CS计算机代考程序代写 scheme data structure discrete mathematics AI algorithm Approximate Distance Oracles

Approximate Distance Oracles

Mikkel Thorup
AT&T Labs – Research

180 Park Avenue
Florham Park, NJ 07932, USA

.com

Uri Zwick

School of Computer Science
Tel Aviv University

Tel Aviv 69978, Israel

.ac.il

ABSTRACT
LetG = (V;E) be an undirected weighted graph with jV j = n and
jEj = m. Let k � 1 be an integer. We show that G = (V;E) can
be preprocessed in O(kmn1=k) expected time, constructing a data
structure of size O(kn1+1=k), such that any subsequent distance
query can be answered, approximately, in O(k) time. The approxi-
mate distance returned is of stretch at most 2k�1, i.e., the quotient
obtained by dividing the estimated distance by the actual distance
lies between 1 and 2k � 1. We show that a 1963 girth conjecture
of Erdős, implies that
(n1+1=k) space is needed in the worst case
for any real stretch strictly smaller than 2k+1. The space require-
ment of our algorithm is, therefore, essentially optimal. The most
impressive feature of our data structure is its constant query time,
hence the name “oracle”. Previously, data structures that used only
O(n1+1=k) space had a query time of
(n1=k) and a slightly larger,
non-optimal, stretch. Our algorithms are extremely simple and easy
to implement efficiently. They also provide faster constructions of
sparse spanners of weighted graphs, and improved tree covers and
distance labelings of weighted or unweighted graphs.

1. INTRODUCTION
Consider the following interesting problem which is, perhaps,

the most natural formulation of the classical all-pairs shortest paths
problem (APSP). We are given a description of a large network,
such as the Internet, or a large road network, such as the US road
network1, with n nodes and m connections. Each connection has
a length, or weight, associated with it. Usually m � n2. We are
to preprocess the network, so that subsequent distance queries or
shortest path queries could be answered quickly, on-line.

This formulation seems to capture more accurately the real na-
ture of the all-pairs shortest paths problem, as in most applications
we are not really interested in all distances, we just want the ability
to retrieve them quickly, if needed. For example, there are probably
many pairs of addresses in the US whose distance is of interest to

�Work supported in part by the Israel Science Foundation
founded by The Israel Academy of Sciences and Humanities.
1The US road network is a planar network. To get a more interest-
ing non-planar network, assume that the weights attach to the edges
represent travel time, and add flight connections between airports.

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no one. This is precisely the case with “sublinear” algorithms for
static metric space problems. The input to such a problem is often
the shortest paths metric of a graph. A “sublinear” algorithm at-
tempts to solve such problem while querying only some of the dis-
tances. For more details, see Indyk [34] (Note that for the location
problems considered in [34], we now have faster direct solutions
for sparse graphs [53]).

Using an APSP algorithm, we can preprocess the graph in
~O(mn) time, and produce a data structure of size O(n2), an n�n
matrix holding the distances, and perhaps a succinct representation
of shortest paths between all pairs of vertices of the graph. Any
distance query can then be answered in O(1) time.

There are, however, several serious objections to this solution.
First, a preprocessing time of ~O(mn) may be too long. Second,
even if we are willing to wait that long, the n� n matrix produced
may be too large to store efficiently (typically m � n2, and then
this table is much larger than the network itself).

Here, we explore alternative solutions to this problem. We show
that better solutions exist, if the network is undirected, and if we are
willing to settle for approximate distances, instead of exact ones.
The approximate distances produced by our algorithms are of a fi-
nite stretch. An estimate Æ̂(u; v) to the distance Æ(u; v) from u
to v is said to be of stretch t if and only if Æ(u; v) � Æ̂(u; v) �
t � Æ(u; v). Stretched distances may be acceptable under some sce-
narios, while unacceptable in others. Many recent algorithms deal-
ing with finite metric spaces produce only approximate answers,
even if exact distances are used. In particular, this is the case with
the above mentioned sublinear metric space algorithms of Indyk
[34]. Adapting these algorithms to exploit our approximate dis-
tance oracles is therefore a straightforward task.

As stated in the abstract, we describe, for any integer k � 1, a
preprocessing algorithm that runs in O(kmn1=k) time, producing
a data structure of size O(kn1+1=k). Note that the preprocessing
time is almost linear in the size of the network, if k is a large con-
stant, while the size of the data structure produced is almost linear
in the number of nodes. In particular, for dense enough graphs, the
data structure produced is much more compact than the network
itself. Subsequent queries can then be answered, approximately, in
O(k) time. i.e., constant time. The stretch of the approximations
returned is at most 2k � 1. More formally:

THEOREM 1.1. Let G = (V;E) be a weighted undirected
graph with non-negative edge weights with jV j = n, jEj = m.
Let k � 1 be an integer. Then, the graph G can be preprocessed
in O(kmn1=k) expected time, producing a data structure of size
O(kn1+1=k), such that subsequent distance queries can be an-
swered, approximately, in O(k) time. The stretch of the produced
estimates is at most 2k � 1. Paths no longer than the estimates
returned can be produced in constant time per edge.

For k = 1, we simply get the APSP solution. When k = 2,
we get a preprocessing time of O(mn1=2), space O(n3=2), query
time O(1), and stretch at most 3. When k = blog nc, we get a
preprocessing time of O(m log n), space O(n log n), query time
O(log n), and stretch O(log n). Higher values of k do not improve
the space or preprocessing time.

The most interesting feature of our algorithms, we believe, is the
fact that for every fixed k we get a constant query time, hence the
name distance oracles.

The space requirements of our oracles are essentially optimal.
We show that a 1963 girth conjecture of Erdős, and others, im-
plies that
(n1+1=k) bits of storage are needed, in the worst case,
by any oracle, however slow, that gives estimated distances with
stretch strictly less than 2k + 1. This girth conjecture is known to
hold for k = 1; 2; 3; 5. Thus, in particular, any oracle giving stretch
2:99 answers must use, on some graphs, at least
(n2) bits of stor-
age, and any oracle giving stretch 4:99 answers must use, on some
graphs, at least
(n3=2) bits of storage, almost the same amount of
storage used by our stretch 3 oracle.

The oracle model of the shortest paths problem was considered
before, at least implicitly, by Awerbuch et al. [7], Cohen [18], and
by Dor et al. [25]. (See discussion in the next section.) Our re-
sults significantly improve, however, the previously available re-
sults. Most strikingly, using slightly less space, we reduce the query
time from ~O(kn1=k) to O(k), while giving, at the same time, more
accurate distance estimates.

Theorem 1.1 is proved in Section 4. Before that, in Section 3, we
present a simplified version of our algorithm for the the special case
where the input is the complete distance matrix of a finite metric
space. This version of the algorithm is faster (O(n2) time) and
particularly suited for external memory implementation.

As a byproduct of our oracle construction for graphs, we also
get faster algorithms for constructing sparse spanners and compact
tree covers of weighted graphs (see Section 4.4), and near-optimal
distance labelings of graphs (see Section 3.5).

As mentioned in the abstract, all our algorithms are extremely
simple and easy to implement efficiently.

The rest of the paper is organized as follows. In the next section,
we compare our results with previously available results. In Sec-
tion 3 we construct approximate distance oracles for finite metric
spaces. The input in this setting is an n� n matrix giving the dis-
tance between any two points in the space. In Section 4 we adapt
this construction to work on the shortest paths metric of a given
input graph. The input this time is the graph, and not an explicit
representation of all the distances in it. Breaking the description
of our distance oracles in this way allows us to separate the metric
aspects of our constructions from the algorithmic graph techniques
needed for efficient implementation. In Section 5 we describe al-
most matching lower bounds on the space requirements of approx-
imate distance oracles. We also show that essentially no non-trivial
distance oracles are possible for directed graphs. We end, in Sec-
tion 6, with some concluding remarks and open problems.

2. PREVIOUS RESULTS
A summary of previously obtained algorithms for computing

exact or approximate distances in general weighted undirected
graphs, cast in our framework, is given in Table 1.

In more detail, the fastest solution for APSP for directed and
undirected weighted graphs with non-negative weights from an ar-
bitrary (comparison based) domain is to run a single-source shortest
paths (SSSP) algorithm from each node. This takesO(m+n log n)
time using the classical algorithm of Dijkstra [23], implemented
using Fibonacci heaps [31] (see also Cormen et al. [20, Chapter

21]). In this paper, we are only interested in undirected graphs, and
then an improved running time of O(m) can be obtained when the
weights are integer [49] (or floating point [51]). Consequently, the
time bound for APSP is O(mn). The O(m) time bound for SSSP
has been incorporated in the other time bounds below, so an ~O(�)
time bound indicates the presence of logarithmic factors not stem-
ming from Dijkstra’s algorithm.

For completeness, we note that improved time bounds may be
obtained if not all edges are part of shortest paths [35, 43], or if the
graph is dense and all weights are small integers [59, 48].

Zwick [59] has shown that if a stretch of 1 + �, for some fixed
� > 0, is allowed, then APSP can be solved in ~O(n!) time, where
! < 2:376 is the exponent of matrix multiplication. For stretches 2, 7=3, and 3, Cohen and Zwick [19] have shown that APSP can be solved in time ~O(m1=2n3=2), ~O(n7=3), and ~O(n2), respectively. The above ~O(n2) time bound is clearly near-optimal, if we insist on producing a complete table of distances. It is interesting to note, however, that stretch 3 is also the smallest stretch for which one can hope to produce distance oracles that use o(n2) space for all graphs. Indeed, Dor et al. [25] describe a stretch 3 oracle that uses only ~O(m1=3n+n2=m1=3) space, which is always ~O(n5=3), has a preprocessing time of ~O(m2=3n), and O(1) query time. The preprocessing time of this algorithm is o(n2) if m = o(n3=2). The ideas of Dor et al. [25] do not extend, however, to larger stretches. Our new stretch 3 oracle (Theorem 1.1 with k = 2) has a pre- processing time of O(mn1=2), optimal space O(n3=2), and O(1) query time. The new preprocessing time is faster than all the other preprocessing times when m = O(n3=2), in which case it achieves a preprocessing time of O(n2). For general stretch, Awerbuch et al. [7] gave, for every integer k � 1, a stretch 64k oracle with space ~O(kn1+1=k), preprocessing time ~O(mn1=k), and ~O(kn1=k) query time. Cohen [18] signifi- cantly improved this result, reducing the stretch to 2k + � while leaving the other parameters unchanged. In this paper, we reduce the stretch a bit further to 2k�1, which is probably optimal for the amount of space used, and much more importantly, dramatically reduce the query time from ~O(kn1=k) to O(k). Distance oracles are closely related to spanners. A t-spanner of a weighted undirected graph G is a subgraph H of G such that the distances in H are stretch t estimates of the distances in G (see Pe- leg and Schäffer [46]). Clearly, a stretch t oracle, like ours, capable of producing paths witnessing the estimated distances, must explic- itly or implicitly contain a t-spanner. Hence, t-spanners provide a clean mathematical view of compact distance oracles. Indeed, all of the above mentioned results providing o(n2) space bounds [7],[18],[25] can be viewed as producing spanners. The sizes of spanners are closely related to the girth of a graph, which is the size of its smallest simple cycle. Clearly, the girth of a graph is at least t + 2 if and only if no proper subgraph of it is a t-spanner. A classical result in extremal graph theory (see discussion references in Section 5) states that an n-vertex graph with at least n1+1=k edges is of girth at most 2k. As pointed out by Althöfer et al. [5], this implies that every weighted undirected graph on n vertices has a (2k�1)-spanner with O(n1+1=k) edges. Such a spanner can be constructed using an algorithm similar to Kruskal’s algorithm (see [36] or Cormen et al. [20, Chapter 24]) for constructing minimum spanning trees: Building the spanner from scratch, consider the edges of the graph in a non-decreasing order of weight, adding each edge to the spanner if its endpoints are not already connected, in the spanner, by a path using at most 2k � 1 edges. At any stage, the spanner is a (2k� 1)-spanner of the edges already considered, and its unweighted girth is at least 2k + 1, so it has only O(n1+1=k) edges. The fastest implementation of this Stretch Query time Space Preproc. time Reference Space lower-bound 1 O(1) O(n2) O(mn) [49] O(m) O(m) 0 [49] 1 + � O(1) O(n2) ~O(n!) [59] (m) bits 2 O(1) O(n2) ~O(m1=2n3=2) [19] 7=3 O(1) O(n2) ~O(n7=3) [19] O(1) O(n2) ~O(n2) [19] 3 O(1) ~O(m1=3n+ n2=m1=3) ~O(m2=3n) [25] (minfm;n3=2g) bits [47] O(1) O(n3=2) O(mn1=2) This paper 2k � 1 O(n1+1=k) O(n1+1=k) O(mn1+1=k) [5] O(k) O(kn1+1=k) O(kmn1=k) This paper (minfm;n1+1=kg) bits Conjecture [28] 2k + � ~O(kn1=k) ~O(kn1+1=k) ~O(kmn1=k) [18] 64k ~O(kn1=k) ~O(kn1+1=k) ~O(kmn1=k) [7] (minfm;n 1+1=32kg) bits Conjecture [28] Table 1: All available exact and approximate distance oracles for weighted undirected graphs algorithm that we are aware of runs in O(mn1+1=k) time. Our pre- processing algorithm constructs a similar spanner much faster. The construction time is now O(kmn1=k), instead of O(mn1+1=k). It is conjectured by many (e.g., Erdős [28], equation 7 on p. 33, Bondy and Simonovits [12], remark 1 on p. 98 and Bollobás [11], item 13 on p. 164), that there are n-vertex graphs with (n1+1=k) edges that are of girth 2k + 2. This conjecture is proved for k = 1; 2; 3; 5 (see Section 5). Since these graphs have no proper t- spanners, for t < 2k+1, the conjecture would imply that the above mentioned upper bounds are best possible. We show in Section 5 that the conjecture also implies that (n1+1=k) bits are needed, in the worst case, by any oracle giving estimates of stretch smaller than 2k + 1, even if it not required to construct appropriate paths. Some distance oracles were constructed for special classes of graphs. Efficient distance oracles for graphs of small treewidth were obtained by Chaudhuri and Zaroliagis [16]. Exact and ap- proximate distance oracles for planar and Euclidean graphs were considered by Arikati et al. [6], Chiang and Mitchell [17], Djidjev [24], and recently by Thorup [52]. Finally, we mention that there has been some work on approx- imating distances, in unweighted undirected graphs, with additive rather than multiplicative errors (see Aingwoth et al. [1] and Dor et al. [25]), and very recently, on approximating distances with multiplicative and additive errors (see Elkin and Peleg [27] and Elkin [26]). There has also been work on low distortion embed- dings of general metric spaces into some low-dimensional metrics (see Bourgain [13], Linial at al. [40] and Bartal [9]), but these em- beddings lead to stretches of at least (log n), as compared with our 2k � 1. Techniques. Our construction technique is most closely related to the techniques employed by Awerbuch et al. [7] and Cohen [18]. A common feature of these previously used techniques is the con- struction of a family of balls with the property that each vertex is contained in at most of ~O(kn1=k) balls. The returned distance between two vertices is then the smallest diameter of a ball contain- ing them both. To find this ball, they inspect each of the ~O(kn1=k) balls containing the first vertex, and check, in constant time per ball, whether it also contains the second. Though conceptually sim- ple, the use of balls leads to several technical complications. One of them, for example, is an added logarithmic factor paid for the construction of balls with exponentially increasing diameters. The main drawback of this approach, however, is the lack of a quick way of finding the smallest diameter ball containing two given ver- tices. In our construction, we relax the rigid notion of balls with limited diameter, and use instead collections of induced trees that form a tree cover of the graph. Each vertex is contained in only in a small number of trees, and for any pair of vertices, there is a tree in the cover containing a small-stretch path between them. Furthermore, we can identify the appropriate tree in constant time. 3. APPROXIMATE DISTANCE ORACLES FOR METRIC SPACES We begin by presenting approximate distance oracles for general metric spaces. THEOREM 3.1. Let (V; Æ) be a finite metric space represented as an n � n distance matrix. Let k � 1 be an integer. The metric space (V; Æ) can be preprocessed in O(n2) expected time, produc- ing a data structure of O(kn1+1=k) size, such that subsequent dis- tance queries can be answered, approximately, in O(k) time. The stretch of the produced estimates is at most 2k � 1. Our preprocessing and query answering algorithms are given, re- spectively, in Figure 1 and Figure 2. Both are extremely simple and easy to implement. The algorithms and their implementation details are discussed in more detail in the next two subsections. The following two subsections are then devoted to the analysis of the algorithms, showing that they satisfy the requirements of The- orem 3.1. In Section 3.5 we show that our approximate distance oracles also produce, as a byproduct, almost optimal distance la- bels. In Section 3.6 we show that our randomized preprocessing al- gorithm may be derandomized with only a small loss of efficiency. algorithm preprok(V; Æ) A0 V ; Ak ; for i 1 to k � 1 let Ai contain each element of Ai�1, independently, with probability n�1=k . for every v 2 V for i 0 to k � 1, let Æ(Ai; v) minf Æ(w; v) j w 2 Aig, and pi(v) 2 Ai be nearest to v, i.e. Æ(pi(v); v) = Æ(Ai; v). Æ(Ak; v) 1 letB(v) [k�1i=0 fw 2 Ai�Ai+1 j Æ(w; v) < Æ(Ai+1; v)g. Figure 1: Preprocessing a finite metric space algorithm distk(u; v) w u ; i 0 while w 62 B(v) i i+ 1 (u; v) (v; u) w pi(u) return Æ(w; u) + Æ(w; v) Figure 2: Answering a distance query Finally, in the last two subsections of this section we consider more practical issues. 3.1 Preprocessing a finite metric space A description of the preprocessing algorithm preprok(V; Æ) is given in Figure 1. The missing implementation details are ex- plained below. The algorithm receives an n�n matrix representing a finite metric Æ(u; v) on a set V containing n points referred to as vertices. (In the next section, we consider the case in which the input to the preprocessing algorithm is not an explicit n�n matrix that describes the metric on V , but rather a weighted undirected graph G = (V;E) that induces a shortest paths metric on V .) The preprocessing algorithm starts by constructing a non- increasing sequence of sets A0 � A1 � � � � � Ak�1 by a process of repeated sampling. The sequence begins with A0 = V . Each set Ai, where 1 � i < k, is then obtained by taking, roughly, an n�1=k fraction of the elements of Ai�1. More precisely, each element of Ai�1 is placed in Ai, independently, with probability n�1=k . Finally, Ak = ;. The expected size of Ai, for 0 � i � k, is clearly n1�i=k . In the following, for simplicity, we assume that Ak�1 6= ;. This is the case with extremely high probability, and if not, we can just rerun the algorithm. Now, for each vertex v and index i = 0; :::; k� 1, the algorithm computes Æ(Ai; v), the smallest distance from an element of Ai to v. It also lets pi(v) 2 Ai be nearest possible to v, that is, Æ(pi(v); v) = Æ(Ai; v). Note that A0 = V , so Æ(A0; v) = 0, and p0(v) = v, for every v 2 V . Next, the algorithm sets Æ(Ak; v) Æ(;; v) =1, with pk(v) undefined. Finally, for each vertex v 2 V , the algorithm computes a bunch p2(v) v p1(v) Figure 3: The construction of the bunch B(v). B(v) � V as follows. A vertex w 2 Ai�Ai+1 is put in the bunch B(v) if and only if Æ(w; v) < Æ(Ai+1; v), i.e., ifw is strictly closer to v than all the vertices of Ai+1. Note that as Æ(Ak; v) =1, we get that Ak�1 � B(v), for every v 2 V . We show in Section 3.3, that the expected size of the bunch B(v), for every v 2 V , is at most kn1=k . A schematic description of the construction of the bunch B(v), for some v 2 V , is given in Figure 3. It is assumed there that k = 3. The black vertices are the vertices of A2, the grey ones are the vertices of A1�A2, while the white ones are those ofA0�A1. The bunch B(v) is composed of the the vertices pointed to by an arrow from v. Also shown in the figure are p1(v), the vertex of A1 closest to v, and p2(v), the vertex of A2 closest to v. This almost completes the operation of the preprocessing algo- rithm. For each bunch B(v), the preprocessing algorithm con- structs a hash table (see Carter and Wegman [15], or [20, Chap- ter 12]) of size O(jB(v)j) that holds, for every w 2 B(v), the distance Æ(w; v). Using this hash table it can be checked for every w 2 V , in expectedO(1) time, whetherw 2 B(v) and if so what is Æ(w; v). Alternatively, it constructs a 2-level hash table (see Fred- man, Komlos and Szemeredi [30]), again of size O(jB(v)j), using which it is possible to check whetherw 2 B(v), and return Æ(w; v) if so, in O(1) worst case time. For the sake of implementations, we note here that multiplicative hashing has recently been proved uni- versal by Dietzfelbinger et al. [21], and Dietzfelbinger and Hüne [22], and Thorup [50] have found it to be an order of magnitude faster than the standard universal hashing schemes from text books using arithmitic modulo a prime number. The data structure constructed by the preprocessing algorithm stores for each vertex v 2 V , � for 0 � i � k � 1, the witness pi(v) and the corresponding distance Æ(pi(v); v) = Æ(Ai; v). � the (2-level) hash table for the bunch B(v), holding Æ(v; w), for every w 2 B(v). The total size of the data structure is O(kn + P v2V jB(v)j). In Section 3.3, we show that the expected size of B(v), for every v 2 V , is at most kn1=k . The total size of the data structure is therefore O(kn1+1=k). The time complexity of preprok(V; Æ) is clearly O(n2). 3.2 Answering a distance query A description of the very simple query answering algorithm distk(u; v) is given in Figure 2. It uses only four variables: u and v, the two vertices whose distance is to be estimated, a third vertex w and an index i. The algorithm repeatedly swaps u and v. This clearly does not affect their distance. Initially, w = u = p0(u) and i = 0. If w 2 B(v), a condition checked by accessing the (2- level) hash table of B(v), we are done. Otherwise, the algorithm increments i, swaps u and v, and lets w pi(u) 2 Ai. It con- tinues in this way until w 2 B(v). This condition is guaranteed to hold when i = k � 1, if not before, as then w 2 Ak�1 and Ak�1 � B(v) for every v 2 V . When w 2 B(v), the algorithm returns Æ(w;u) + Æ(w; v) as an upper bound on Æ(u; v). The distance Æ(w; u) = Æ(pi(u); u) is read directly from the data structure constructed during the prepro- cessing stage. The distance Æ(w; v) = Æ(v; w) is returned by the (2-level) hash table of B(v) together with the answer to the query w 2 B(v). The complexity of distk(u; v) is clearly O(k). The most time consuming operations are the at most k accesses to the hash tables to test whether w 2 B(v), returning Æ(w; v) if so. The stretch of the estimate produced by distk(u; v) is analyzed in Section 3.4. 3.3 Analysis of the preprocessing algorithm We have shown already, in Section 3.1, that the running time of the preprocessing algorithm is O(n2) and that the size of the data structure produced by it is O(kn + P v2V jB(v)j). All that remains, therefore, is to analyze the expected sizes of the bunches B(v), for v 2 V . LEMMA 3.2. For every v 2 V , we have E[jB(v)j] � kn1=k . PROOF. Let v 2 V . We prove the lemma by showing, for every 0 � i � k�1, that the expected size of B(v)\Ai is at most n 1=k . For i = k � 1, the statement is trivial as E[jAk�1j] = n 1=k . Assume, therefore, that i < k � 1. We show that the expected size of B(v) \ Ai, for i < k � 1, is stochastically dominated by a geometric random variable with parameter p = n�1=k. Let w1; w2; : : : ; w` be the elements of Ai, arranged in any non-decreasing order of distance from v. If wi 2 B(v), then Æ(wi; v) < Æ(Ai+1; v), and thus w1; w2; : : : ; wi�1 62 Ai+1. Note that p = Pr[w 2 Ai+1 j w 2 Ai] = n �1=k , for i < k � 1. Thus Pr[wi 2 B(v)] � (1 � p) i�1 and the expected size of B(v) \Ai is at most X̀ i=1 Pr[wi 2 B(v)] � X̀ i=1 (1� p) i�1 < p �1 = n 1=k : This completes the proof of the lemma. As described, the preprocessing algorithm has, therefore, a worst case running time of O(n2) and the data structure produced has an expected size of O(kn1+1=k). We can get a data structure of sizeO(kn1+1=k) by rerunning the algorithm until the data structure produced is small enough. By Markov’s inequality, the expected number of repetitions required is constant, so the expected running time of this version of the algorithm is still O(n2), as stated in Theorem 3.1. 3.4 Analysis of the query answering algorithm We next obtain an upper bound of 2k � 1 on the stretch of the estimated distance returned by distk(u; v). LEMMA 3.3. distk(u; v) � (2k � 1)Æ(u; v). PROOF. Clearly, the swapping of u and v does not change their distance � = Æ(u; v). Before the while loop starts, w = u so Æ(w; u) = 0. We want to show that each iteration increases Æ(w; u) by at most �. Since Ak�1 � B(v), there are at most k � 1 iter- ations, so we will then end up with Æ(w; u) � (k � 1)�. Now, Æ(w; v) � Æ(w; u) + Æ(u; v) � (k � 1)� + � � k�, so the estimated distance returned is at most (2k � 1)�. All that remains, therefore, is to show that Æ(w; u) increases, in each iteration, by at most � = Æ(u; v). Let ui; vi and wi be the values of the variables u; v and w assigned with a given value of i. Then v0 and u0 are the original values of u and v, and then w0 = u0, so Æ(w0; u0) = 0. We want to show that Æ(wi; ui) � Æ(wi�1; ui�1)+� if the i-th iteration passes the test of the while-loop. Then wi�1 62 B(vi�1), so Æ(wi�1; vi�1) � Æ(Ai; vi�1) = Æ(pi(vi�1); vi�1). How- ever, vi�1 = ui and wi = pi(ui), so we get Æ(wi; ui) = Æ(pi(ui); ui) = Æ(pi(vi�1); vi�1) � Æ(wi�1; vi�1) � Æ(wi�1; ui�1) +�, as required. This completes the proof of Theorem 3.1. 3.5 Distance labels Let G = (V;E) be a weighted undirected graph on n-vertices with integer edge weights. Let � be the diameter of G, and let k � 1 be an integer. Peleg [45] describes a way of assigning each vertex v 2 V of the graph G = (V;E) an O(kn1=k log n log �)- bit label, denoted label(v), such that for any u; v 2 V , a stretch 8k estimate of the distance Æ(u; v) may be obtained just by looking at label(u) and label(v). Computing this estimate, in Peleg’s scheme, may take (n1=k) time. We obtain the following improvement to Peleg’s result: THEOREM 3.4. Let (V; Æ) be a metric space on n points with integral distances with diameter �. Let 1 � k � log n be an integer. Then, it is possible to assign to each point v 2 V an O(n1=k log1�1=k n log(n�))-bit label, denoted label(v), such given label(u) and label(v), for any two points u; v 2 V , it is pos- sible to compute, in O(k) time, an approximation to the distance Æ(u; v) with a stretch of at most 2k � 1. As would follow from the results of Section 5, this result is es- sentially optimal. Lower bounds on the size of labels in various kinds of labeling schemes are also obtained by Gavoille et al. [32]. In our labeling scheme, label(v), for each v 2 V , is composed of the the witnesses pi(v) and the distances Æ(Ai; v), for 1 � i < k, as well as the (2-level) hash table that holds, for every w 2 B(v), the distance Æ(w; v). It is easy check that all the information needed by the query answering algorithm distk(u; v) is contained in label(u) or in label(v). Thus, a stretch 2k � 1 estimate of the distance Æ(u; v) may be obtained in O(k) time just by examining label(u) and label(v). It follows from Lemma 3.2 that the expected size of label(v), for any v 2 V , is O(kn1=k) words, where each word holds either a name of a vertex or a distance. As there are n vertices in the graph, and as the diameter of the graph is �, each word contains at most log(n�) bits. We are interested, here, however, in the maximum size of a label, not its expected size. It is not difficult to show, using arguments similar to arguments used below, that with high probability, the size of every bunch B(v), for v 2 V , is O(n1=k log n). This yields, therefore, a dis- tance labeling scheme with O(n1=k log n log(n�))-bit labels. A factor of about log1=k n may be gained by slightly changing the sampling probability used by the preprocessing algorithm: s n1=k(lnn+ 1)1�1=k A0 V ; Ak ; for i 1 to k � 1 for every v 2 V , let Ni(v) contain the s vertices of Ai�1 closest to v. let Ai be a subset of Ai�1 of size at most jAi�1j s (lnn+ 1) that hits Ni(v), for every v 2 V . Figure 4: Deterministic construction of the samples. LEMMA 3.5. If the sampling probability used by preprok(V; Æ) is changed from n�1=k to (n= lnn)�1=k , then with high prob- ability, the size of every bunch B(v), for v 2 V , is O(n1=k log1�1=k n). The proof of the lemma would appear in the full version. 3.6 Derandomization The preprocessing algorithm prepro(V; Æ) given in Section 3.1 is randomized. In it not difficult, however, to derandomize it, with only a small loss in efficiency. Randomization is only used by prepro(V; Æ) in the selection of the samples A0 � A1 � � � � � Ak, and in the construction of the (2-level) hash tables. A deterministic way of constructing a sequence of samples with all the desired properties is given in Figure 4. The sets Ai are con- structed one by one. The set A0 is simply V . Suppose that Ai�1, for 1 � i < k, was already constructed. The algorithm lets Ni(v), for every v 2 V , be the set of the n1=k ln1�1=k n vertices of Ai�1 that are closest to v. Ties are broken arbitrarily. Then, the algo- rithm chooses a set Ai of size at most n 1�i=k(lnn+1)i=k that hits all the neighborhoods Ni(v), for v 2 V . To construct the set Ai, the algorithm relies on the following well known lemma, which is a slight modification of Theorem 2.2 of Alon and Spencer [4, p. 6]: LEMMA 3.6. Let N1; : : : ; Nn � U be a collection of sets with jU j = u and jNij � s, for 1 � i � n. Then, a setA of size at most u s (ln ns u +1) � u s (lnn+1) such that Ni\A 6= ;, for 1 � i � n, can be found, deterministically, in O(u+ Pn i=1 jNij) time. The setA, whose existence is claimed in Lemma 3.6, is obtained by repeatedly adding to A elements of U that hit as many unhit sets as possible, until only u s sets are unhit. The construction of A is then completed by adding an element from each one of the unhit sets. For more details, see Alon and Spencer [4, p. 6]. (Lemma 3.6 is slightly more general than Theorem 2.2 of [4] that assumes u = n.) We now claim: THEOREM 3.7. If the random sampling used by preprok(V; Æ) is replaced by the deterministic sampling procedure described in Figure 4, then the size of each bunch B(v), for v 2 V , is at most kn1=k(lnn+ 1)1�1=k . PROOF. Let v 2 V . Note that B(v) = [k�1i=0 Bi(v), where Bi(v) = fw 2 Ai � Ai+1 j Æ(w; v) < Æ(Ai+1; v)g, for 1 � i � k. We claim that jBi(v)j � s = n 1=k(lnn + 1)1�1=k , for 0 � i < k � 1, otherwise Ni(v) � Bi(v), and by the con- struction of Ai+1, we have Ai+1 \ Bi(v) 6= ;, a contradiction. Finally, Bk�1(v) = Ak�1, and it is easy to show by induction that jAk�1j � n 1=k(lnn+ 1)1�1=k . We have thus lost only a factor of about log1�1=k n with respect to the expected bunch size of the randomized algorithm, and only algorithm preprok(V;E) A0 V ; Ak ; for i 1 to k � 1 let Ai contain each element of Ai�1, independently, with probability n�1=k. Æ(Ak; v) 1 for i k � 1 downto 0 for every v 2 V , compute Æ(Ai; v) and find pi(v) 2 Ai such that Æ(pi(v); v) = Æ(Ai; v). if Æ(Ai; v) = Æ(Ai+1; v) then pi(v) pi+1(v) (*) for every w 2 Ai �Ai+1, grow a shortest path tree T (w) from w spanning C(w) = f v 2 V j Æ(w; v) < Æ(Ai+1; v) g. for every v 2 V , let B(v) fw 2 V j v 2 C(w)g. Figure 5: Preprocessing a graph a factor of about k with respect to the maximum bunch size of the randomized algorithm with the slightly modified sampling. The preprocessing algorithm preprok(V;E) also has to con- struct a 2-level hash table for each bunch B(v), where v 2 V . (This step is not explicit in the description of the algorithm given in Figure 1.) The linear time algorithm given by Fredman et al. [30] for the construction of such tables is randomized. Their algo- rithm is derandomized, however, by Alon and Naor [3]. To con- struct a perfect hash table over q = ~O(n1=k) elements from a universe of size n, without assuming that k is constant, they use O(q log q log n) = ~O(n1=k) time. Hence constructing the hash table for all B(v) takes ~O(n1+1=k) time, so this derandomization does not affect the overall running time of O(n2) for the prepro- cessing algorithm. 4. APPROXIMATE DISTANCE ORACLES FOR GRAPHS In the previous section, we assumed that metric Æ(u; v) is given to us explicitly. Here, we consider the more realistic situation in which the metric that we are supposed to process is the shortest paths metric of a weighted undirected graph. The graph, and not the metric, is given to us this time. We can, of course, begin by solving the APSP problem for the input graph and then use the algorithms of the previous section to preprocess the metric obtained. This solution is wasteful, however, both in terms of running time and in terms of space. It is much more efficient to directly process the graph that induces the metric. The new preprocessing algorithm is described next, in Sec- tion 4.1. A modification to the query answering algorithm that al- lows it to return paths, and not just approximate distances, is then described in Section 4.2. The analysis of the modified preprocess- ing algorithm is given in Section 4.3. Finally, in Section 4.4 we show that our preprocessing algorithm is also a very efficient algo- rithm for constructing sparse spanners and compact tree covers. 4.1 Preprocessing a graph A description of the preprocessing algorithm preprok(V;E) is given in Figure 5. It receives as input a weighted undirected graph Figure 6: Constructing the clusters. G = (V;E). The preprocessing algorithm is similar to the pre- processing algorithm given in Figure 1. In particular, the sets Ai and the bunches B(v) would be exactly the same. The implemen- tation details, this time, are less trivial, as distances Æ(u; v) have to be computed, instead of just being read from an input matrix. This is why we introduce the new sets C(�) before computing the bunches B(�). The algorithm starts again by constructing the samples A0 � A1 � � � � � Ak�1 � Ak, where A0 = V and Ak = ;. The elements of Ai are sometimes referred to as i-centers. The operation of the algorithm is then composed of k iterations. The i-th iteration starts by computing the distances Æ(Ai; v), for every v 2 V , where Æ(Ai; v) = minf Æ(w; v) j w 2 Aig. This is done by adding to G = (V;E) a new source vertex s, and edges (s; w) of weight 0, for every w 2 Ai, and by computing the dis- tances from the new source s to all the other vertices of the graph. The distances are found in O(m) time by running the single-source shortest paths algorithm of Thorup [51]. It is easy to check that for every v 2 V , the distance from s to v in the new graph is indeed Æ(Ai; v). Furthermore, the shortest paths tree constructed by the al- gorithm supplies, for every v 2 V , a witness pi(v) 2 Ai such that Æ(pi(v); v) = Æ(Ai; v). Indeed, if v is in the branch of the short- est paths tree that starts with the edge (s; w), where w 2 Ai, then Æ(Ai; v) = Æ(w; v) and we can set pi(v) w. All the witnesses are easily found, therefore, in O(m) time. The minimum distance Æ(Ai; v) may be attained by several ver- tices of Ai. The preprocessing algorithm of the previous section lets pi(v) be an arbitrary vertex of Ai satisfying Æ(pi(v); v) = Æ(Ai; v). The new statement (*) trivially preserves this property. In addition, it also ensures that the following property, that plays a crucial role in the construction of small-stretch paths, as described in Section 4.2, also holds: LEMMA 4.1. For any v 2 V and 0 � i � k � 1, we have pi(v) 2 B(v). PROOF. We prove the claim by induction on i from above. The claim trivially holds when i = k � 1, as then pk�1(v) 2 Ak�1 � B(v), for every v 2 V . Suppose therefore that i < k � 1, and that pi+1(v) 2 B(v). If the test of (*) fails, we get pi(v) = pi+1(v) 2 B(v). Otherwise, Æ(pi(v); v) = Æ(Ai; v) < Æ(Ai+1; v), and then pi(v) 2 B(v) by definition of B(v). Next, the algorithm constructs a cluster C(w) around each i- center w 2 Ai � Ai+1. The cluster C(w) is composed of all the vertices that are closer to w than to any (i � 1)-center. In other words, C(w) = f v 2 V j Æ(w; v) < Æ(Ai+1; v) g. Note that for every w 2 Ak�1 we have C(w) = V , as Æ(Ak; v) =1, for every v 2 V . It is easy to see that the bunches of the previous section and the clusters of this section are ‘inverses’ of each other, in that w 2 B(v) if and only if v 2 C(w) for any v; w 2 V . Thus, the bunches constructed by the final loop of the preprocessing algorithm are identical to the bunches that would have been constructed by the preprocessing algorithm of the previous section. The construction of clusters is reminiscent of the construction of Voronoi diagrams. An important difference here, however, is that each i-center w 2 Ai � Ai+1 captures all vertices that are closer to it than to all the (i+1)-centers, and not to all the i-centers as the definition of Voronoi diagrams would suggest. In particu- lar, the clusters at a particular iteration are not necessarily disjoint. A schematic description of the clustering construction process is given in Figure 6. The filled vertices there are (i + 1)-centers. The two large unfilled vertices are i-centers and the two polygons depict the clusters associated with them. (In Figure 6, it is implic- itly assumed that the distances between the vertices are Euclidean. This is done for illustration purposes only. Our algorithms work on general weighted graphs.) Each cluster C(w) is computed by running a slightly modified version of Thorup’s SSSP algorithm from w [49]. Since this algo- rithm is rather complicated, we describe instead a modified version of Dijkstra’s classical SSSP algorithm [23] (see also [20, Chapter 25]). The changes to Thorup’s algorithm are very similar. Modifying Dijkstra’s algorithm. Dijkstra’s algorithm with source w maintains for each vertex v an upper bound d(v) on the distance Æ(w; v). If d(v) has not been assigned yet, it is in- terpreted as infinite. Initially, we just set d(w) = 0, and we have no visited vertices. At each iteration, we select an unvis- ited vertex u with the smallest finite d(u), visit it, and relax all its edges, that is, for each incident edge (u; v) 2 E, we set d(v) minf d(v); d(u) + `(u; v)g. We continue in this way until no unvisited vertex v has a finite d(v). Our simple modification of Dijkstra’s algorithm is that we relax the edge (u; v) only if d(u) + `(u; v) < Æ(Ai+1; v). Note that Æ(Ai+1; v) was computed in the previous iteration so the test takes constant time. LEMMA 4.2. The modified Dijkstra’s algorithm visits exactly the vertices of C(w), assigning each the correct distance from w. PROOF. The proof is similar to correctness proof of Dijkstra’s original algorithm. Suppose w 2 Ai � Ai+1. The essential new point is the following easily verified claim: if v 2 C(w) and v0 lies on a shortest path from w to v, then Æ(w; v0) < Æ(Ai+1; v 0), so v0 2 C(w). By definition, v 2 C(w) if and only if Æ(w; v) < Æ(Ai+1; v), but then Æ(w; v 0) = Æ(w; v)�Æ(v; v0) < Æ(Ai+1; v)� Æ(v; v0) � Æ(Ai+1; v 0), as desired. Note that we only relax an edge (u; v) if d(u) + `(u; v) < Æ(Ai+1; v). Consequently, a vertex v 62 C(w) is never assigned a finite distance, and hence it is never visited. We now show that all vertices visited are assigned correct dis- tances from w. The proof is by induction. Suppose that v 2 C(w) is about to be visited and that all previously visited ver- tices were assigned the correct distance. Let p be a shortest path from w to v. Let u0 be the last visited vertex on this path, and let v0 be the next vertex on the path. By the above claim, Æ(w; v0) < Æ(Ai+1; v 0). Moreover, by the induction hypothe- sis, when u0 was visited d(u0) = Æ(w; u0), and then Æ(w; v0) = Æ(w; u0)+`(u0; v0) = d(u0)+`(u0; v0). Hence d(u0)+`(u0; v0) < Æ(Ai+1; v 0), so the edge (u0; v0) was relaxed, setting d(v0) Æ(u; v0). As v is about to be visited before v0, we must have d(v) � d(v0) = Æ(w; v0) � Æ(w; v) � d(v), so d(v) = Æ(w; v). Finally, we want to show that all v 2 C(w) are visited. Suppose for a contradiction that v 2 C(w) is not visited. Let (u0; v0) be the last edge on a shortest path to v with u0 visited. From above, we know that u0 got assigned the correct distance, and the same anal- ysis as above implies that (u0; v0) got relaxed when u0 was visited, but then v0 will be visited eventually, contradicting the choice of (u0; v0). Thus, the modified version of Dijkstra’s algorithm that we de- scribed does construct C(w). It is easy to arrange that it would also produce a shortest path tree T (w) spanning the cluster C(w). This would not affect the running time of the algorithm. For a simple, yet relatively efficient, implementation of Dijk- stra’s algorithm, we can just use William’s heap [57] (see also [20, Chapter 7]) to store the finite distances d(v) of the unvisited vertices. We can then both find the v minimizing d(v) and de- crease some d(v) in O(log n) time. The former is done at most n � 1 times, and the latter is done at most m times, so the to- tal running time of the unmodified algorithm, from a given source vertex, is O((m + n) log n). It is easy to see that in the modi- fied version of the algorithm, all the edges relaxed are edges that touch vertices of C(w). Thus, the time spent on the construc- tion of C(w) is O(jE(C(w))j log n), where E(C(w)) is the set of edges touching vertices of C(w). The complexity is reduced to O(jE(C(w))j+ jC(w)j log n) if the more sophisticated Fibonacci heaps of Fredman and Tarjan [31] (see also [20, Chapter 20]) are used. The same conditional relaxation can be applied to Thorup’s SSSP algorithm [49]. We first spend, once and for all, O(m) time on constructing a so-called component hierachy. Afterwards, each cluster C(w), for w 2 V , can be computed in O(jE(C(w))j) time. When all the clusters C(w), for w 2 V are constructed, they are used to generate the bunches B(v), for v 2 V . Recall that, by definition, w 2 B(v) if and only if v 2 C(w). The conversion can clearly be done in O( P w2V jC(w)j) = O( P v2V jB(v)j) time. Finally, the algorithm constructs (2-level) hash tables for the bunches B(v), for v 2 V , and outputs the witnesses pi(v), the distances Æ(pi(v); v) = Æ(Ai; v), and the hash tables of B(v), for every 1 � i � k and v 2 V . In addition to that, the preprocess- ing algorithm also outputs, for every w 2 V , the shortest paths tree T (w) that spans the cluster C(w). The sum of the sizes of all the trees T (w), for w 2 W , is the same as the sum of the sizes of all the clusters, which is also the sum of the sizes of all the bunches. Thus, the size of the data struc- ture produced is, asymptotically, the same as the size of the data structure that would have been produced, had the preprocessing al- gorithm of the previous section been applied to the shortest paths metric of the graph. Thus, the expected size of the produced data structure is O(kn1+1=k). All that remains, therefore, is to analyze the running time of preprok(V;E). This is done in Section 4.3. As a final remark, we note that instead of constructing a sepa- rate hash table for each bunch B(v), for v 2 V , we can construct a single (2-level) hash table of size O( P v2V jB(v)j) that holds Æ(w; v), for every w; v 2 V such that w 2 B(v). The access time would still be O(1). 4.2 Answering a path query As all the data structures returned by the metric preprocessing algorithm of Section 3.1 are also returned by the graph preprocess- ing algorithm of Section 4.1, the query answering algorithm from Figure 2, detailed in Section 3.2, may be used, without any modifi- cation, to answer approximate distance queries. We next describe how to augment the distance query algo- rithm if it is to return not just an estimated distance distk(u; v) of stretch at most 2k � 1, but also a path from u to v of length at most distk(u; v). When the distance query algorithm terminates, w 2 B(v) so v 2 C(w). Moreover, by Lemma 4.1, w = pi(u) 2 B(u), so we also have u 2 C(w). Hence, the path between u and v in T (w), the shortest paths tree of C(w), is of length at most Æ(w; u)+ Æ(w; v). To report the edges on this path in constant time per edge, we move in parallel from u and v towards the root w, stopping as soon as we reach, from one of u and v, a vertex w0 that was already reached from the other. (This vertex is the least common ancestor of u and v in the tree.) We then output the edges on the path from u to w0 and, in reversed order, the edges on the path from v to w0. The above solution constructs the small-stretch path from u to v in amortized constant time per edge. Using techniques from [54] it is possible to construct the path in worst case constant time per edge. We do not elaborate on this here. 4.3 Analysis of graph preprocessing As mentioned, the complexity of constructing the cluster C(w) isO(jE(C(w))j) (orO((jE(C(w))j+ jC(w)j) log n) if the simple modification of Dijkstra’s algorithm is used). Recall that E(C(w)) is the set of edges that touch vertices of C(w). Let E(v) be the set of edges that touch the vertex v. The total cost of construct- ing all clusters is asymptotically bounded by P w2V jE(C(w))j �P w2V;v2C(w) jE(v)j = P v2V;w2B(v) jE(v)j = P v2V (jB(v)j � jE(v)j). By Lemma 3.2, the expected size of jB(v)j is at most kn1=k , for any v 2 V , so by linearity of expectation, the expected total cost is asymptotically bounded by X v2V (n 1=k jE(v)j) = 2kmn 1=k : Since all other operations in preprok(V;E) take onlyO(km) time, its total complexity is O(kmn1=k). As in the last section we note that it is only the expected size of the data structure constructed which is O(kn1+1=k). To ob- tain a data structure of size O(kn1+1=k), we may have to run preprok(V; E) several times, but the expected number of repeti- tions is constant, so the total expected preprocessing time is still O(kmn1=k), as specified in Theorem 1.1. 4.4 Sparse spanners and tree covers As described in Section 4.2, the query answering algorithm may actually find a stretched path between u and v in some tree T (w). We get, therefore, the following interesting corollary: COROLLARY 4.3. The collection of shortest paths trees T (w), for w 2 V , constructed by algorithm preprok(V;E), forms a (2k � 1)-spanner of the graph G = (V;E). The expected size of this (2k � 1)-spanner is O(kn1+1=k) and it can be constructed in O(kmn1=k) time. As mentioned, the fact that every weighted graph on n-vertices has a (2k � 1)-spanner with O(n1+1=k) edges is not new. The corollary gives, however, a much faster algorithm for construct- ing such spanners. The fastest running time known before, for weighted graph, was O(mn1+1=k) [5]. For unweighted graphs, there is linear time algorithm for constructing such spanners (see also Exercise 3 on page 188 of Peleg [44], attributed to [33]). Combining Corollary 4.3 with Lemma 3.5 we also get the fol- lowing corollary: girth number of edges lower-bound references 4 �(n2) complete bipartite graphs 6 �(n3=2) [47],[29],[14],[56] 8 �(n4=3) [55],[10],[56] 10 (n6=5) ; O(n5=4) [55],[10],[37] 12 �(n6=5) [55],[10],[56],[37] 14 (n9=8) ; O(n7=6) [38],[39] 16 (n10=9) ; O(n8=7) [58],[38] 4r ; r � 5 (n 1+ 1 3(r�1) ) ; O(n 1+ 1 2r�1 ) [38],[39] 4r + 2 ; r � 4 (n 1+ 1 3r�1 ) ; O(n 1+ 1 2r ) [38],[39] Table 2: Best known bounds on the maximum number of edges in an n-vertex graph with a given girth. COROLLARY 4.4. The collection of shortest paths trees T (w), forw 2 V , constructed by algorithm preprok(V;E), with the sam- pling probability changed from n�1=k to (n= lnn)�1=k , forms a tree cover of the graph G = (V;E) with the following proper- ties: (i) With high probability, every vertex is contained in only O(n1=k log1�1=k n) trees. (ii) For every two vertices u; v 2 V , there is a tree T (w) in this collection that contains a path between u and v that is of stretch at most 2k � 1. Furthermore, the corre- sponding tree can be identified in O(k) time. A deterministic algorithm for constructing such tree covers may be obtained using the technique of Section 3.6. Our tree cover con- struction improves a construction implicit in Awerbuch and Peleg [8] (see also Peleg [44, Chapter 15]). In [54], we use our tree cover construction, together with other ideas, to obtain routing schemes for weighted undirected networks that exhibit an essentially opti- mal tradeoff between the size of the routing tables used and the stretch of the resulting routes. 5. SPACE LOWER BOUND A simple argument shows that for any integer k, any graph on n vertices with at least n1+1=k edges contains a cycle of size at most 2k. (For a proof that 1 2 n1+1=k edges are in fact enough, see Alon et al. [2].) This result is conjectured by Erdős [28], Bondy and Simonovits [12] and Bollobás [11] to be tight. Namely, it is conjectured that for any k � 1, there are graphs with (n1+1=k) edges and girth greater than 2k. As any graph contains a bipar- tite subgraph with at least half the edges, the conjecture actually implies the existence of graphs with (n1+1=k) edges and girth at least 2k + 2. This conjecture was proved, however, only for k = 1; 2; 3; 5 (see references below). Let mg(n) be the maximal number of edges in an n-vertex graph with girth g. The girth conjecture says that m2k+2(n) = (n1+1=k). Note, as mentioned above, that m2k+2(n) = �(m2k+1(n)). The best bounds on mg(n), for even girth g, are given in Table 2. (Several references are given for each result. This is either because the result was independently discovered by sev- eral authors, or because there are several variants of the construc- tion. Some of the references, e.g., Wenger [56], were added as they are more accessible than the older references.) The results for g = 6 follow from constructions of finite projective geometries. The constructions of Lazebnik et al. [38],[39] slightly improve re- sults obtained by Margulis [42] and the results obtained using the Ramanujan graphs of Lubotzky et al. [41]. PROPOSITION 5.1. Let k be an integer, and let t < 2k + 1. Then, any stretch t distance oracle for graphs with n vertices and m edges must use at least minfm;m2k+2(n)g bits of storage on at least one input graph. PROOF. Let O be a stretch t distance oracle for graphs with n vertices and m edges. For any graph H of this size, let OH be the data structure produced by O by preprocessing H . Let OH(v; w) be the approximate distance returned by the oracle for the query (v; w). Note that ÆH(v; w) � OH(v; w) � t ÆH(v; w). Let G be a girth 2k + 2 unweighted graph on n vertices with m0 = minfm;m2k+2(n)g edges. (If m < m2k+2(n) we can simply pick an m-edge subgraph of a girth 2k + 2 graph with m2k+2(n) edges.) Let H be any subgraph of G. Consider any edge (v; w) of G. If (v; w) is in H , then OH(v; w) � t < 2k + 1. But, if (v; w) is not in H , the shortest path from v to w in H has at least 2k + 1 edges, so OH(v; w) � 2k + 1. Consequently, all the 2 m0 subgraphs of G have different tables, and hence at least one requires m0 bits. Proposition 5.1 holds even if the oracle is only required to pro- duce estimated distances, without being required to produce corre- sponding paths. We point out, however, that there is still a loga- rithmic gap of �(k log n) between this lower bound and our upper bound, even if the girth conjecture holds, as our algorithms use O(kn1+1=k) words while the lower bound is (n1+1=k) bits. Finally, we point out that no space efficient approximate distance oracles are possible for directed graphs: PROPOSITION 5.2. For any finite stretch distance oracle for di- rected graphs must use at least (n2) bits of storage on at least one n-vertex graph. The simple proof would appear in the full version of the paper. 6. CONCLUDING REMARKS We presented approximate distance oracles with fast preprocess- ing times, essentially optimal space requirements, and constant query time. Our construction is extremely simple. It yields, as byproducts, improved algorithms for constructing sparse spanners, more compact tree covers, and more concise distance labelings. Due to their basic nature, we expect our ideas to prove useful in many other contexts. Some interesting open questions remain. First, our basic prepro- cessing algorithm is randomized. While it was easy to derandomize it when the full distance matrix was available, it is not clear how to do it in o(mn) time in the graph setting. It seems that new ideas would be needed to achieve that. Our oracles are almost optimal, in all respects, when the param- eter k is large. It remains an interesting open problem, however, to reduce the preprocessing times of small stretch oracles. The sit- uation for stretch 3 is especially intriguing. We show here that a stretch 3 oracle with a space requirement of O(n3=2) can be con- structed in O(mn1=2) time. Cohen and Zwick [19] have shown that a stretch 3 oracle that uses O(n2) space can be constructed in O(n2 log n) time. Could these results be combined, i.e., is it pos- sible to construct a stretch 3 oracle that uses only O(n3=2) space in ~O(n2) time? As mentioned in Section 4.4, the results of this paper, com- bined with some other ingredients, yield essentially optimal rout- ing schemes for weighted undirected networks. More on this can be found in [54]. 7. 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