A tight bound on approximating arbitrary metrics by tree ...guyk/metric.pdfJittat Fakcharoenphol,...
Transcript of A tight bound on approximating arbitrary metrics by tree ...guyk/metric.pdfJittat Fakcharoenphol,...
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A tight bound on approximating arbitrary metrics by tree metric
Reference :[FRT2004]Jittat Fakcharoenphol, Satish Rao, Kunal Talwar
Journal of Computer & System Sciences, 69 (2004), 485-497
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A simple but fundamental problem
Given : an undirected graph G = (V, E).
compute a spanning tree T ⊆ E which serves as approximate shortest path tree
for all vertices.
∀(u, v) ∈ EdT (u, v)
dT (u, v)is small
Is it possible ??
Counterexample : G a cycle
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A simple but fundamental problem
Given : an undirected graph G = (V, E).
compute a spanning tree T ⊆ E which serves as approximate shortest path tree
for all vertices.
∀(u, v) ∈ E,dT (u, v)
dG(u, v)is small
Is it possible ??
Counterexample : G a cycle
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A simple but fundamental problem
Given : an undirected graph G = (V, E).
compute a spanning tree T ⊆ E which serves as approximate shortest path tree
for all vertices.
∀(u, v) ∈ E,dT (u, v)
dG(u, v)is small
Is it possible ??
Counterexample : a G cycle
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A simple but fundamental problem
Given : an undirected graph G = (V, E).
compute a spanning tree T ⊆ E which serves as approximate shortest path tree
for all vertices.
∀(u, v) ∈ E,dT (u, v)
dG(u, v)is small
Not possible ??
Counterexample : when G is a cycle
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A simple and fundamental problem
Given : an undirected graph G = (V, E).
compute a spanning tree T ⊆ E which minimizes
1
|E|
∑
(u,v)∈E
δT (u, v)
δG(u, v)
Known Results :
1. Elkin et al. [STOC 2005] : O(log2 n log log n))
2. Bartal et al. [FOCS 2008] : O(log n log log n)
Lower bound : Ω(log n)
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A simple and fundamental problem
Given : an undirected graph G = (V, E).
compute a spanning tree T ⊆ E which minimizes
1
|E|
∑
(u,v)∈E
δT (u, v)
δG(u, v)
Known Results :
1. Elkin et al. [STOC 2005] : O(log2 n log log n))
2. Bartal et al. [FOCS 2008] : O(log n log log n)
Lower bound : Ω(log n)
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A simple and fundamental problem
Given : an undirected graph G = (V, E).
compute a spanning tree T ⊆ E which minimizes
1
|E|
∑
(u,v)∈E
δT (u, v)
δG(u, v)
Known Results :
1. Elkin et al. [STOC 2005] : O(log2 n log log n))
2. Bartal et al. [FOCS 2008] : O(log n log log n)
Lower bound : Ω(log n)
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[FRT2004] solves the problem if
we remove the restriction for T to be a subgraph of G
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[FRT2004] solves the problem if
we remove the restriction for T to be a subgraph of G
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What is a metric ?
A metric (space) is an ordered pair (V, d) where d : V × V → R such that
1. d(x, y) ≥ 0
2. d(x, y) = 0 if and only if x = y.
3. d(x, y) = d(y, x)
4. d(x, y) ≤ d(x, z) + d(y, z)
A useful view : a metric (V, d) as a complete graph with length of edge (u, v)
equal to d(u, v).
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Tree Metric
A tree spanning the points V where the distance d between any two vertices is
defined by the length of the path between them in the tree.
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When does one metric dominate another metric ?
metric (V ′, d′) is said to dominate another metric (V, d) if
1. V ⊆ V ′
2. d′(u, v) ≥ d(u, v) for each u, v ∈ V
Ideally we would like d′(u, v) ≤ αd(u, v).
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α-probabilistically approximation for metric (V, d)
Let S be a collection of tree metrics over V andD be a probability distribution
over them. Then (S,D) is said to α-probabilistically approximate (V, d) if
1. Each metric in S dominates (V, d)
2. Ed′∈(S,D)[d′(u, v)] ≤ αd(u, v)
α is usually called the distortion/stretch
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Result [FRT 2004]
Given any metric (V, d), there exists a distribution over tree metrics which
approximates (V, d) probabilistically with distortion O(log n).
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Outline of the algorithm
1. A deterministic construction of a tree metric which dominates (V, d).
2. Randomization is added to ensure expected stretch O(log n) for each edge.
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Construction of a tree metric which dominates (V, d)
Let smallest distance =1, Let the diameter ∆ is equal to 2δ for some δ > 0.
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Construction of a tree metric which dominates (V, d)
Let smallest distance =1, Let the diameter ∆ is equal to 2δ for some δ > 0.
S1 S2 S3
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Construction of a tree metric which dominates (V, d)
Let smallest distance =1, Let the diameter ∆ is equal to 2δ for some δ > 0.
S1 S2 S3
Diam(S1)2
Diam(S2)2
Diam(S3)2
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Construction of a tree metric which dominates (V, d)
Let smallest distance =1, Let the diameter ∆ is equal to 2δ for some δ > 0.
S = V
S1 S2 S3
Diam(S)2
Diam(S1)2
Diam(S2)2
Diam(S3)2
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Construction of level δ − 1
π = x1, x2, ...xn
C2 : vertices within distance 2δ−2 from x2
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Construction of level δ − 1
π = x1, x2, ...xn
C1
C1 : vertices within distance 2δ−2 from x1
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Construction of level δ − 1
π = x1, x2, ...xn
C1
C1
C2 : vertices within distance 2δ−2 from x2
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Construction of level δ − 1
π = x1,x2, ..., xn
C1
C1 C2
C2 : vertices within distance 2δ−2 from x2
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Construction of level δ − 1
π = x1,x2, ..., xn
C1
C1
C2
C2
C2 : vertices within distance 2δ−2 from x2
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Construction of level δ − 1
...
π = x1, x2, ...,xn
C1
C1
C2
C2
C3
C4
C5
C6
C6
C2 : vertices within distance 2δ−2 from x2
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Construction of level δ − 1 is complete now
...
π = x1, x2, ..., xn
C1
C1
C2
C2
C3
C4
C5
C6
C6
2δ−12δ−1
C2 : vertices within distance 2δ−2 from x2
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Construction of level δ − 1
The highest level is δ, and consists of cluster S = V .
π = x1, x2, ..., xn.
For j = 1 to n do
1. Create a new cluster consisting of all unassigned vertices of S which are at
distance≤ 2δ−2 from vertex xj .
2. Assign length 2δ−1 to the edge.
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Construction of level δ − 2
...
π = x1, x2, ..., xn
C1
C1
C2
C2
C3
C4
C5
C6
C6
2δ−12δ−1
C2 : vertices within distance 2δ−2 from x2
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Construction of level δ − 2
...
π = x1, x2, ..., xn
C1
C1 C2
C2
C3
C4
C5
C6
C6
C11
C11
2δ−12δ−1
C11 : vertices of C1 within distance 2δ−3 from x1
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Construction of level δ − 2
...
...
π = x1, x2, ...xn
C1 C2
C2
C3
C4
C5
C6
C6
C12
C12
C1
C1
C13
C13
2δ−12δ−1
2δ−2
C2 : vertices within distance 2δ−2 from x2
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Deterministic Construction of tree metric : Top Down approach
The highest level is δ, and consists of cluster S = V .
i← δ − 1;
π = x1, x2, ..., xn;
While (i >= 0) do
βi ← 2i−1;
For each cluster S at level i + 1 do
For j = 1 to n do
1. Create a new cluster consisting of all unassigned vertices of S which are with
in distance βi from vertex xj .
2. Assign length 2βi to the edge between the node for S and the child node
corresponding to the new cluster.
This defines the level i of the tree.
i← i− 1
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Analysing the algorithm from perspective of an edge (u, v)
...
π = x1, x2, ..., xn
u
u
v
v
δ
i + 1
2δ−12δ−1
only vertices within distance 2i−1 from (u, v) matter
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Analysing the algorithm from perspective of an edge (u, v)
x u v
2i−1
Case 1
Case 1 : ??? edge (u, v) is retained at level i as well !
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Analysing the algorithm from perspective of an edge (u, v)
x u v
2i−1
Case 1
Case 1 : the edge (u, v) is retained at level i as well !
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Analysing the algorithm from perspective of an edge (u, v)
x
x
u
u v
v
2i−1
2i−1
Case 1
Case 2
Case 2 : ??? edge (u, v) is cut at level i !!
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Analysing the algorithm from perspective of an edge (u, v)
x
x
u
u v
v
2i−1
2i−1
Case 1
Case 2
Case 2 : the edge (u, v) is cut at level i !!
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Analysing the algorithm from perspective of an edge (u, v)
Case 2 : Careful look
...
π = x1, x2, ..., xn
u
u
v
v
δ
i + 1
i2i
2δ−12δ−1
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Analysing the algorithm from perspective of an edge (u, v)
Case 2 : what is dT (u, v) = ???
...
π = x1, x2, ..., xn
u
u
v
v
δ
i + 1
i2i
2δ−12δ−1
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Analysing the algorithm from perspective of an edge (u, v)
Case 2 : dT (u, v) = 2∑
j≤i 2j = 2i+2
...
π = x1, x2, ..., xn
u
u
v
v
δ
i + 1
i2i
2δ−12δ−1
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Some more Observations
1. In the tree T , exactly one vertex cuts the edge (u, v).
2. If a vertex cuts the edge (u, v) at level i, then dT (u, v) ≤ 2i+2.
3. A vertex w has potential to cut (u, v) if and only if ∃j,
d(w, u) ≤ 2j < d(w, v).
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What can be dT (u,v)d(u,v)
?
huge if d(u, v) <<< 2i+2
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What can be dT (u,v)d(u,v)
?
huge if d(u, v) <<< 2i+2
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Part II : adding some randomization to the construction
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Main obstacle :
How to ensure that the vertices which are nearer to (u, v) have higher chances
to cut the edge.
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A small probability exercise
There are two persons a and b aiming to shoot an arrow at a target which is a red strip of length τ . An
arrow is said to shoot the target if it hits anywehre within the red strip. They aim at the center of the
strip. Both of them are sharp shooters so that if everything goes fine, they will hit the center of the
strip. However, due to randomness of wind speed, the arrow may miss its target. Assume that if the
actual target is at distance x from the point from where it is shot, then it may land any where with in
the interval [x, 2x] from the point where it was shot. If a and b are at distance α and β respectively
from the center of thetarget strip, then who is more likely to shoot the target. (see the following slides
for more details)
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A small probability exercise
who has more chances to win ?
αβ
abτ
vertex closer to the target are more likely to hit the target
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A small probability exercise
who has more chances to win ?
αβ
abτ
vertex closer to the target are more likely to hit the target
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A small probability exercise
who has more chances to win ?
αα
ββ
abτ
vertex closer to the target are more likely to hit the target
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A small probability exercise
who has more chances to hit the target ?
αα
ββ
abτ
vertex closer to the target are more likely to hit the target
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A small probability exercise
who has more chances to hit the target ?
αα
ββ
abτ
person closer to the target is more likely to hit the target
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The randomized construction
1. Generate the permutation π uniformly randomly
2. Select a random number y uniformly randomly in the interval [1, 2] and
replace βi by 2i−1y.
Distribution of βi ???[2i−1]
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The randomized construction
1. Generate the permutation π uniformly randomly
2. Select a random number y uniformly randomly in the interval [1, 2] and
replace βi by 2i−1y.
Distribution of βi ???[2i−1]
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The randomized construction
1. Generate the permutation π uniformly randomly
2. Select a random number y uniformly randomly in the interval [1, 2] and
replace βi by 2i−1y.
Distribution of βi : uniform in [2i−1, 2i]
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Observations after randomization
1. In a tree T computed, exactly one vertex cuts the edge (u, v).
2. If a vertex cuts the edge (u, v) at level i, then dT (u, v) ≤ 2i+3.
3. For all trees in (S,D), there exists only two possible levels at which a vertex
can potentially cut the edge.
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Analysis
w u v
2k
2k−1
2k−2
Every vertex has potential to cut an edge (u, v)!!
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Analysis for an edge (u, v) and a vertex w
Let Xw be the indicator random variable which takes value 1 if vertex w cuts the
edge (u, v) in T for T ∈ (S,D).
iw : one of the two levels at which w can cut (u, v).
What is P[Xw = 1] ?
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The necessary condition for w to cut (u, v) at level iw ?
w u v
2iw
2iw−1
2iw+1
1. ??? d(w, u) ≤ βiw< d(w, v) : Probability = d(w,v)−d(w,u)
2iw−1
2. w must precede all vertices closer to (u, v) in π
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The necessary conditions for w to cut (u, v) at level iw ?
w u v
2iw
2iw−1
2iw+1
1. d(w, u) ≤ βiw< d(w, v) : Probability = d(w,v)−d(w,u)
2iw−1
2. w must precede all vertices closer to (u, v) in π
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The necessary conditions for w to cut (u, v) at level iw ?
w u v
2iw
2iw−1
2iw+1
βiw
1. d(w, u) ≤ βiw< d(w, v) : Probability = ???
d(w,v)−d(w,u)2iw−1
2. w must precede all vertices closer to (u, v) in π
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The necessary conditions for w to cut (u, v) at level iw ?
w u v
2iw
2iw−1
2iw+1
βiw
1. d(w, u) ≤ βiw< d(w, v) : Probability = d(w,v)−d(w,u)
2iw−1
2. w must precede all vertices closer to (u, v) in π
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The necessary conditions for w to cut (u, v) at level iw ?
w u v
2iw
2iw−1
2iw+1
βiw
1. d(w, u) ≤ βiw< d(w, v) : Probability = d(w,v)−d(w,u)
2iw−1
2. ???w must precede all vertices closer to (u, v) in π
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The necessary conditions for w to cut (u, v) at level iw ?
w u v
2iw
2iw−1
2iw+1
1. d(w, u) ≤ βiw< d(w, v) : Probability = d(w,v)−d(w,u)
2iw−1
2. w must precede all vertices closer to (u, v) in π
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If w is sth vertex closest to edge (u, v)
P[Xw = 1] ≤1
s
d(w, v)− d(w, u)
2iw−1
≤1
s
d(u, v)
2iw−1
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If w is sth vertex closest to edge (u, v)
P[Xw = 1] ≤1
s
d(w, v)− d(w, u)
2iw−1
≤1
s
d(u, v)
2iw−1
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What is expected value of dT (u, v), T ∈ (S,D) ?
E[dT (u, v)] =∑
w
P[Xw = 1] 2iw+3
≤∑
w
1
s
d(u, v)
2iw−12iw+3
≤∑
w
161
sd(u, v)
= 16d(u, v)∑
s
1
s
≤ 16 log2 n.
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What is expected value of dT (u, v), T ∈ (S,D) ?
E[dT (u, v)] =∑
w
P[Xw = 1] 2iw+3
≤∑
w
1
s
d(u, v)
2iw−12iw+3
≤∑
w
161
sd(u, v)
= 16d(u, v)∑
s
1
s
≤ 16 log2 n.
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What is expected value of dT (u, v), T ∈ (S,D) ?
E[dT (u, v)] =∑
w
P[Xw = 1] 2iw+3
≤∑
w
1
s
d(u, v)
2iw−12iw+3
≤∑
w
161
sd(u, v)
= 16d(u, v)∑
s
1
s
≤ 16 log2 n.
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What is expected value of dT (u, v), T ∈ (S,D) ?
E[dT (u, v)] =∑
w
P[Xw = 1] 2iw+3
≤∑
w
1
s
d(u, v)
2iw−12iw+3
≤∑
w
161
sd(u, v)
= 16d(u, v)∑
s
1
s
≤ 16 log2 n.
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What is expected value of dT (u, v), T ∈ (S,D) ?
E[dT (u, v)] =∑
w
P[Xw = 1] 2iw+3
≤∑
w
1
s
d(u, v)
2iw−12iw+3
≤∑
w
161
sd(u, v)
= 16d(u, v)∑
s
1
s
≤ 16 log2 n.
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Derandomization
average stretch =2
n(n− 1)
∑
(u,v)
dT (u, v)
d(u, v)
To compute a tree metric T which achieves average stretch O(log n).
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What if we want a T ⊆ G to achieve it ?
Open problem : A simpler and/or tight (randomized) construction ?
72