Distance oracles for unweighted graphs :
breaking the quadratic barrier with constant
additive error
Surender Baswana
⋆
, Akshay Gaur
∗∗
, Sandeep Sen
∗∗
, and Jayant Upadhyay
⋆⋆
Abstract. Thorup and Z wick, in the seminal paper [Journal of ACM,
52(1), 2005, pp 1-24], showed that a weighted undirected graph on n
vertices can be preprocessed in subcubic time to design a data struc-
ture which occupies only subquadratic space, and yet, for any pair of
vertices, can answer distance query approximately in constant time. The
data stru cture is termed as approximate distance oracle. Subsequently,
there has been improvement in their preprocessing time, and presently
the best known algorithms [4, 3] achieve expected O ( n
2
) preprocessing
time for these oracles. For a class of graphs, these algorithms indeed run
in Θ(n
2
) time. In this paper, we are able to break this quadratic barrier
at the expense of introducing a (small) constant additive error for un-
weighted graphs. In achieving this goal, we have been able to preserve
the optimal size-stretch trade offs of the oracles. One of our algorithms
can be extended to weighted graphs, where the additive error becomes
2 · w
max
(u, v) - here w
max
(u, v) is the heaviest edge in the sh ortest path
between vertices u, v.
1 Introduction
Let G = (V, E) be a graph on |V | = n vertices and |E| = m edges, and δ(u, v) de-
note the distance between any pair of vertices u, v ∈ V in graph G. The all-pairs
shortest paths (APSP) pr oblem requires preprocessing the given graph G so a s to
build a data structure using which we can retrieve distance or the s hortest path
between any pa ir of vertices efficiently. APSP is undoubtedly one o f the most
fundamental a lgorithmic graph pro blems of computer science. Despite being a
classical problem with widespread applications, there exists a huge gap between
the lower bound Ω(n
2
) and the worst case upper bound O(n
3
/ log
2
n) (due to
Chan [6]) of the time complexity of APSP problem. Furthermore, Θ(n
2
) space
requirement is a major bottleneck for graphs in many large scale applications.
These two factors have motivated researchers to desig n e fficient algorithms (or
data structures) for reporting approximate distances. In the last fifteen years,
⋆
Department of Comp. Sc. & Engg. Indian Institute of Technology Kanpur, Kan-
pur - 208016, I ndia. Email : sbaswana@iitk.ac.in. The work was supported by a
fellowship from Research I Found ation, CSE, IIT Kanpur.
⋆⋆
Department of Comp. Sc. & Engg., Indian Institute of Technology Delhi, New Delhi-
110016, India. Email : {manu,ssen,jayant}@cse.iitd.ernet.in
many novel algorithms [1, 8, 7, 2, 11] have been designed which work for undi-
rected g raphs. However, among all these algorithms the approximate distance
oracles designed by Thorup and Zwick [12] dese rve spec ial mention. They showed
that any given weighted undirected graph on n vertices can be preprocessed in
sub-cubic time for any integer t ≥ 3 to build a data structur e of sub-quadratic
size which for any pair of vertices u, v reports t-approximate distance - at least
δ(u, v) and at most tδ(u, v). There are two very impressive features of their data
structure. First, the trade-off between stretch t and the size of data structur e is
essentially optimal a ssuming a 196 3 girth lower bound conjecture of Erd˝os [9]
and second, in spite of its s ub- quadratic size their data structure c an answer
any dista nce query in constant time, hence the name “oracle”. More precisely,
Thorup and Zwick achieved the following result.
Theorem 1. [12] For any integer k ≥ 1, an undirected weighted graph on n
vertices and m edges can be preprocessed in expected O(kmn
1/k
) time to build
a data structure of size O(kn
1+1/k
) that can answer any (2k − 1)-approximate
distance query in O(k) time.
Having achieved optimal size-stretch trade offs, and es sentially constant query
time, it is only the preprocessing time of these oracles which may be improved.
The preprocessing time has been improved to O(min(n
2
, kmn
1/k
)) for unweighted
graphs [4], and recently for weighted graphs as well [3]. Therefore, a natural
question is whether it is possible to achieve O (m + n
2−ǫ
) - a subquadratic upp e r
bound for constructing approximate distance oracles. Note that any approximate
all pairs shortest path algorithm takes Ω(n
2
) steps beca us e of the output size.
Therefore, a sub-quadr atic time oracle construction provides a cle ar advantage
over such algorithms when we are not interested in all the pair-wise distances.
The main objective here is to achieve sub-quadratic preprocessing time for ap-
proximate distance oracles without violating the size-stre tch trade off. It may be
noted that the quadratic upper bound of the existing preprocess ing a lgorithms
[12][4] for these oracles is indeed tight - there exists a family of graphs on which
these algorithms would execute in Θ(n
2
) time.
In this paper, we design approximate distance oracles which, at the expense
of constant additive e rror, are constructable in sub-quadratic time and preserve
size stretch trade-off optimally. More precisely, we show the following. For any
k > 1, there is a data-structure which occupies O(kn
1+1/k
) space such that for
any pair of vertices u, v ∈ V , it takes O(k) time to return
ˆ
δ(u, v) satisfying
δ(u, v) ≤
ˆ
δ(u, v) ≤ (2k − 1)δ(u, v) + c
k
where c
k
= 2 for k ≥ 3 and c
2
= 8
As a natural ex tension of (2k−1)-approximate distance oracle of [12], we denote
the above oracle by (2k−1, c
k
)-approximate distance oracle, where the first term
(2k−1) is the stretch (multiplicative error) and c
k
is the surplus (a dditive error).
The expected preproces sing time for (2k − 1, c
k
) oracle is O(m + kn
2−α
k
), where
α
k
takes value in the interval [
1
12
,
1
2
) - takes value
1
12
for k = 2 and approaches
1
2
steadily as k increases (see Table 1).
In short, the small additive error has allowed us to break the quadratic barrier
of preproc e ssing o f approximate distance oracles . It would be very important to
Stretch Space Preprocessing Time Reference
(2k − 1, 0) O(kn
1+1/k
) O(min(n
2
, kmn
1/k
)) [12, 3, 4]
(3, 8) O(n
3/2
) O(min(m + n
23
12
, m
√
n)) this paper
(2k −1, 2), k ≥ 3 O(k n
1+1/k
) O(min(m + kn
3
2
+
1
2k
+
1
2k−2
, kmn
1
k
))
this paper
Table 1. Comparing the new algorithms with the existing algorithms for approximate
distance oracles.
explore the limits to which the preprocess ing time can be further improved. The
result of this paper can be viewed as the first significant step in this direction.
1.1 Overview of the new algorithms
The observation which forms the basis of our algorithms is the simple fact that
the O(kmn
1/k
) time complexity of the algorithm of Thorup and Zwick [12] is
already sub-quadratic provided the graph is sparse enough. In order to utilize
this observation, we use the idea of pa rtitioning the graph into sparse and dense
subgraphs. Previously this idea was used by the algorithms which compute all-
pairs approximate distance with purely additive error only [1, 8]. Using a random
sample S ⊆ V of vertices, we define a sparse subgraph with o(n
2−1/k
) edges,
and execute Thorup and Zwick algorithm on this sub graph. This algorithm will
execute in o(n
2
) time and will easily take care of the case when the shortest
paths between a pair o f vertices is fully preserved in the sparse graph. Novelty
of our algorithms is to handle the other case. Our algorithms make use of a
combination of old and new ideas which enables achieving sub-quadratic space
without compromising the optimal size-stretch trade-off. In o rder to make these
ideas work, our a lgorithms effectively use suitable emulators and spanners which
are sufficiently sparse.
Definition 1 An (α, β)-spanner of a graph G = (V, E) is a subgraph (V, E
′
), E
′
⊆
E with the property that distance bet ween any two vertices u, v ∈ V in the span-
ner is at least δ(u, v) and at most α(δ(u, v)) + β.
Definition 2 An (α, β)-emulator of a graph G = (V, E) is a weighted graph
(V, E
∗
) such that the distance δ
∗
(u, v) between any two vertices u, v ∈ V in the
emulator is at least δ(u, v) and at most αδ(u, v) + β.
In the following section we describe the nota tions and lemmas which will be used
throughout this paper. In section 3, we describe our (3, 8)-approximate distance
oracle. In sec tion 4, we des c ribe (2k−1, 2)-approximate distance oracle for k > 2.
2 Preliminaries
For a given graph G = (V, E), and any subs e t S ⊆ V , we shall use the following
notations :
– N (v) : the set consisting of v and every neighbor of v in the graph G.
– N (S) : ∪
v∈S
N (v).
– p
S
(v) : the vertex from set S which is nearest to v (break the tie arbitrarily
in case there are multiple nearest vertices).
– δ(v, S) : distance between v and p
S
(v).
– E(v) : the set of edges in G which are incident on v.
– E
S
(v) : the se t E(v ) if v is not adjacent to any vertex of set S, and ∅ otherwise.
– E
S
: ∪
v∈V
E
S
(v).
– G
S
: the subgraph (V, E
S
).
– O
G
t
: the t-approximate distance oracle of Thorup and Zwick [12] created on
a subgraph G of G.
Our data structure will s tore information about p
S
(v) and δ(v, S) for each vertex
in the given graph G. To compute this information, the gra ph G can be processed
in just O(m) time as follows : insert a dummy vertex o in to the graph, connect
it to all the vertices of set S, and perform a BFS traversal on the graph starting
from o. We shall use T
S
to denote the set of edges of this BFS tree excluding
the edges incident on the dummy vertex.
Lemma 1. The edge set T
S
preserves the shortest path between v and p
S
(v) for
all v ∈ V . The size of T
S
is O(n).
Now we redefine an important concept (due to Thorup and Zwick [12]) of ball
around a vertex.
Definition 3 [12] For a vertex u ∈ V and a set S ⊆ V in a graph G = (V, E),
we define ball(u, V, S) as the sub graph induced by all t hose vertices v ∈ V which
satisfy δ(u, v) < δ(u, S) (i.e., for u, it is v which is n earer than p
S
(u)).
We now state the following Lemma about the number of vertices and edges in
ball(u, V, S) when S is formed by random sampling.
Lemma 2. [12, 4] For a given graph G = (V, E), let S ⊆ V be a set formed
by selecting each vertex from V independently with probability q > 0. Then the
expected nu m ber of vertices and expected number of edges in ball(u, V, S) are
O(1/q) and O(1/q
2
) respectively.
We shall now state a few imp ortant Lemmas about the sparse subgraph (V, E
S
).
Lemma 3. If set S ⊆ V is formed by selecting each vertex independently with
probability q > 0, the expected size of the set E
S
would be O(n/q).
Lemma 4. If on the shortest path between any two vertices u, v ∈ V in the
graph G = (V, E) there are no two consecutive vertices in set N (S), then the
shortest path between u and v is preserved exactly in the subgraph (V, E
S
).
The above property of the edge set E
S
will prove to be very useful in our con-
struction. For the other case we observe the following.
Lemma 5. If the shortest path between u and v in the graph G contains at least
2 consecutive vertices from N (S), then δ(u, S) + δ(v , S) ≤ δ(u, v) + 1.
Proof. Suppose on the shortest path betwe e n u and v, u
′
be the vertex from
the set N (S) nearest to u, and v
′
be the vertex from the s e t N (S) nearest to
v. Since u
′
∈ N (S) either u
′
belongs to S or some neighbor of u
′
belongs to S.
This implies that δ(u, u
′
) ≥ δ(u, S) − 1. Similarly δ(v, v
′
) ≥ δ(v, S) − 1. Also
note that δ(u , u
′
) + δ(u
′
, v
′
) + δ(v
′
, v) = δ(u, v). Furthermore, δ(u
′
, v
′
) ≥ 1 since
there are at least two vertices fr om N (S) on the shortest path between u and v.
Therefore, δ(u, u
′
) + δ(v
′
, v) + 1 ≤ δ(u , v). This along with the lower bounds o n
δ(u, u
′
) and δ(v, v
′
) derived above imply that δ(u, S) + δ(v, S) ≤ δ(u, v) + 1.
For constr uction of our (2k −1, 2)-oracle for k > 2, we shall employ the following
result on spanners.
Theorem 2. [10] For a given u nweighted graph G = (V, E) and any integer
k > 1, t here exists an O(m) time algorithm for computing a (2k − 1)-spanner of
size O(n
1+1/k
).
3 A (3, c)-approximate distance oracle in expected
O(n
2−
1
12
) time
Let G be the given undirected unweighted graph. Let S be a set for med by
selecting each vertex independently with probability n
−
5
12
. Our preprocessing
algorithm for (3, c)-approximate distance oracle, where c = 8, will employ the
sparse subgraph (V, E
S
) and an emulator of the given graph G. We shall need
a (3, 2)-emulator which also s atisfies some additional properties which are very
crucial (see Lemma 7). We describe the construction and properties of this em-
ulator in the following subsection first.
3.1 The e mulator (V, E
∗
) : its construction and properties
In the construction of the emulator, we shall e mploy the (3, 2)-spanner designed
by Baswana et al. [5 ].
Theorem 3. [5] For a given graph G = (V, E), let S
′
be a set formed by select-
ing each vertex from V independently with probability p = n
−
1
3
. It takes expected
O(m) time t o construct a (3, 2)-spanner of size O(n
4/3
) t hat satisfies the follow-
ing additional properties for each u ∈ V .
1. If u ∈ V has no neighbor from set S
′
in G, then every edge incident onto u
will be in the spanner.
2. If u has one or more neighbors from set S
′
in G, then for some unique
neighbor among them, denoted by c(u), the following assertions hold true.
(a) t he edge (u, c(u)) is present in the spanner also.
(b) for each edge (u, v) ∈ E not present in the spanner, there is a path
between c(u) and c(v) in the spann er with length at most 3.
