Highway and VC Dimensions:
from Practice to Theory and Back
Ittai Abraham Daniel Delling1 Amos Fiat2
Andrew V. Goldberg1 Renato F. Werneck1
1 Microsoft
Research – Silicon Valley
2 Tel
A.V. Goldberg (MSR-SVC)
Aviv University
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6/11/2013
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Highway Dimension 3.0
Original HD Definition
A.V. Goldberg (MSR-SVC)
Highway and VC Dimensions
6/11/2013
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Highway Dimension 3.0
Original HD Definition
New HD Definition
more intuitive results
close relationship of HD to doubling dimension
analysis of the transit node algorithm
better query bounds
A.V. Goldberg (MSR-SVC)
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Outline
1
Motivation
2
Transit Node Algorithm
3
VC-dimension and Shortest Paths
4
Highway Dimension and Shortest Path Covers (SPCs)
5
Computing SPCs
6
Analyzable Preprocessing
7
TN Algorithm Analysis
8
Hub Labeling Algorithm
9
Experimental Results
10
Concluding Remarks
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Shortest Paths: Recent Developments
Continent-sized road networks have 10s of millions intersections.
Dijkstra’s algorithm: ≈ 2 s
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Shortest Paths: Recent Developments
Continent-sized road networks have 10s of millions intersections.
Dijkstra’s algorithm: ≈ 2 s
Routing in Road Networks
arc flags [Lauther 04, Köhler et al. 06, Bauer & Delling 08]
A∗ with landmarks [Goldberg & Harrelson 05]
reach [Gutman 04], [Goldberg et al. 06]
highway hierarchies [Sanders & Schultes 05]
contraction hierarchies [Geisberger et al. 08]
transit nodes [Bast et al. 06]
DIMACS Shortest Paths Implementation Challenge (2005–2006)
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Shortest Paths: Recent Developments
Continent-sized road networks have 10s of millions intersections.
Dijkstra’s algorithm: ≈ 2 s
Routing in Road Networks
arc flags [Lauther 04, Köhler et al. 06, Bauer & Delling 08]
A∗ with landmarks [Goldberg & Harrelson 05]
reach [Gutman 04], [Goldberg et al. 06]
highway hierarchies [Sanders & Schultes 05]
contraction hierarchies [Geisberger et al. 08]
transit nodes [Bast et al. 06]
DIMACS Shortest Paths Implementation Challenge (2005–2006)
Greatly improved performance: << 1 ms, ≈ 0.1 s on a mobile device
Only a few hundred or less intersections searched
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Definitions and Model
Input
graph G = (V, E) with unique shortest paths
|V | = n, |E| = m
for this talk, assume G is undirected
weight function ` (length, transit time, fuel consumption, ...)
static problem, G and ` incorporate all modeling information
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Definitions and Model
Input
graph G = (V, E) with unique shortest paths
|V | = n, |E| = m
for this talk, assume G is undirected
weight function ` (length, transit time, fuel consumption, ...)
static problem, G and ` incorporate all modeling information
Query (multiple times for the same input network)
given origin s and destination t, find optimal path from s to t
exact algorithms help modeling and debugging
A.V. Goldberg (MSR-SVC)
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Definitions and Model
Input
graph G = (V, E) with unique shortest paths
|V | = n, |E| = m
for this talk, assume G is undirected
weight function ` (length, transit time, fuel consumption, ...)
static problem, G and ` incorporate all modeling information
Query (multiple times for the same input network)
given origin s and destination t, find optimal path from s to t
exact algorithms help modeling and debugging
Algorithms with preprocessing
two phases: practical preprocessing and real-time queries
preprocessing output not much bigger than the input
preprocessing may use more resources than queries
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Outline
1
Motivation
2
Transit Node Algorithm
3
VC-dimension and Shortest Paths
4
Highway Dimension and Shortest Path Covers (SPCs)
5
Computing SPCs
6
Analyzable Preprocessing
7
TN Algorithm Analysis
8
Hub Labeling Algorithm
9
Experimental Results
10
Concluding Remarks
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Transit Node (TN) Algorithm
[Bast et al. 06]
For a region, there is a small set of nodes such that all sufficiently long
shortest paths out of the region pass a node in the set
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TN Preprocessing
Basic concepts
divide a map into regions (a few thousand)
for each region, optimal paths to far away places pass
through one of a small number of access nodes (≈ 10 on
the average)
the union of access nodes is the set of transit nodes
(≈ 10 000)
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TN Preprocessing
Basic concepts
divide a map into regions (a few thousand)
for each region, optimal paths to far away places pass
through one of a small number of access nodes (≈ 10 on
the average)
the union of access nodes is the set of transit nodes
(≈ 10 000)
Empirical observation: small number of access/transit nodes
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TN Preprocessing
Basic concepts
divide a map into regions (a few thousand)
for each region, optimal paths to far away places pass
through one of a small number of access nodes (≈ 10 on
the average)
the union of access nodes is the set of transit nodes
(≈ 10 000)
Empirical observation: small number of access/transit nodes
Preprocessing algorithm
find access nodes for every region
connect each vertex to its access nodes
compute all pairs of shortest paths between transit nodes
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TN Query
Long-range query algorithm
the shortest path has the form
s – access(s) – access(t) – t
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TN Query
Long-range query algorithm
the shortest path has the form
s – access(s) – access(t) – t
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TN Query
Long-range query algorithm
the shortest path has the form
s – access(s) – access(t) – t
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TN Query
Long-range query algorithm
the shortest path has the form
s – access(s) – access(t) – t
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TN Query
Long-range query algorithm
the shortest path has the form
s – access(s) – access(t) – t
table look-up for the
(access(s), access(t)) node
pairs
Remarks
very fast: 10 × 10 table look-ups per long-range query
local queries: another method or multilevel approach
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Theoretical Results
Practice
intuitive and practical algorithms, but...
why do they work well on road networks?
what is a road network (formally)?
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Theoretical Results
Practice
intuitive and practical algorithms, but...
why do they work well on road networks?
what is a road network (formally)?
Theory [Abraham et al. 10, 11]
define highway dimension (HD)
good time bounds for transit nodes, highway hierarchies, and
reach algorithms assuming HD is small
even better bound for the hub labeling (HL) algorithm [Gavoille et
al. 01, Cohen et al. 02]
improved practical performance [Abraham et al. 11]
analysis highlights algorithm similarities
generative model of small HD networks (road network formation)
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Outline
1
Motivation
2
Transit Node Algorithm
3
VC-dimension and Shortest Paths
4
Highway Dimension and Shortest Path Covers (SPCs)
5
Computing SPCs
6
Analyzable Preprocessing
7
TN Algorithm Analysis
8
Hub Labeling Algorithm
9
Experimental Results
10
Concluding Remarks
A.V. Goldberg (MSR-SVC)
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6/11/2013
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VC-Dimension [Vapnik & Chervonenkis 71]
base set X, collection of subsets R, set system (X, R)
for Y ⊆ X, Y|R = (Y, {R ∩ Y|R ∈ R})
R shatters Y if Y|R = 2Y
(X, R) has VC-dimension d if d is the smallest integer such that no
d + 1 subset of X can be shattered
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VC-Dimension [Vapnik & Chervonenkis 71]
base set X, collection of subsets R, set system (X, R)
for Y ⊆ X, Y|R = (Y, {R ∩ Y|R ∈ R})
R shatters Y if Y|R = 2Y
(X, R) has VC-dimension d if d is the smallest integer such that no
d + 1 subset of X can be shattered
Example (points and half-planes)
X is a plane, R is the set of all half-planes, VC-dimension is 3
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VC-Dimension [Vapnik & Chervonenkis 71]
base set X, collection of subsets R, set system (X, R)
for Y ⊆ X, Y|R = (Y, {R ∩ Y|R ∈ R})
R shatters Y if Y|R = 2Y
(X, R) has VC-dimension d if d is the smallest integer such that no
d + 1 subset of X can be shattered
Example (points and half-planes)
X is a plane, R is the set of all half-planes, VC-dimension is 3
A.V. Goldberg (MSR-SVC)
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VC-Dimension [Vapnik & Chervonenkis 71]
base set X, collection of subsets R, set system (X, R)
for Y ⊆ X, Y|R = (Y, {R ∩ Y|R ∈ R})
R shatters Y if Y|R = 2Y
(X, R) has VC-dimension d if d is the smallest integer such that no
d + 1 subset of X can be shattered
Example (points and half-planes)
X is a plane, R is the set of all half-planes, VC-dimension is 3
A.V. Goldberg (MSR-SVC)
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VC-Dimension [Vapnik & Chervonenkis 71]
base set X, collection of subsets R, set system (X, R)
for Y ⊆ X, Y|R = (Y, {R ∩ Y|R ∈ R})
R shatters Y if Y|R = 2Y
(X, R) has VC-dimension d if d is the smallest integer such that no
d + 1 subset of X can be shattered
Example (points and half-planes)
X is a plane, R is the set of all half-planes, VC-dimension is 3
A.V. Goldberg (MSR-SVC)
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VC-Dimension [Vapnik & Chervonenkis 71]
base set X, collection of subsets R, set system (X, R)
for Y ⊆ X, Y|R = (Y, {R ∩ Y|R ∈ R})
R shatters Y if Y|R = 2Y
(X, R) has VC-dimension d if d is the smallest integer such that no
d + 1 subset of X can be shattered
Example (points and half-planes)
X is a plane, R is the set of all half-planes, VC-dimension is 3
A.V. Goldberg (MSR-SVC)
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6/11/2013
12 / 38
VC-Dimension [Vapnik & Chervonenkis 71]
base set X, collection of subsets R, set system (X, R)
for Y ⊆ X, Y|R = (Y, {R ∩ Y|R ∈ R})
R shatters Y if Y|R = 2Y
(X, R) has VC-dimension d if d is the smallest integer such that no
d + 1 subset of X can be shattered
Example (points and half-planes)
X is a plane, R is the set of all half-planes, VC-dimension is 3
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USP Systems
Definition
Unique Shortest Path (USP) Systems
G is a graph with unique shortest paths, X = V
R contains (some) sets of vertices on shortest paths
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USP Systems
Definition
Unique Shortest Path (USP) Systems
G is a graph with unique shortest paths, X = V
R contains (some) sets of vertices on shortest paths
Strange question: VC-dimension of a USP system?
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USP Systems
Definition
Unique Shortest Path (USP) Systems
G is a graph with unique shortest paths, X = V
R contains (some) sets of vertices on shortest paths
Strange question: VC-dimension of a USP system?
Theorem
VC-dimension of a USP system is at most two
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USP Systems
Definition
Unique Shortest Path (USP) Systems
G is a graph with unique shortest paths, X = V
R contains (some) sets of vertices on shortest paths
Strange question: VC-dimension of a USP system?
Theorem
VC-dimension of a USP system is at most two
Proof:
three points not on a shortest path cannot be shattered
three points on a (unique) shortest path cannot be shattered
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Outline
1
Motivation
2
Transit Node Algorithm
3
VC-dimension and Shortest Paths
4
Highway Dimension and Shortest Path Covers (SPCs)
5
Computing SPCs
6
Analyzable Preprocessing
7
TN Algorithm Analysis
8
Hub Labeling Algorithm
9
Experimental Results
10
Concluding Remarks
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r-Significant Paths
Given a scale r, which paths are significant?
r < `(P) ≤ 2r?
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r < `(P)?
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r-Significant Paths
Given a scale r, which paths are significant?
r < `(P) ≤ 2r?
r < `(P)?
continuous graph yields more intuitive results
r
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r-Significant Paths
Given a scale r, which paths are significant?
r < `(P) ≤ 2r?
r < `(P)?
continuous graph yields more intuitive results
r
Definition
P is r-significant if P with optional single-edge extensions at one or
both ends is a shortest path of length greater than r
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r-Significant Paths
Given a scale r, which paths are significant?
r < `(P) ≤ 2r?
r < `(P)?
continuous graph yields more intuitive results
r
Definition
P is r-significant if P with optional single-edge extensions at one or
both ends is a shortest path of length greater than r
if P = (v, w) is a single-edge path with `(P) > r, then trivial paths
{v} and {w} are r-significant
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More Definitions
assume that G is connected and every edge is a shortest path
network diameter D
Pr : the set of all r-significant paths
Br (v) = {w : dist(v, w) ≤ r} (a ball)
hitting set for (X, R): H ⊆ X : ∀R ∈ R, H ∩ R 6= ∅
dist(v, P) = min{dist(v, w) : w ∈ P}
Sr (v) = {P ∈ Pr : dist(v, P) ≤ 2r} (neighborhood system)
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Highway Dimension Motivation
For a region, there is a small set of nodes such that all sufficiently long
shortest paths out of the region pass a node in the set
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Highway Dimension Definition
Highway dimension (HD) h
For every r > 0 and v ∈ V, ∃ a hitting set of size
at most h covering all r-significant P :
dist(v, P) ≤ 2r
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v
B2r(v)
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Highway Dimension Definition
Highway dimension (HD) h
For every r > 0 and v ∈ V, ∃ a hitting set of size
at most h covering all r-significant P :
dist(v, P) ≤ 2r
v
B2r(v)
Lemma
If C is a hitting set for a “nice” set S of r-significant paths and v is an
endpoint of P ∈ S, then ∃u ∈ P ∩ C with dist(v, u) ≤ r
Proof:
v u
r
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Relationship to Doubling Dimension
Doubling dimension (DD)
A metric space has a doubling dimension d if ∀r > 0, every ball
of radius 2r can be covered by 2d balls of radius r
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Relationship to Doubling Dimension
Doubling dimension (DD)
A metric space has a doubling dimension d if ∀r > 0, every ball
of radius 2r can be covered by 2d balls of radius r
Theorem
If G, ` has HD h, then the metric space induced by G, ` has DD
log(h + 1)
Proof:
cover the ball B2r (v)
r-ball centers: S =
{v} ∪ {hitting set for r-significant paths at distance ≤ 2r from v}
|S| ≤ h + 1
w ∈ Br (v) is covered
w ∈ B2r (v) − Br (v): v to w s.p. contains u ∈ S : dist(u, w) ≤ r
w is covered
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Star Graph
n-vertex star graph, unit edge lengths
doubling dimension: log n
HD (old definition): 1
HD (new definition): n
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New/Improved Query Bounds
Lemma
A graph with HD h has the maximum degree of at most h
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New/Improved Query Bounds
Lemma
A graph with HD h has the maximum degree of at most h
Improved query bounds
algorithm
old bound
transit nodes NA
(global)
reach
O((∆ + h log D)(h log D))
contraction
O((∆ + h log D)(h log D))
hierarchies
hub labels
O(h log D)
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new bound
O ( h2 )
O((h log D)2 )
O((h log D)2 )
O(h log D)
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Shortest Path Covers (SPCs)
(r, k) Shortest path cover ((r, k)-SPC):
All paths in Pr can be hit by a sparse set
A set C such that C is an r-hitting set for Pr and
∀v ∈ V, |C ∩ B2r (v)| ≤ k
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P
v
B 2r(v)
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Shortest Path Covers (SPCs)
(r, k) Shortest path cover ((r, k)-SPC):
All paths in Pr can be hit by a sparse set
A set C such that C is an r-hitting set for Pr and
∀v ∈ V, |C ∩ B2r (v)| ≤ k
P
v
B 2r(v)
Theorem
If G has HD h, then ∀ r ∃ an (r, h)-SPC.
Proof:
Show S∗ , the smallest r-hitting set for Pr , is an (r, h)-SPC
P
Let S0 (v) = S∗ ∩ B2r (v)
Suppose |S0 (v)| > h
v
S0 (v) hits only paths in Sr (v)
B 2r(v)
Let H be the r-hitting set of Sr such that |H | ≤ h
Replacing S0 (v) by H gives a smaller set S∗
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Computing Approximate SPCs
Finding S∗ is NP-hard. Efficient construction?
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Computing Approximate SPCs
Finding S∗ is NP-hard. Efficient construction?
Greedy gives an O(log n) approximation [Abraham et al. 10].
Approximation independent of n?
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Computing Approximate SPCs
Finding S∗ is NP-hard. Efficient construction?
Greedy gives an O(log n) approximation [Abraham et al. 10].
Approximation independent of n?
Theorem
Suppose we have a poly-time, (c log h)-approximation algorithm for
hitting set. If G has HD h, then for any r we can construct, in
polynomial time, an (r, O(h log h))-SPC.
Proof: Similar to the previous proof. Maintain a hitting set C. If for
some v, |C ∩ B2r | > ch log h, compute a hitting set for (V, Sr ) of size at
most ch log h and get a smaller hitting set C.
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Hitting Sets in Low VC-dimension
[Clarkson 88, Brönnimann & Goodrich 95, Even et al. 95]
Theorem
For a set system with optimal hitting set of size h and VC-dimension d,
an O(hd log(hd)) hitting set is poly-time computable
Corollary
For an HD h graph, we can efficiently compute an O(h log h)-size
hitting set for (V, Sr (v))
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Outline
1
Motivation
2
Transit Node Algorithm
3
VC-dimension and Shortest Paths
4
Highway Dimension and Shortest Path Covers (SPCs)
5
Computing SPCs
6
Analyzable Preprocessing
7
TN Algorithm Analysis
8
Hub Labeling Algorithm
9
Experimental Results
10
Concluding Remarks
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Multi-scale SPCs
k = h or O(h log h) depending on preprocessing complexity
Definition
Ci = (2i−1 , k)-SPC for 0 ≤ i ≤ log D
Can construct “dense” Ci s:
C0 = V
|Clog D | ≤ k
|Ci | ≤ k|Ci+1 |
Basis for our preprocessing algorithms
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Outline
1
Motivation
2
Transit Node Algorithm
3
VC-dimension and Shortest Paths
4
Highway Dimension and Shortest Path Covers (SPCs)
5
Computing SPCs
6
Analyzable Preprocessing
7
TN Algorithm Analysis
8
Hub Labeling Algorithm
9
Experimental Results
10
Concluding Remarks
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TN Algorithm Analysis
Preprocessing
√
pick T so that T2 is not too big (e.g., T = m)
pick smallest t : |Ct | ≤ T
let Ct be the set of transit nodes
compute distances between transit nodes
compute access nodes A(v): build s.p. tree from v, include
w ∈ Ct iff the v to w tree path has no other elements of Ct
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TN Algorithm Analysis
Preprocessing
√
pick T so that T2 is not too big (e.g., T = m)
pick smallest t : |Ct | ≤ T
let Ct be the set of transit nodes
compute distances between transit nodes
compute access nodes A(v): build s.p. tree from v, include
w ∈ Ct iff the v to w tree path has no other elements of Ct
Lemma
∀v, |A(v)| ≤ k
Proof. If v ∈ Ct , then |A(v)| = 1
Otherwise ∀w ∈ A(v), dist(v, w) ≤ 2 · 2t−1 , so w ∈ B2·2t−1 (v)
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TN Algorithm Analysis
Preprocessing
√
pick T so that T2 is not too big (e.g., T = m)
pick smallest t : |Ct | ≤ T
let Ct be the set of transit nodes
compute distances between transit nodes
compute access nodes A(v): build s.p. tree from v, include
w ∈ Ct iff the v to w tree path has no other elements of Ct
Lemma
∀v, |A(v)| ≤ k
Proof. If v ∈ Ct , then |A(v)| = 1
Otherwise ∀w ∈ A(v), dist(v, w) ≤ 2 · 2t−1 , so w ∈ B2·2t−1 (v)
Theorem
TN global queries take O(k2 ) time
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TN: Global vs. Local
Global query correctness
let g(v, w) be the global query value
if dist(v, w) > 2t−1 , then g(v, w) = dist(v, w)
if dist(v, w) ≤ 2t−1 , g(v, w) may be too big
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TN: Global vs. Local
Global query correctness
let g(v, w) be the global query value
if dist(v, w) > 2t−1 , then g(v, w) = dist(v, w)
if dist(v, w) ≤ 2t−1 , g(v, w) may be too big
how big can g(v, w) get in this case?
∀u ∈ A(v), dist(v, u) ≤ 2t
g(v, w) ≤ 2t + 2t−1 + 2t = 5 · 2t−1
if g(v, w) > 5 · 2t−1 , the global query is correct
otherwise, run a local query algorithm
A.V. Goldberg (MSR-SVC)
Highway and VC Dimensions
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TN: Global vs. Local
Global query correctness
let g(v, w) be the global query value
if dist(v, w) > 2t−1 , then g(v, w) = dist(v, w)
if dist(v, w) ≤ 2t−1 , g(v, w) may be too big
how big can g(v, w) get in this case?
∀u ∈ A(v), dist(v, u) ≤ 2t
g(v, w) ≤ 2t + 2t−1 + 2t = 5 · 2t−1
if g(v, w) > 5 · 2t−1 , the global query is correct
otherwise, run a local query algorithm
distance oracle without Euclidean distances
different such oracle (independently discovered): [Arz et al. 13]
A.V. Goldberg (MSR-SVC)
Highway and VC Dimensions
6/11/2013
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Outline
1
Motivation
2
Transit Node Algorithm
3
VC-dimension and Shortest Paths
4
Highway Dimension and Shortest Path Covers (SPCs)
5
Computing SPCs
6
Analyzable Preprocessing
7
TN Algorithm Analysis
8
Hub Labeling Algorithm
9
Experimental Results
10
Concluding Remarks
A.V. Goldberg (MSR-SVC)
Highway and VC Dimensions
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Hub Labeling Algorithm
[Gavoille et al. 01, Cohen et al. 02]
Generic preprocessing
∀v ∈ V precompute a label L(v) ⊆ V
∀w ∈ L(v) precompute distances dist(v, w)
Cover property: ∀s, t, a shortest s-t path intersects L(s) ∩ L(t)
A.V. Goldberg (MSR-SVC)
Highway and VC Dimensions
6/11/2013
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Hub Labeling Algorithm
[Gavoille et al. 01, Cohen et al. 02]
Generic preprocessing
∀v ∈ V precompute a label L(v) ⊆ V
∀w ∈ L(v) precompute distances dist(v, w)
Cover property: ∀s, t, a shortest s-t path intersects L(s) ∩ L(t)
Query algorithm
Find and return minu∈L(s)∩L(t) (dist(s, u) + dist(u, t))
A.V. Goldberg (MSR-SVC)
Highway and VC Dimensions
6/11/2013
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Hub Labeling Algorithm
[Gavoille et al. 01, Cohen et al. 02]
Generic preprocessing
∀v ∈ V precompute a label L(v) ⊆ V
∀w ∈ L(v) precompute distances dist(v, w)
Cover property: ∀s, t, a shortest s-t path intersects L(s) ∩ L(t)
Query algorithm
Find and return minu∈L(s)∩L(t) (dist(s, u) + dist(u, t))
Remarks
very simple query, efficient if the labels are small
A.V. Goldberg (MSR-SVC)
Highway and VC Dimensions
6/11/2013
31 / 38
HL for Low HD
Low HD preprocessing
for i = 0, 1, . . . , log D compute (2i−1 , k)-SPC Ci
∀v ∈ V set L(v) =
S
i (Ci
∩ B2·2i (v))
Lemma: ∀v ∈ V, |L(v)| = O(k log D)
Correctness
let P be the shortest s-t path and 2i < `(P) ≤ 2i+1
P ⊆ B2·2i (s) and P ⊆ B2·2i (t)
∃u ∈ P ∩ Ci , for such u we have u ∈ L(s) ∩ L(t)
Theorem
HL preprocessing can be poly-time and queries take O(k log D) time
A.V. Goldberg (MSR-SVC)
Highway and VC Dimensions
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Practical Implementation
Query
∀v, sort hubs by vertex ID
merge-sort like search for label intersection
good locality
A.V. Goldberg (MSR-SVC)
Highway and VC Dimensions
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Practical Implementation
Query
∀v, sort hubs by vertex ID
merge-sort like search for label intersection
good locality
distance queries < 300 ns.
A.V. Goldberg (MSR-SVC)
Highway and VC Dimensions
6/11/2013
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Practical Implementation
Query
∀v, sort hubs by vertex ID
merge-sort like search for label intersection
good locality
distance queries < 300 ns.
Preprocessing
theoretical preprocessing impractical
use theory-guided heuristics
new algorithmic ideas and careful engineering
preprocessing time: 4 min. [Abraham et al. 11]
A.V. Goldberg (MSR-SVC)
Highway and VC Dimensions
6/11/2013
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Practical Implementation
Query
∀v, sort hubs by vertex ID
merge-sort like search for label intersection
good locality
distance queries < 300 ns.
Preprocessing
theoretical preprocessing impractical
use theory-guided heuristics
new algorithmic ideas and careful engineering
preprocessing time: 4 min. [Abraham et al. 11]
average label size 85 (recently improved to 69)
A.V. Goldberg (MSR-SVC)
Highway and VC Dimensions
6/11/2013
33 / 38
Practical Implementation
Query
∀v, sort hubs by vertex ID
merge-sort like search for label intersection
good locality
distance queries < 300 ns.
Preprocessing
theoretical preprocessing impractical
use theory-guided heuristics
new algorithmic ideas and careful engineering
preprocessing time: 4 min. [Abraham et al. 11]
average label size 85 (recently improved to 69)
A.V. Goldberg (MSR-SVC)
Highway and VC Dimensions
6/11/2013
33 / 38
Outline
1
Motivation
2
Transit Node Algorithm
3
VC-dimension and Shortest Paths
4
Highway Dimension and Shortest Path Covers (SPCs)
5
Computing SPCs
6
Analyzable Preprocessing
7
TN Algorithm Analysis
8
Hub Labeling Algorithm
9
Experimental Results
10
Concluding Remarks
A.V. Goldberg (MSR-SVC)
Highway and VC Dimensions
6/11/2013
34 / 38
Experimental Results
query [ns]
space [GB]
prepr [h:m]
CH
79 373
0.4
0:14
TNR
1 711
2.5
0:21
TNR-AF
922
5.7
2:00
HL
556
18.8
0:37
HL-gl
256
17.7
60:00
HLC
2 554
1.8
0:50
Random queries, Western Europe, n = 18M, m = 42M
CH: [Geisberger et al. 08]
TNR: [Arz et al. 13]
TNR-AF: [Bauer et al. 10]
HL: [Abraham et al. 12]
HL-gl: [Abraham et al. 12]
HLC: [Delling et a. 13]
Good HL performance motivated further work (HHL, HLC, HLDB)
A.V. Goldberg (MSR-SVC)
Highway and VC Dimensions
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Concluding Remarks
Research Directions
complexity of computing SPCs in small HD networks
(NP-hard?)
practical algorithms for SPCs for large road networks
other applications of HD and SPCs
Complexity of hierarchical hub labels.
A.V. Goldberg (MSR-SVC)
Highway and VC Dimensions
6/11/2013
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Concluding Remarks
Research Directions
complexity of computing SPCs in small HD networks
(NP-hard?)
practical algorithms for SPCs for large road networks
other applications of HD and SPCs
Complexity of hierarchical hub labels.
Science of Algorithmics
experimental observations: several algorithms are very fast, small
number of transit nodes
theory: highway dimension and sublinear query bounds
prediction: the hub labeling algorithm is fast
verification: practical implementation and experiments
A.V. Goldberg (MSR-SVC)
Highway and VC Dimensions
6/11/2013
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Thank You!
SPA (Shortest Path Algorithms) project page
http://research.microsoft.com/en-us/projects/SPA/
A.V. Goldberg (MSR-SVC)
Highway and VC Dimensions
6/11/2013
37 / 38
Questions?
1
Motivation
2
Transit Node Algorithm
3
VC-dimension and Shortest Paths
4
Highway Dimension and Shortest Path Covers (SPCs)
5
Computing SPCs
6
Analyzable Preprocessing
7
TN Algorithm Analysis
8
Hub Labeling Algorithm
9
Experimental Results
10
Concluding Remarks
A.V. Goldberg (MSR-SVC)
Highway and VC Dimensions
6/11/2013
38 / 38