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l1-Embeddings and Algorithmic Applications

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l1-Embeddings and Algorithmic Applications

1. 1. 1 -embeddings and algorithmic applications Grigory Yaroslavtsev (proofs from “The design of approximation algorihms” by Williamson and Shmoys) Pennsylvania State University March 12, 2012Grigory Yaroslavtsev (PSU) March 12, 2012 1 / 17
2. 2. Metric embeddings and tree metrics A ﬁnite metric space is a pair (V , d), where V is a set of n points and d : X × X → R+ is a distance function (three axioms). A metric embedding of (V , d) is a metric space (V , d ), such that V ⊆ V and for all u, v ∈ V we have du,v ≤ du,v . Distortion = max du,v /du,v . u,v ∈V A tree metric is a shortest path metric in a tree.Theorem (Fakcharoenphol, Rao, Talwar)Given a distance metric (V , d), there is a randomized polynomial-timealgorithm that produces a tree metric (V , T ), V ⊆ V , such that for allu, v ∈ V , duv ≤ Tu,v and E[Tuv ] ≤ O(log n)duv . Grigory Yaroslavtsev (PSU) March 12, 2012 2 / 17
3. 3. Metric embeddings and tree metricsTheorem (Fakcharoenphol, Rao, Talwar)Given a distance metric (V , d), there is a randomized polynomial-timealgorithm that produces a tree metric (V , T ), V ⊆ V , such that for allu, v ∈ V , duv ≤ Tu,v and E[Tuv ] ≤ O(log n)duv . With a single tree Ω(n) distortion for a cycle (Steiner vertices don’t help). √ Distribution on trees [Alon, Karp, Peleg, West]: O(2 log n log logn ). With Steiner points [Bartal]: O(log n log log n). Lower bound for any tree metric [Bartal]: Ω(log n).With 1 -embeddable metrics (more general), distributions and Steinerpoints are not needed. Grigory Yaroslavtsev (PSU) March 12, 2012 3 / 17
4. 4. Embeddings into Rk and 2 -embeddings 1/p kEmbedding of (V , d) into (Rk , p ): d p (x, y ) = i=1 |xi − yi |p .Some facts about 2 -embeddings: If (V , d) is exactly 2 -embeddable ⇒ it is exactly p -embeddable for 1 ≤ p ≤ ∞. Distortion: O(log n) [Bourgain’85] (dimension n is enough). Minimum distortion embedding can be computed via SDP. Lower bound Ω(log n) via dual SDP (for expander graphs). Dimension reduction: n-point 2 -metric can be embedded into log n O 2 R with distortion 1 + [Johnson, Lindenstrauss ’84]. Dimension above is optimal ([Jayram, Woodruﬀ, SODA’11]). Multiple applications. Grigory Yaroslavtsev (PSU) March 12, 2012 4 / 17
5. 5. 1 -embeddingsSome facts about 1 -embeddings: Embedding with distortion O(log n) and dimension O(log2 n) (later). JL-like dimension reduction impossible [Brinkman, Charikar; Lee, 2 Naor]: for distortion D dimension nΩ(1/D ) is needed. Any tree metric is 1 -embeddable, converse is false. Representable as a convex combination of cut metrics (later). Grigory Yaroslavtsev (PSU) March 12, 2012 5 / 17
6. 6. 1 -embeddings and cut metricsDeﬁnition (Cut metric)For S ⊆ V , a cut metric is χS (u, v ) = 1 if |{u, v } ∩ S| = 1, otherwiseχS (u, v ) = 0.LemmaIf (V , d) is an 1 -embeddable metric with an embedding f , then thereexist λS ≥ 0 for all S ⊆ V such that for all u, v ∈ V , f (u) − f (v ) 1 = λS · χS (u, v ) S⊆VIf f is an embedding into Rm then ≤ mn of the λS are non-zero. Grigory Yaroslavtsev (PSU) March 12, 2012 6 / 17
7. 7. 1 -embeddings and cut metricsIf (V , d) is an 1 -embeddable metric with an embedding f , then thereexist λS ≥ 0 for all S ⊆ V such that for all u, v ∈ V , f (u) − f (v ) 1 = λS · χS (u, v ) S⊆VIf f is an embedding into Rm then ≤ mn of the λS are non-zero.Proof. If m = 1, then f embeds V into n points on a line. Let xi = f (i) and assume that x1 ≤ · · · ≤ xn . Consider cuts Si = {1, . . . , i}. j−1 Let λSi = xi+1 − xi , then |xi − xj | = k=i λSk . n−1 |xi − xj | = k=1 λSk χSk (i, j). If m > 1, do the same for each coordinate separately ⇒ ≤ mn non-zero λS , which can be computed eﬃciently. Grigory Yaroslavtsev (PSU) March 12, 2012 7 / 17
8. 8. Computing an 1 -embeddingTheorem (Bourgain; Linial, London, Rabinovich)Any metric (V , d) embeds into 1 with distortion O(log n). The 2embedding f : V → RO(log n) can be computed w.h.p. in polynomial time.Theorem (Aumann, Rabani; Linial, London, Rabinovich)Given a metric (V , d) and k pairs of terminals si , ti ∈ V , we can compute 2in polynomial time an embedding f : V → RO(log k) such that w.h.p: 1 f (u) − f (v ) 1 ≤ r · O(log k) · duv , for all u, v ∈ V , 2 f (si ) − f (ti ) 1 ≥ r · dsi ti , for all 1 ≤ i ≤ k,for some r > 0.Second theorem is more general ⇒ O(log k) approximation for sparsestcut (later today). Grigory Yaroslavtsev (PSU) March 12, 2012 8 / 17
9. 9. Fr´chet embedding eDeﬁnition (Fr´chet embedding) eFor a metric space (V , d) and p subsets A1 , . . . , Ap ⊆ V a Fr´chet eembedding f : V → Rp is deﬁned for all u ∈ V as: f (u) = (d(u, A1 ), . . . , d(u, Ap )) ∈ Rp ,where d(u, S) = minv ∈S d(u, v ) for a subset S ⊆ V .LemmaFor a Fr´chet embedding f : V → Rp of (V , d), we have e f (u) − f (v ) 1 ≤ pdu,v for all u, v ∈ V .Proof.For each 1 ≤ i ≤ p, we have |d(u, Ai ) − d(v , Ai )| ≤ duv . Grigory Yaroslavtsev (PSU) March 12, 2012 9 / 17
10. 10. Proof of the main theoremIdea: pick O(log2 k) sets Aj randomly, such that w.h.p.: f (si ) − f (ti ) 1 = Ω(log k)dsi ti , for all (si , ti ),then by taking r = Θ(log k) we’re done by the previous lemma. Let size of T = ∪i {si , ti } be a power of two and τ = log2 (2k). Let L = q log k for some constant q. Let At, for 1 ≤ t ≤ τ , 1 ≤ ≤ L be sets of size 2k/2t , chosen randomly with replacement from T . We have Lτ = O(log2 k) sets. Will show: f (si ) − f (ti ) 1 ≥ Ω(Ldsi ti ) = Ω(log k) · dsi ti w.h.p. Grigory Yaroslavtsev (PSU) March 12, 2012 10 / 17
11. 11. Proof of the main theoremWant to show: f (si ) − f (ti ) ≥ Ω(Ldsi ti ) w.h.p. 1 (Open) ball B o (u, r ) = {v ∈ T |du,v < r } ≤ Let rt be minimum r , such that |B(si , r )| ≥ 2t and |B(ti , r )| ≥ 2t . Let ˆ = minimum t, such that rt ≥ 1 dsi ti . t 4Will show: for any 1 ≤ ≤ L, 1 ≤ t ≤ ˆ we have (w.l.o.g.): t Pr[(At ∩ B(si , rt−1 ) = ∅) ∧ (At ∩ B o (ti , rt ) = ∅)] ≥ const LBy Chernoﬀ: =1 |d(si , At ) − d(ti , At )| ≥ Ω(L(rt − rt−1 )), w.h.p. ˆBecause f (si ) − f (ti ) 1 ≥ t t=1 L =1 |d(si , At ) − d(ti , At )|, we have: ˆ t f (si ) − f (ti ) 1 ≥ Ω(L(rt − rt−1 )) = Ω(Lrˆ) = Ω(Ldsi ti ) t . t=1 Grigory Yaroslavtsev (PSU) March 12, 2012 11 / 17
12. 12. Proof of the main theoremWant to show: for any 1 ≤ ≤ L, 1 ≤ t ≤ ˆ we have (w.l.o.g.): t Pr[(At ∩ B(si , rt−1 ) = ∅) ∧ (At ∩ B(ti , rt ) = ∅)] ≥ const Let event Et = (At ∩ B(si , rt−1 ) = ∅) ∧ (At ∩ B(ti , rt ) = ∅). Let G = B(si , rt−1 ), B = B o (ti , rt ) and A = At . Pr[E t ] = Pr[A ∩ B = ∅ ∧ A ∩ G = ∅] = Pr[A ∩ G = ∅|A ∩ B = ∅] · Pr[A ∩ B = ∅] ≥ Pr[A ∩ G = ∅] · Pr[A ∩ B = ∅]. Recall, that |A| = 2τ −t , |B| < 2t and |G | ≥ 2t−1 . |A| |B| τ −t Pr[A ∩ B = ∅] = 1 − |T | ≥ (1 − 2τ −t )2 ≥ 1. 4 |A| |G | Pr[A ∩ G = ∅] = 1 − 1 − |T | ≥ 1 − e −|G ||A|/|T | ≥ 1 − e −1/2 . Grigory Yaroslavtsev (PSU) March 12, 2012 12 / 17
13. 13. Approximation for sparsest cutSparsest cut: given an undirected graph G (V , E ), costs ce ≥ 0 for e ∈ Eand k pairs (si , ti ) with demands di , ﬁnd S, which minimizes: e∈δ(S) ce ρ(S) = . i:|S∩{si ,ti }|=1 diLP relaxation (variables yi ≥ 0, 1 ≤ i ≤ k and xe ≥ 0, ∀e ∈ E ): minimize: ce xe e∈E k subject to: di yi = 1, i=1 xe ≥ yi ∀P ∈ Pi , 1 ≤ i ≤ k, e∈Pwhere Pi is the set of all si − ti paths. Grigory Yaroslavtsev (PSU) March 12, 2012 13 / 17
14. 14. Approximation for sparsest cutLP relaxation (variables yi ≥ 0, 1 ≤ i ≤ k and xe ≥ 0, ∀e ∈ E ): minimize: ce xe e∈E k subject to: di yi = 1, i=1 xe ≥ yi ∀P ∈ Pi , 1 ≤ i ≤ k, e∈Pwhere Pi is the set of all si − ti paths.Intended solution: if we separate pairs D = {di1 , . . . , dit } with a cut S: χS (e) 1D (i) xe = , yi = . t dit t dit Grigory Yaroslavtsev (PSU) March 12, 2012 14 / 17
15. 15. Approximation for sparsest cut: rounding Given a solution {xe }, deﬁne a shortest path metric dx (u, v ). 2 Find an embedding f : (V , dx ) → RO(log k) with distortion O(log k). 2 Find ≤ O(n log k) values λS : f (u) − f (v ) 1 = S⊆V λS χS (u, v ). Return S ∗ , such that ρ(S ∗ ) = min ρ(S). S : λS >0 e∈δ(S) ce ce χS (e) ρ(S ∗ ) = min = min e∈E S : λS >0 i : |S∩{si ,ti }|=1 di S : λS >0 i di χS (si , ti ) S⊆V λS e∈E ce χS (e) e∈E ce S⊆V λS χS (e) ≤ = S⊆V λS i di χS (si , ti ) i di S⊆V λS χS (si , ti ) e=(u,v )∈E ce f (u) − f (v ) 1 r · O(log k) e=(u,v )∈E ce dx (u, v ) = ≤ . i di f (si ) − f (ti ) 1 r· i di dx (si , ti ) Grigory Yaroslavtsev (PSU) March 12, 2012 15 / 17
16. 16. Approximation for sparsest cutLP relaxation (variables yi ≥ 0, 1 ≤ i ≤ k and xe ≥ 0, ∀e ∈ E ): minimize: ce xe (1) e∈E k subject to: di yi = 1, (2) i=1 xe ≥ yi ∀P ∈ Pi , 1 ≤ i ≤ k, (3) e∈Pwhere Pi is the set of all si − ti paths. e=(u,v )∈E ce dx (u, v ) (3) e=(u,v )∈E ce xe ρ(S ∗ ) ≤ O(log k) ≤ O(log k) i di dx (si , ti ) i di y i (2) (1) = O(log k) ce xe ≤ O(log k)OPT . e∈E Grigory Yaroslavtsev (PSU) March 12, 2012 16 / 17
17. 17. ConclusionWhat we saw today: 2 1 -embedding into RO(log n) with distortion O(log n). O(log k)-approximation for sparsest cut.Extensions: Cut-tree packings, approximating cuts by trees [R¨cke; Harrelson, a Hildrum, Rao]. √ Balanced sparsest cut: O( log n)-approximation [Arora, Rao, Vazirani]. Grigory Yaroslavtsev (PSU) March 12, 2012 17 / 17