Coarse differentiation and multi-flows in planar graphs

Coarse differen tiation a nd m ulti-flows in planar graphs James R. Lee ∗ Prasad Ra gha v endra † Abstract W e show that the m ulti-commo dity max-flow/min-cut gap for series-pa r allel graphs c a n b e as bad as 2, matching a recent upp er b ound [8] for this class , and r esolving one side of a conjecture of Gupta, Newman, Ra binovic h, and Sinclair. This also improv es the largest kno wn g ap for planar graphs fro m 3 2 to 2, yielding the first low e r b ound that do e sn’t follow from elementary calculations . Our approach uses the c o arse differ entiation metho d of Eskin, Fisher, and Wh yte in order to lo w er b ound the distortion for embedding a particular fa mily of shortest-path metrics into L 1 . 1 In tro duction Since the app earance of [26] and [4], lo w -distortion metric emb eddings hav e b ecome an incr easingly p o w erful tool in s tu dying the relatio nship b et ween cuts a nd m ulticommod it y flo ws in graphs. F or bac kground on the fi eld of metric embedd ings and their applications in theoretical computer science, w e r efer to Matou ˇ sek’s b o ok [28, Ch. 15], the surveys [20, 25], and the comp endium of op en problems [27]. One of the central connections lies in the corresp ondence b et ween lo w-distortion L 1 em b eddings, on the one han d , and the Sp arsest Cut pr oblem (see, e.g. [26, 4, 3, 2]) and c oncurr ent multi- c ommo dity flows (see, e.g. [19, 14]) on the other. This relationship allo ws one to bring sophisticated geometric and analytic tec hn iqu es to b ear on classic al pr ob lems in graph partitioning and in the theory of net w ork flows. In the p resen t pap er, w e show ho w tec hniques dev elop ed initially in geometric g roup theory can b e used to shed new ligh t on the connections betw een sparse cuts and m ulti-commodity flo ws in p lanar graphs. Multi-commo dit y flo ws a nd sparse cuts. Let G = ( V , E ) b e an und irected graph, with a c ap acity C ( e ) ≥ 0 asso ciated to ev er y edge e ∈ E . Assume that we are giv en k pairs of ve rtices ( s 1 , t 1 ) , ... , ( s k , t k ) ∈ V × V and D 1 , . . . , D k ≥ 1. W e think of the s i as sour c es , the t i as tar ge ts , and the v alue D i as the demand of the terminal p air ( s i , t i ) for c ommo dity i . In the MaxFlow problem the ob jectiv e is to maximize the fr action λ of the demand that can b e shipp ed sim ultaneously for all the commo dities, sub ject to the capacit y constrain ts. Denote this maxim um by λ ∗ . A straigh tforward up p er b ound on λ ∗ is th e sp arsest cut r atio . Giv en any subs et S ⊆ V , w e write Φ( S ) = P uv ∈ E C ( uv ) · | 1 S ( u ) − 1 S ( v ) | P k i =1 D i · | 1 S ( s i ) − 1 S ( t i ) | , ∗ Universit y of W ashington, Seattle. Research partially supp orted by NSF CA REER aw ard CCF-064403 7. P art of this researc h was completed while th e aut hor was a p ostdoctoral fello w at t he Institute for Ad v anced Stud y , Princeton. † Universit y of W ashington, Seattle. Researc h supp orted in part by NSF grant CCF-0343672. 1 where 1 S is the c haracteristic function of S . The v alue Φ ∗ = min S ⊆ V Φ( S ) is the minim um ov er all cuts (partitions) of V , of the ratio b et ween the total capacit y crossing the cut and the total demand crossing the cut. In the case of a single commo dity (i.e. k = 1) the classical MaxFlo w- MinCut theorem sta tes that λ ∗ = Φ ∗ , but in general this is no longer the case. It is kn o wn [26, 4] that Φ ∗ = O (log k ) λ ∗ . Th is r esult is p erhaps th e first striking application o f m etric e m b eddin gs in com b inatorial optimiza tion (sp ecifically , it uses Bourgain’s em b edding theorem [6]). Indeed, the connection b et ween L 1 em b eddings and m ulti-commodity flo w/cut gaps can b e made qu ite pr ecise. F or a graph G , let c 1 ( G ) r epresen t the largest distortion necessary to em b ed an y s hortest-path metric on G in to L 1 (i.e. the maxim u m o ve r all p ossible assignmen ts of n on - negativ e lengths to the edges of G ). Then c 1 ( G ) giv es an u pp er b ound on th e ratio b et wee n the sparsest cut r atio and th e maxim um flo w for any m ulti-commod it y flo w instance on G (i.e. with an y c hoices of capacities and demands) [26, 4]. F urthermore, this connection is tigh t in the sense that there is alw a ys a m ulti-commo dity fl o w instance on G that ac hiev es a gap of c 1 ( G ) [19]. Despite signifi can t progress [30, 19, 14, 8, 7], some fun damen tal questions are still left u nan- sw ered. As a p r ime example, consid er the well-kno wn plana r emb e dding c onje ctur e [1 9, 20, 25, 27]: Ther e e xists a c onstant C such that every planar gr aph metric emb e ds into L 1 with distortion at most C . In initiating a systematic stud y of L 1 em b eddings [19] for minor-closed families, Gupta, New- man, Rabin o vic h, and Sin clair put forth the follo wing v ast generalizat ion of this conjecture (w e refer to [16] for the relev an t graph theory). Conjecture 1 (Minor-close d em b edding co njecture) . If F is any non-trivial minor-close d family, then sup G ∈F c 1 ( G ) < ∞ . Lo w er b ounds on the m ulti-commodity max-flo w/min-cut ratio in planar graphs. While tec hn iques for p ro ving upp er b ounds on the L 1 -distortion requ ired to em b ed such families has steadily improv ed, p rogress on low er b ounds has b een significan tly slo w er, and recen t breakthroughs in lo wer boun ds f or L 1 em b eddings of discrete met ric spaces that r ely on discrete F ourier analysis [22, 21] do not apply to excluded-min or families. The b est previous lo w er b ound on c 1 ( G ) when G is a planar graph occurred for G = K 2 ,n , i.e. the complete 2 × n b ipartite graph. By a straightfo rw ard generalizatio n of the low er b ound of Ok am ura and Seymour [30], it is p ossible to sho w that c 1 ( K 2 ,n ) → 3 2 as n → ∞ (see also [1] for a simple proof of this fact in the dual setting). W e sho w that, in fact there is an infinite family of series-parallel (and hence, planar) graphs { G n } suc h that lim n →∞ c 1 ( G n ) = 2; this is not only a new lo wer b ou n d for planar graphs , but yields an optimal lo we r boun d on the L 1 -distortion (and hence the flo w/cut gap) for series-parall el graphs. The matc hing upp er b ound was recen tly pro ved in [8]. 1.1 Results and technique s Generalizati ons of classical differentia tion theo ry hav e pla ye d a p r ominen t role in proving the non- existence of bi-Lipsc hitz em b eddings b etw een v arious spaces, when th e target space Z is suffi ciently nice (e.g. if Z is a Banac h space with the Radon-Nikodym prop ert y); see, for instance [31, 9, 24, 5, 12]. But this app roac h does not apply to targets lik e L 1 whic h don’t ha ve the Radon-Nik o dym 2 prop erty; in particular, ev en Lipsc hitz mapp ings f : R → L 1 are not guarant eed to b e differen tiable in t he classica l sense. More recen tly , h o w ever, Cheeger and Kleiner [10, 11] ha v e successfully applied weak er notions of differen tiabilit y to th e study of L 1 em b eddings of the Heisenberg group. Their approac h reduces the non-embedd abilit y problem to understanding the structur e of sets of finite p erimeter, f or which they b uild on the local regularit y theory of [18]. See [24] for the relev ance to inte gralit y gaps for the Sparsest Cut problem. Our lo w er b ounds are also insp ired by d ifferen tiatio n theory . W e use th e c o arse differ entia- tion tec hniqu e of Es kin, Fisher, and Whyte [17], wh ic h giv es a discrete appr oac h to findin g lo cal regularit y in distorted paths. Our basic app roac h is simple; w e kno w that c 1 ( K 2 ,n ) ≤ 3 2 for ev ery n ≥ 1. But consid er s, t ∈ V ( K 2 ,n ) whic h constitute the partition of size 2. Sa y that a cu t S ⊆ V ( K 2 ,n ) is mono tone with r esp e c t to s a nd t if every simp le s - t path in K 2 ,n has at most one edge crossing th e cut ( S, ¯ S ). It is not difficult to sho w that if an L 1 em b edding is comp osed ent irely of cuts whic h are monotone with resp ect to s and t , then that embedd ing must hav e distortion a t lea st 2 − 2 n . Consider no w the r ecursiv ely defined family of grap h s K ⊘ k 2 ,n , where K ⊘ 1 2 ,n = K 2 ,n and K ⊘ k 2 ,n arises by replacing ev ery edge of K ⊘ k − 1 2 ,n with a copy of K 2 ,n . The family { K ⊘ k 2 , 2 } k ≥ 1 are the w ell- kno wn diamond gr aphs of [29, 19]. W e sho w that in an y lo w-d istortion em b edd ing of K ⊘ k 2 ,n in to L 1 , for k ≥ 1 large enough, it is p ossible to find a (metric) cop y of K 2 ,n for whic h the indu ced em b edding is comp osed almost entirely of monotone cuts. The clai med distortion b ound f ollo ws, i.e. lim n,k →∞ c 1 ( K ⊘ k 2 ,n ) = 2. In S ection 5, we exhibit em b eddings wh ic h show that for ev ery fixed n , l im k →∞ c 1 ( K ⊘ k 2 ,n ) < 2, th us it is necessary to ha v e the base graph s g ro w in size. The abilit y to fi nd these monotone copies o f K 2 ,n inside a low-distortio n L 1 em b edding of K ⊘ k 2 ,n arises from t w o s ou r ces. The first is the coa rse differen tiat ion tec hnique men tioned earlier; this is carried out in Section 3. The second asp ect is the relationship b et w een regularit y and monotonicit y for L 1 em b eddings whic h is exp ounded up on in Section 3. 2, a nd r elies o n the w ell-known fact th at ev ery L 1 em b edding d ecomp oses in a certain wa y in to a distribution o ver cuts. W e remark that a s imilar app roac h wa s d isco vered indep en d en tly b y Cheeger and K leiner (in the significan tly more sophistica ted setting of the He isen b erg group). Section 1.8 of [10] describes an alte rnate p r o of of of the non-emb eddabilit y of the Heisenberg group int o L 1 whic h uses metric differen tiation in the sense of [23, 32], and a classificatio n of monotone su bsets of the Heisen b erg group. T his is carried out in full detail in [1 3]. 2 Preliminaries F or a g raph G , we w ill use V ( G ) , E ( G ) to denote the sets of v ertices and edges o f G , respectiv ely . Sometimes w e will equip G with a n on -n egativ e length function len : E ( G ) → R + , and we let d len denote the shortest-path (semi-)metric on G . W e sa y that len is a r e duc e d length if d len ( u, v ) = len ( u, v ) for ev ery ( u, v ) ∈ E ( G ). All length functions consid ered in the p resen t pap er will b e reduced. W e will write d G for the path metric on G if the length fun ction is implicit. F or an intege r n , l et K 2 ,n denote the complete bipartite graph with 2 v ertices on one side, and n on the ot her. 3 Figure 1: A single edge H , H ⊘ K 2 , 3 , and H ⊘ K 2 , 3 ⊘ K 2 , 2 . 2.1 s - t graphs and ⊘ -pro ducts An s - t graph G is a g raph which h as t wo distinguished ve rtices s, t ∈ V ( G ). F or an s - t graph, w e use s ( G ) and t ( G ) to d enote the ve rtices lab ele d s and t , respectiv ely . Throughout this article, t he graphs K 2 ,n are considered s - t grap h s in the natural w a y (the t wo v ertices forming on e side of the partition are lab eled s and t ). Definition 2.1 (Comp osition of s - t graphs) . Given two s - t gr aphs H and G , define H ⊘ G to b e the s - t gr aph obtaine d by r eplacing e ach e dge ( u, v ) ∈ E ( H ) by a c opy of G (se e Figur e 1 ). F ormal ly, • V ( H ⊘ G ) = V ( H ) ∪ ( E ( H ) × V ( G ) \ { s ( G ) , t ( G ) } ) . • F or every e dge e = ( u, v ) ∈ E ( H ) , ther e ar e | E ( G ) | e dges, n ( e, v 1 ) , ( e, v 2 )  | ( v 1 , v 2 ) ∈ E ( G ) and v 1 , v 2 / ∈ { s ( G ) , t ( G ) } o ∪ n u, ( e, w )  | ( s ( G ) , w ) ∈ E ( G ) o ∪ n ( e, w ) , v  | ( w , t ( G )) ∈ E ( G ) o • s ( H ⊘ G ) = s ( H ) and t ( H ⊘ G ) = t ( H ) . If H and G ar e e quipp e d with length functions len H , len G , r esp e ctiv ely, we define len H ⊘ G as fol lows. Using the pr e c e ding notation, for eve ry e dge e = ( u, v ) ∈ E ( H ) , len ( ( e, v 1 ) , ( e, v 2 )) = len H ( e ) d len G ( s ( G ) , t ( G )) len G ( v 1 , v 2 ) len ( u, ( e, w )) = len H ( e ) d len G ( s ( G ) , t ( G )) len G ( s ( G ) , w ) len (( e, w ) , v ) = len H ( e ) d len G ( s ( G ) , t ( G )) len G ( w, t ( G )) . This choic e implies that H ⊘ G c ontains an isometric c opy of ( V ( H ) , d len H ) . Observe that ther e is some ambiguit y in the defin ition ab o ve, as th ere are tw o w a ys to subs titute an ed ge of H with a co p y of G , thus we assume t hat ther e exists some arbitrary orien tation of the edges of H . Ho wev er, for our pu rp oses the graph G will b e symmetric, and thus the orienta tions are irrelev ant. Definition 2.2 (Recursiv e comp osition) . F or an s - t gr aph G and a numb er k ∈ N , we define G ⊘ k inductively by letting G ⊘ 0 b e a si ng le e dge of unit length, and setting G ⊘ k = G ⊘ k − 1 ⊘ G . The foll o w ing result is straigh tforw ard. 4 Lemma 2.3 (Associativit y of ⊘ ) . F or any thr e e gr aphs A, B , C , we have ( A ⊘ B ) ⊘ C = A ⊘ ( B ⊘ C ) , b oth gr aph-the or etic al ly and as metric sp ac es. Definition 2.4. F or two g r aphs G , H , a subset of vertic es X ⊆ V ( H ) is said to b e a c opy of G if ther e exists a bije ction f : V ( G ) → X with distortion 1, i.e. d H ( f ( u ) , f ( v )) = C · d G ( u, v ) for som e c onstant C > 0 . No w we mak e the follo wing t wo simple observ ations ab out c opies of H and G in H ⊘ G . Observ ation 2.5. The gr aph H ⊘ G c onta ins | E ( H ) | distinguishe d c opies of the gr aph G , one c opy c orr esp onding to e ach e dge in H . Observ ation 2.6. The subset of vertic es V ( H ) ⊆ V ( H ⊘ G ) form an isometric c opy of H . F or any graph G , we can wr ite G ⊘ N = G ⊘ k − 1 ⊘ G ⊘ G N − k . By obser v ation 2.5, there are | E ( G ⊘ k − 1 ) | = | E ( G ) | k − 1 copies of G in G ⊘ k − 1 ⊘ G . No w u sing observ ation 2.6, we obtain | E ( G ) | k − 1 copies of G in G ⊘ N . W e refer to th ese as the level- k c opies of G , and th eir vertices as level- k vertic es. In the case of K ⊘ N 2 ,n , we will use a compact n otation to refer to the copies of K 2 ,n . F or t w o lev el- k v ertice s x, y ∈ V ( K N 2 ,n ), we will use K ( x,y ) 2 ,n to denote the cop y of K 2 ,n for which x and y are the s - t p oin ts. Note that suc h a cop y does not exist b et w een all pairs of lev el- k vertice s. 2.2 Cuts and L 1 em b eddings Cuts. A cut of a graph is a partition of V in to ( S, ¯ S )—we somet imes refer to a subset S ⊆ V as a cut as wel l. A cut giv es rise to a semi-metric; usin g ind icator functions, we can wr ite the cut semi-metric as ρ S ( x, y ) = | 1 S ( x ) − 1 S ( y ) | . A fact central to ou r pro of is that embed d ings of fi nite metric sp aces into L 1 are equiv alen t to sums of p ositiv ely w eigh ted cut metrics o v er that set (for a simple proof of this see [1 5]). A cut me asur e on G is a function µ : 2 V → R + for whic h µ ( S ) = µ ( ¯ S ) for eve ry S ⊆ V . Eve ry cut measure giv es rise to an embedd ing f : V → L 1 for whic h k f ( u ) − f ( v ) k 1 = Z | 1 S ( u ) − 1 S ( v ) | dµ ( S ) , (1) where the int egral is o v er all cuts ( S, ¯ S ). Con ve rsely , to ev ery em b eddin g f : V → L 1 , w e can asso ciate a cut m easure µ suc h that (1) h olds . W e will use this corresp ondence freely in what follo ws. When V is a finite set (as it will b e throughout), for A ⊆ 2 V , we d efine µ ( A ) = P S ∈ A µ ( S ). Em b eddings and dist ort ion. If ( X , d X ) , ( Y , d Y ) are metric spaces, and f : X → Y , then w e write k f k Lip = su p x 6 = y ∈ X d Y ( f ( x ) , f ( y )) d X ( x, y ) . If f is injectiv e, then the distortion of f is defined b y di st ( f ) = k f k Lip · k f − 1 k Lip . A map with distortion D will sometimes b e referred to as D -bi-lipschitz. If d Y ( f ( x ) , f ( y )) ≤ d X ( x, y ) for ev er y x, y ∈ X , w e say t hat f is non-exp ansive. F or a met ric space X , w e use c 1 ( X ) to denote the least distortion required to em b ed X int o L 1 . 5 3 Coarse differen tiation In the present sect ion, w e stud y the regularit y of paths un der bi-lipsc hitz mappings in to L 1 . Ou r main to ol is based o n differen tiation [17]. First, we need a discrete analog o f b ounded v ariati on. Definition 3.1. A se quenc e { x 1 , x 2 , . . . , x k } ⊆ X in a metric sp ac e ( X , d ) is said to ǫ -efficient if d ( x 1 , x k ) ≤ k − 1 X i =1 d ( x i , x i +1 ) ≤ (1 + ǫ ) d ( x 1 , x k ) Of co urse the left inequalit y follo ws trivially f rom the triangle inequalit y . Definition 3.2. A function f : Y → X b etwe en two metric sp ac es ( X , d ) and ( Y , d ′ ) , is said to b e ǫ -efficien t on P = { y 1 , y 2 , . . . , y k } ⊆ Y if the se que nc e f ( P ) = { f ( y 1 ) , f ( y 2 ) , . . . f ( y k ) } is ǫ -effici ent in X . F or the s ak e of simp licit y , we fir st p resen t the coarse differen tiation argumen t for a function on [0 , 1]. Let f : [0 , 1] → X b e a n on-expansiv e map in to a metric s p ace ( X , d ). Let M ∈ N b e giv en, and for eac h k ∈ N , let L k = { j M − k } M k j =0 ⊆ [0 , 1] b e the set of level- k p oints , and let S k =  ( j M − k , ( j + 1) M − k ) : j ∈ { 1 , . . . , M k − 1 }  b e the set of level- k p airs . F or an int erv al I = [ a, b ], f | I denotes the restriction of f to the interv al I . No w w e s ay that f | I is ε -e fficient at gr anularity M if M − 1 X j =0 d  f  a + ( b − a ) j M  , f  a + ( b − a )( j + 1) M  ≤ (1 + ε ) d ( f ( a ) , f ( b )) . F urther, w e sa y t hat a fun ction f is ( ε, δ ) -inefficient at level k if    ( a, b ) ∈ S k : f | [ a,b ] is not ε -efficien t at granularit y M    ≥ δ M k . In other w ords, the probabilit y that a randomly c hosen lev el k restricti on f | [ a,b ] is n ot ε -efficien t is at least δ . Oth erwise, w e say th at f is ( ε , δ ) -e fficient at level k . The main theorem o f this sectio n follo ws. Theorem 3.3 (Coarse differen tiatio n) . If a non-exp ansive map f : [0 , 1] → X is ( ε, δ ) -ine ffic i ent at an α -fr action of levels k = 1 , 2 , . . . , N , then dist ( f | L N +1 ) ≥ 1 2 εαδ N . Pr o of. Let D = dist ( f | L N +1 ), a nd let 1 ≤ k 1 < · · · < k h ≤ N b e the h ≥ ⌊ αN ⌋ lev els at whic h f is ( ε, δ )-inefficien t. Let us consider the fi rst leve l k 1 . Let S ′ k 1 ⊆ S k 1 b e a subset of size | S ′ k 1 | ≥ ⌊ δ | S k 1 |⌋ for whic h ( a, b ) ∈ S ′ k 1 = ⇒ f | [ a,b ] is n ot ε - efficien t at granularit y M F or an y suc h ( a, b ) ∈ S ′ k 1 , w e kno w that M − 1 X j =0 d  f  a + j M − k 1 − 1  , f  a + ( j + 1) M − k 1 − 1  > (1 + ε ) d ( f ( a ) , f ( b )) ≥ d ( f ( a ) , f ( b )) + ε M − k 1 D . 6 b y the defin ition of (not b eing) ε -efficien t, and the fact that d ( f ( a ) , f ( b )) ≥ | a − b | /D . F or all segmen ts ( a, b ) ∈ S k 1 \ S ′ k 1 , the triangle inequalit y yields M − 1 X j =0 d  f  a + j M − k 1 − 1  , f  a + ( j + 1) M − k 1 − 1  ≥ d ( f ( a ) , f ( b )) By summing the ab o v e inequalities o ve r all the segmen ts in S k 1 , w e get X ( u,v ) ∈ S k 1 +1 d ( f ( u ) , f ( v )) ≥ X ( a,b ) ∈ S k 1 d ( f ( a ) , f ( b )) + εδ 2 D , where the e xtra fact or 2 in the denominator on t he RHS just comes from remo ving the flo or f rom | S ′ k 1 | ≥ ⌊ δ | S k 1 |⌋ . S imilarly , for eac h of the lev els k 2 , . . . , k h , we will pick up an exce ss term of εδ / (2 D ). W e conclude th at 1 ≥ X ( u,v ) ∈ S N +1 d ( f ( u ) , f ( v )) ≥ εδ h 2 D , where the LHS comes from the fact that f is non-expansiv e. Simplifying ac h iev es the desired conclusion. 3.1 Differen t iation for famili es of geo desics Let G = ( V , E ) b e an unw eigh ted graph, and let P denote a family of geo desics (i.e. sh ortest-paths) in G . F ur thermore, assum e th at eve ry γ ∈ P has length M r for some M , r ∈ N . Let f : ( V , d G ) → X b e a non-expansive map in to some metric sp ace ( X, d ). F or the sake of con v enience, we w ill index the vertices along the paths us in g num b ers from [0 , 1]. Sp ecifically , we will refer to the i th v ertex along the path γ ∈ P b y γ  i M r  . F or ind ices a, b , w e will use γ [ a, b ] to denote th e sub path starting at γ ( a ) and endin g at γ ( b ). W e w ill also use f | γ [ a,b ] to denote the restriction of f to the path γ [ a, b ]. As earlier, th e fu nction f | γ [ a,b ] is said to b e ǫ -efficient at g ran ularit y M if M − 1 X j =0 d  f  γ  a + M − 1 ( b − a ) j   , f  γ  a + M − 1 ( b − a )( j + 1)   ≤ (1 + ε ) d ( f ( a ) , f ( b )) . Let the sets L k and S k b e defin ed as b efore. T h us a lev el- k segmen t of a path γ ∈ P is γ [ a, b ] for some ( a, b ) ∈ S k . W e sa y that f is ( ǫ, δ ) inefficie nt at level k for the family of p aths P if the follo win g holds:    ( a, b ) ∈ S k , γ ∈ P : f | γ [ a,b ] is n ot ε - efficien t at gran ularit y M    ≥ δ M k |P | . A straig h tforw ard v ariation of the p ro of of Theorem 3.3 yields the f ollo wing. Theorem 3.4. If a non-exp ansive map f : V → X is ( ε, δ ) -i nefficient at an α - fr action of levels k = 1 , 2 , . . . , N , then d ist ( f ) ≥ 1 2 εαδ N . 7 Pr o of. Let D = dist ( f ), and let 1 ≤ k 1 < · · · < k h ≤ N b e the h ≥ ⌊ αN ⌋ lev els for whic h f is ( ε, δ )-inefficien t at lev el k i . Let us consider the first lev el k 1 . Let S ′ k 1 ⊆ P × S k 1 b e a subset of size | S ′ k 1 | ≥ ⌊ δ | S k 1 ||P |⌋ for whic h  γ , ( a, b )  ∈ S ′ k 1 = ⇒ f | γ [ a,b ] is not ε -efficien t at granularit y M . F or an y suc h  γ , ( a, b )  ∈ S ′ k 1 , w e kno w that M − 1 X j =0 d  f  γ ( a + j M − k 1 − 1 )  , f  γ ( a + ( j + 1) M − k 1 − 1 )  > (1 + ε ) d ( f ( γ ( a )) , f ( γ ( b ))) ≥ d ( f ( γ ( a )) , f ( γ ( b ))) + ε M N − k 1 D . b y the definition of (not b eing) ε -efficien t, and the fact that d ( f ( γ ( a )) , f ( γ ( b ))) ≥ M N | a − b | /D . In particular, summ ing b oth sid es o v er all the s egmen ts γ [ a, b ] o v er all paths γ and segments [ a, b ] ∈ S k 1 (and replac ing the pr eceding inequ alit y by the triangle inequality if ( a, b ) / ∈ S ′ k 1 ), w e get X γ ∈P X ( u,v ) ∈ S k 1 +1 d ( f ( γ ( u )) , f ( γ ( v ))) ≥ X γ ∈P X ( a,b ) ∈ S k 1 d ( f ( γ ( a )) , f ( γ ( b ))) + εδ M N |P | 2 D , Similarly , for eac h of th e lev els k 2 , . . . , k h , we will pic k up an excess term of εδM N |P | / (2 D ). W e conclude that M N |P | ≥ X γ ∈P X ( u,v ) ∈ S N +1 d ( f ( γ ( u )) , f ( γ ( v ))) ≥ εδ hM N |P | 2 D , The desired conclusion follo ws. 3.2 Efficien t L 1 -v alued maps and monotone cuts Finally , w e relate monotonicit y of L 1 -v alued mappings to prop erties o f their cut decomp ositions. Definition 3.5. A se qu enc e P = { x 1 , x 2 , . . . , x k } ⊆ X is said to b e monoto ne with resp ect to a cu t ( S, S ) (wher e X = S ⊎ ¯ S ) if S ∩ P = { x 1 , x 2 , . . . , x i } or ¯ S ∩ P = { x 1 , x 2 , . . . , x i } for some 1 ≤ i ≤ k . If µ is a cut measure on a finite set X and x, y ∈ X , we define the sep ar ation me asur e µ x | y as follo ws: F or ev ery S ⊆ X , let µ x | y ( S ) = µ ( S ) | 1 S ( x ) − 1 S ( y ) | . Lemma 3.6. L et ( X , d ) b e a finite metric sp ac e, and let P = { x 1 , x 2 , . . . , x k } ⊆ X b e a finite se quenc e. Given a mapping f : X → L 1 , let µ b e the c orr e sp onding cut me asur e (se e (1) ). If f is ǫ -efficient on P , then µ x 1 | x k  S : P is mon otone with r esp e ct to ( S, ¯ S )  ≥ (1 − ǫ ) k f ( x 1 ) − f ( x k ) k 1 . Pr o of. If th e sequen ce P is not monotone with resp ect to a cut ( S, S ), then k − 1 X i =1 | 1 S ( x i ) − 1 S ( x i +1 ) | ≥ 2 | 1 S ( x 1 ) − 1 S ( x k ) | . 8 No w , let E = { S : P is not monotone with r esp ect t o ( S, ¯ S ) } , and f or the sak e of con tr ad iction, assume t hat µ x 1 | x k ( E ) > ε k f ( x 1 ) − f ( x k ) k 1 , then k − 1 X i =1 k f ( x i ) − f ( x i +1 ) k 1 = k − 1 X i =1  Z E | 1 S ( x i ) − 1 S ( x i +1 ) | dµ ( S ) + Z ¯ E | 1 S ( x i ) − 1 S ( x i +1 ) | dµ ( S )  ≥ 2 Z E | 1 S ( x 1 ) − 1 S ( x k ) | dµ ( S ) + Z ¯ E | 1 S ( x 1 ) − 1 S ( x k ) | dµ ( S ) = 2 µ x 1 | x k ( E ) + µ x 1 | x k ( ¯ E ) > (1 + ǫ ) k f ( x 1 ) − f ( x k ) k 1 , where we observ e that k f ( x 1 ) − f ( x k ) k 1 = µ x 1 | x k ( E ) + µ x 1 | x k ( ¯ E ) . This is a con tradiction, since f is assumed to b e ǫ -efficien t on P . 4 The distortion lo w er b ound Our lo wer b ound examples a re the recursiv ely defined family of graphs { K ⊘ k 2 ,n } ∞ k =1 . W e recall that the graphs K ⊘ k 2 , 2 are kno wn as diamond graphs [19, 29]. Lemma 4.1. L et G b e an s - t gr aph with a uniform length f u nction, i.e. len ( e ) = 1 for every e ∈ E ( G ) . Then f or every ǫ, D > 0 , ther e exists an inte ger N = N ( G, ε, D ) such that the fol lowing holds: F or any non-exp ansive map f : G ⊘ N → X with dis t ( f ) ≤ D , ther e exi sts a c opy G ′ of G in G ⊘ N such that f is ǫ - efficient on al l s - t ge o desics in G ′ . Pr o of. Let M = d G ( s, t ), an d let P G denote the family of s - t ge o desics in G . Fix δ = 1 |P G | , α = 1 2 and N = 8 D ǫδ . Let P d enote the f amily of all s - t geo desics in G ⊘ N . Eac h path in P is of length M and consists of M N edges. F r om th e c h oice of parameters, observ e that 1 2 ǫαδ N > D . Applying Th eorem 3.4 to the family P , an y n on-expansiv e map f with dist ( f ) ≤ D is ( ǫ, δ )-efficien t at an α = 1 2 -fraction of lev els k = 1 , 2 , . . . N . Sp ecifically , there exists a le v el k suc h that f is ( ǫ, δ )-efficien t at lev el k . F or a u niformly ran d om choice of path γ ∈ P , and lev el- k segmen t ( a, b ) of γ , f | γ [ a,b ] is not ǫ -efficien t at granularit y M with p r obabilit y at m ost δ . In c ase of th e family P , eac h of the lev el- k segmen ts is nothing more than an s - t geodesic in a lev el- k cop y of G . If, for at least one of the level- k copies of G , f is ǫ -efficient on all the s - t geo desics in that copy , the p r o of is complete. On the con trary , supp ose eac h level - k cop y has an s - t geo desic on which f is not ǫ -efficien t. Then in eac h lev el- k cop y at least a δ = 1 P G -fraction of the s - t geo desics are ǫ -inefficien t. As the lev el- k copies partition the set of all lev el- k seg men ts, this implies that at least a δ -fr action of the segmen ts are ǫ -inefficien t. T his con tradicts the fact th at f is ( ǫ, δ )-efficien t at lev el k . Although we will not need it , the same t y p e of argument pro v es the follo win g ge neralization to w eigh ted graphs G . The idea is th at in G ⊘ N for N large enough, there exists a cop y of a sub division of G with eac h edge finitely sub d ivid ed. P a ying small distortion, we can app ro ximate G (up to uniform sca ling) b y this sub d ivided cop y , where the latter is equipp ed with uniform edge lengths. Lemma 4.2. L et G b e an s - t gr aph with with arbitr ary non-ne gative e dge lengths len : E ( G ) → R + . Then for every ǫ, D > 0 , ther e exists an inte ger N = N ( G, ε, D , len ) such that the fol lowing holds: 9 F or any non-exp ansive map f : G ⊘ N → X with dist ( f ) ≤ D , ther e e xists a c opy G ′ of G in G ⊘ N such that f is ǫ - efficient on al l s - t ge o desics in G ′ . In the graph K 2 ,n , w e will refer to the n v ertices other than s, t b y M = { m i } n i =1 . Lemma 4.3. F or ǫ < 1 2 and any function f : V ( K 2 ,n ) → L 1 that is ǫ/n -efficient with r esp e ct to e ach of the ge o desics s - m i - t , for 1 ≤ i ≤ n , we have dist ( f ) ≥ 2 − 2 n − 2 ǫ . Pr o of. Let µ b e the cut measure corresp onding to f . By scaling, w e ma y a ssume that k f ( s ) − f ( t ) k 1 = µ { S : 1 S ( s ) 6 = 1 S ( t ) } = 1 . Let V = V ( K 2 ,n ). Without loss of generalit y , w e assume that µ is supp orted on 2 V \ {∅ , V } . Let γ i b e th e geo desic s - m i - t i for i ∈ { 1 , 2 , . . . , n } . Define E =  S : ( S, ¯ S ) is not monot one with r esp ect to γ i for so me i ∈ [ n ]  . Applying Lemma 3.6 , by a union b ound and the fact that f is ǫ/n efficien t on eve ry γ i , w e see that µ ( E ) ≤ ǫ . Consider a cut ( S, S ) that is mon otone with resp ect to all the γ i geod esics, an d suc h that µ ( S ) > 0. Let us refer to these cuts as go o d cuts. By monotonicit y , and the fact that S / ∈ { 0 , V } , w e k n o w that | 1 S ( s ) − 1 S ( t ) | = 1. Th u s for a go o d c ut ( S, S ), we hav e X i,j ∈ [ n ] | 1 S ( m i ) − 1 S ( m j ) | = 2( | S | − 1)( n − | S | − 1) ≤ n 2 2 . (2) It follo ws that, X i,j ∈ [ n ] k f ( m i ) − f ( m j ) k 1 = Z E X i,j ∈ [ n ] | 1 S ( m i ) − 1 S ( m j ) | dµ ( S ) + Z ¯ E X i,j ∈ [ n ] | 1 S ( m i ) − 1 S ( m j ) | dµ ( S ) ≤ µ ( ¯ E ) n 2 2 + µ ( E ) n 2 ≤ (1 − ǫ ) n 2 2 + ǫ n 2 = (1 + ǫ ) n 2 2 k f ( s ) − f ( t ) k 1 . where in the first inequalit y , we ha v e used (2), and w e r ecall that k f ( s ) − f ( t ) k 1 = 1. Con trasting th is with the fact th at X i,j ∈ [ n ] d K 2 ,n ( m i , m j ) = n ( n − 1) d K 2 ,n ( s, t ) yields dist ( f ) ≥ n ( n − 1) (1+ ǫ ) n 2 2 = 2 1 + ǫ  1 − 1 n  ≥ 2 − 2 n − 2 ǫ. 10 Theorem 4.4. F or any n ≥ 2 , lim k →∞ c 1 ( K ⊘ k 2 ,n ) ≥ 2 − 2 n . Pr o of. F or an y ǫ ′ > 0, let N b e the in teger obtained by applying Lemma 4.1 to K 2 ,n with ǫ = ǫ ′ /n, D = 2 and G = K 2 ,n . W e will show that for an y map f : K ⊘ N 2 ,n → L 1 , dist ( f ) ≥ 2 − 2 n − 2 ǫ ′ . Without loss of generalit y , assume that f is non-expansiv e. If dist ( f ) ≤ 2, then from Lemma 4.1 there exists a co p y of K 2 ,n in whic h f is ǫ ′ n on all the s - t geo desics. Using Lemm a 4. 3, w e see that on this cop y o f K 2 ,n w e get dist ( f | K 2 ,n ) ≥ 2 − 2 n − 2 ǫ ′ . The result follo ws b y taking ǫ ′ → 0. 5 Em b eddings of K ⊘ k 2 ,n In this section, w e sh o w that for ev ery fixed n , lim k →∞ c 1 ( K ⊘ k 2 ,n ) < 2. A next-em b edding op erator. Let T b e a r andom v ariable ranging o ver subsets of V ( K ⊘ k 2 ,n ), and let S b e a random v ariable ranging o v er subsets of V ( K 2 ,n ). W e define a random subset P S ( T ) ⊆ V ( K ⊘ k +1 2 ,n ) as follo ws. One mov es from K ⊘ k 2 ,n to K ⊘ k +1 2 ,n b y replacing ev ery edge ( x, y ) ∈ E ( K ⊘ k 2 ,n ) with a cop y of K 2 ,n whic h we w ill call K ( x,y ) 2 ,n . F or ev ery edge ( x, y ) ∈ K ⊘ k 2 ,n , let S ( x,y ) b e an in d ep end ent cop y of the cut S (whic h r an ges o ver subsets of V ( K 2 ,n )). W e form the cut P S ( T ) ⊆ V ( K ⊘ k +1 2 ,n ) as follo ws. If ( x, y ) ∈ E ( K ⊘ k 2 ,n ), then for v ∈ V ( K ( x,y ) 2 ,n ), w e put 1 P S ( T ) ( v ) =    1 P S ( T )  s ( K ( x,y ) 2 ,n )  if 1 S ( x,y ) ( v ) = 1 S ( x,y )  s ( K ( x,y ) 2 ,n )  1 P S ( T )  t ( K ( x,y ) 2 ,n )  otherwise W e note that, s trictly s p eaking, the op erator P S dep end s on n and k , but w e allo w these to b e implicit paramete rs. 5.1 Em b eddings for small n Consider the graph K 2 ,n with v ertex set V = { s, t } ∪ M . An em b edding in the st yle o f [19] would define a random su bset S ⊆ V by selecting M ′ ⊆ M to co n tain eac h v ertex from M indep enden tly with p robabilit y 1 2 , and then setting S = { s } ∪ M ′ . The resu lting embedd ing has distortion 2 since, for ev ery pair x, y ∈ M , w e ha v e Pr[ 1 S ( x ) 6 = 1 S ( y )] = 1 2 . T o do sligh tly b etter, w e c ho ose a uniformly rand om subset M ′ ⊆ M of size ⌊ n 2 ⌋ and set S = { s } ∪ M ′ or S = { s } ∪ ( M \ M ′ ) eac h with probabilit y half. In this case, we hav e Pr[ 1 S ( x ) 6 = 1 S ( y )] = ⌊ n 2 ⌋ · ⌊ n +1 2 ⌋  n 2  > 1 2 , resulting in a distortion slight ly b etter than 2. A recursive applicatio n of these id eas results in lim k →∞ c 1 ( K ⊘ k 2 ,n ) < 2 for every n ≥ 1, though the calculation is complicated by the fact that th e w orst distortion is incur red for a pair { x, y } with x ∈ M ( H ) and y ∈ M ( G ) where H is a copy of K ⊘ k 1 2 ,n and G is a copy of K ⊘ k 2 2 ,n , and the relationship b etw een k 1 and k 2 dep end s on n . (F or instance, c 1 ( K 2 , 2 ) = 1 w hile lim k →∞ ( K ⊘ k 2 , 2 ) = 4 3 .) Theorem 5.1. F or any n, k ∈ N , we have c 1 ( K ⊘ k 2 ,n ) ≤ 2 − 2 2 ⌈ n 2 ⌉ +1 . 11 s t x y u v (a) Case I s t x y u v (b) Ca se I I Figure 2: The t wo cases of Th eorem 5.1 Pr o of. F or simplicit y , we pro v e the b ound for K ⊘ k 2 , 2 n . A similar analysis h olds f or K ⊘ k 2 , 2 n +1 . W e define a random cut S k ⊆ V ( K ⊘ k 2 , 2 n ) ind uctiv ely . F or k = 1 , c ho ose a uniformly r an d om partition M ( K ⊘ 1 2 , 2 n ) = M s ∪ M t with | M s | = | M t | = n , and le t S 1 = { s ( K ⊘ 1 2 , 2 n ) } ∪ { M s } . Th e k ey fact which causes the distortion to b e less than 2 is the follo w ing: F or an y x, y ∈ M ( K ⊘ 1 2 , 2 n ), w e ha v e Pr[ 1 S 1 ( x ) 6 = 1 S 1 ( y )] = n 2  2 n 2  = n 2 n − 1 > 1 2 . (3) This follo ws b ecause th ere are  2 n 2  pairs { x, y } ∈ M ( K ⊘ 1 2 , 2 n ) and n 2 are separate d b y S 1 . Assume now that w e ha v e a random sub set S k ⊆ V ( K ⊘ k 2 , 2 n ). W e set S k +1 = P S 1 ( S k ) where P S 1 is th e op erator defined ab ov e, which maps random subsets of V ( K ⊘ k 2 , 2 n ) to random sub sets of V ( K ⊘ k +1 2 , 2 n ). I n ot her w ords S k = P k − 1 S 1 ( S 1 ). Let s 0 = s ( K ⊘ k 2 , 2 n ) and t 0 = t ( K ⊘ k 2 , 2 n ). It is easy to see that the cut S = S k defined ab o v e is alw a ys monotone with resp ect to every s 0 - t 0 shortest p ath in K ⊘ k 2 , 2 n , thus ev ery such p ath has exactly one edge cut b y S k , and f urthermore the cut edge is uniformly c hosen from along the path, i.e. Pr[ 1 S ( x ) 6 = 1 S ( y )] = 2 − k for every ( x, y ) ∈ E ( K ⊘ k 2 , 2 n ). In particular, it follo ws that if u, v ∈ V ( K ⊘ k 2 , 2 n ) lie along the same simple s 0 - t 0 path, th en Pr[ 1 S ( u ) 6 = 1 S ( v )] = 2 − k d ( u, v ). No w consid er any u, v ∈ V ( K ⊘ k 2 , 2 n ). F ix some s h ortest path P from u to v . By symmetry , w e ma y assume that P go es left (to w ard s 0 ) and then righ t (to w ard t 0 ). Let s b e the le ft-most p oint of P . In this case, s = s ( H ) for some subgraph H wh ic h is a cop y of K ⊘ k ′ 2 , 2 n with k ′ ≤ k , and suc h that u, v ∈ V ( H ); w e let t = t ( H ). W e also hav e d ( u, v ) = d ( u, s ) + d ( s, v ). Let M = M ( H ), and fix x, y ∈ M wh ic h lie along the s - u - t and s - v - t shortest-paths, resp ectiv ely . Without loss of generalit y , w e ma y a ssume that d ( s, v ) ≤ d ( s, y ). W e need to consider tw o cases (see Figure 5.1). Case I: d ( u, s ) ≤ d ( x, s ). F or an y p air a, b ∈ V ( K ⊘ k 2 , 2 n ), w e let E a,b b e the eve n t { 1 S ( a ) 6 = 1 S ( b ) } . In this case, w e hav e Pr[ E u,v ] = Pr [ E s,t ] · Pr[ E u,v | E s,t ] . Since s, t clearly lie on a shortest s 0 - t 0 path, we h a ve Pr[ E s,t ] = 2 − k d ( s, t ). F or an y ev en t E , w e le t µ [ E ] = Pr[ E | E s,t ]. No w w e calculate using (3), µ [ E u,v ] ≥ µ [ E x,y ] ( µ [ E x,s | E x,y ] µ [ E u,s | E x,s , E x,y ] + µ [ E x,t | E x,y ] µ [ E v,s | E x,t , E x,y ]) = n 2 n − 1  1 2 · d ( u, s ) d ( x, s ) + 1 2 · d ( v , s ) d ( y , s )  = n 2 n − 1 d ( u, v ) d ( s, t ) . 12 Hence i n this case, Pr[ 1 S ( u ) 6 = 1 S ( v )] ≥ n 2 n − 1 · 2 − k d ( u, v ). Case I I: d ( u, s ) ≥ d ( x, s ). Here, we need to b e more careful ab out b ound in g µ [ E u,v ]. I t will b e helpf ul to in tro duce the notation a 7→ b to r ep resen t the ev en t { 1 S ( a ) = 1 S ( b ) } . W e hav e, µ [ E u,v ] = µ [ x 7→ t, y 7→ s ] + µ [ x 7→ t, y 7→ t, v 7→ s ] + µ [ x 7→ s , y 7→ s , u 7→ t ] + µ [ x 7→ s, y 7→ t, u 7→ t, v 7→ s ] + µ [ x 7→ s, y 7→ t, u 7→ s, v 7→ t ] = 1 2 n 2 n − 1 + n − 1 2 n − 1 d ( v , y ) + d ( u, x ) d ( s, t ) + 1 2 n 2 n − 1  d ( u, x ) d ( v , y ) + d ( u, t ) d ( v , s ) d ( x, t ) d ( y , s )  If w e set A = d ( v,s ) d ( s,t ) and B = d ( u,x ) d ( s,t ) , then d ( u,v ) d ( s,t ) = 1 2 + A + B and simplifying the expression ab o ve, w e h a v e µ [ E u,v ] = 1 2 + B + A 2 n − 1 − 4 n 2 n − 1 AB Since the shortest path from u to v go es thr ou gh s b y assumption, w e m ust ha v e A + B ≤ 1 2 . Thus w e are in terested in the minimum of µ [ E u,v ] / ( 1 2 + A + B ) su b ject to the constraint A + B ≤ 1 2 . It is easy to see that th e minim um is achiev ed at A + B = 1 2 , thus setting B = 1 2 − A , w e are left to find min 0 ≤ A ≤ 1 2  1 − 2 A + 4 nA 2 2 n − 1  = 2 n + 1 4 n . (The minim um o ccurs at A = 1 2 − 1 4 n .) S o in this c ase, Pr[ 1 S ( u ) 6 = 1 S ( v )] ≥ 2 n +1 4 n 2 − k d ( u, v ). Com bining the ab o ve tw o cases, w e conclude that the distribution S = S k induces an L 1 em b edding of K ⊘ k 2 , 2 n with distortion at most m ax { 2 n − 1 n , 4 n 2 n +1 } = 2 − 2 2 n +1 . A similar calculation yields c 1 ( K ⊘ k 2 , 2 n +1 ) ≤ min 0 ≤ A ≤ 1 2  1 − 2 A + 4( n + 1) A 2 2 n + 1  ! − 1 = 2 − 2 2 n + 3 . Ac kno w ledgmen ts W e thank Alex Eskin for exp laining coarse differen tiat ion to us, Y uri Rabino vich for relating the distortion 2 conjecture, and K ost ya and Y ury Mak a yr c hev for a n um b er of fruitful discussions. References [1] A. And oni, M. Deza, A. Gup ta, P . Indyk, and S. Raskho dn ik o v a. Lo w er b ounds for em b edding of ed it distance into normed spaces. In Pr o c e e dings of the 14th annual ACM-SIAM Symp osium on Discr ete A lgorithms , 2003. [2] S . Arora, J. R. L ee, and A. Nao r. Euclidean d istortion and th e Sparsest Cu t. J. Amer. Math. So c. , 21( 1):1–2 1, 200 8. 13 [3] S . Arora, S. Rao, and U. V azirani. Expand er flows, geometric em b eddings, and graph p arti- tionings. In 36th Annual Symp osium on the The ory of Computing , pages 222–23 1, 2004. T o app ear, J. A CM . [4] Y. Aumann and Y. Rabani. An O (log k ) appro ximate min-cut max-flo w theorem and ap p ro x- imation a lgorithm. SIAM J. Comput. , 27( 1):291 –301 (electronic), 1998. [5] Y. Ben y amini and J . Lindenstraus s. Ge ometric nonline ar functional analysis. Vol. 1 , volume 48 of A meric an Mathematic al So ciety Col lo qu i um P ublic ations . American Ma thematical Societ y , Pro vidence, RI , 20 00. [6] J . Bourgain. O n Lipsc hitz emb edding of fin ite metric spaces in Hilb ert sp ace. Isr ael J. Math. , 52(1-2 ):46–5 2, 1985. [7] B. Brinkman , A. Karagiozo v a, and J. R. Lee. V ertex cuts, rand om wa lks, and dimension reduction in series-parallel graphs. In 39th Annual Symp osium on the The ory of Computing , pages 6 21–63 0, 2007. [8] A. Chakrabarti, A. J affe, J. R. Lee, and J. Vincen t. Embed d ings, flo ws, and cuts in 2 -sums of graphs. Submitted, 2008. [9] J . Cheeger. Differen tiabilit y of Lipsc h itz functions on metric measure spaces. Ge om. F unct. Ana l. , 9(3):428–5 17, 1999 . [10] J. Cheeger and B. Kleiner. Diffe ren tiating maps in to L 1 and the geometry of BV functions. arXiv:math.MG/06 11954 , 2006. [11] J. Cheeger and B. Kleiner. Generalized differenti al and bi-Lipsc hitz nonem b edding in L 1 . C. R. Math. A c ad. Sci. Paris , 3 43(5):2 97–30 1, 2006. [12] J. Cheeger and B. Kleiner. On the differentia bilit y of Lipsc hitz maps from metric measur e spaces to Banac h space s. In Inspir e d by S. S. Chern , vol ume 11 of Nankai T r acts Math. , pages 129–1 52. W orld Sci. Pu b l., Hac k ensac k, NJ, 2006. [13] J. Cheeger and B. Kleiner. Metric d ifferen tiatio n, monotonicit y and maps to L 1 . arXiv:0907 .3295, 2009. [14] C. Chekuri, A. Gu pta, I. Ne wman, Y. Rabino vic h, and A. Sinclair. Em b edding k -outerp lanar graphs in to l 1 . SIAM J. Discr ete Math. , 20(1): 119–1 36, 2006 . [15] M. M. Deza and M. Laurent. Ge ometry of cuts and metrics , volume 15 of A lgorithm s and Combinatorics . S pringer-V erlag, Berlin, 1 997. [16] R. Diestel. Gr aph the ory , v olume 173 of Gr aduate T exts in Mathematics . Sp ringer-V erlag, Berlin, third edition, 2005. [17] A. Es kin, D. Fisher, and K. Wh yte. Quasi-isometries and r igidity of solv able groups. Preprint, 2006. [18] B. F ranc hi, R . Serapioni, and F. Serra Cassano. Rectifiabilit y and p erimeter in the Heisenberg group. M ath. Ann. , 321(3): 479–5 31, 2001 . 14 [19] A. Gu pta, I. Newman, Y. Rab in o vic h, and A. Sinclair. Cu ts, trees and l 1 -em b eddings of graphs. Com binatoric a , 24(2): 233–2 69, 2004 . [20] P . Indyk. Algorithmic applications of low-distortio n geometric em b eddings. In 42nd Ann ual Symp osium on F oundatio ns of Computer Scienc e , pages 10–33. IEEE Computer So ciet y , 2001. [21] S. Khot and A. Naor. Nonem b eddabilit y theorems via Four ier analysis. Math. Ann. , 334(4 ):821– 852, 200 6. [22] S. Khot and N. Vishnoi. Th e uniqu e games conjecture, in tegral it y gap for cut problems and em b eddabilit y of negativ e t yp e m etrics in to ℓ 1 . In 46th Annual Symp osium on F oundations of Computer Scienc e , pages 53–62. IEEE Computer So c., Los Alamitos, CA, 20 05. [23] B. Kirc hheim. Rectifiable metric spaces: lo cal structure a nd regularit y of the Ha usdorff mea- sure. Pr o c . Amer. M ath. So c. , 121(1) :113–1 23, 1994 . [24] J. R. Lee and A. Naor. L p metrics on the Heisen b erg group and the Go emans-Lin ial conjecture. In 47th Annual Symp osium on F oundations of Computer Scienc e . IEEE Computer So c., Los Alamitos, CA, 2006. [25] N. Linial. Finite m etric-spaces—com binatorics, geometry and algorithms. In Pr o c e e dings of the International Congr ess of Mathematicians, V ol. III (Beijing, 2002) , pages 573 –586, Beijing, 2002. Higher Ed. Press. [26] N. Linial, E. London, and Y. Rabino vic h. The geometry of graph s and some of its algorithmic applications. Combinat oric a , 15(2):2 15–24 5, 1995. [27] J. Matou ˇ sek. O p en problems on lo w-distortion em b eddings of finite metric sp aces. Online: h ttp://k am.mff.cuni.cz/ ∼ matousek/met rop.ps. [28] J. Matou ˇ sek. L e ctur es on discr ete ge ometry , v olume 212 of Gr aduate T exts in Mathematics . Springer-V erlag, New Y ork, 2002. [29] I. Newman and Y. Rabino vich. A lo we r b ound on the distortion of em b edding planar metrics in to Eu clidean space. Discr ete Comput. Ge om. , 29 (1):77 –81, 2003. [30] H. O k am ura and P . D. S eymour. Multicommo d it y flows in planar grap h s. J. Combin. The ory Ser. B , 31 (1):75 –81, 1981. [31] P . P ans u . M ´ etriques d e Carnot-Carath ´ eodory et quasiisom ´ etries des espaces sym ´ etriques de rang u n. A nn. of Math. (2) , 1 29(1): 1–60, 1989. [32] S. D. P auls. The large scale geometry of n ilp oten t Lie grou p s. Comm. Anal. Ge om. , 9(5):951– 982, 2 001. 15