l1-Embeddings and Algorithmic Applications
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Transcript of l1-Embeddings and Algorithmic Applications
`1-embeddings and algorithmic applications
Grigory Yaroslavtsev(proofs from “The design of approximation algorihms”
by Williamson and Shmoys)
Pennsylvania State University
March 12, 2012
Grigory Yaroslavtsev (PSU) March 12, 2012 1 / 17
Metric embeddings and tree metrics
A finite metric space is a pair (V , d), where V is a set of n points andd : X × X → R+ is a distance function (three axioms).
A metric embedding of (V , d) is a metric space (V ′, d ′), such thatV ⊆ V ′ and for all u, v ∈ V we have du,v ≤ d ′u,v .
Distortion = maxu,v∈V
d ′u,v/du,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
Metric embeddings and tree metrics
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 .
With a single tree Ω(n) distortion for a cycle (Steiner vertices don’thelp).
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
Embeddings into Rk and `2-embeddings
Embedding of (V , d) into (Rk , `p): d`p(x , y) =(∑k
i=1 |xi − yi |p)1/p
.
Some facts about `2-embeddings:
If (V , d) is exactly `2-embeddable ⇒ it is exactly `p-embeddable for1 ≤ 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
RO(
log n
ε2
)with distortion 1 + ε [Johnson, Lindenstrauss ’84].
Dimension above is optimal ([Jayram, Woodruff, SODA’11]).
Multiple applications.
Grigory Yaroslavtsev (PSU) March 12, 2012 4 / 17
`1-embeddings
Some facts about `1-embeddings:
Embedding with distortion O(log n) and dimension O(log2 n) (later).
JL-like dimension reduction impossible [Brinkman, Charikar; Lee,Naor]: for distortion D dimension nΩ(1/D2) 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
`1-embeddings and cut metrics
Definition (Cut metric)
For S ⊆ V , a cut metric is χS(u, v) = 1 if |u, v ∩ S | = 1, otherwiseχS(u, v) = 0.
Lemma
If (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 =∑S⊆V
λS · χS(u, v)
If f is an embedding into Rm then ≤ mn of the λS are non-zero.
Grigory Yaroslavtsev (PSU) March 12, 2012 6 / 17
`1-embeddings and cut metrics
If (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 =∑S⊆V
λS · χS(u, v)
If 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.Let λSi = xi+1 − xi , then |xi − xj | =
∑j−1k=i λSk
.
|xi − xj | =∑n−1
k=1 λSkχSk
(i , j).
If m > 1, do the same for each coordinate separately ⇒ ≤ mnnon-zero λS , which can be computed efficiently.
Grigory Yaroslavtsev (PSU) March 12, 2012 7 / 17
Computing an `1-embedding
Theorem (Bourgain; Linial, London, Rabinovich)
Any metric (V , d) embeds into `1 with distortion O(log n). The
embedding f : V → RO(log2 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
in polynomial time an embedding f : V → RO(log2 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
Frechet embedding
Definition (Frechet embedding)
For a metric space (V , d) and p subsets A1, . . . ,Ap ⊆ V a Frechetembedding 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 .
Lemma
For a Frechet embedding f : V → Rp of (V , d), we have‖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
Proof of the main theorem
Idea: 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 = ∪isi , 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 , chosenrandomly 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
Proof of the main theorem
Want to show: ‖f (si )− f (ti )‖1 ≥ Ω(Ldsi ti ) w.h.p.
(Open) ball Bo(u, r) = v ∈ T |du,v<≤ r
Let rt be minimum r , such that |B(si , r)| ≥ 2t and |B(ti , r)| ≥ 2t .
Let t = minimum t, such that rt ≥ 14 dsi ti .
Will show: for any 1 ≤ ` ≤ L, 1 ≤ t ≤ t we have (w.l.o.g.):
Pr[(At` ∩ B(si , rt−1) 6= ∅) ∧ (At` ∩ Bo(ti , rt) = ∅)] ≥ const
By Chernoff:∑L
`=1 |d(si ,At`)− d(ti ,At`)| ≥ Ω(L(rt − rt−1)), w.h.p.
Because ‖f (si )− f (ti )‖1 ≥∑t
t=1
∑L`=1 |d(si ,At`)− d(ti ,At`)|, we have:
‖f (si )− f (ti )‖1 ≥t∑
t=1
Ω(L(rt − rt−1)) = Ω(Lrt) = Ω(Ldsi ti ) .
Grigory Yaroslavtsev (PSU) March 12, 2012 11 / 17
Proof of the main theorem
Want to show: for any 1 ≤ ` ≤ L, 1 ≤ t ≤ t we have (w.l.o.g.):
Pr[(At` ∩ B(si , rt−1) 6= ∅) ∧ (At` ∩ B(ti , rt) = ∅)] ≥ const
Let event Et` = (At` ∩ B(si , rt−1) 6= ∅) ∧ (At` ∩ B(ti , rt) = ∅).
Let G = B(si , rt−1), B = Bo(ti , rt) and A = At`.
Pr[E t`] = Pr[A ∩ B = ∅ ∧ A ∩ G 6= ∅]= Pr[A ∩ G 6= ∅|A ∩ B = ∅] · Pr[A ∩ B = ∅]≥ Pr[A ∩ G 6= ∅] · Pr[A ∩ B = ∅].
Recall, that |A| = 2τ−t , |B| < 2t and |G | ≥ 2t−1.
Pr[A ∩ B = ∅] =(
1− |B||T |)|A|≥ (1− 2τ−t)2τ−t ≥ 1
4 .
Pr[A ∩ G 6= ∅] = 1−(
1− |G ||T |)|A|≥ 1− e−|G ||A|/|T | ≥ 1− e−1/2.
Grigory Yaroslavtsev (PSU) March 12, 2012 12 / 17
Approximation for sparsest cut
Sparsest 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:
ρ(S) =
∑e∈δ(S) ce∑
i :|S∩si ,ti|=1 di.
LP relaxation (variables yi ≥ 0, 1 ≤ i ≤ k and xe ≥ 0, ∀e ∈ E ):
minimize:∑e∈E
cexe
subject to:k∑
i=1
diyi = 1,∑e∈P
xe ≥ yi ∀P ∈ Pi , 1 ≤ i ≤ k,
where Pi is the set of all si − ti paths.Grigory Yaroslavtsev (PSU) March 12, 2012 13 / 17
Approximation for sparsest cut
LP relaxation (variables yi ≥ 0, 1 ≤ i ≤ k and xe ≥ 0, ∀e ∈ E ):
minimize:∑e∈E
cexe
subject to:k∑
i=1
diyi = 1,∑e∈P
xe ≥ yi ∀P ∈ Pi , 1 ≤ i ≤ k,
where Pi is the set of all si − ti paths.Intended solution: if we separate pairs D = di1 , . . . , dit with a cut S :
xe =χS(e)∑
t dit
, yi =1D(i)∑
t dit
.
Grigory Yaroslavtsev (PSU) March 12, 2012 14 / 17
Approximation for sparsest cut: rounding
Given a solution xe, define a shortest path metric dx(u, v).
Find an embedding f : (V , dx)→ RO(log2 k) with distortion O(log k).
Find ≤ O(n log2 k) values λS : ‖f (u)− f (v)‖1 =∑
S⊆V λSχS(u, v).
Return S∗, such that ρ(S∗) = minS : λS>0
ρ(S).
ρ(S∗) = minS : λS>0
∑e∈δ(S) ce∑
i : |S∩si ,ti|=1 di= min
S : λS>0
∑e∈E ceχS(e)∑i diχS(si , ti )
≤∑
S⊆V λS∑
e∈E ceχS(e)∑S⊆V λS
∑i diχS(si , ti )
=
∑e∈E ce
∑S⊆V λSχS(e)∑
i di∑
S⊆V λSχS(si , ti )
=
∑e=(u,v)∈E ce‖f (u)− f (v)‖1∑
i di‖f (si )− f (ti )‖1≤
r · O(log k)∑
e=(u,v)∈E cedx(u, v)
r ·∑
i didx(si , ti ).
Grigory Yaroslavtsev (PSU) March 12, 2012 15 / 17
Approximation for sparsest cut
LP relaxation (variables yi ≥ 0, 1 ≤ i ≤ k and xe ≥ 0, ∀e ∈ E ):
minimize:∑e∈E
cexe (1)
subject to:k∑
i=1
diyi = 1, (2)∑e∈P
xe ≥ yi ∀P ∈ Pi , 1 ≤ i ≤ k, (3)
where Pi is the set of all si − ti paths.
ρ(S∗) ≤ O(log k)
∑e=(u,v)∈E cedx(u, v)∑
i didx(si , ti )
(3)
≤ O(log k)
∑e=(u,v)∈E cexe∑
i diyi
(2)= O(log k)
∑e∈E
cexe(1)
≤ O(log k)OPT .
Grigory Yaroslavtsev (PSU) March 12, 2012 16 / 17
Conclusion
What we saw today:
`1-embedding into RO(log2 n) with distortion O(log n).
O(log k)-approximation for sparsest cut.
Extensions:
Cut-tree packings, approximating cuts by trees [Racke; Harrelson,Hildrum, Rao].
Balanced sparsest cut: O(√
log n)-approximation [Arora, Rao,Vazirani].
Grigory Yaroslavtsev (PSU) March 12, 2012 17 / 17