Joint work with Michael Elkin BenGurion

1
Lower-Stretch Spanning Trees
Daniel A. Spielman
Joint work with Michael Elkin (Ben-Gurion)
Yuval Emek (Weizmann)
Shang-Hua Teng (BU)

2
Spanning tree
T
G

• subgraph
• connected
• no cycles

3
Stretch of a spanning tree
v
v
u
u
T
G
stretchT(u,v) length of uv path in T
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Average stretch of a spanning tree
v
v
2
3
1
u
2
u
1
2
1
1
1
2
T
G
stretchT(u,v) length of uv path in T
ave-stretch(T) ave(u,v) in G stretchT(u,v)
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Example n-cycle
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Example n-cycle
n-1
1
1
1
1
1
1
1
Every spanning tree has some edge of stretch n-1.
ave-stretch
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How low average stretch?
Alon-Karp-Peleg-West 92
This paper
Conjecture
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Why?
AKPW Analysis of k-server algorithm
Bartal 95 Approximate metrics by tree metrics
Polynomial approximation
algorithms
Boman-Hendrickson 01 Solve diagonally
dominant linear systems in time
S-Teng 03 Solve in time
9
Why?
AKPW Analysis of k-server algorithm
Bartal 95 Approximate metrics by tree metrics
Polynomial approximation
algorithms
Boman-Hendrickson 01 Solve diagonally
dominant linear systems in time
S-Teng 03 Solve in time
10
Once more
v
v
2
3
1
u
2
u
1
2
1
1
1
2
T
G
stretchT(u,v) length of uv path in T
ave-stretch(T) ave(u,v) in G stretchT(u,v)
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Star decomposition at x0
V1
V7

• Partition V into

V2
x0
V6
V0
V3

• x0 in V0

V5
V4

• each Vi touches V0

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d-star decomposition at x0
For
Require
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Cost of star decomposition
cost number of edges between components
V1
x1
V7
x7
V2
x2
x0
x6
V6
V0
x3
V3
x4
x5
V5
V4
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Star decomposition theorem
Build a d-star decomposition at x0 of
cost where in time
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Star decomposition theorem
Build a d-star decomposition at x0 of
cost where in time
large r decreases cost small d increases cost
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Trees from star decompositions
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Trees from star decompositions
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Radius of the tree
If we choose
Also true in each component
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Total stretch of the tree
Stretch of edges cut at first level
Sum over levels
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Total stretch of the tree
Stretch of edges cut at first level
Sum over levels
improve to
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How to build a low-cost star decomposition
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Classic technique ball growing (Awerbuch 84)
For
r
Such that for
bdry decreases as r increases
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Classic technique ball growing (Awerbuch)
Grow level-by-level until small
Boundary at one level is inside interior at
next All boundaries big implies ball grows
quickly
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Classic technique ball growing (Awerbuch)
Theorem
Such that for
we have
Proof. If
for
then
contradiction
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Star-decomposition, first attempt
Grow ball, remove, grow ball in
remaining graph, etc.
Wont be star-connected!
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Star-decomposition, idea
Cone from x1 everything reach from x1,
moving away from x0 v some shortest path to
x0 goes though x1
x1
v
x0
Removing cone 1. Sets touch 2. Do not
increase dist of any node to x0
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Star-decomposition, second attempt
Grow ball, remove, cut remaining graph
into cones from boundary
Could cut too many edges!
x1
x0
Cone from x1 everything reach from x1,
moving away from x0
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Star decomposition technique
Grow cones!
Cone(S) all can reach from S, moving away from
x0 v some shortest path
to x0 goes though S
x1
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Cone growing theorem
For
r
r0
x1
r-r0
dr
Such that for
And,
can go up to dr steps back, so at most drr-r0
forward
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Remove grown cones until nothing left
x3
x2
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Star decomposition theorem
Build a d-star decomposition at x0 of cost in
time
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Low-stretch tree from d-star decompositions
ave-stretch
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Edges with lengths
Issue shrinking radius does not
bound recursion depth
Solution contract all edges of length lt
radius/n2 each edge active in
at most
rounds
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How to improve
such that for
If V1 very small, lowers recursion depth If
V1 big, grow another dr levels, for smaller
boundary
ave-stretch
35
what if planar?
36
Implementation
Heuristic (Jan 04)
Star-decomposition
linsolve
2-4x faster
huh??