Andrew Goldberg. Highway Dimension and Provably Efficient Shortest Path Algorithms

Highway Dimension and
Provably Eﬃcient
Shortest Paths Algorithms

Andrew V. Goldberg
Microsoft Research – Silicon Valley
www.research.microsoft.com/∼goldberg/

Joint with Ittai Abraham, Amos Fiat, and Renato Werneck

Theory & Practice

[Anonymous]
• Theory is when you know something, but it doesn’t work.
• Practice is when something works, but you don’t know why.
• Programmers combine theory and practice: Nothing works
and they don’t know why.

Highway dimension 1

Theory & Practice

[Anonymous]

Algorithm Engineering:
Know something ⇒ make it work.

Reverse Engineering:
Something works ⇒ explain why (this talk).

Highway dimension 2

Theory & Practice

[Anonymous]

Algorithm Engineering:
Know something ⇒ make it work.

Reverse Engineering:
Something works ⇒ explain why (this talk).

Eﬃcient driving directions: How and why modern routing al-
gorithms work.

Highway dimension 3

Motivation

Computing driving directions.
Recent shortest path algorithms (exact):

• Arc ﬂags [Lauther 04, K¨hler et al. 06].
o
• A∗ with landmarks [Goldberg & Harrelson 05].
• Reach [Gutman 04, Goldberg et al. 06].
• Highway [Sanders & Schultes 05] and contraction [Geisberger
08] hierarchies.
• Transit nodes [Bast et al. 06].
• Combinations (pruning and directing algorithms).

Good performance on continent-sized road networks:
Tens of millions of vertices, a few hundred vertices scanned,
under a millisecond on a server, about 0.1 second on a mobile
device. Used in practice.

Highway dimension 4

Result Summary

Elegant and practical algorithms, but...

Until now: No analysis for non-trivial network classes and no
understanding of what makes the algorithms work.

Our results:
• We deﬁne highway dimension (HD).
• Give provably eﬃcient implementations of several recent al-
gorithms under the small HD assumption.
• Analysis highlights algorithm similarities.
• Give a generative model of small HD networks (road network
formation).

Compare to the small world model [Milgram 67, Kleinberg 99].

Highway dimension 5

Outline

• Dijkstra’s algorithm review.
• Shortcuts and CH algorithm.
• Reach and RE algorithm.
• Transit nodes and TN algorithm.
• Highway dimension and sparse covers.
• Analysis of RE and CH.
• Generative model.
• Concluding remarks.

Highway dimension 6

Deﬁnitions

• Undirected graph G = (V, E), |V | = n, |E| = m.
• ℓ : E → N , minimum length 1, diameter D.
• Maximum degree ∆. Small for road networks, assume con-
stant for this talk.
• Ball Bv,r induced by vertices within distance r from v.
• For a path P , |P | denotes the length of P .
• Query: ﬁnd the shortest path from s to t.

Algorithms with preprocessing:
• Two phases, preprocessing and query.
• Preprocessing output not much bigger than the input.
• Preprocessing may use more resources than queries.
• Practical preprocessing, fast queries.
Highway dimension 7

Three Intuitive Methods

• Reach pruning (RE): Local intersections far from origin/destination
can be ignored.
• Transit nodes (TN): All long shortest paths to/from a region
pass through a small number of nodes.
• Highway/contraction hierarchies (CH): Shortest path goes
from local roads to local highways to global highways to
local highways to local roads.

These ideas can be mathematically formalized and lead to prov-
ably correct algorithms which work very well on road networks.

HD intuition based on TN; all preprocessing is based on CH;
query analysis in this talk is for RE.

Highway dimension 8

Scanning Method

• For each vertex v maintain its distance label ds(v) and status
S(v) ∈ {unreached, labeled, scanned}.

• Unreached vertices have ds(v) = ∞.

• If ds(v) decreases, v becomes labeled.

• To scan a labeled vertex v, for each arc (v, w),
if ds(w) > ds(v) + ℓ(v, w) set ds(w) = ds(v) + ℓ(v, w).

• Initially all vertices are unreached.

• Start by decreasing ds(s) to 0.

• While there are labeled vertices, pick one and scan it.

• Diﬀerent selection rules lead to diﬀerent algorithms.

Highway dimension 9

Dijkstra’s Algorithm

[Dijkstra 1959], [Dantzig 1963].
• At each step scan a labeled vertex with the minimum label.
• Stop when t is selected for scanning.

Work almost linear in the visited subgraph size.

Reverse Algorithm: Run algorithm from t in the graph with all
arcs reversed, stop when s is selected for scanning.

Bidirectional Algorithm
• Run forward Dijkstra from s and backward from t.
• Stopping criteria (careful!).

Highway dimension 10

Example Graph

1.6M vertices, 3.8M arcs, travel time metric.


Dijkstra’s Algorithm

Searched area

Bidirectional Algorithm

forward search/ reverse search


Contraction Hierarchies

[Geisberget et al. 08] Based on preprocessing that orders vertices
and adds shortcuts. Fast and simple!

Shortcuts:

100
101 110

1 10

11

A shortcut arc can be omitted if redundant (alternative path
exists).


CH Preprocessing

(1) Heuristically order vertices.
(2) Shortcut vertices in that order.
(3) To the original graph, add all shortcuts introduced in step 2.

For road networks, a good heuristic ordering is fast and increases
the number of arcs by a small constant factor.

Intuition: Higher-order vertices are more important (“global”).
Query is bidirectional, each search goes from lower to higher
order vertices.


CH Query

Let (V, E ∪ E +) be the graph augmented by shortcuts, recall that
the vertices are ordered. Let Gf be the directed graph induced
by
(v, w) : (v, w) ∈ E ∪ E +, v < w
and Gr – by
(w, v) : (v, w) ∈ E ∪ E +, v > w.

Given a query, run a modiﬁed bidirectional Dijkstra’s algorithm,
with the forward search in Gf and the reverse search in Gr . (Both
searches go “up”.)

Upon termination dist(s, t) = minv (d(s, v) + d(v, t)).

Experiments show that for road networks queries look at a very
small subgraph, even in the worst case.


Reach Intuition

Identify local intersections and prune them when searching far
from s and t.


Reach Deﬁnition

[Gutman 04]
• Consider a vertex v that splits a path P into P1 and P2.
rP (v) = min(ℓ(P1), ℓ(P2)).
• r(v) = maxP (rP (v)) over all shortest paths P through v.

Using reaches to prune Dijkstra:

LB(w,t)
d(s,v) v w

s t

If r(w) < min(d(v) + ℓ(v, w), LB(w, t)) then prune w.


Lower Bounds for Nothing

Bidirectional search gives implicit bounds (Rt below).

LB(w,t)
d(s,v) v w

s Rt t

Reach-based query algorithm is Dijkstra’s algorithm with prun-
ing based on reaches. Given a lower-bound subroutine, a small
change to Dijkstra’s algorithm.


Shortcuts

• Consider the graph below.
• Many vertices have large reach.

1000 1000

10 10 10 10 10 10 10 10
s 1000 1010 1020 1030 1040 1030 1020 1010 1000 t


Shortcuts

• Add a shortcut arc, break ties by the number of hops.

80
1000 1000

10 10 10 10 10 10 10 10
s t


Shortcuts

• Reaches decrease.

s 1000 60 50 40 30 40 50 60 1000 t


Shortcuts

• Repeat.

s 1000 20 10 20 30 20 10 20 1000 t


Reach Algorithm

• Repeat.
• A small number of shortcuts can greatly decrease many reaches.

RE algorithm: reaches with shortcuts, bidirectional Dijkstra,
balance the searches by the search radius.

s 1000 0 10 0 30 0 10 0 1000 t


Reach Algorithm


TN Algorithm: Intuition

For a region, there is a small set of nodes such that all suﬃciently
long shortest paths out of the region pass a vertex in the set.


Transit Node (TN) Algorithm

[Bast et. al 06]
• Divide a map into regions.
• For each region, paths to far away places pass through a
small number of access points.
• The union of all access nodes is the set of transit nodes.
• Preprocessing: ﬁnd all access points, connect each vertex
to its access nodes, compute all pairs of shortest paths be-
tween transit nodes.
• Query: Look for a “local” shortest path or a shortest path
of the form origin – transit node – transit node – destination.
• Various ways to speed up local queries.
• 10 access nodes per region on the average.
• Leads to the fastest algorithms.

Highway Dimension: Intuition

TN algorithm works because for a region, there is a small set
of nodes such that all suﬃciently long shortest paths out of the
region pass a vertex in the set.


Highway Dimension

P

u

4r

Highway dimension (HD) h:

∀ r ∈ ℜ, ∀u ∈ V, ∃S ⊆ Bu,4r , |S| ≤ h, such that
∀ v, w ∈ Bu,4r ,
if P is a SP: |P (v, w)| > r and P (v, w) ⊆ Bu,4r ,
then P (v, w) ∩ S = ∅.
Locally, a small set covers all long SPs.


Shortest Path Cover

P

u
2r

(r, k) Shortest path cover ((r, k)-SPC): a set C such that
∀ SP P : r < |P | ≤ 2r ⇒
P ∩ C = ∅ and
∀u ∈ V, |C ∩ Bu,2r | ≤ k
All SPs in a range can be covered by a sparse set.

Can use constants diﬀerent from 4 and 2, but the constants are
related.

Highway vs. Doubling Dimension

A metric space has a doubling dimension α if every ball or radius
r can be covered by 2α balls of radius r/2.

Line: Small HD

v

A line has HD 7 and doubling dimension 1.


Grid: High HD

√
A grid has HD Θ( n) and the doubling dimension 2.


HD vs. SPC

Theorem. If G has highway dimension h, then for any r there
exists an (r, h)-SPC.
Proof idea: Show that S ∗, the smallest set that covers all short-
est paths P : r < |P | ≤ 2r, is an (r, h)-SPC.

Finding S ∗ is NP-hard. Eﬃcient construction?

Theorem. If G has highway dimension h, then for any r we can
construct, in polynomial time, an (r, O(h log n))-SPC.
Proof idea: Use the greedy set-cover algorithm to get an O(log n)
approximation of S ∗.

Proofs depend on the choice of constants in the deﬁnitions.


Generic Preprocessing

Let k denote h or O(h log n) (for poly-time preprocessing).
• Let S0 = V . For 1 ≤ i ≤ log D build (2i, k)-SPC covers Si.
• Let Li = Si − log D Sj (vertex layers).
i+1
• Order vertices so that Li comes before Li+1; ordering inside
layers is arbitrary.
• Do shortcutting in this order to get E +.

Lemma. Let v ∈ Li and j ≥ i. Then the number of (v, w) ∈ E +
with w ∈ Lj is at most k.
Proof. (v, w) corresponds to P with internal vertices less than
v, w. Thus w ∈ Bv,2·2i . The SPC deﬁnition implies the lemma.

Theorem. In (V, E ∪ E +) vertex degrees are bounded by
∆ + k log D and |E ∪ E +| = O(m + nk log D).


Additional RE Preprocessing

Reach bounds are in the graph with shortcuts.

Lemma: If v ∈ Li then reach(v) ≤ 2 · 2i.
Proof: Suppose the reach is greater. Then there is a shortest
path P that v divides into P1 and P2 with |P1|, |P2| > 2 · 2i. Both
P1 and P2 contain vertices in Lj with j > i, so there is a shortcut
from P1 to P2. But then P is not a shortest path.

Additional work is linear.


RE Query Bound

Theorem: RE query takes O((k log D)2) time.
Proof: Consider a forward search from s. In Bs,2·2i , the search
scans only vertices of Li in Bs,2·2i . Thus O(k log D) scans.

• Shortest path can be extracted in time linear in the number
of its arcs.
• Similar analysis for CH yields the same bound.
• Also develop a faster version of TN: O(k log D) query.


Network Formation

Natural networks with constant highway dimension?

Attempt to model road networks:
• Build on the Earth surface (low doubling dimension).
• Build in decentralized and incremental manner.
• Highways are faster than local roads.
Capture some, but not all, real-life properties.

Setting:
• Metric space (M, d), doubling dimension log α, diameter D.
• Speedup parameter δ ∈ (0, 1).
• Edge {v, w} has length d(v, w) and transit time τ (v, w) =
d1−δ (v, w). (Long roads are fast.)
• On-line network formation, adversarial (or random) vertex
placement.

Network Formation (cont.)

In the spirit of dynamic spanners [Gottlieb & Roditty 08].
• Adversary adds vertices, we connect them.
• Intuition: connect a new village to all nearby villages and to
the closest town.
• Formally: maintain covers Ci for 0 ≤ i ≤ log D.
C0 = V , Ci+1 ⊆ Ci, vertices in Ci are at least 2i apart.
• When adding a new vertex v, add v to C0, . . . , Ci for appro-
priate i. (The ﬁrst vertex added to all C’s.)
• For 0 ≤ j ≤ i, connect v to Cj ∩ Bv,6·2j .
• If i < log D, connect v to the closest element of Ci+1.

Theorem: The network has highway dimension of αO(1).


Concluding Remarks

• Highway dimension explains why intuitive algorithms work
well on a non-trivial graph class.
• Uniﬁed view of several diﬀerent-looking algorithms.
• Extensions for directed graphs.
• Additional assumptions, such as small separators, may lead
to better bounds.
• For real road networks, small HD condition may not hold for
some r and some v.
• Does low HD make other problems simpler?
• Computing HD dimension of road networks (work in progress
with Daniel Delling): USA/Europe highway dimension is ≈
1, 000, access dimension ≈ 20.


Thank You!


Andrew Goldberg. Highway Dimension and Provably Efficient Shortest Path Algorithms

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