Approximate Distance Oracles and Spanners with sublinear surplus

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Approximate Distance OraclesandSpanners with
sublinear surplus

• Mikkel ThorupATT Research
• Uri ZwickTel Aviv University

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Approximate Distance Oracles (TZ01)
n by ndistancematrix
APSPalgorithm
mn timen2 space
Compact datastructure
Weightedundirected graph
mn1/k timen11/k space
Stretch-Space tradeoff is essentially optimal!
u,v
d(u,v(
O(1) query time stretch 2k-1
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An estimated distance ?(u,v) is of stretch t
iff ?(u,v) ? ?(u,v) ? t ?(u,v)
Approximate Shortest Paths
Let ?(u,v) be the distance from u to v.
Multiplicativeerror
Additiveerror
An estimated distance ?(u,v) is of surplus t
iff ?(u,v) ? ?(u,v) ? ?(u,v) t
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Spanners
Given an arbitrary dense graph, can we always
find a relatively sparse subgraph that
approximates all distances fairly well?
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Spanners PU89,PS89
Let G(V,E) be a weighted undirected graph.
A subgraph G(V,E) of G is said to be a
t-spannerof G iff dG (u,v) t dG (u,v) for
every u,v in V.
Theorem Every weighted undirected graph has a
(2k-1) -spanner of size O(n11/k). ADDJS 93
Furthermore, such spanners can be constructed
deterministically in linear time. BS 04 TZ
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The size-stretch trade-off is essentially
optimal.(Assuming there are graphs with
?(n11/k) edges of girth 2k2, as conjectured by
Erdös and others.)
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Additive Spanners
Let G(V,E) be a unweighted undirected graph.
A subgraph G(V,E) of G is said to be an
additivet-spanner if G iff dG (u,v) dG (u,v)
t for every u,v ?V.
Theorem Every unweighted undirected graph has an
additive 2-spanner of size O(n3/2). ACIM 96
DHZ 96
Theorem Every unweighted undirected graph has an
additive 6-spanner of size O(n4/3). BKMP 04
Major open problem Do all graphs have additive
spanners with only O(n1e) edges, for every egt0 ?
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Spanners with sublinear surplus
Theorem For every kgt1, every undirected graph
G(V,E) on n vertices has a subgraph G(V,E)
with O(n11/k) edges such that for every u,v?V,
if dG(u,v)d, then dG(u,v)dO(d1-1/(k-1)).
d
dO(d1-1/(k-1))
Extends and simplifies a result of Elkin and
Peleg (2001)
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All sorts of spanners
A subgraph G(V,E) of G is said to be a
functionalf-spanner if G iff dG (u,v) f(dG
(u,v)) for every u,v ?V.
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Part I Approximate Distance Oracles
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Approximate Distance Oracles TZ01A hierarchy
of centers
A0?V Ak ?? Ai ?sample(Ai-1,n-1/k)
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Bunches
A0A1A2
p2(v)
v
p1(v)
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Lemma EB(v) kn1/k
Proof B(v)?Ai is stochastically dominated by a
geometric random variable with parameter pn-1/k.
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The data structure

• Keep for every vertex v?V
• The centers p1(v), p2(v),, pk-1(v)
• A hash table holding B(v)

For every w?V, we can check, inconstant time,
whether w?B(v), and if so, what is ?(v,w).
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Query answering algorithm
Algorithm distk(u,v) w?u , i?0 while
w?B(v) i ?i1 (u,v) ?(v,u) w
?pi(u) return ?(u,w) ?(w,v)
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Query answering algorithm
w3p3(v)?A3
w2p2(u)?A2
w1p1(v)?A1
u
v
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Analysis
wipi(u)?Ai
Claim 1 d(u,wi) i? , i evend(v,wi) i? , i
odd
wi-1pi-1(v)?Ai-1
(i1)?
i?
i?
Claim 2 d(u,wi) d(wi,v) (2i1)?
(i-1)?
(2k-1)?
v
u
?
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Where are the spanners?
Define clusters, the dual of bunches.
For every u?V, include in the spanner a tree of
shortest paths from u to all the vertices in the
cluster of u.
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Clusters
w
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Bunches and clusters
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Part II Spanners with sublinear surplus
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The construction used above, when applied to
unweighted graphs, produces spanners with
sublinear surplus!
We present a slightly modified construction with
a slightly simpler analysis.
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Balls
p2(v)
v
p1(v)
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The modified construction
The original construction
Select a hierarchy of centers A0??A1??Ak-1.
For every u?V, add to the spanner a shortest
paths tree of Clust(u).
Select a hierarchy of centers A0??A1??Ak-1.
For every u?V, add to the spanner a shortest
paths tree of Ball(u).
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Spanners with sublinear surplus
Select a hierarchy of centers A0??A1??Ak-1.
For every u?V, add to the spanner a shortest
paths tree of Ball(u).
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The path-finding strategy
Suppose we are at u?Ai and want to go to v.
Let ? be an integer parameter.
If the first xi?i-?i-1 edges of a shortest path
from u to v are in the spanner, then use them.
Otherwise, head for the (i1)-center ui1 nearest
to u.
? The distance to ui1 is at most xi. (As
u?Ball(u).)
/
ui1?Ai1
u?Ai
u
v
xi
xi
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The path-finding strategy
We either reach v, or at least make xi?i-?i-1
steps in the right direction.
Or, make at most xi?i-?i-1 steps, possibly in a
wrong direction, but reach a center of level i1.
If ik-1, we will be able to reach v.
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The path-finding strategy
After at most ?i steps
ui
either we reach v
or distance to v decreased by ?i -2?i-1
xi-1
xi?i-?i-1
xi-2
?i-1
x1
x0
u0
v
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The path-finding strategy
After at most ?i steps
Surplus 2?i-1
either we reach v
or distance to v decreased by ?i -2?i-1
Stretch
The surplus is incurred only once!
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Sublinear surplus
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Open problems

• Arbitrarily sparse additive spanners?
• Distance oracles with sublinear surplus?