Embedding -Outerplanar Graphs Into ½

k `

Embedding -Outerplanar Graphs into ½

Ý Þ Þ Ü Chandra Chekuri£ Anupam Gupta Ilan Newman Yuri Rabinovich Alistair Sinclair

Abstract nal metric into a nice metric, solve the problem for the nice metric, and ﬁnally translate the solution back to the original We show that the shortest-path metric of any -outerplanar metric. graph, for any ﬁxed , can be approximated by a probability distribution over tree metrics with constant distortion, and When the optimization problem is monotone and scal-

able in the associated metric (as is usually the case), it is nat- hence also embedded into ½ with constant distortion. These graphs play a central role in polynomial time approximation ural to restrict one’s attention to nice metrics which dominate schemes for many NP-hard optimization problems on gen- the original metric, i.e., in which no distances are decreased.

The maximum factor by which distances are stretched in the

eral planar graphs, and include the family of weighted planar grids. approximating metric is called the distortion of the embed- This result implies a constant upper bound on the ratio ding. Typically, the distortion translates more or less directly between the sparsest cut and the maximum concurrent ﬂow into the approximation factor that one has to pay in trans- forming the problem from one metric to the other, so obvi- in multicommodity networks for -outerplanar graphs, thus extending a classical theorem of Okamura and Seymour [26] ously we seek an embedding with low distortion. The num- for outerplanar graphs, and of Gupta et al. [17] for treewidth- ber of applications of this paradigm has exploded in the past 2 graphs. In addition, we obtain improved approximation few years, and it has become a versatile and standard part of the algorithm designer’s toolkit: see the recent survey by In- ratios for -outerplanar graphs on various problems for which approximation algorithms are based on probabilistic dyk [19] and the forthcoming book by Matouˇsek [23, Chap- tree embeddings. We also conjecture that our random tree ter 10]. These applications have also given impetus to the study of the underlying theory of ﬁnite metric spaces. embeddings for -outerplanar graphs can serve as a building

In this paper we will be concerned with embedding ﬁ-

block for more powerful ½ embeddings in future.

nite metric spaces into ½ , i.e., real space endowed with the

1 Introduction ½ (or Manhattan) metric. Low distortion embeddings into

½ have been recognized, along with embeddings into Eu-

1.1 Background Many optimization problems on graphs

½ clidean space ¾ and into low-dimensional , to be of fun- and related combinatorial objects involve some ﬁnite metric damental importance in applications of the above paradigm, associated with the object. In particular, the shortest-path as well as for the underlying theory. One of several com-

metric on the vertices of an undirected graph with nonnega-

pelling reasons for studying ½ -embeddingscomes from their tive weights on the edges frequently plays an important role. intimate connection with the maxﬂow-mincut ratio in a mul- While for general metric spaces such an optimization prob- ticommodity ﬂow network. Namely, if every shortest-path lem can be intractable, it is often possible to identify a subset metric on a given graph with arbitrary edge lengths can be

of “nice” metricsfor which the problemis easy. Thus, a natu-

embedded into ½ with distortion at most , then the ratio be- ral approach to such problems — and one which has proved tween the sparsest cut and the maximum concurrent ﬂow for highly successful in many cases — is to embed the origi- any set of capacities and demands on the graph is bounded 1

by [22, 2, 17] . For more details on the sparsest cut prob-

£ Lucent Bell Labs, 600 Mountain Avenue, Murray Hill NJ 07974. lem, its relation to embeddings, and its application to the Email: [email protected] design of a host of divide-and-conquer algorithms, see the Ý Lucent Bell Labs, 600 Mountain Avenue, Murray Hill NJ 07974. survey by Shmoys [31]. Email: [email protected]. Part of this work was Equally important in algorithmic applications are cer-

done when the author was visiting the University of California, Berkeley, supported by a US-Israeli BSF grant #19999325. tain special ½ embeddings known as embeddings into ran-

Þ Computer Science Department, University of Haifa, Haifa 31905, dom (dominating) trees, whereby the given metric is approx- g Israel. Email: filan, yuri @cs.haifa.ac.il. Supported in part imated by a probability distribution over tree metrics. Since by a US-Israeli BSF grant #19999325. every tree metric can be embedded isometrically (i.e., ex- Ü Computer Science Division, University of California, Berkeley CA 94720-1776. Email: [email protected]. Supported in part

1 ` by NSF grants CCR-9820951 and CCR-0121555 and by a US-Israeli BSF Indeed, the distortion of an optimal ½ embedding is exactly the worst grant #19999325. cut-ﬂow ratio for any choice of capacities and demands.

actly, or with distortion 1) into ½ , approximating a metric by constant distortion strictly larger than 1. (For example, the

random trees with expected distortion immediately yields graph ¾ has treewidth 2 but is not isometrically embed-

an embedding into ½ with distortion . As has been recog- dable; see [12, Example 6.3.2] for a simple proof of this nized in the work of Bartal and others [1, 5, 6], random tree fact.) embeddings have many additional applications to online and Some, but not all of the above results carry over to

approximation algorithms that are not enjoyed by arbitrary the more restrictive setting of embedding into random trees.

½ embeddings. In [17] it is shown how to embed outerplanar graphs into

For general metrics the question of embeddability random trees with small constant distortion (note that the

into ½ is essentially resolved: Bourgain [9] showed that isometric embedding of Okamura and Seymour is not a tree

any -point metric can be embedded into ½ with embedding); on the other hand, in the same paper it is shown

distortion, and a matching lower bound was established for that even series-parallel graphs incur a distortion the shortest-path metrics of unit-weighted expander graphs for tree embeddings. Despite this limitation, it is worth

in [22]. For embeddings into random trees, a construction pointing out that the random tree embeddings of outerplanar

of Bartal [6] yields a distortion of for an graphs played a key role in the development of constant

arbitrary -point metric. distortion ½ embeddings of series-parallel graphs in [17]: However, tight bounds are still not known for many im- the trick was to combine the special structure of the tree portant classes of graphs, including planar graphs and graphs embeddings with judicious use of random cuts. with bounded treewidth; many such restricted classes are conjectured to be embeddable with constant distortion. In- 1.2 Results In this paper, we extend the above line of

deed, the general question of how the topology of a graph research to a much wider class of planar graphs, namely

affects its embeddability into ½ , and into random trees, is -outerplanar graphs for arbitary constant . Informally,

one of the most important open issues in the area of met- a planar graph is -outerplanar if it has an embedding

ric embeddings. (See, e.g., the tutorial by Indyk in the with disjoint cycles properly nested at most deep. A last FOCS [19].) In addition to its inherent mathematical in- formal deﬁnition is given in Section 2, while Figure 4.2

terest, this question impacts the design of approximation al- shows a simple example; a canonical example of a -

gorithms for many problems on restricted families of graphs outerplanar family is the sequence of rectangular

and networks. grids. -outerplanar graphs play a central role in polynomial

Some limited but interesting progress has been made on time approximation schemes for many NP-hard optimization embedding restricted metrics into ½ . Recently, Rao [27] problems on general planar graphs (see, e.g., [4]). Our main

showed that the shortest-path metric of any graph that result is the following:

excludes a is embeddable into with distortion

. This beats the lower bound for gen-

THEOREM 1.1. Any shortest-path metric of a -outerplanar

eral graphs for any constant , and also gives dis- graph can be embedded into a probability distribution over

tortion embeddings for the classes of planar and bounded-

trees, and hence into ½ , with distortion for some treewidth graphs. However, Rao’s approach (of ﬁrst embed-

absolute constant . Moreover, such an embedding can be

ding these graphs into ¾ and then using isometric embed- found in randomized polynomial time.

dings of ¾ into ) was shown to be tight in [24], where a

lower bound of distortion was shown for embed-

Thus, not only do such graphs embed well into ½ , ding even treewidth-2 (and hence also planar) graphs into ¾ . but they even embed well into dominating trees. This is

Approaching the question from the other direction, a

in contrast to the lower bound of for treewidth-2 celebrated theorem of Okamura and Seymour [26] implies graphs [17]. that any outerplanar metric can be embedded isometrically

2 Our result immediately implies a constant maxﬂow- into ½ . However, it has been shown that outerplanar graphs

mincut ratio for arbitrary multicommodity ﬂow problems on are essentially the only graphs (with the exception of 4 )

-outerplanar graphs. This is the ﬁrst progress in this di- that are isometrically embeddable into ½ [25]. More re- rection in the two decades since the Okamura-Seymour recently, Gupta et al. [17] showed a constant distortion em-

sult [26], which proves a ratio of 1 for 1-outerplanar graphs.

bedding into ½ for treewidth-2 graphs (which are essentially

Additionally, because our ½ -embeddings are in fact random series-parallel graphs, and hence also planar). This was the tree embeddings, we also obtain as an immediate byproduct ﬁrst natural class of graphs shown to be embeddable with improved approximation ratios for a number of algorithms

for problems on -outerplanar graphs, including the buy-at- 2Their result deals more generally with the cut/ﬂow ratio in planar

networks where all terminals lie on a single face; this and other results where bulk problem [3] and the group Steiner problem [15]. For

restrictions are placed on both the ﬂow network and the demand structure any ﬁxed , the improvement in each case is by a can be found in surveys by Frank [14] and Schrijver [29]. factor; we defer the details to the full version of the paper.

We should also note that our result is the first demonstra- The rest of the paper is organized as follows. We first tion of constant distortion ½ embeddingsfor a natural family fix notation and give essential definitions in Section 2. In of graphs with arbitrarily large (but bounded) treewidth3. In- Section 3 we show how to embed Halin graphs into random

deed, -outerplanar graphs are a natural parameterized fam- trees with constant distortion. This is extended to obtain

ily of planar graphs having bounded treewidth. (Note that al- constant distortion embeddings for all -outerplanar graphs

though all treewidth-2 graphs are planar, treewidth-3 graphs in Section 4. In the interests of clarity of exposition, we make

¿ include non-planar examples such as ¿ .) no attempt to optimize the constants that arise in the various Finally, recall that constant distortion random tree em- steps of our procedure.

beddings of 1-outerplanar graphs were a key ingredient in the construction of good ½ embeddings of series-parallel 2 Notation and Preliminaries graphs in [17]. We are therefore optimistic that, with the addition of suitably chosen cuts, our new tree embeddings of Metrics: For general background on ﬁnite metrics and

-outerplanar graphs may become a building block for con- embeddings, see the book of Deza and Laurent [12]. Given

two metric spaces, and , and a map stant distortion ½ embeddings of wider classes of graphs,

such as bounded treewidth graphs or planar graphs. , deﬁne the following quantities.

1.3 Techniques We start with the approach of trying to

extend the random tree embeddings of outerplanar graphs

[17] to -outerplanargraphs. We do not know a way to solve

this problem directly. The ﬁrst main idea in the paper is to

identify a subclass of -outerplanar graphs that are easier to

embed, namely Halin graphs. Informally, a “Halin graph” We say that has contraction , expansion and

is obtained by embedding a tree in the plane and attaching distortion . The distortion between

a cycle around the leaves. (The formal deﬁnition can be and is at most if there exists with

foundin Section 2). Halin graphsare useful for the following . We often consider two metrics and over

reason. Given a -outerplanar graph, we can use the random the same vertex set ; in such cases, we assume that is

embedding of [17] to embed the inner outerplanar graphs the identity map. Metric is said to dominate if for all

obtained by removing the outer face(s) into a random tree , .

(in fact a forest). If we now add the outer face to this random Let be an undirected graph. A metric

tree we get a graph which is (very similar to) a Halin graph. is supported on (or generated by) if it is the shortest-path

Hence, if we can embed Halin graphs we can embed - metric of w.r.t. some nonnegative weighting of the edges

outerplanar graphs. We are then able to extend this approach . Given a graph with edge weights , denotes

to embed any -outerplanar graph by peeling off the outer the shortest path metric of , and we assume that the edge

layer and recursively embedding the inner layers. weights satisfy for unless

The second main idea is a technique for embedding otherwise stated.

Halin graphs. We note that even this deceptively simple For , the cut metric is deﬁned to be if

subclass of 2-outerplanar graphs had so far resisted attempts , and otherwise. It is known that a metric at constant distortion embeddings. This is derived by a is embeddable into ½ iff it can be written as a non-negative subtle modiﬁcation of the algorithm of Gupta [16] which linear combination of cut metrics [12].

showed how to remove Steiner vertices from a tree metric A metric supported on a graph is -

with only a constant factor distortion in distances between probabilistically approximated by a distribution over trees

the remaining vertices. Though seemingly unrelated to our if (1) each tree in the distribution has ;

problem (since we have no Steiner vertices), this algorithm (2) for all and in the distribution, dom-

can nonetheless be applied (with suitable modiﬁcations) to inates , i.e., ; and (3) for all

℄

the tree in the Halin graph, with the effect of reducing the , the expected distance D .

Halin graph to an outerplanar graph on its leaves. This we We shall also refer to this as an embedding of with dis-

can once again embed into random trees using [17]. tortion into random trees. (The fact that the distortion is only in expectation will often not be mentioned.) It is known

3 that general graphs can be embedded into random trees with

¢´k µ

The maximum treewidth among k -outerplanar graphs is .

4Given an induced metric deﬁned on a subset of vertices of a graph, we distortion [20, 1, 5, 6, 10, 11]. call the vertices not belonging to this subset the Steiner vertices. Although we are interested only in the metric space induced on the non-Steiner Graph-Theoretic Terms: Most graph-theory concepts vertices, the Steiner vertices might be necessary in order to deﬁne the which we use, such as treewidth, minors, and planarity, are distances between the non-Steiner vertices. covered in standard text-books (see, e.g., [13, 33]). An embedding of a graph is outerplanar (or 1- done incurring a constant expected distortion. outerplanar) if it is planar, and all vertices lie on the un-

Note that the previous step was almost what we wanted bounded face. An embedding of a graph is -outerplanar if it is planar, and deleting all the vertices on the unbounded — we just have to get rid of the Steiner vertices.

The second part, given in Section 3.1.2, eliminates the

face leaves a -outerplanar embedding of the remaining ¼

Steiner vertices of by contracting some of its edges,

graph. A graph is -outerplanar if it has a -outerplanar em-

´½¾µ ´½µ thus converting into . This process is shown bedding. It is known that a -outerplanar graph has treewidth

to incur a further distortion of a constant factor.

[8, 28]; other properties of these graphs and related

concepts can be found in [4, 8]. Given a planar graph, a -

´½¾µ

3.1.1 Processing I: Getting the tree In this sec- outerplanar embedding for which is minimal can be found

in polynomial time [7]. tion, we will show how to convert the tree into the tree

´½¾µ A Halin graph [18] is obtained by taking a planar while incurring only a constant distortion. The al-

gorithm Process-Tree to perform this processing cuts off a

embedding of a tree and attaching a cycle

subtree ¼ of which contains the root but not the leaves,

around the leaves of the tree (in order).

recursively acts on the subtrees thus created, makes a new

denotes the set of leaves of , and hence . (Note

that there may be vertices on the cycle that are not leaves of root vertex and adds edges from it to the roots of each of the

processed subtrees, and ﬁnally hangs ¼ off this new root.

℄

.) A Halin graph is 2-outerplanar and has treewidth 3. Many algorithmic problems can be solved (See Figure 3.1.) efﬁciently on these graphs (see, e.g., [32] and the references Before we make Process-Tree concrete, we deﬁne the

therein). auxiliary procedure Cut-Midway which cuts a random set of

edges to separate the root from all the leaves of . It returns

3 Embedding a Halin Graph a special tree ¼ containing the root of and none of its

leaves, and a set of subtrees (for ), each rooted

Given a Halin graph, we will embed it into random trees

at some vertex . We say that an edge is at a distance

thus: we ﬁrst take the tree from the Halin

from a vertex if is in the cut deﬁned by the set of vertices ´½µ

graph and process it to give a random dominating tree , whose distance from is at most .

which approximates distances in to within a constant (in

´½µ

expectation). Furthermore, has a speciﬁc structure: it Algorithm Cut-Midway( )

¼¼ ¼¼

consists of a tree on just the leaves of while there is a path from to a leaf in

the original tree , and the rest of the vertices in are let distance to closest leaf

¼¼ ¼¼

attached to vertices in . Also, the tree is a minor of let set of leaves at distance from in

¼¼

, and so attaching the cycle back to the vertices in let be the union of paths from to vertices in

gives us an outerplanar graph. This outerplanar graph is then choose uniformly

embedded into a random tree using known techniques [17] to edges in at distance from

give the following theorem, which is the main result of this delete edges in from

section: end while

¼ component of containing root but no leaves of

THEOREM 3.1. Any metric generated by a Halin graph can

¾ ½ other components of

be embedded into a distribution over dominating trees with

let value of when edge connecting to was cut.

constant distortion.

½ ½ ¾ ¾ return ¼ .

3.1 Processing the tree Let us assume that the tree has Now we can formally state Process-Tree:

a root vertex , which imposes an ancestor-

descendant relation between the vertices in . Each vertex Algorithm Process-Tree ( )

naturally deﬁnes a tree , namely the subtree induced by apply Cut-Midway( ) to get

½ ½ ¾ ¾

the vertices that are descendents of . For a vertex , let ¼

be the leaf in closest to , and be the distance of let be a new vertex, called the “Steiner twin” of

from in . Note that these functions are ﬁxed given the attach to with edge of length ¼

input tree . The processing algorithm works in two parts. for // We don’t have to work on ¼

if is just a single vertex (hence ) then

The ﬁrst step of the algorithm, given in Section 3.1.1,

´½¾µ

´½¾µ ¼

returns a tree . This tree consists of a tree else

deﬁned on the vertices of and some extra (or Steiner)

´½¾µ

Process-Tree ( )

vertices, and the vertices of hang off the vertices

´½¾µ

¼ ¼

of in the form of (possibly several) subtrees. This is let be root of

// is the Steiner twin of , the root of We claim that the cut made at this point will lie below ;

¼ ¼ ££

add edge with length i.e., . Indeed, such a cut made from at

££

end for a distance at least from it, where

´½¾µ ¼

return tree with as its root . Hence taking distances from , this cut is at

distance at least

´½¾µ

Recall that we had mentioned that would have a

. But this distance is greater than , and hence

portion called ; this is formed by the new edges added always lies above this next cut. Thus, when this next cut

¼ ¼

between and (for ) during the various

is made, either will be deleted (if fell below this cut), or

recursive calls to Process-Tree. (Note that this does not the cut will fall below and the edge will never again be a

include the edges added between and , i.e., between the candidate to be cut, proving the lemma.

original roots and their Steiner twins.) Hence includes Before we end, let us note that the portion of that

££ ££

all the leaves of , plus all the Steiner twins created. For lies in distance interval is disjoint from

an example, see Figure 3.1, where Cut-Midway performed the portion considered earlier, and has a length of at most

three cuts, and Process-Tree resulted in the tree on the right.

. As before, multiplying this by

££

The solid edges belong to , the dashed ones to , and the gives the probability that is cut if it is considered a

edge is shown as a faint line. second time. ¾

Let us call an edge a candidate to be cut at some step

if it has a non-zero probability of being cut at that step. We Let denote the length of edge in .

can now show the following bound on the expected distortion

incurred by the above procedure: LEMMA 3.2. If an edge is cut by Cut-Midway

with parameter , the expected distance between and in

¾µ

THEOREM 3.2. The (expected) distortion introduced by ´½

is at most .

procedure Process-Tree is at most .

Proof. Consider an edge of length which

Proof. Before we prove this, let us give a high-level sketch.

is cut in some iteration of Cut-Midway, and let be the It can be veriﬁed to see that distances are never contracted

value of the parameter at this point. Consider the distance

¾µ

by Process-Tree, and hence it sufﬁces to bound the expected ´½

=¾ ½ between and in the resulting tree .

expansion. We show this via two lemmas: ﬁrstly, Lemma 3.1

The vertex will be in ¼ and the vertex is the root

¾µ

shows that an edge is a candidate to be cut on at most two ´½

of for some and hence will be in when ,

(consecutive) occasions. Lemma 3.2 then shows that when

rooted at , is processed. From the description of

an edge is a candidate to be cut, it suffers only a constant

´½=¾µ ´½=¾µ

Process-Tree we see that can

¼ ¼

expected expansion. Combining these two results then gives ¼

´½=¾µ ´½=¾µ

be expressed as

¼ us the result. ¼

´½=¾µ ´½=¾µ

. From our construction,

¼ ¼ ¼

´½=¾µ ´½=¾µ

and and . We

LEMMA 3.1. No edge is a candidate for cutting more than

twice during the entire run of the algorithm Process-Tree. observe that for all , and that

. The latter is true because for to be cut, is on

Proof. Let be an edge, where is the ancestor of

the path from to a leaf in of length at most . Putting . Consider the ﬁrst instant when an edge is a candidate these observations together we obtain that

to be cut in a call to Cut-Midway. Let be the root at

¼ ¼ ¼

´½=¾µ ´½=¾µ ´½=¾µ

this point, and be the value of the parameter in the

while loop of this call to Cut-Midway. In this call of ¼

´½=¾µ

Cut-Midway, it is clear that cannot be considered again.

Indeed, after the cut, will not lie on any path from to

a leaf. A fact that will be useful later is that the portion

£ £

of that lies in the distance interval from

£ £

is , and that

value multiplied by is the probability that is cut at

this time. ¾

The edge will never be considered again if the cut fell

“below” , or if it passed through , so let us assume that the Now we complete the proof of Theorem 3.2: by

cut was above and lies in one of the trees with root Lemma 3.1, the edge is cut at most twice. The

. Clearly, it lies in some path from to a leaf, and hence ﬁrst time it is considered, it is cut with probability

££ £ £ £

it will be part of the tree at some point in the call to , and

Cut-Midway from . the expected length is at most . The second time r T r’

Cut 1 r r’ Cut 2 1 Cut 3 1r

T1 T’1 T’t

T’2

Figure 3.1: One call of the procedure Process-Tree.

£ ££ ¼¼ ´½¾µ ¼

the chance is ¾ , tree . (Since consists of with several subtrees

££ ¼¼

and the expected length is . Finally, with probabil- attached to it via cut-edges, attaching those subtrees to

´½µ

ity ½ , the length remains . Thus the expected will give us a new tree .) The argument in this section

distance between and is at most is similar in spirit to that in [16]. The Steiner twin vertices

´½¾µ

from are removed in the same order in which they

£ ££

¼ ¼

½ ¾ ½ ¾

were created. Consider , the root of ; it was created as

£ ££

the Steiner twin of vertex . We now identify all vertices

½ ¾

£ £

on the path between and with . This process is

performed on each of the Steiner twin vertices in turn (in

℄

order of their creation), causing each of them to be identiﬁed

´½µ

with some vertex in . Call the resulting tree .

This has the vertex set , since we removed all the Steiner which implies an expected distortion of at most , and vertices we created in the previous section. The following proves the theorem. ¾ lemma proves the main result of this section:

We close this subsection with a further observation LEMMA 3.3. This edge-contraction procedure ensures that

´½µ about the tree T’ constructed by the procedure Process-Tree. the distance between each pair of vertices of in is no

shorter than its distance in . ¼

CLAIM 1. The tree constructed as above is a minor of

the tree . Proof. To show that there is no contraction, it sufﬁces to ´½µ

check that no edge in is shorter than the distance

Proof. In each call to Process-Tree, we progressively con- between its endpoints in . There are just three kinds of

´½µ

¼ struct by removing the tree and replacing it with a star

edges remaining in : those which belong to the trees ¼

connecting to the various (for ). But this star in the various invocations of Process-Tree, those between

could equivalently be obtained by contracting all but the leaf

some and , and those between and . Note

edges of the tree ¼ . (Of course, we are placing new lengths that the edges of this last type are the only edges that exist

on these edges, but this does not affect the structure.)

between and , since such edges (w.l.o.g.) must be

caused by being the root at some invocation of Process- ¼

This claim also shows that the tree with the cycle

Tree and being one of the ’s created at this step, and

around its leaves is still a Halin graph, since Halin graphs

later being identiﬁed with .

are closed under taking minors.

Clearly, the edges in the trees ¼ are not changed at

all. Now consider an edge between a vertex and .

3.1.2 Processing II: Removing the Steiner vertices In

¼ ¼

5 Ö

this section, we remove the Steiner vertices in the tree These edges were added between Ö and , and the latter has been

´Ö µ that were created during runs of Process-Tree, giving us a identiﬁed with Ð . embeddable in the plane such that removing the vertices

on the outermost face times deletes the graph. Before we begin, let us state two simple lemmas (whose proofs we omit) that allow us to replace a subgraph by its tree embedding, and to give embeddings of graphs in terms of

their blocks.

PROPOSITION 1. Let be a subgraph of

. Let be a graph on

such that for all

. Then in the graph ,

for all .

PROPOSITION 2. Let the graph have a cut-edge whose

Figure 4.2: A 3-outerplanar graph from [4]. The layers are

removal results in a tree and a graph . If can be prob- A-G, a-g, and 1-8.

abilistically approximated by tree metrics with distortion ,

then so can .

The main result of this section, and of the paper is the

(This edge is created since was identiﬁed with .) The following:

¼¼

length of this edge in is just , which is also the

THEOREM 4.1. There is a universal constant such that

distance between and in . Finally, for an edge

´½µ any metric generated by a -outerplanar graph can be

between and in , the length is just .

embedded into random trees with distortion at most .

However, the distance between these points in is at most

which we upper bound next. Let i

Proof. The proof is by induction on ; however, the induc-

be the value of when was separated from in the tion hypothesis required is stronger than the statement of the

procedure Cut-Midway. Then it follows that

theorem. We will assume that is given along

£ £

and .

i i

with its -outerplanar embedding, and ¼ is the set of

Hence the distance between and in is at most

vertices on the outer face of . (In the sequel, we will often

£ £

, however , therefore the distance is at abuse notation and blur the distinction between a face and

most . the vertices that lie on it.)

Induction Hypothesis: Let be a connected - ´½µ

3.2 Wrapping it all up Since the distances in are at

outerplanar graph with ¼ as the outer face in some -

least those in , andatmost times those in , this gives us

outerplanar embedding of . Then, the shortest-path metric a total (expected) distortion of . Furthermore, we can now

of can be probabilistically approximated by a collection

add back the cycle on the vertices of , giving the graph

of trees on with expected distortion at most such that

´½µ

. This consists of an outerplanar graph on , along

each tree in the distribution has the following

with vertices of in the form of subtrees attached to properties:

¼¼

vertices of . To see that forms an outerplanar graph,

¼ ¼¼

note from Claim 1 that was a minor of , and hence (i) the subgraph of induced by is a minor of ; ¼

obtained by contracting some edges in still leaves us with and

a minor of , so the planar embedding of induces a

¼¼ ¼¼ ¼

(ii) the subgraph of induced by is a forest,

planar embedding of . Furthermore, is deﬁned

and each tree in the forest is connected to ¼ by a

on the vertices of , so all the vertices lie on the outer face. single edge. But now we can invoke the procedure of [17, Theo-

Informally, we will require that the random tree for ´½µ

rem 5.2]to get a randomsubtree of which approximates be embeddable in the plane even when the vertices on the

´½µ distances (in expectation) in to within a factor of , and

outer face of are “pinned down” to the plane. Clearly, if

hence those in to within a factor of . This

the trees are subgraphs of , then this is trivially satisﬁed. completes the proof of Theorem 3.1. The reader can verify that the Halin graphembeddingof Sec- tion 3 producesgraphs which, though they are not subgraphs,

4 Onto -outerplanar graphs nevertheless satisfy the above property.

In this section, we extend the construction of the previous The base case for the induction is when is an

section to -outerplanar graphs. Recall that these are graphs outerplanar graph. For outerplanar graphs [17, Theorem 5.2]

shows an embedding of into trees that are subgraphs of the hypothesis; in fact is a minor of . Furthermore,

with constant distortion. Hence the auxiliary conditions are , i.e., vertices of not in , induces a forest

trivially satisﬁed. in that is connected via cut-edges to . Also note that

there are no edges between ¼ and , since the

F F graph is planar, and the layer ¼ separates these two sets

3 5 of vertices. Using Proposition 2, we can eliminate vertices

¼ F F G in (for ) from . It now sufﬁces

1 4 5

to embed the resulting graph, which we call core , into G3 G trees with expected distortion at most . 1 F 2 The key claim that essentially reduces this problem to G4 the embeddingsof Halin graphs given in the previous section is the following:

G2 ¼

F ¼

6 CLAIM 2. Let be obtained by taking the tree , and

adding the vertices and all the edges incident on in

℄

. Then is a Halin graph.

Proof. By the induction hypothesis, the tree is a minor

Figure 4.3: Partitioning of a -outerplanar graph into

of , and hence the planar embedding of induces a

-outerplanar graphs ½ . The bold lines ¼

natural planar embedding of . Furthermore, from our

indicate , the graph induced by the outer face.

earlier assumption, each vertex of has at most one edge

¼ ¼

to ; let be the set of these edges. It follows that

along with these edges still forms a tree. Finally, the

For the induction step, let us assume that is two

edges along the face form a cycle around this tree, and

vertex connected, else we can work with the blocks of

¼ ¾

hence is a Halin graph as claimed.

independently. Let be the subgraph of induced by

¼ , the vertices on its outer face; clearly is an

¼ ¼

Note that our current graph core is simply .

outerplanar graph. (See Figure 4.3.) Let ½

Since each is a minor of , we obtain the following

¼ be the internal faces of , the subset of

result.

lying inside the face , and the induced graph on .

We assume without loss of generality that is connected

PROPOSITION 3. The graph core is a minor of .

for otherwise we can work with its connected components

Now since each is a Halin graph (with as its outer

separately. We make the following assumption for technical

face), we can apply the procedure of Section 3 to it. The

reasons: for any vertex there is at most one vertex

¼¼

resulting graph, which we call , will be an outerplanar

such that . This is without loss of

graph on , with the vertices of inducing a forest,

generality, since if this does not hold for a vertex ,

we can split into a path of vertices (with the edges between the trees of which are connected to vertices of via cut-

them of length 0) and connect each one to a unique vertex edges. Using Proposition 2 again, we can remove these

¼ ¼

hanging trees to obtain the graph core . of without violating planarity. Note the following fact,

which allows us to use the induction hypothesis: Note that the procedure in Section 3 guarantees that

¼¼ ¼ ¼ ¼

core is a minor of . Furthermore, each core

FACT 4.1. For , is a -outerplanar graph.

is an outerplanar graph on the face of the outerplanar

Now applying the induction hypothesis, each can graph . These two facts together imply that

½ ¼ ¼

be -probabilistically approximated by trees satisfying core is also an outerplanar graph. We can embed

conditions (i) and (ii). We now give a procedure to extend into subtrees of with constant expected distortion

the embeddings of the various to an embedding of . For following [17, Theorem 5.2]. (We assume that the distortion

, we pick a tree from the distribution over tree is at most by choosing sufﬁciently large.) This establishes

metrics for . Let be the graph obtained by adding the that can be embedded with expected distortion at most .

vertices of ¼ and the edges incident to them (in ) to It just remains to show that the conditions (i) and (ii) are

the trees ½ . Proposition 1 implies that the metric met for the trees produced by this procedure. The ﬁnal step

¼ ½

induced by is within an expected distortion of , is an embedding of whose vertex set is . It can be

and hence approximating by tree metrics with an expected seen that is a minorof ; indeed, Proposition 3 shows that

¼ ¼

distortion of will prove the induction hypothesis for . core is a minor of , and as observed above,

¼ ¼¼ ¼

Let be the subtree of induced by . The each is aminorof . Finally, using the procedurein [17]

fact that it is a subtree is guaranteed by condition (i) of to embed gives subtrees of which are clearly minors of e d e d 5 5 8 8 c c 4 7 4 7 3 3 1 6 1 6

2 2

a b a b

Figure 4.4: Returning from the induction: the bold lines denote .

, and thus of . Hence our procedure guarantees that each spaces in Hilbert space. Israel Journal of Mathematics, 52(1-

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