1. 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

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-time
algorithm that produces a tree metric (V , T ), V ⊆ V , such that for all
u, v ∈ V , duv ≤ Tu,v and E[Tuv ] ≤ O(log n)duv .
Grigory Yaroslavtsev (PSU) March 12, 2012 2 / 17

3. Metric embeddings and tree metrics
Theorem (Fakcharoenphol, Rao, Talwar)
Given a distance metric (V , d), there is a randomized polynomial-time
algorithm that produces a tree metric (V , T ), V ⊆ V , such that for all
u, 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 Steiner
points are not needed.
Grigory Yaroslavtsev (PSU) March 12, 2012 3 / 17

4. Embeddings into Rk and 2 -embeddings
1/p
k
Embedding 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. 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,
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. 1 -embeddings and cut metrics
Definition (Cut metric)
For S ⊆ V , a cut metric is χS (u, v ) = 1 if |{u, v } ∩ S| = 1, otherwise
χS (u, v ) = 0.
Lemma
If (V , d) is an 1 -embeddable metric with an embedding f , then there
exist λS ≥ 0 for all S ⊆ V such that for all u, v ∈ V ,
f (u) − f (v ) 1 = λS · χS (u, v )
S⊆V
If f is an embedding into Rm then ≤ mn of the λS are non-zero.
Grigory Yaroslavtsev (PSU) March 12, 2012 6 / 17

7. 1 -embeddings and cut metrics
If (V , d) is an 1 -embeddable metric with an embedding f , then there
exist λS ≥ 0 for all S ⊆ V such that for all u, v ∈ V ,
f (u) − f (v ) 1 = λS · χS (u, v )
S⊆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}.
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. Computing an 1 -embedding
Theorem (Bourgain; Linial, London, Rabinovich)
Any metric (V , d) embeds into 1 with distortion O(log n). The
2
embedding 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
2
in 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 sparsest
cut (later today).
Grigory Yaroslavtsev (PSU) March 12, 2012 8 / 17

9. Fr´chet embedding
e
Definition (Fr´chet embedding)
e
For a metric space (V , d) and p subsets A1 , . . . , Ap ⊆ V a Fr´chet
e
embedding 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 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. 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 = ∪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. Proof of the main theorem
Want 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 4
Will 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
L
By Chernoff: =1 |d(si , At ) − d(ti , At )| ≥ Ω(L(rt − rt−1 )), w.h.p.
ˆ
Because 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. Proof of the main theorem
Want 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. Approximation for sparsest cut
Sparsest cut: given an undirected graph G (V , E ), costs ce ≥ 0 for e ∈ E
and k pairs (si , ti ) with demands di , find S, which minimizes:
e∈δ(S) ce
ρ(S) = .
i:|S∩{si ,ti }|=1 di
LP 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∈P
where Pi is the set of all si − ti paths.
Grigory Yaroslavtsev (PSU) March 12, 2012 13 / 17

14. Approximation for sparsest cut
LP 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∈P
where 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. 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. Approximation for sparsest cut
LP 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∈P
where 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. Conclusion
What 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