Fast Approximation Algorithms for the Diameter and
Radius of Sparse Graphs
Liam Roditty
∗
Bar Ilan University
liam.roditty@biu.ac.il
Virginia Vassilevska Williams
†
UC Berkeley and Stanford University
virgi@eecs.berkeley.edu
ABSTRACT
The diameter and the radius of a graph are fundamental
topological parameters that have many important practi-
cal applications in real world networks. The fastest com-
binatorial algorithm for both par am eters works by solving
the all-pairs shortest paths problem (APSP) and has a run-
ning time of
e
O (m n ) in m-edge, n-node graphs. In a seminal
paper, Aingworth, Chekuri, Indyk and Motwani [SODA’96
and SICOMP’99] presented an algorithm that computes in
e
O (m
√
n + n
2
) time an estimate
ˆ
D for the diameter D, such
that ⌊2/3D⌋ ≤
ˆ
D ≤ D. Their paper spawned a long line of
research on approximate APSP. For the specific problem of
diameter approximation, however, no improvement has b een
achieved in over 15 years.
Our paper presents the first improvement over the diame-
ter approximation algorithm of Aingworth et al. , produc in g
an algorithm with the same estimate but with an expected
running time of
e
O(m
√
n). We thus show that for all sparse
enough graphs, the diameter can be 3/2-ap p r oximated in
o(n
2
) time. Our algorithm is obtain ed using a surprisingly
simple method of neighborhood depth estimation that is
strong enough to also approximate, in the same running
time, the radius and more generally, all of the eccentrici-
ties, i.e. for every node the distance to its furthest node.
We also provide strong evidence that our diameter approx-
imation result may be hard to improve. We show that if for
some constant ε > 0 there is an O(m
2−ε
) time (3/2 − ε)-
approximation alg o r ith m for the d ia meter of undirected un-
weighted graphs, then there is an O
∗
((2 − δ)
n
) time algo-
rithm for CNF-SAT on n variables for co n s ta nt δ > 0, and
the strong exponential time hypothesis of [Impagliazzo, Pa-
turi, Zane JCSS’01] is false.
∗
Work supported by the Israel Science Foundation (grant
no. 822/10) .
†
Partially supported by NSF Grants CCF-0830797 and
CCF-1118083 at UC Berkeley, and by NSF Grants IIS-
0963478 and IIS-09 0 4 3 2 5 , and an AFOSR MURI Grant, at
Stanford University.
Permission to make digital or hard copies of all or part of this work for
personal or classroom use is granted without fee provided that copies are
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permission and/or a fee.
STOC’13, June 1 - 4, 2013, Palo Alto, California, USA.
Copyright 2013 ACM 978-1-4503-2029-0/13/06 ...$15.00.
Motivated by this negative result, we give several im-
proved diameter approximation algorithms for special cases.
We show for instance that for unweighted gra p h s of constant
diameter D not divisible by 3, there is an O(m
2−ε
) time al-
gorithm that gives a (3/2 − ε) approximation for constant
ε > 0. This is interesting since the diameter approximation
problem is hardest to so lve for small D.
Categories and Subject Descriptors
F.2.2 [Nonnumerical Algorithms and Problems]; G.2.2
[Graph Theory]: Graph algorithms
Keywords
graph diameter; approximation algorithm; shortest paths
1. INTRODUCTION
The diameter and the radius are two of the most basic
graph parameters. The diameter of a graph is the largest
distance between its vertices. The center of a graph is a ver-
tex that minimizes the ma ximum distance to all other nodes,
and the radius is the distance from the center to th e node
furthest from it. Being able to compute the diameter, center
and radius of a graph effic iently has become an increasingly
important problem in the analysis of large networks [35].
The diameter of the web graph for instance is the largest
number of clicks necessary to get from one document to an-
other, and Albert et al. were ab le to show experimentally
that it is ro u g h ly 19 [2]. The prob lem of computing a center
vertex and the radius of a graph is o ften studied as a fa cility
location problem for networks: pick a single vertex facility
so that the maximum dista n ce from a demand point (client)
in the network is minimized.
The algorithmic complexity of the diameter and radius
problems is very well-studied. For special classes of graphs
there are efficient algorit h ms [21, 19, 15, 11, 12, 5]. E.g. the
radius in chordal graphs can be found in linear time. How-
ever, for general graphs with arbitrary edge weights, the
only known algorithms computing the diameter and radius
exactly compute the distance between every pair of vertices
in the graph, thus solving the all-pairs shortest paths prob-
lem (APSP).
For dense directed unweighted graphs, one can co mp u t e
both th e diameter and the radius using fast matrix multipli-
cation (this is folklore; for a recent simple algorithm see [17 ]),
thus obtaining
˜
O(n
ω
) time algorithms, where ω < 2.38 is the
matrix multiplication exponent [14, 33, 34] and n is the num-
ber of nodes in the graph. It is not known whether APSP
in such graphs can be solved in
˜
O(n
ω
) time – the best al-
gorithm is by Zwick [36] running in O(n
2.54
) time [25], and
hence for directed unweighted graphs diameter and radius
can be solved somewhat faster than APSP. For un d irec ted
unweighted graphs th e best known algorithm for diameter
and radius is Seidel’s
˜
O (n
ω
) time APSP algorithm [3 2 ].
For sparse directed or und irec ted unweighted graphs , the
best known algorithm (ignoring poly-lo g a ri th mic factors)
1
for APSP, diameter and radius, does breadt h -fi r st search
(BFS) f ro m every node and hence runs in O(mn) time,
where m is the number of edges in the grap h . For sparse
graphs with m = O(n), the running time is Θ(n
2
) which
is natural for APSP since the algorithm needs to output n
2
distances. However, for the diameter a nd the radius the ou t -
put is a single integer, and it is not immediately clear why
one should spend Ω(n
2
) time to compute t h em.
A nat u r al question is whether one can get substantially
faster diameter and radius a lg o rit h ms by settling for an ap-
proximation. It is well-known that a 2-approximation for
both the diameter and the radius in an undirected graph
is easy to achieve in O(m + n) time using B FS from an
arbitrary node. On the other hand, for APSP, Dor et al.
[18] show that any (2 − ε)-approximation algorithm in un-
weighted un d irec ted graphs running in T (n) time would im-
ply an O(T (n)) time algorithm for Boolean matrix multi-
plication (BMM). Hence apriori it could be that (2 − ε)-
approximating the diameter and radius of a graph may also
require solving BMM.
In a seminal paper from 1996, Aingworth et al. [1] showed
that it is in fact possible to get a subcubic (2 − ε) - approx-
imation algorithm for the diameter in both directed and
undirected graphs without reso rt in g to fast matrix multi-
plication. They designed an
˜
O (m
√
n + n
2
) time algorithm
computing an estimate
ˆ
D that satisfies ⌊2D/3⌋ ≤
ˆ
D ≤ D.
Their algorithm has several important and interesting prop-
erties. It is the only known algorithm for approximating
the diameter polynomially faster than O(mn) for every m
that is superlinear in n. It always runs in truly subcubic
time even in dense graphs, and does not explicitly compute
all-pairs approximate shortest paths.
For the radiu s problem, Berman and Ka s iviswanathan [6]
showed that the approach of Aingworth et al. can be used
to obtain in
˜
O(m
√
n + n
2
) time an estimate ˆr that satisfies
r ≤ ˆr ≤ 3/2r, where r is the radius of the graph. Thus
both radius and diameter admit
˜
O (m
√
n + n
2
) time 3/2-
approximations.
Aingworth et al. also presented an algorithm that c o m-
putes an additive 2-app roximation for the APSP problem
in
e
O(n
2.5
) time, that is for every u, v ∈ V the algorithm re-
turns a value
ˆ
d(u, v) such that d(u, v) ≤
ˆ
d(u, v) ≤ d(u, v)+2,
where d(u, v) is the distance between u and v. Their pa-
per spawned a long line of research on distance approxi-
mation. However, none of the following works con si d ered
the specific problems of diameter a n d radius approxima-
tion, but rather focused on approximation algorithms for
APSP. Dor, Helperin, and Zwick [18] presented an additive
2-approximation for APSP in unweighted undirected graphs
with a running time of
e
O(min{n
3/2
m
1/2
, n
7/3
}), thus im-
proving o n Aingworth et al. ’s APSP approximation algo-
rithm. Baswana et al. [3] presented an algorithm for un-
1
Chan [10] and Blello ch et al. [8] presented algorithms with
O (m n / poly log n) running times.
weighted undir ect ed graphs with an expected running time
of O(m
2/3
n log n + n
2
) that computes an approximation of
all distances with a multiplicative error of 2 and an a d d it ive
error of 1. Elkin [20] presented an algorithm for unweighted
undirected graphs with a running time of O(mn
ρ
+ n
2
ζ)
that approximates t h e distances with a multiplicative error
of (1 + ε) and an additive error that is a function of ζ, ρ
and ε. Cohen and Zwick [13] extended the results of [18]
to weighted graphs. B as wana and Kavitha [4] presented an
e
O(m
√
n+n
2
) time multiplicative 2-approximation algorithm
and an
e
O(m
2/3
n+n
2
) time 7/3-approximation algorithm for
APSP in weighted un d ir ect ed graphs.
Since Aingworth et al. ’s paper, the only paper that con-
siders the diameter approximation problem direct ly is by
Boitmanis et al. [9]. They presented an algorithm with
e
O(m
√
n) running time that computes the diameter with an
additive error of
√
n. Although such an additive error could
be small for graphs with large diameter, it is prohibitive
when it comes to gr a p h s with small diameter.
A simple rando m sampling argument shows that for all
graphs with diameter at least n
δ
, there is an
e
O(mn
1−δ
/ε)
time (1 + ε)-app roximation algorithm for all ε > 0. Hence
diameter approximation is hardest for graphs with small di-
ameter. For such graphs the additive approximation of Boit-
manis et al. presents no significant a p p r oximation guarantee.
Our contributions.
We give the first improvement over the diameter approxi-
mation algorithm of Aingworth et al. for sparse graphs. We
present an a lg o r ith m with a slightly better approximation
and an expected running time of
e
O(m
√
n). This is always
faster than runtime of [1] for m = o(n
1.5
).
Theorem 1. Let G = (V, E) be a directed or an undi-
rected unweighted graph with diameter D = 3h + z, where
h ≥ 0 and z ∈ {0, 1, 2}. In
e
O(m
√
n) expected time one can
compute an estimate
ˆ
D of D such that 2h + z ≤
ˆ
D ≤ D for
z ∈ {0, 1} and 2h + 1 ≤
ˆ
D ≤ D for z = 2.
We obtain our efficient algorithm by a surprisin g ly simple
node sa mp lin g technique that allows us to replace an expen-
sive neighborhood computation with a cheap estimate.
The diameter and radius are the maximum and minimum
eccentricities in the graph, respectively. In an unweighted
graph, the eccentricity of a vertex is the distance to its fur-
thest node. Our techniques are general enough that we ca n
obtain good estimates of a ll n eccentricities in an und irec ted
unweighted graph in
e
O (m
√
n) time. We prove:
Theorem 2. Let G = (V, E) be an undirected unweighted
graph with diameter D and radius r. In
e
O (m
√
n) expected
time one can compute for every node v ∈ V an estimate ˆe(v)
of its eccentricity ecc(v) such that:
max{r, 2/3ecc(v)} ≤ ˆe(v) ≤ min{D, 3/2ecc(v)}.
We note that until now the only known approximation al-
gorithm for all node eccentricities that runs in o(n
2
) time
for sparse graphs is the simple 2-approximation algorithm
for radius and diameter that runs BFS from a single node.
That algorithm only achieves estimates ˆe(v) for which
max{r, ecc(v)/2} ≤ ˆe(v) ≤ min{D, 2ecc(v) } .
Our approximation algorithm f or radius follows directly
from Theorem 2 by taking ˆr = min
v
ˆe(v). We obtain:
Theorem 3. In
e
O(m
√
n) expected time one can compute
an estimate ˆr of the radius r of an undirected unweighted
graph such that r ≤ ˆr ≤ 3/2r.
Our diameter, radius and eccentricity algorithms natu-
rally extend to graphs with nonnegative edge weights, simi-
lar to the algorithm of Aingworth et al.
A natural question is whether there is an almost linear
time approximation scheme for the diameter problem: an al-
gorithm that for any constant ε > 0 runs in
˜
O(m) time and
returns an estimate
ˆ
D such that (1 − ε)D ≤
ˆ
D ≤ D. Bern-
stein [7] showed that related problems in directed graphs
such as the second shortest path between two nodes and
the replacement paths problem admit such approximation
schemes. Such an algorithm for diameter would be of im-
mense interest, and has not so far been explicitly ruled out,
even conditionally.
Here we give strong evidence that a fast (3/2 − ε) - di-
ameter approximation algorithm may be very hard to find,
even for undirected unweighted graphs. We prove:
Theorem 4. Suppose there is a constant ε > 0 so that
there is a (3/2 − ε)-approximation algorithm for the diam-
eter in m-edge undirected unweighted graphs that runs in
O (m
2−ε
) time for every m. Then, SAT for CNF formulas
on n variables can be solved in O
∗
((2 − δ)
n
) time for some
constant δ > 0.
The fastest known algorithm for CNF-SAT is the exhaus-
tive search algorithm that runs in O
∗
(2
n
) time by trying all
possible 2
n
assignments to the variables. I t is a major open
problem whether there is a faster algorithm. Several other
NP-hard p ro blems are known to be equivalent to CNF-SAT
so that if one of these problems has a faster algorithm than
exhaustive s ear ch, then all of them do [16]. Hence, our result
has the following surprising implica tio n : if the diameter can
be approximated fast enough, then problems su ch as Hitting
Set, Set Splitting, or NAE-SAT, all seemingly unrelated to
the diameter, can be solved faster than exhaustive search.
The strong exponential time hypothesis (SETH) of Im-
pagliazzo, Paturi, and Zane [23, 24] imp lies that there is no
improved O
∗
((2 − δ)
n
) time algorithm for CNF-SAT, and
hence our result also implies that there is no (3/2 − ε)-
approximation algorithm for the diameter approximation
running in O(m
2−ε
) time un les s SETH fails. (We elaborate
on this hypothesis later on in the paper.)
We prove Theorem 4 by showing that any O(n
2−ε
) time
algorithm that distinguishes whether the diameter of a g iven
sparse (m = O(n)) undirec ted unweighted graph is 2 or a t
least 3 would imply an imp r oved CNF-SAT algorithm. Th is
implies that unless SETH fails, O(n
2
) time is essentially re-
quired to get a (3/2−ε)-approximation algorithm for the di-
ameter in sparse graphs, within n
o(1)
factors. Hence, within
n
o(1)
factors, the time for (3/2 − ε)-approximating the di-
ameter in a sparse graph is the same as the time required
for computing APSP exactly!
In their paper, Aingworth et al. showed that one can d is -
tinguish between graphs of diameter 2 and 4 in
e
O (m
√
n)
time, whereas we show that distinguishing between 2 and
3 fast may be difficult. We further explore which graph
diameters can be efficiently distinguished, and prove the fol-
lowing two theo rems that improve upon the a p p r oximation
of Aingworth et al. algorithm.
Theorem 5. Let G = (V, E) be a directed or undirected
unweighted graph with diameter D = 3h + z, where h ≥ 0
and z ∈ {0, 1, 2}. There is an
˜
O(m
2/3
n
4/3
) time algorithm
that reports an estimate
ˆ
D such that 2h + z ≤
ˆ
D ≤ D.
Theorem 6. There is an
˜
O (m
2/3
n
4/3
) time algorithm that
when run on an undirected unweighted graph with diameter
D , reports an estimate
ˆ
D with ⌊4D/5⌋ ≤
ˆ
D ≤ D.
Theorem 5 shows for instance that one can efficiently dis-
tinguish between directed or un directed graph s of diameter
3 an d 5, and Theorem 6 obtains a 5/4-approximation for
the diameter that runs in O(mn/n
ε
) time for some const a nt
ε > 0 in a ll undirected gr a p h s with a superlinear number of
edges. The previous best approximation quality achievable
polynomially faster than O(mn) time for such graph s was
Aingworth et al. ’s 3/2-approximation.
We further investigate whether one ca n ever obtain a (3/2−
ε)-approximation for the diameter in O(m
2−ε
) time, and
show that this is indeed possible for graphs with constant
diameter that is not divisible by 3. This is intriguing since,
as we pointed o u t earlier, the diameter approximation prob-
lem is hardest for gr aphs with small diameter. We prove:
Theorem 7. There is an
˜
O(m
2−1/(2h+3)
) time determin-
istic algorithm that computes an estimate
ˆ
D with ⌈ 2 D/3⌉ ≤
ˆ
D ≤ D for all m-edge unweighted graphs of diameter D =
3h+z with h ≥ 0 and z ∈ {0, 1, 2}. In particular,
ˆ
D ≥ 2h+z.
Notation.
Let G = (V, E) denote a graph. It ca n be directed or
undirected; this will be specified in each context. If the
graph is weighted, then there is a function on the edges
w : E → Q
+
∪ {0} . Unless explicitly specified, the graph s
we consider are unweighted.
For any u, v ∈ V , let d(u, v) denote the distance from u
to v in G. Let BF S
in
(v) and BF S
out
(v) be the incoming
and outgoing breadth-first search (BFS) trees of v, respec-
tively, that is the BFS trees starting at v in G and in G with
the edges reversed. Let d
in
(v) be the depth of BF S
in
(v),
i.e. the larg est distance from a vertex of BF S
in
(v) to v.
Similarly, let d
out
(v) be the depth of BF S
out
(v).
In an unweighted graph, the eccentricity of a vertex v de-
noted with ecc(v) is the depth of its BFS tree BF S(v). In
a weighted graph, the eccentricity ec c( v) of v is the max-
imum over all u ∈ V of d(v, u). The radius of a graph is
r = min
v∈V
ecc(v), and the diameter is D = max
v∈V
ecc(v).
For h ≤ d
in
(v), let BF S
in
(v, h) be the vertices in the
first h levels of BF S
in
(v). Similarly, for h ≤ d
out
(v),
let BF S
out
(v, h) be the vertices in the first h levels of
BF S
out
(v).
Let N
in
s
(v) (N
out
s
(v)) be the set of the s closest inco min g
(outgoing) vertices of v, where ties are broken by taking the
vertex with the smaller id. We assume throughout the paper
that for each v and each s ≤ n, |N
in
s
(v)| = |N
out
s
(v)| = s,
as otherwise the diameter of the graph would be ∞, and this
can be checked with two BFS runs from a n d to an arbitrary
node. Fo r undirected graphs N
s
(v) = N
IN
s
(v) = N
OU T
s
(v).
Let d
in
s
(v) be the largest distance from a vertex o f N
in
s
(v)
to v, and d
out
s
(v) be the largest distance from v to a ver-
tex of N
out
s
(v). Let d
in
s
= max
v∈V
d
in
s
(v) and d
out
s
=
max
v∈V
d
out
s
(v).
For a set S ⊆ V and a vertex v ∈ V we define p
S
(v) to
be a vertex of S such that d ( v, p
S
(v)) ≤ d(v, w) for every
w ∈ S, i.e. the closest vertex of S to v.
For a degree ∆ we defin e p
∆
(v) to be the closest vertex
to v of degree at least ∆, that is, d(v, p
∆
(v)) ≤ d(v, w) for
every w ∈ V of degree at least ∆.
We use the following standard notation fo r runn in g times.
For a function of n, f(n),
˜
O(f(n)) denotes O(f (n)poly lo g n)
and O
∗
(f(n)) denot es O(f (n)poly(n)).
We write whp to mean with high proba b ility, i.e. with
probability at least 1 − 1/poly(n).
2. DIAMETER
In this section we present the pro o f of Theorem 1. We
first revisit th e algorithm of A in g worth et al. and tighten
its approximation analysis. We then present our n ew neigh-
borhoo d est ima tio n approach that is at the basis of our im-
proved algorithm.
2.1 The algorithm of Aingworth et al.
The algorithm of Aingworth, C h eku ri, Indyk and Mot-
wani [1], computes a (roughly) 3/2-a p p r oximation of the di-
ameter of a directed (or undirected) graph in
e
O(m
√
n + n
2
)
time. Let s be a given parameter in [1, n]. The algorithm
works as follows. First, it computes N
out
s
(v) for every v ∈
V . Then, for a vertex w, where d
out
s
(w ) = d
out
s
it com-
putes BF S
out
(w ) and for every u ∈ N
out
s
(w ) it computes
BF S
in
(u). Next, it com p u tes a set S that hits N
out
s
(v) for
every v ∈ V and for every u ∈ S it co mp u t es BF S
out
(u).
As an estimate, the algorithm returns the depth of the deep-
est computed BFS tree. The next lemma appears in [1]. We
state it for completeness.
Lemma 1. The algorithm runtime is
e
O(ns
2
+(n/s+s)m).
Aingworth et al. set s =
√
n and obtain their running
time. We note that if one sets s = m
1/3
instead, one can
get a runtime of
e
O(m
2/3
n) that is better for sparse graphs;
we later show that both of these runtimes can be improved
using our new method.
We now analyze the quality of the estimate returned by
the algorithm. Aing worth et al. [1] proved that this estimate
is at least ⌊2D/3⌋ in gr a p h s with diameter D. Here we
present a tighter ana lysis .
Lemma 2. Let G = (V, E) be a directed graph with diam-
eter D = 3h +z, where h ≥ 0 and z ∈ {0, 1, 2}. Let
ˆ
D be the
estimate returned by the algorithm. For z ∈ {0, 1}, we have
2h + z ≤
ˆ
D ≤ D. For z = 2, we have that 2h + 1 ≤
ˆ
D ≤ D.
Proof. Let a, b ∈ V such that d(a, b) = D. First notice
that the algorithm always retu r n s the depth of some shortest
paths tree and hence
ˆ
D ≤ D.
If d
out
s
(w ) ≤ h then also d
out
s
(a) ≤ h and as S hits
N
out
s
(a), one of the BFS trees computed for vertices of S
has depth at least 2h + z. Hence, assume that d
out
s
(w ) > h.
We can also assume that d
out
(w ) < 2 h + z as otherwise
when we c o mp u t e BF S
out
(w ), the estimate would become
at least 2h + z.
As d
out
(w ) < 2h + z, also d(w , b) < 2h + z. Sin ce
d
out
s
(w ) > h, we have that BF S
out
(w , h) ⊆ N
out
s
(w ).
Hence there is a vertex w
′
∈ N
out
s
(w ) on the path from w to
b such that d(w, w
′
) = h and hence d(w
′
, b) < h + z. Since
d(a, b) = 3h+z, we must have that d(a, w
′
) ≥ 2h+1. As the
algorithm computes BF S
in
(u) for every u ∈ N
out
s
(w ), in
particular, it computes BF S
in
(w
′
), and returns an estimate
≥ 2h + 1. For z ∈ {0 , 1}, d(a, w
′
) ≥ 2h + 1 ≥ 2h + z and
hence the final estimate returned is always at least 2h + z.
For z = 2 we only have that d(a, w
′
) ≥ 2h + 1 and if the al-
gorithm returns d(a, w
′
) as an estimate, it may return 2h+1
instead of 2h + z. 2
2.2 Improving the running time
The algorit h m of Aingworth et al. [1] runs in
e
O (n s
2
+
(n/s + s)m). In this section we show how to get rid of
the ns
2
term wit h some randomizat io n , while keeping the
quality of the estimate unchanged. B y choosing s =
√
n, we
get an algorithm running in
e
O(m
√
n) time.
The term of n s
2
in the running t ime comes from the com-
putation of N
out
s
(v) for every v ∈ V . This c o mp u t a tio n is
done to accomplish two tasks. One t as k is to ob ta in d
out
s
(v)
for every v ∈ V and then to use it to find a vertex w s u ch
that d
out
s
(w ) = d
out
s
. A second task is to obtain, deter-
ministically, a hitting set S of size
e
O (n / s ) that hits the set
N
out
s
(v) of every v ∈ V .
Our main idea is to accomplish these two tasks without
explicitly computing N
out
s
(v) for every v ∈ V . The major
step in our a p p r o a ch is to completely mod ify the first task
above by picking a different type o f vertex to play the role of
w. Making the second task above fast can be accomplished
easily with randomization. We elaborate on this below.
Our a lg o rit h m works as follows. First, it computes a hit-
ting set by using randomization, that is, it picks a random
sample S of the vertices of size Θ(n/s log n ) . This g u a r a n -
tees that with high pr o b a b ility (at least 1 − n
−c
, for s om e
constant c), S ∩ N
out
s
(v) 6= ∅, for every v ∈ V . This ac-
complishes the second task above in
˜
O (n ) time, with high
probability. Similarly to the algorithm of Aingworth et al.
[1], our algorithm computes BF S
out
(v), for every v ∈ S.
We now explain the main idea of our algorithm, i.e. how
to replace the first task above w it h a much faster step. First,
for every v ∈ V our algorithm computes the closest node of
S, p
S
(v), to v, by creating a new graph as follows. It adds
an add itio n a l vertex r with edges (u, r), for every u ∈ S. It
computes BF S
in
(r) in this graph. It is easy to see that for
every v ∈ V the last vertex before r on the shortest path
from v to r is p
S
(v). This step t a kes O(m) time.
Now, the crucial point of our algorithm is that, as op-
posed to the algorithm of Aingworth et al. that picks a
vertex w such that d
out
s
(w ) = d
out
s
, our algorithm finds
a vertex w ∈ V that is furthest away from S: i.e. such
that d(w, p
S
(w )) ≥ d(u, p
S
(u)), for every u ∈ V . The vertex
w plays the same role as its counterpart in [1]: Our algo-
rithm computes BF S
out
(w ) and obtains N
out
s
(w ) from it.
Finally, it computes BF S
in
(u) for every u ∈ N
out
s
(w ). As
an estimate, the algo r ith m returns the d ep th of the deepest
BFS tree that it h a s computed.
In the next Lemma we analyze the running time of the
algorithm.
Lemma 3. The algorithm runtime is
e
O (( n / s + s)m).
Proof. A hittin g set S is formed in
˜
O (n ) time. With a
single BFS computation , in O(m) time, we find p
S
(v) for
every v ∈ V , and hence also find w. The cost of computing
a BFS tree for every v ∈ S ∪ N
out
s
(w ) is
e
O((n/s + s)m). 2
Next, we show that the estimate p r oduc ed by our algo-
rithm is of the same quality as the estimate produced by
Aingworth et al. algorithm, whp.
Lemma 4. Let G = (V, E) be a directed (or undirected)
graph with diameter D = 3h + z, where h ≥ 0 and z ∈
{0, 1, 2}. Let
ˆ
D be the estimate returned by the above algo-
rithm. With high probability, 2h + z ≤
ˆ
D ≤ D whenever
z ∈ {0, 1}, and 2h + 1 ≤
ˆ
D ≤ D whenever z = 2.
Proof. Let a, b ∈ V such that d( a, b) = D. Let w be a
vertex that satisfies d(w, p
S
(w )) ≥ d(u, p
S
(u)), ∀u ∈ V .
If d(w, p
S
(w )) ≤ h then also d(a, p
S
(a)) ≤ h. As the
algorithm computes BF S
out
(v) for every v ∈ S, it follows
that BF S
out
(p
S
(a)) is computed as well and its depth is at
least 2h + z as required. Hence, assume that d(w, p
S
(w )) >
h. We can assume also that d
out
(w ) < 2h + z since the
algorithm computes BF S
out
(w ) and if d
out
(w ) ≥ 2h + z
then it computes a BF S tree of depth at least 2h + z.
Since d
out
(w ) < 2h + z it follows that d(w, b) < 2h + z.
Moreover, since d (w, p
S
(w )) > h and S hits N
out
s
(w ) whp,
we must have that N
out
s
(w ) contains a no d e at distance > h
from w, and hence BF S
out
(w , h) ⊆ N
out
s
(w ). This implies
that there is a vertex w
′
∈ N
out
s
(w ) on the path from w to
b such that d(w, w
′
) = h and hence d(w
′
, b) < h + z. Since
d(a, b) = 3h + z, we also have tha t d(a, w
′
) ≥ 2h + 1.
The algorithm computes BF S
in
(u) for every u ∈ N
out
s
(w ),
and in particular, it computes BF S
in
(w
′
), thus returning an
estimate at least d(a, w
′
) ≥ 2h + 1. Hence for z ∈ {0, 1} the
final estimate is always ≥ 2h + z, and for z = 2 the estimate
could be 2h + 1 but no less . 2
We n ow turn to prove Theorem 1 from the introduction.
Reminder of Theorem 1 Let G = (V, E) be a directed
or an undirected graph with diameter D = 3h + z, where
h ≥ 0 and z ∈ {0, 1, 2}. In
e
O (m
√
n) expected time one can
compute an estimate
ˆ
D of D such that 2h + z ≤
ˆ
D ≤ D for
z ∈ {0, 1} and 2h + 1 ≤
ˆ
D ≤ D for z = 2.
Proof. From Lemma 3 we have th at if we set s =
√
n the
algorithm runs in
e
O (m
√
n) worst case time. From Lemma 4
we have that wh p , the algorithm returns an estimate of the
desired quality. We now show how to convert the algorithm
into a Las-Vegas one so that it always returns an estimate
of the desired quality but the running time is
e
O(m
√
n) in
expectation.
Randomization is used only in order to obtain a set that
hits N
out
s
(v) for every v ∈ V . The only place that the
hitting set affects the quality of the approximation is in
Lemma 4 where we used the fac t that, wh p , S contains a
node of N
out
s
(w ), so th a t if d(w, S) > h, N
out
s
(w ) contains
a node at distance > h from w.
Algorithm 1: Approx-Ecc(G)
Let S be a random sample of Θ(n/s log n) nodes.
Let w be such that d(w, p
S
(w )) ≥ d(u, p
S
(u)) for all
u ∈ V .
foreach x ∈ N
s
(w ) ∪ S do
BFS(x).
foreach v ∈ V do
if d(v, v
t
) ≤ d(v
t
, w) then
ˆe(v) = max{max
q∈S
d(v, q), d(v, w), ecc(v
t
)}
else
ˆe(v) =
max{max
q∈S
d(v, q), d(v, w), min
q∈S
ecc(s)}
Note that the algorithm computes N
out
s
(w ) and we can
check whether S intersects it in
e
O (s ) time. If it does not,
we can r eru n the algorithm until we have verified that S ∩
N
out
s
(w ) 6= ∅. In each run, S ∩N
out
s
(w ) = ∅ holds with very
small probability: S is large enough so that whp it intersects
the s-neighborhoods of all n vertices of the graph . Thus, th e
expected running time of the algorithm is
e
O(m
√
n) and its
estimate is guaranteed to have th e required quality. 2
Just as in [1], our algorithm works for grap h s with non-
negative weights as well by replacing every use of BFS with
Dijkstra’s algorithm. The p r oofs are analogous, th e running
time is increased by at most a log n factor, and the quality of
the approximation only suffers an additive W term, where
W is the maximum edge weight in the graph. (The same
approximation quality is achieved by Aingworth et al. b u t
with an
e
O(m
√
n + n
2
) running time.) We obtain:
Theorem 8. Let G = (V, E) be a directed or an undi-
rected graph with nonnegative edge weights at most W and
diameter D. In
e
O(m
√
n) expected time one can compute an
estimate
ˆ
D of D such that ⌊2D/3 − W ⌋ <
ˆ
D ≤ D.
3. ECCENTRICITIES
In this section we show that our method can be generalized
to compute for every vertex v in an undirected unweighted
graph, a good approximation ˆe(v) of its eccentricity ecc(v).
We prove Theorem 2.
Reminder of Theorem 2 Let G = (V, E) be an undirected
graph with diameter D and radius r. In
e
O (m
√
n) expected
time one can compute for every node v ∈ V an estimate ˆe(v)
of its eccentricity ecc(v) such that:
max{r, 2/3ecc(v)} ≤ ˆe(v) ≤ min{D, 3/2ecc(v)}.
We note that our ec centricities algorithm can also be ma d e
to work for undirected graphs with nonnegative weights at
most W by again using Dijkstra’s algorithm in place of
BFS. T h en the runnin g time is still
e
O (m
√
n) and t h e ap-
proximation quality becomes 2/3ecc(v) − 2W < ˆe(v) <
3/2ecc(v) + W .
One can immediately obtain our 3/2-approximation of the
radius in unweighted undirected graphs stated in Th eo r em 3
as a corollary to Theorem 2 by taking ˆr = min
v
ˆe(v). For
this choice, ˆr ≥ r, and ˆr ≤ min
v
3/2ecc(v) = 3/2r.