to create a truncated embedding 111An embedding $f:V\rightarrow\ell_{1}$ is truncated if for every vertex $v$, $\|f(v)\|_{1}$ is bounded by some formerly specified number.. Specifically, consider a cluster $C$ where all the terminals lie on a single face $F$. Let $\mathcal{B}=V\setminus C$ be the boundary of $C$, which is the set of vertices outside $C$. We construct a Lipschitz embedding $f$ of $F$ into $\ell_{1}$ such that the norm $\|f(v)\|_{1}$ of every vertex $v\in F$ is bounded by its distance to the boundary $d_{G}(v,\mathcal{B})$, while $f$ has constant contraction for pairs far enough from the boundary. Our final embedding is defined as a concatenating of the embedding for the PSPD with a truncated embedding for every cluster, providing a constant contraction on all pairs and $O(\sqrt{\log\gamma})$ expansion.

Our truncated embedding does not uses the embedding of [OS81]. As a middle step, given a parameter $t>0$, we provide a uniformly truncated embedding 222In uniformly truncated embedding $\|f(v)\|_{1}$ is bounded for all the vertices by a global single parameter. $f_{t}$ such that $f_{t}$ is Lipschitz, the norm $\|f_{t}(v)\|_{1}$ of every vertex $v\in F$ is exactly $t$, and $f_{t}$ provides constant contraction for pairs at distance at most $t$. The construction of the uniformly truncated embedding goes through a stochastic embedding into trees. To create the non-uniformly truncated embedding we combine uniformly truncated embeddings for all possible truncation scales.

1.2 Related Work

The notion of face cover $\gamma(G,K)$ was extensively studied in the context of Steiner tree problem [EMJ87, Ber90, KNvL20], cuts and (multicommodity) flows [MNS85, CW04], all pairs shortest path [Fre91, Fre95, CX00] and cut sparsifiers [KR20, KPZ19]. Given a drawing and a terminal set $K$, $\gamma(G,K)$ can be found in $2^{O(\gamma(G,K))}\cdot{\rm poly}(n)$ time, but generally it is known to be NP-hard [BM88]. Frederickson [Fre91] (Lemma 7.1) presented a polynomial-time approximation scheme (PTAS) for the problem of finding a face cover of minimum size. Specifically, given a planar graph with a drawing, Frederickson’s algorithm finds a face cover of size at most $(1+\varepsilon)\cdot\gamma(G,K)$ in $O(2^{\frac{3}{\varepsilon}}\cdot n)$ time. Denote by $\gamma^{*}(G,K)$ the minimal size of a face cover over all planar drawings of $G$. It is known that computing $\gamma^{*}(G,K)$ is NP-hard [BM88]. Frederickson [Fre91] presented a $4$-approximation for $\gamma^{*}(G,K)$ in the special case where $K=V$, i.e. the terminals are the entire set $V$. However, for general $K\subseteq V$, to the best of the author’s knowledge, no approximation is known.

It is well known that Euclidean metrics, as well as distributions over trees, embed isometrically into $\ell_{1}$ (See [Mat02]). Therefore, in order to construct a bounded distortion embedding to $\ell_{1}$, it is enough to embed into either $\ell_{2}$ or a distribution over trees.

Outerplanar graphs are $1$-outerplanar. A graph is called $k$-outerplanar, if by removing all the vertices on the outer face, the graph becomes $k-1$-outerplanar. Chekuri et al. [CGN*+*06] proved that $k$-outerplanar graphs embed into distribution over trees with $2^{O(k)}$ distortion.

Next consider minor-closed graph families. Following [GNRS04], Chakrabarti *et al. * [CJLV08] showed that every graph with treewidth-$2$ (which excludes $K_{4}$ as a minor) embeds into $\ell_{1}$ with distortion $2$ (which is tight, as shown by [LR10]). Already for treewidth-$3$ graphs, it is unknown whether they embed into $\ell_{1}$ with a constant distortion. Abraham et al. [AFGN22] showed that every graph with pathwidth $k$ embeds into $\ell_{1}$ with distortion $O(\sqrt{k})$ (through $\ell_{2}$), improving a previous result of Lee and Sidiropoulos [LS13] who showed a $(4k)^{k^{3}+1}$ distortion (via embedding into trees). Graphs with treewidth $k$ are embeddable into $\ell_{2}$ with distortion $O(\sqrt{k\log n})$ [KLMN05]. For genus $g$ graphs, [LS10] showed an embedding into Euclidean space with distortion $O(\log g+\sqrt{\log n})$. Finally, for $H$-minor-free graphs, combining the results of [AGG*+*19, KLMN05] provides Euclidean embeddings with $O(\sqrt{|H|\log n})$ distortion.

For other notions of distortion, Abraham et al. [ABN11] showed that $\beta$-decomposable metrics (which include planar graphs as well as all other families mentioned in this section), for fixed $\beta$, embed into $\ell_{2}$ with scaling distortion $O(\sqrt{\log\frac{1}{\varepsilon}})$. This means that for every $\varepsilon\in(0,1)$ all but an $\varepsilon$ fraction of the pairs in ${V\choose 2}$ have distortion at most $O(\sqrt{\log\frac{1}{\varepsilon}})$. Bartal et al. [BFN19] proved that $\beta$-decomposable metrics (for fixed $\beta$) embed into $\ell_{2}$ with prioritized distortion $O(\sqrt{\log j})$. In more detail, given a priority order $\{v_{1},\dots,v_{n}\}$ over the vertices, the pair $\{v_{i},v_{j}\}$ for $j\leq i$, will have distortion at most $O(\sqrt{\log j})$.

1.3 Follow-Up Work

In a followup work, Kumar [Kum22] showed that every planar graph can be embedded into $\ell_{1}$ with a non expansive embedding, such that every pair of vertices $u,v$ that lie on the same face have contraction at most $c$, for some universal constant $c$. Compering to our work, the distortion in [Kum22] is constant regardless of the number of faces. However, the distortion guarantee is only for vertices lying on the same face.

Since our paper, there been also exciting developments in embedding planar and minor free graphs into bounded treewidth graphs. Previously, it was known that every deterministic embedding of planar graphs into bounded-treewidth graph with constant distortion requires treewidth $\Omega(\sqrt{n})$ [CG04], while any stochastic embedding into graphs with constant treewidth requires distortion distortion $\Omega(\log n)$ [CG04, CJLV08] (a distortion which already can be obtain by embedding general metrics into trees [FRT04]). Furthermore, every stochastic embedding into graphs with treewidth $O(\frac{\log^{\nicefrac{{1}}{{3}}}n}{\log\log n})$ requires expected distortion $\Omega(\frac{\log^{\nicefrac{{1}}{{3}}}n}{\log\log n})$ [FL22]. Recently, Cohen-Addad et al. [CLPP23] showed that every fixed minor free graph, can be embedded with stochastic distortion $1+\varepsilon$, into a distribution over graphs with treewidth ${\rm poly}(\frac{1}{\varepsilon},\log n,\log\Phi)$, where $\Phi=\frac{\max_{x,y}d_{G}(x,y)}{\min_{x,y}d_{G}(x,y)}$ is the aspect ratio.

Another new exciting line of work is regarding metric embeddings with additive stretch. Here we are given a graph $G$ with diameter $D$, and the guarantee is that the pairwise distances do not decrease, while they can increase by at most an $+\varepsilon\cdot D$ additive factor. Improving over [FKS19], Filtser and Le [FL22], and Chang et al. [CCL*+*23] showed that every planar graph can be deterministically embedded with additive distortion $+\varepsilon\cdot D$ into a graph with treewidth $O\left(\min\{\varepsilon^{-1}\cdot(\log\log n)^{2},\varepsilon^{-4}\}\right)$. Filtser and Le [FL22] (improving over [CFKL20]) showed that every $K_{r}$ minor free graph can be stochastically embedded with expected additive distortion $+\varepsilon\cdot D$ into a distribution of treewidth $O_{r}(\varepsilon^{-2}\cdot(\log\log n)^{2})$ graphs. There are also metric embeddings with alternative guarantees of additive distortion (clan, Ramsey) [FL21, FL22].

2 Preliminaries

Graphs. We consider connected undirected graphs $G=(V,E)$ with edge weights $w:E\to\mathbb{R}_{\geq 0}$. Let $d_{G}$ denote the shortest path metric in $G$. For a vertex $x\in V$ and a set $A\subseteq V$, let $d_{G}(x,A):=\min_{a\in A}d(x,a)$, where $d_{G}(x,\emptyset):=\infty$. For a subset of vertices $A\subseteq V$, let $G[A]$ denote the induced graph on $A$. Let $G\setminus A:=G[V\setminus A]$ be the graph after deleting the vertex set $A$ from $G$.

See Section 1 for definitions of embedding, distortion, contraction, expansion and Lipschitz. We say that an embedding is dominating if it is non-contractive. Given a graph family $\mathcal{F}$, a stochastic embedding of $G$ into $\mathcal{F}$ is a distribution $\mathcal{D}$ over pairs $(H,f_{H})$ where $H\in\mathcal{F}$ and $f_{H}$ is an embedding of $G$ into $H$. We say that $\mathcal{D}$ is dominating if for every $(H,f_{H})\in{\rm supp}(\mathcal{D})$, $f_{H}$ is dominating. We say that a dominating stochastic embedding $\mathcal{D}$ has expected distortion $t$, if for every pair $u,v\in V$ it holds that

[TABLE]

A terminated planar graph $G=(V,E,w,K)$ is a planar graph $(V,E,w)$, with a subset of terminals $K\subseteq V$. A graph G is outerplanar if there is a drawing of $G$ in the plane such that all the vertices lie on the unbounded face. A face cover is a set of faces such that every terminal lies on at least one face from the cover. Given a graph $G$ with a drawing in the plane, denote by $\gamma(G,K)$ the minimal size of a face cover. In the special case where all the terminals are covered by a single face, i.e. $\gamma(G,K)=1$, we say that $G$ is an Okamura-Seymour graph, or O-S graph for short. See illustration on the right (using the notation from Figure 1).

A tree decomposition of a graph $G=(V,E)$ is a tree $T$ with nodes $B_{1},\dots,B_{s}$ (called bags) where each $B_{i}$ is a subset of $V$ such that: (1) For every edge $\{u,v\}\in E$, there is a bag $B_{i}$ containing both $u$ and $v$. (2) For every vertex $v\in V$, the set of bags containing $v$ forms a connected subtree of $T$. The width of a tree decomposition is $\max_{i}\{|B_{i}|-1\}$. The treewidth of $G$ is the minimal width of a tree decomposition of $G$. It is straightforward to verify that every tree graph has treewidth $1$.

Given a set of $s$ embeddings $f_{i}:V\rightarrow\mathbb{R}^{d_{i}}$ for $i\in\{1,\dots,s\}$, the concatenation of $f_{1},\dots,f_{s}$, denoted by $\oplus_{i=1}^{s}f_{i}$, is a function $f:V\rightarrow\mathbb{R}^{\sum_{i}d_{i}}$, where the coordinates from $1$ to $d_{1}$ correspond to $f_{1}$, the coordinates from $d_{1}+1$ to $d_{1}+d_{2}$ correspond to $f_{2}$, etc.

3 Partial Shortest Path Decomposition

Abraham et al. [AFGN22] defined shortest path decompositions (SPDs) of “low depth”. Every (weighted) path graph has an SPDdepth $1$. A graph $G$ has an SPDdepth $k$ if there exist a shortest path $P$, such that every connected component in $G\setminus P$ has an SPDdepth $k-1$. In other words, given a graph, in SPD we hierarchically delete shortest paths from each connected component, until no vertices remain. In this paper we define a generalization called partial shortest path decomposition (PSPD), where we remove the requirement that all the vertices will be deleted. See the formal definition below. In Section 6 we will argue that every terminated planar graph with face cover of size $\gamma$ has a PSPD of depth $O(\log\gamma)$ such that in each connected component in the lower level of the hierarchy, all the terminals lie on a single face.

A partial partition $\mathcal{X}$ of a set $X$, is a disjoint set of subsets of $X$. In other words, for every $A\in\mathcal{X}$, $A\subseteq X$, and for every different subsets $A,B\in\mathcal{X}$, $A\cap B=\emptyset$.

Definition 1** (Partial Shortest Path Decomposition (PSPD)).**

Given a weighted graph $G=(V,E,w)$, a PSPD of depth $k$ is a pair $\left\{\mathcal{X},\mathcal{P}\right\}$, where $\mathcal{X}$ is a collection $\mathcal{X}_{1},\dots,\mathcal{X}_{k+1}$ of partial partitions of $V$, $\mathcal{P}$ is a collection of sets of paths $\mathcal{P}_{1},\dots,\mathcal{P}_{k}$, and:

$\mathcal{X}_{1}=\{V\}$. 2. 2.

For every $1\leq i\leq k$ and every cluster $X\in\mathcal{X}_{i}$, there exist a unique path $P_{X}\in\mathcal{P}_{i}$ such that $P_{X}$ is a shortest path in $G[X]$. 3. 3.

For every $2\leq i\leq k+1$, $\mathcal{X}_{i}$ consists of all connected components of $G[X\setminus P_{X}]$ over all $X\in\mathcal{X}_{i-1}$.

The remainder of the PSPD $\left\{\mathcal{X},\mathcal{P}\right\}$ is a pair $\left\{\mathcal{C},\mathcal{B}\right\}$ where $\mathcal{C}=\mathcal{X}_{k+1}$ is the set of connected components in the final level of the PSPD, and $\mathcal{B}=\bigcup_{i=1}^{k}\cup\mathcal{P}_{i}$ is the set of all the vertices in the removed paths. $\mathcal{B}$ is also called the boundary.

Under Definition 1, SPD is a special case of PSPD where $\mathcal{C}=\emptyset$ (and $\mathcal{B}=V$). The main theorem in [AFGN22] states that if a graph $G$ has SPD of depth $k$, then it is embeddable into $\ell_{1}$ with distortion $O(\sqrt{k})$ 333In fact [AFGN22] proved a more general result, stating that $G$ is embeddable into $\ell_{p}$ for $p\in[1,\infty]$, with distortion $O(k^{\min\{\frac{1}{2},\frac{1}{p}\}})$. Similarly, in Theorem 2 can replace $\ell_{1}$ with $\ell_{p}$ and the expansion $\sqrt{k}$ with $k^{\min\{\frac{1}{2},\frac{1}{p}\}}$. The contraction condition and values remains the same.. [AFGN22] construct a different embedding for each level of the decomposition. Each such embedding is Lipschitz, while for every pair of vertices $u,v\in V$ there is some level $i$ such that the embedding for this level has constant contraction w.r.t. $u,v$. Specifically, the level $i$ with the bounded contraction guarantee is the first level in which either $u,v$ are separated or the distance between $\{u,v\}$ to a deleted path is at most $\frac{d_{G}(u,v)}{12}$. In particular, given a PSPD, by using the exact same embedding from [AFGN22] (w.r.t. the existing levels in the decomposition), we get the following theorem.

Theorem 2** (Embedding using PSPD).**

Let $G=(V,E,w)$ be a weighted graph, and let $\left\{\mathcal{X},\mathcal{P}\right\}$ be a PSPD of depth $k$ with remainder $\left\{\mathcal{C},\mathcal{B}\right\}$. There is an embedding $f:V\rightarrow\ell_{1}$ with the following properties:

For every $u,v\in V$, $\left\|f(v)-f(u)\right\|_{1}\leq O(\sqrt{k})\cdot d_{G}(u,v)$. 2. 2.

For every $u,v\in V$ which are either separated by $\mathcal{C}$ (that is $u,v$ do not belong to the same cluster in $\mathcal{C}$), or such that $\min\left\{d_{G}(v,\mathcal{B}),d_{G}(u,\mathcal{B})\right\}\leq\frac{d_{G}(u,v)}{12}$, it holds that $\left\|f(v)-f(u)\right\|_{1}\geq d_{G}(u,v)$.

4 Uniformly Truncated Embedding

In this section we construct a uniformly truncated embedding for O-S graphs into $\ell_{1}$. Specifically, given a truncation parameter $t$, we show how to embed O-S graphs into $\ell_{1}$ via a Lipschitz map such that the norm of all the vectors is exactly $t$, and it is non-contractive for terminals at distance at most $t$. We will use two previous results on stochastic embeddings. The following theorem was proven by Englert et al. [EGK*+*14] (Thm. 12) in a broader sense. Lee et al. [LMM15] (Thm. 4.4) observed that it implies embedding of O-S graphs into outerplanar graphs.

Theorem 3**.**

Consider a weighted planar graph $G=(V,E,w)$ with $F\subseteq V$ being a face. There is a stochastic embedding of $F$ into dominating outerplanar graphs with expected distortion $O(1)$.

The following theorem was proven by Gupta et al. [GNRS04] (Thm. 5.4).

Theorem 4**.**

Consider a weighted outerplanar graph $G=(V,E,w)$. There is a stochastic embedding of $G$ into dominating trees with expected distortion $O(1)$.

As it was already observed in [KLR19], we conclude:

Corollary 2**.**

Consider a planar graph $G=(V,E,w)$ with a face $F$. There is a stochastic embedding of $F$ into dominating trees with expected distortion $O(1)$.

Proof.

Let $\mathcal{D}_{OP}$ be the distribution over outerplanar graphs from Theorem 3. For every $G^{\prime}\in{\rm supp}(\mathcal{D}_{OP})$, let $\mathcal{D}_{G^{\prime}}$ be the distribution over trees from Theorem 4 (w.r.t. $G^{\prime}$). We define a distribution $\mathcal{D}$ of embeddings of $F$ into trees as follows. First sample an outerplanar graph $G^{\prime}$ using $\mathcal{D}_{OP}$. Then sample a tree $T$ using $\mathcal{D}_{G^{\prime}}$. We argue that the distribution $\mathcal{D}$ has the desired properties.

All the embeddings in the support of $\mathcal{D}$ are dominating metrics as both $\mathcal{D}_{OP}$ and $\mathcal{D}_{G^{\prime}}$ (for all $G^{\prime}$) have this property. Consider $v,u\in V$, it holds that

[TABLE]

∎

A first step towards truncated embedding of O-S graphs will be a truncated embedding of trees.

Lemma 1**.**

Let $T=(V,E,w)$ be some tree and let $t>0$ be a truncation parameter. There exists an embedding $f:T\rightarrow\ell_{1}$ such that the following holds:

Sphere surface: for every $v\in V$, $\left\|f(v)\right\|_{1}=t$. 2. 2.

Lipschitz: for every $u,v\in V$, $\left\|f(v)-f(u)\right\|_{1}\leq 4\cdot d_{T}(v,u)$. 3. 3.

Bounded contraction: for every $u,v\in V$, $\left\|f(v)-f(u)\right\|_{1}\geq\min\left\{d_{T}(v,u),t\right\}$.

Proof.

Add a new vertex $v_{t}$ to $T$ with edges of weight $\frac{t}{2}$ to all the other vertices. Call the new graph $T_{t}$. Notice that $T_{t}$ has treewidth $2$. According to Chakrabarti *et al. * [CJLV08], there is an embedding $f_{{\rm tw}}$ of $T_{t}$ into $\ell_{1}$ with distortion $2$. By rescaling, we can assume that the contraction is $1$ (and the expansion is at most $2$). Additionally, by shifting, we can assume that $f_{{\rm tw}}(v_{t})=\vec{0}$. Let $f$ be the embedding $f_{{\rm tw}}$ with an additional coordinate. The value of every $v\in V$ in the new coordinate equals to $t-\|f_{{\rm tw}}(v)\|_{1}$. We argue that $f$ has the desired properties.

The first property follows as for every vertex $v\in V$, the distance in $T$ to $v_{t}$ is exactly $\frac{t}{2}$, therefore $\left\|f_{{\rm tw}}(v)\right\|_{1}=\left\|f_{{\rm tw}}(v)-f_{{\rm tw}}(v_{t})\right\|_{1}\leq 2\cdot d_{T_{t}}(v,v_{t})=t$. Therefor $\left\|f(v)\right\|_{1}=\left\|f_{{\rm tw}}(v)\right\|_{1}+\left|t-\left\|f_{{\rm tw}}(v)\right\|_{1}\right|=t$.

The second property follows as $f_{{\rm tw}}$ has expansion $2$, and distances in $T_{t}$ can only decrease w.r.t. distances in $T$. Thus for every $u,v\in V$, $\left\|f_{{\rm tw}}(v)-f_{{\rm tw}}(u)\right\|_{1}\leq 2\cdot d_{T_{t}}(v,u)\leq 2\cdot d_{T}(v,u)$. By the triangle inequality,

[TABLE]

For the third property, consider some pair $u,v\in V$. As every shortest path containing the new vertex $v_{t}$ will be of weight at least $t$, it holds that $d_{T_{t}}(v,u)=\min\{d_{T}(v,u),t\}$. We conclude that $\left\|f(v)-f(u)\right\|_{1}\geq\left\|f_{{\rm tw}}(v)-f_{{\rm tw}}(u)\right\|_{1}\geq d_{T_{t}}(v,u)=\min\{d_{T}(v,u),t\}$. ∎

Next, we construct an embedding of O-S graphs into the sphere of radius $t$ in $\ell_{1}$.

Corollary 3**.**

Let $G=(V,E,w)$ be a planar graph, $F$ a face, and $t>0$ a truncation parameter. There exists an embedding $f:F\rightarrow\ell_{1}$ such that the following holds:

Sphere surface: for every $v\in F$, $\left\|f(v)\right\|_{1}=t$. 2. 2.

Lipschitz: for every $u,v\in F$, $\left\|f(v)-f(u)\right\|_{1}\leq O(d(v,u))$. 3. 3.

Bounded Contraction: for every $u,v\in F$, $\left\|f(v)-f(u)\right\|_{1}\geq\min\left\{d_{G}(v,u),t\right\}$.

Proof.

Let $\mathcal{D}$ be the distribution over dominating trees guaranteed in Corollary 2. For every $T\in{\rm supp}(\mathcal{D})$, let $f_{T}$ be the embedding of $T$ into $\ell_{1}$ from Lemma 1 with parameter $t$. Our embedding is constructed by concatenating all $f_{T}$, scaled by their probabilities. That is, $f=\oplus\left\{\Pr\left[T\right]\cdot f_{T}\mid T\in{\rm supp}(\mathcal{D})\right\}$.

The first property follows as for every $v\in X$ and $f_{T}$, $\left\|f(v)\right\|_{1}=t$. Similarly, the third property follows as for every $u,v\in F$ and $T\in{\rm supp}(\mathcal{D})$, $\left\|f_{T}(v)-f_{T}(u)\right\|_{1}\geq\min\left\{d_{T}(v,u),t\right\}\geq\min\left\{d(v,u),t\right\}$.

The second property follows as for every $v,u\in V$,

[TABLE]

∎

Remark 1**.**

*Efficient construction: the support of the distribution $\mathcal{D}$ might be of exponential size. Nevertheless, we can bypass this barrier by carefully sampling polynomially many trees.

Denote by $m$ the number of vertices on $F$. For a pair $v,u$, by Markov inequality, the probability that a sampled tree has distortion larger than $m^{3}$ on $u,v$ is $O(m^{-3})$. We say that a tree is bad if it has distortion $m^{3}$ on some pair, otherwise it is good. By the union bound, the probability for sampling a bad tree is $O(1/m)$. Let $\mathcal{D}^{\prime}$ be the distribution $\mathcal{D}$ restricted to good trees only. $\mathcal{D}^{\prime}$ is a distribution over dominating trees with constant expected distortion, and worst case distortion $m^{3}$. Sample $m^{6}$ trees $T_{1},\dots,T_{m^{6}}$ from $\mathcal{D}^{\prime}$. By Hoeffding 444See https://sarielhp.org/misc/blog/15/09/03/chernoff.pdf, Theorem 7.4.3 . and union bound inequalities, w.h.p. the average distortion of all pairs will be constant. Define the embedding $f=\circ\left\{m^{-6}\cdot f_{T_{i}}\right\}_{i=1}^{m^{6}}$. The proof above still goes through.*

5 Non-Uniformly Truncated Embedding

In this section we generalize Corollary 3. Instead of a uniform truncation parameter $t$ for all the vertices, we will allow a somewhat customized truncation. See Figure 3 for illustration of the guarantees in the lemma.

Lemma 2**.**

Let $G=(V,E,w)$ be a planar graph with a given drawing on the plane. Let $F,\mathcal{I},\mathcal{B}\subseteq V$ such that $F\subseteq\mathcal{I}$, $\mathcal{I}\cup\mathcal{B}=V$, $\mathcal{I}\cap\mathcal{B}=\emptyset$, and $F$ is a face in $G[\mathcal{I}]$. Then there is an embedding $f:F\rightarrow\ell_{1}$ such that the following holds:

For every $v\in F$, $\|f(v)\|_{1}=d_{G}(v,\mathcal{B})$. 2. 2.

Lipschitz: for every $u,v\in F$, $\left\|f(v)-f(u)\right\|_{1}\leq O(d_{G}(v,u))$. 3. 3.

Bounded Contraction: for every $c\geq 1$, and every $u,v\in F$ such that $\min\left\{d_{G}(v,\mathcal{B}),d_{G}(u,\mathcal{B})\right\}\geq\frac{d_{G}(u,v)}{c}$ it holds that $\left\|{f}(v)-{f}(u)\right\|_{1}\geq\frac{d_{G}(v,u)}{12\cdot c}$.

Proof.

We will construct the non-uniformly truncated embedding by a smooth combination of uniformly truncated embeddings for all possible truncation scales. A similar approach was applied in [AFGN22]. Assume (by scaling) w.l.o.g. that the minimal weight of an edge in $G$ is $1$. Let $M\in\mathbb{N}$ be minimal integer such that the diameter of $G$ is strictly bounded by $2^{M}$.

Consider the graph $G[\mathcal{I}]$ induced by $\mathcal{I}$. Note that $G[\mathcal{I}]$ is an O-S graph w.r.t. $F$. For every distance scale $t\in\{0,1,\dots,M\}$ let $f_{t}$ be the embedding of $F$ w.r.t. the shortest path metric induced by $G[\mathcal{I}]$ from Corollary 3 with truncation parameter $2^{t}$. For a vertex $v\in\mathcal{I}$, let $t_{v}\in\mathbb{N}$ be such that $d_{G}(v,\mathcal{B})\in[2^{t_{v}},2^{t_{v}+1})$. Set $\lambda_{v}=\frac{d_{G}(v,\mathcal{B})-2^{t_{v}}}{2^{t_{v}}}$. Note that $0\leq\lambda_{v}<1$. For $t\in\{0,\dots,M\}$ we define a function ${\tilde{f}}_{t}:F\rightarrow\ell_{1}$,

[TABLE]

Define $f$ to be the concatenation of ${\tilde{f}}_{0}(v),\dots,{\tilde{f}}_{M}(v)$.

For every $v\in F$, according to Corollary 3 it holds that

[TABLE]

Next we prove that $f$ is Lipschitz. Consider a pair of vertices $u,v\in\mathcal{F}$. If $d_{G}(u,v)<d_{G[\mathcal{I}]}(u,v)$, then the shortest path from $u$ to $v$ in $G$ has to go through the boundary $\mathcal{B}$. It follows that $d_{G}(u,\mathcal{B})+d_{G}(v,\mathcal{B})\leq d_{G}(u,v)$. We conclude

[TABLE]

Otherwise, $d_{G}(u,v)=d_{G[\mathcal{I}]}(u,v)$. It follows from Corollary 3 that for every scale parameter $t$, $\left\|{f}_{t}(v)-{f}_{t}(u)\right\|\leq O\left(d_{G[\mathcal{I}]}(u,v)\right)=O\left(d_{G}(u,v)\right)$. We will prove a similar inequality for $\tilde{f}_{t}$. As $\tilde{f}_{t}(u)$ and $\tilde{f}_{t}(v)$ combined might be nonzero in at most $4$ different scales, the bound on expansion will follow.

Denote by $p_{t}$ the scaling factor of $v$ in ${\tilde{f}}_{t}(v)$. That is, $p_{t_{v}+1}=\lambda_{v}$, $p_{t_{v}}=1-\lambda_{v}$, and $p_{t}=0$ for $t\notin\{t_{v},t_{v}+1\}$. Similarly, define $q_{t}$ for $u$. First, observe that for every $t$,

[TABLE]

It suffice to show that $\left|p_{t}-q_{t}\right|=O(d_{G}(u,v)/2^{t})$. Indeed, for indices $t\notin\left\{t_{u},t_{u}+1,t_{v},t_{v}+1\right\}$, $p_{t}=q_{t}=0$, and in particular $\left|p_{t}-q_{t}\right|=0$. Let us consider the other cases. W.l.o.g. assume that $d_{G}\left(v,\mathcal{B}\right)\geq d_{G}\left(u,\mathcal{B}\right)$ and hence $t_{v}\geq t_{u}$. We proceed by case analysis.

• •

$\boldsymbol{t_{u}=t_{v}:}$ In this case, $\left|p_{t_{v}}-q_{t_{v}}\right|=\left|(1-\lambda_{v})-(1-\lambda_{u})\right|=|\lambda_{v}-\lambda_{u}|=\left|p_{t_{v}+1}-q_{t_{v}+1}\right|$. The value of this quantity is bounded by

[TABLE]

Hence, we get that $\left|p_{t}-q_{t}\right|=O(d_{G}(u,v)/2^{t})$ for all $t\in\{t_{v},t_{v}+1\}$.

• •

$\boldsymbol{t_{u}=t_{v}-1}:$ It holds that

[TABLE]

We conclude:

[TABLE]

• •

$\boldsymbol{t_{u}<t_{v}-1}:$ By the definition of $t_{v}$ and $t_{u}$, $d_{G}(v,u)\geq d_{G}(v,\mathcal{B})-d_{G}(u,\mathcal{B})\geq 2^{t_{v}}-2^{t_{u}+1}\geq 2^{t_{v}-1}$. It follows that for every $t\leq t_{v}+1$, $\left|p_{t}-q_{t}\right|\leq 1\leq\frac{d_{G}(u,v)}{2^{t_{v}-1}}=O\left(\frac{d_{G}(u,v)}{2^{t}}\right)$.

Next we argue that $f$ has small contraction for pairs far enough form the boundary. Consider a pair of vertices $u,v\in F$, and let $c\geq 1$ such that $\min\left\{d_{G}(v,\mathcal{B}),d_{G}(u,\mathcal{B})\right\}\geq\frac{d_{G}(u,v)}{c}$. It holds that $2^{t_{v}}>\frac{1}{2}\cdot d_{G}(v,\mathcal{B})\geq\frac{1}{2c}\cdot d_{G}(u,v)$. For every $t\geq t_{v}$, by the contraction property of Corollary 3, it holds that

[TABLE]

where the last inequality follows as $c\geq 1$. Assume w.l.o.g. that $d_{G}(v,\mathcal{B})\geq d_{G}(u,\mathcal{B})$, thus $t_{v}\geq t_{u}$ and let $t\in\{t_{v},t_{v}+1\}$ such that $p_{t}\geq\frac{1}{2}$. Set $S=12\cdot c$. We consider two cases:

• •

If $|p_{t}-q_{t}|>\frac{d_{G}(v,u)}{2^{t}\cdot S}$ then

[TABLE]

• •

Else $|p_{t}-q_{t}|\leq\frac{d_{G}(v,u)}{2^{t}\cdot S}$. First, assume that $p_{t}\geq q_{t}$. As $\frac{d_{G}(v,u)}{2^{t}\cdot S}\leq\frac{2c}{S}=\frac{1}{6}$, it holds that $q_{t}\geq\frac{1}{3}$. We conclude

[TABLE]

The case where $q_{t}>p_{t}$ is symmetric.

∎

Remark 2**.**

In Lemma 2 we used uniformly truncated embeddings in order to create a non-uniformly truncated embedding. Such a transformation might be relevant in other contexts as well. Consider a case where each vertex $v$ has some truncation parameter $s_{v}$, and there exists a uniformly truncated embedding for every parameter $t$. As long as for every $v,u$, $|s_{v}-s_{u}|=O(d_{G}(u,v))$, following the same construction as above, one can create a similar non-uniformly truncated embedding where $\|f(v)\|=s_{v}$.

6 Construction of a PSPD for Planar Graphs

We begin this section by proving the following separator theorem. Even though similar statements to Theorem 5 already appeared in the literature, we provide a proof for completeness.

Theorem 5**.**

Let $G=(V,E,w,K)$ be a weighted terminated planar graph. Suppose that $\gamma(G,K)=\gamma$. Then there are two shortest paths $P_{1},P_{2}$ in $G$, such that for every connected component $C$ in $G\setminus\{P_{1}\cup P_{2}\}$ it holds $\gamma(G[C],K\cap C)\leq\frac{2}{3}\gamma+1$.

Proof.

A planar cycle separator theorem for vertices have already became folklore [Mil86, Tho04]. Specifically, given a weight function $\omega:V\rightarrow\mathbb{R}_{+}$ over the vertices, and a root vertex $v\in V$, one can efficiently find a cycle $S$, that consists of two shortest paths rooted at $v$, such that the total weight of the vertices in each connected component of $G[V\setminus S]$ is at most $\frac{2}{3}\sum_{v\in V}\omega(v)$.

We start by defining a weight function $\omega$. Let $\mathcal{F}$ be a face cover of size $\gamma$. For every face $F\in\mathcal{F}$, let $v_{F}\in F$ be an arbitrary vertex (not necessarily unique). Initially the weight of all the vertices is [math]. For every $F\in\mathcal{F}$, add a single unit of weight to $v_{F}$. Note that the total weight of all the vertices is $\gamma$, while for every $F\in\mathcal{F}$, the total weight of the vertices in $F$ is at least $1$. See Figure 4 for an illustration.

Let $v$ be an arbitrary vertex on the outer face. We use the planar cycle separator theorem w.r.t. the weight function $\omega$ and the root vertex $v$. As a result, we get a pair of shortest paths $P_{1},P_{2}$ rooted in $v$. Let $C$ be a connected component in $G\setminus\{P_{1}\cup P_{2}\}$. The total weight of all the vertices in $C$ is bounded by $\frac{2}{3}\gamma$. Consider the drawing of $C$ obtained by removing all other vertices from the drawing of $G$. Next we define a face cover $\mathcal{F}_{C}$. For every $F\in\mathcal{F}$, if $v_{F}\in C$ then add $F$ to $\mathcal{F}_{C}$ (or the new face containing the remainder of $F$). Additionally, add the outer face in the drawing of $C$ to $\mathcal{F}_{C}$.

It is straightforward that $|\mathcal{F}_{C}|\leq\frac{2}{3}\gamma+1$. We argue that $\mathcal{F}_{C}$ is a face cover for $K\cap C$. Indeed, let $u\in K\cap C$. Let $F\in\mathcal{F}$ be some face s.t. $u\in F$. If $v_{F}\in C$, then $F$ (or its remainder) is in $\mathcal{F}_{C}$, and therefore $u$ is covered. Otherwise, $v_{F}\notin C$. Therefore $v_{F}$ and $u$ were separated by the deletion of $P_{1},P_{2}$. Necessarily some vertex of $F$ belongs to $P_{1}\cup P_{2}$. We conclude that $u$ is now part of the outer face, and therefore covered. ∎

If the size of the face cover is $2$, we can reduce this size to $1$ by removing a single shortest path containing vertices from both faces, the remaining graph will be O-S. Similarly, if the size of the face cover is $3$ we can reduce to $1$ by removing a pair of shortest paths. We can invoke Theorem 5 repeatedly in order to hierarchically partition $G$, reducing the size of the face cover in each iteration. After $O(\log\gamma)$ iterations, the size of the face cover in each connected component will be at most $1$.

Corollary 4**.**

Let $G=(V,E,w,K)$ be a weighted terminated planar graph such that $\gamma(G,K)=\gamma$. Then there is an PSPD $\left\{\mathcal{X},\mathcal{P}\right\}$ of depth $O(\log\gamma)$ and remainder $\left\{\mathcal{C},\mathcal{B}\right\}$, such that for every cluster $C\in\mathcal{C}$, all the terminals in $C$ lie on a single face (i.e. $\forall C\in\mathcal{C},~{}~{}\gamma(C,K\cap C)\leq 1$).

7 Embedding Parametrized by Face Cover: Proof of Theorem 1

Given the PSPD $\left\{\mathcal{X},\mathcal{P}\right\}$ from Corollary 4, with remainder $\left\{\mathcal{C},\mathcal{B}\right\}$, we are ready to define the embedding of Theorem 1. Let $f_{\textsf{PSPD}}$ be the embedding of $G$ into $\ell_{1}$ from Theorem 2, restricted to $K$. For every cluster $C\in\mathcal{C}$ which contains a terminal, let $\mathcal{B}_{C}=V\setminus C$ and $F_{C}$ be the single face in the face cover of $C$. Let $f_{C}$ be the embedding from Lemma 2 with parameters $F_{C},C,\mathcal{B}_{C}$. Let $\hat{f}_{C}$ be the embedding $f_{C}$ restricted to $K\cap C$, and extended to $K$ by sending every $v\in K\setminus C$ to $\vec{0}$. The final embedding $f$ will be a concatenation of $f_{\textsf{PSPD}}$ with $\hat{f}_{C}$ for all $C\in\mathcal{C}$.

Expansion

Consider a pair of vertices $v,u\in V$. By Theorem 2, $\left\|f_{\textsf{PSPD}}(v)-f_{\textsf{PSPD}}(u)\right\|_{1}=O\left(\sqrt{\log\gamma}\right)\cdot d_{G}(v,u)$. On the other hand, for every $C\in\mathcal{C}$, using Lemma 2, if $u,v\in C$ then $\left\|f_{C}(v)-f_{C}(u)\right\|_{1}=O\left(1\right)\cdot d_{G}(v,u)$. Otherwise if $v\in C,u\notin C$, then $\left\|f_{C}(v)-f_{C}(u)\right\|=d_{G}(v,\mathcal{B})\leq d_{G}(v,u)$ (similarly for $v\notin C,u\in C$). As each vertex is nonzero only in a single function $f_{C}$, the $O(\sqrt{\log\gamma})$ bound on the expansion follows.

Contraction

Consider a pair of terminal vertices $v,u$. If either $u,v$ are separated by $\mathcal{C}$ or $\min\left\{d_{G}(v,\mathcal{B}),d_{G}(u,\mathcal{B})\right\}\leq\frac{d_{G}(u,v)}{12}$, it holds that $\left\|f_{\textsf{PSPD}}(v)-f_{\textsf{PSPD}}(u)\right\|_{1}\geq d_{G}(u,v)$ and we are done. Otherwise, there must exist a cluster $C\in\mathcal{C}$ such that $u,v\in F_{C}$ and $\min\left\{d_{G}(v,\mathcal{B}_{C}),d_{G}(u,\mathcal{B}_{C})\right\}\geq\frac{d_{G}(u,v)}{12}$. By Lemma 2 $\left\|\hat{f}_{C}(v)-\hat{f}_{C}(u)\right\|_{1}=\left\|f_{C}(v)-f_{C}(u)\right\|_{1}\geq\frac{d_{G}(u,v)}{12\cdot 12}$.

Theorem 1 now follows. Bellow we discuss the implementation details of the embedding.

7.1 Polynomial Implementation

Given a planar graph with a drawing in the plane, using the PTAS of Frederickson [Fre91] we can find a face cover of size $2\cdot\gamma(G,K)$ in linear time (see Section 1.2). Note that using a cover of size $2\cdot\gamma(G,K)$ instead of $\gamma(G,K)$ is insignificant for our $O(\sqrt{\log(\gamma(G,K))}\ )$ upper bound. Next, construct a PSPD for this face cover using cycle separators. Since we construct at most $n$ separators, the construction of the PSPD also takes polynomial time. Given a PSPD, the embedding of [AFGN22] is efficiently computed.

After the creation of $f_{\textsf{PSPD}}$ we are left with a remainder $\{\mathcal{C},\mathcal{B}\}$. For every $C\in\mathcal{C}$ we start by computing uniformly truncated embeddings (see Remark 1). Given an O-S graph with $k$ terminals, there are at most $2k$ truncation scales (as each terminal participates in two scales only). Thus in polynomial time we can compute the embedding for all truncation scales, and thus compute the non-uniformly truncated embedding.

Acknowledgments

The author would like to thank Ofer Neiman for helpful discussions.