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New Results on Fault Tolerant Geometric Spanners

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We investigate the problem of constructing spanners for a given set of points that are tolerant for edge/vertex faults. Let ℝd be a set of n points and let k be an integer number. A k-edge/vertex fault tolerant spanner for S has the property that after the deletion of k arbitrary edges/vertices each pair of points in the remaining graph is still connected by a short path. Recently it was shown that for each set S of n points there exists a k-edge/vertex fault tolerant spannerwith O(k 2n) edges which can be constructed in O(n log n + k 2n) time. Furthermore, it was shown that for each set S of n points there exists a k-edge/vertex fault tolerant spannerwhose degree is bouned by O(c k+1) for some constant c. Our first contribution is a construction of a k-vertex fault tolerant spanner with O(kn) edges which is a tight bound. The computation takes O(nlogd−1n + kn log log n) time. Then we show that the same k-vertex fault tolerant spanner is also k-edge fault tolerant. Thereafter, we construct a k-vertex fault tolerant spanner with O(k 2n) edges whose degree is bounded by O(k 2). Finally, we give a more natural but stronger definition of k-edge fault tolerance which not necessarily can be satisfied if one allows only simple edges between the points of S. We investigate the question whether Steiner points help.We answer this question affirmatively and prove Θ(kn) bounds on the number of Steiner points and on the number of edges in such spanners.
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New Results on Fault Tolerant Geometric Spanners
Tam´as Lukovszki
Heinz Nixdorf Institute,
University of Paderborn, D-33102 Paderborn, Germany
tamas@hni.uni-paderborn.de
Abstract. We investigate the problem of constructing spanners for a given set of
points that are tolerant for edge/vertex faults. Let S
IR
d
be a set of n points and
let k be an integer number. A k-edge/vertex fault tolerant spanner for S has the
property that after the deletion of k arbitrary edges/vertices each pair of points in
the remaining graph is still connected by a short path.
Recently it was shown that for each set S of n points there exists a k-edge/vertex
fault tolerant spanner with O k
2
n edges which can be constructed in O nlogn
k
2
n time. Furthermore, it was shown that for each set S of n points there exists a
k-edge/vertex fault tolerant spanner whose degree is bouned by O c
k
1
for some
constant c.
Our first contribution is a construction of a k-vertex fault tolerant spanner with
O kn edges which is a tight bound. The computation takes O nlog
d
1
n
knloglogn time. Then we show that the same k-vertex fault tolerant spanner
is also k-edge fault tolerant. Thereafter, we construct a k-vertex fault tolerant
spanner with O
k
2
n edges whose degree is bounded by O k
2
. Finally, we give
a more natural but stronger definition of k-edge fault tolerance which not neces-
sarily can be satisfied if one allows only simple edges between the points of S.
We investigate the question whether Steiner points help. We answer this question
affirmatively and prove Θ kn bounds on the number of Steiner points and on the
number of edges in such spanners.
1 Introduction
Geometric spanners have many applications in various areas of the computer science.
They have been studied intensively in recent years. Let S be a set of n points in IR
d
,
where d is an integer constant. Let G
S E be a graph whose edges are straight
line segments between the points of S. For two points p q IR
d
, let dist
2
p q be the
Euclidean distance between p and q. The length length e of an edge e a b E is
defined as dist
2
a b . For a path P in G the length length P is defined as the sum of
the length of the edges of P. A path between two points p q S is called a pq-path.
Let t 1 be a real number. The graph G is a t-spanner for S if for each pair of points
p q S there is a pq-path in G such that the length of the path is at most t times the
Euclidean distance dist
2
p q between p and q. We call such a path a t-spanner path
and t is called the stretch factor of the spanner. If G is a directed graph and G contains a
Partially supported by EU ESPRIT Long Term Research Project 20244 (ALCOM-IT)and DFG
Graduiertenkolleg ”Parallele Rechnernetzwerke in der Produktionstechnik” Me872/4-1.
directed t-spanner path between each pair of points then G iscalled a directed t-spanner.
In order to distinguish the edges of a directed from an undirected graph we use a b
to denote an edge between the vertices a and b in a directed and a b in an undirected
graph.
Spanners were introduced by Chew [6]. They have applications in motion plan-
ing [7], they were used for approximating the minimum spanning tree [17], to solve a
special searching problem which appears in walkthrough systems [8], and to a polyno-
mial time approximation scheme for the traveling salesman and related problems [13].
The problem of constructing a t-spanner for a real constant t 1, that has O n
edges, has been investigated by many researchers. Keil [10] gave a solution for this
problem introducing the θ-graph
1
, which was generalized by Ruppert and Seidel [14]
and Arya et al [2] to any fixed dimension d. These authors gave also an O nlog
d
1
n
time algorithm to construct the θ-graph. Chen et al [5] proved that the problem of
constructing any t-spanner for t 1 takes nlogn time in the algebraic computation
tree model [3]. Callahan and Kosaraju [4], Salowe [15] and Vaidya [16] gave optimal
O nlogn time algorithm for constructing t-spanners. Several interesting quantities re-
lated to spanners were studied by Arya et al [1]. They gave constructions for bounded
degree spanners, spanners with low weight, spanners with low diameter, and for span-
ners having more than one of these properties. The weight w G of a graph G is the sum
of the length of its edges.
Fault tolerant spanners were introduced by Levcopoulos et al [12]. For the formal
definition we need the following notions. For a set S IR
d
of n points let K
S
denote the
complete Euclidean graph with vertex set S. If G S E is a graph and E E then
G
E denotes the graph G S E E . Similarly, if S S then the graph G S is the
graph with vertex set S S and edge set p q E : p q S S . Let t 1 be a real
number and k be an integer, 1 k n 2.
A graph G S E is called a k-edge fault tolerant t-spanner for S, or k t -EFTS,
if for each E E, E k, and for each pair p q of points of S, the graph G E
contains a pq-path whose length is at most t times the length of a shortest pq-path
in the graph K
S
E .
Similarly, G S E is called a k-vertex fault tolerant t-spanner for S, or k t -
VFTS, if for each subset S S, S k, the graph G S is a t-spanner for S S .
Levcopoulos et al [12] presented an algorithm with running time O nlogn k
2
n
which constructs a k t -EFTS/VFTS with O k
2
n edges for any real constantt 1. The
constants hidden in the O-notation are
d
t 1
O d
if t 1. They also showed that kn
is a lower bound for the number of edges in such spanners. This follows from the obvi-
ous fact that each k-edge/vertex fault tolerant spanner must be k-edge/vertex connected.
Furthermore, they gave another algorithm with running time O nlogn c
k 1
n , for
some constant c, which constructs a k t -VFTS whose degree is bounded by O c
k 1
and whose weight is bounded by O c
k 1
w MST .
1
Yao [17] and Clarkson [7] used a similar construction to solve other problems.
1.1 New results
We consider directed and undirected fault tolerant spanners. Our first contribution is
a construction of a k t -VFTS with O kn edges in O nlog
d
1
n
knloglogn time.
Then we showthat the same k-vertex fault tolerantspanner is also a k-edge fault tolerant
spanner. Our bounds for the number of edges in fault tolerant spanners are optimal up
to a constant factor and they improvethe previous O k
2
n bounds significantly. Further-
more, we construct a k-vertex fault tolerant spanner with O k
2
n edges whose degree is
bounded by O k
2
which also improves the previous O c
k 1
bound.
Then we study Steinerized fault tolerant spanners that are motivated by the follow-
ing. In the definition of k t -EFTS we only require that after deletion of k arbitrary
edges E in the remaining graph each pair of points p q is still connected by a path
whose length is at most t times the length of the shortest pq-path in K
S
E . Such a path
can be arbitrarily long, much longer than dist
2
p q . To see this consider the following
example. Let r 1 be an arbitrarily large real number. Let p q S be two points such
that dist
2
p q 1 and let the remaining n 2 points of S be placed on the ellipsoid
x IR
d
: dist
2
p x dist
2
q x r t . Clearly, each t-spanner G for S contains the
edge between p and q, because each path which contains any third point s S p q
has a length at least r t. Therefore, if the edge p q E then the graph G E can
not be a t-spanner for S. However, G E can contain a path satisfying the definition
of the k-edge fault tolerance. In some applications one would need a stronger property.
After deletion of k edges a pq-path would be desirable whose length is at most t times
dist
2
p q . In order to solve this problem we extend the original point set S by Steiner
points. Then we investigate the question how many Steiner points and how many edges
do we need to satisfy the following natural but stronger condition of edge fault toler-
ance. Let t
1 be a real number and k IN.
The graph G V E with S V is called a k-edge fault tolerant Steiner t-spanner
for S, or k t -EFTSS, if for each E E, E k and for each two points p q S,
there is a pq-path P in G E such that length P t dist
2
p q .
Similarly, G V E with S V is a k-vertex fault tolerant Steiner t-spanner for S,
or k t -VFTSS, if for each V V, V k and for each two points p q S V ,
there is a pq-path P in G V such that length P t dist
2
p q .
To our knowledge, fault tolerant Steiner spanners have not been investigated before.
First we show that for each set S of n points, t 1 real constant, and k IN, a k t -
EFTSS/VFTSS for S can be constructed which contains O kn edges and O kn Steiner
points. Then we show that there is a set S of n points in IR
d
, d 1, such that for each
t 1 and k IN, each k t -EFTSS for S contains kn edges and kn Steiner
points. In this paper we assume that the dimension d is a constant.
2 A k-vertex fault tolerant t-spanner with O kn edges
The construction of a k-vertex fault tolerant t-spanner with O kn edges is based on a
generalization of the θ-graph [17, 7, 10, 11, 14,12]. First we introduce the notion of ith
order θ-graph of the point set S, for 1 i n 1. Then we prove that for appropriate
θ, the k 1 th order θ-graph is a k t -VFTS for the given set of points.
2.1 The ith order θ-graph
For the formal description we need the notion of simplicial cones. We assume that
the points of IR
d
are represented by coordinate vectors. Let p
0
p
1
p
d
be points in
IR
d
such that the vectors p
i
p
0
, 1 i d, are linearly independent. Then the set
p
0
d
i
1
λ
i
p
i
p
0
: λ
i
0 for all i is called a simplicial cone and p
0
is called the
apex of the cone (see, e.g , in [9]). Let θbe a fixed angle 0 θ π andC be a collection
of simplicial cones such that
1. each cone c C has its apex at the origin,
2.
c C
c IR
d
,
3. for each cone c C there is a fixed halfline l
c
having the endpoint at the origin such
that for each halfline l, which has the endpoint at the origin and is contained in c,
the angle between l
c
and l is at most θ 2.
We call such a collection C of simplicial cones a θ-frame
2
. Yao [17] and Ruppert
and Seidel [14] showed methods how a θ-frame C of
d
θ
O d
cones can be constructed.
Assuming that the dimension d and the angle θ are constant we obtain a constant num-
ber of cones. In the following, the number of cones in C is denoted by C .
Let 0 θ π be an angle and C be a corresponding θ-frame. For a simplicial cone
c C and for a point p IR
d
, let c p be the translated cone x p : x c and let l
c
p
be the translated cone axis x p : x l
c
. For c C and p q IR
d
such that q c p ,
let dist
c
p q denote the Euclidean distance between p and the orthogonal projection
of q to l
c
p .
Now we define the ith order θ-graph G
θ
i
S for a set S of n points in IR
d
and for
an integer 1 i n 1 as follows. For each point p S and each cone c C, let
S
c
p
: c p S p , i.e , S
c
p
is the set of points of S p that are contained in
the cone c p . For any integer i, 1 i n 1, let N
i
c
p S
c
p
be the set of the
min i S
c
p
-nearest neighbors of p in the cone c p w.r.t the distance dist
c
, i.e , for
each q N
i
c
p and q S
c
p
N
i
c
p holds that dist
c
p q dist
c
p q . Let G
θ
i
S
be the directed graph with vertex set S such that for each point p S and each cone
c C there is a directed edge p q to each point q N
i
c
p .
2.2 The vertex fault tolerant spanner property
In [14] it is proved that for 0 θ π 3, the graph G
θ
1
S is a spanner for S with
stretch factor t
1
1 2sin θ 2
. The proof is based on the following lemma which will be
also crucial to show the fault tolerant spanner property of G
θ
i
S for i 1.
Lemma 1. [14] Let 0 θ π 3. Let p IR
d
be a point and c C be a cone. Fur-
thermore, let q and r be two points in c p such that dist
c
p r dist
c
p q . Then
dist
2
r q dist
2
p q 1 2sin θ 2 dist
2
p r .
Theorem 1. Let S IR
d
be a set of n points. Let 0 θ π 3 and 1 k n 2 be
an integer number. Then the graph G
θ
k 1
S is a directed k
1
1 2sin θ 2
-VFTS for
S. G
θ
k 1
S contains O C kn edges and it can be constructed in O C nlog
d
1
n
knloglogn time.
2
The notion of θ-frame was introduced by Yao [17]. We use a slightly modified θ-frame defini-
tion which is suggested by Ruppert and Seidel [14].
Proof. Let S S be a set of at most k points. We show that for each two points p q
S S there is a (directed) pq-path P in G
θ
k 1
S S such that the length of P is at
most
1
1 2sin θ 2
dist
2
p q . The proof is similar to the proof of Ruppert and Seidel [14].
Consider the path constructed in the following way. Let p
0
: p, i : 0 and let P contain
the single point p
0
. If the edge p
i
q is present in the graph G
θ
k 1
S S then add
the vertex q to P and stop. Otherwise, let c p
i
be the cone which contains q. Choose
an arbitrary point p
i
1
N
k
1 c
p
i
as the next vertex of the path P and repeat the
procedure with p
i
1
.
Consider the ith iteration of the above algorithm. If
p
i
q G
θ
k 1
S then the al-
gorithm terminates. Otherwise, if p
i
q G
θ
k 1
S then by definition the cone c p
i
contains at least k 1 points that are not further from p
i
than q w.r.t the distance
dist
c
. Hence, in the graph G
θ
k 1
S the point p
i
has k 1 neighbors in c p
i
and,
therefore, in the graph G
θ
k 1
S S it has at least one neighbor in c p
i
. Conse-
quently, the algorithm is well defined in each step. Furthermore, Lemma 1 implies that
dist
2
p
i
1
q dist
2
p
i
q and hence, each point is contained in P at most once. There-
fore, the algorithm terminates and finds a pq-path P in G
θ
k 1
S S . The bound on the
length of P follows by applying Lemma 1 iteratively in the same way as in [14]: Let
p
0
p
m
be the vertices on P, p
0
p and p
m
q. Then
0
i m
dist
2
p
i
1
q
0
i m
dist
2
p
i
q 1 2sin θ 2 dist
2
p
i
p
i
1
Rearranging the sum we get
0
i m
dist
2
p
i
p
i
1
1
1 2sin θ 2
0
i m
dist
2
p
i
q dist
2
p
i
1
q
1
1 2sin θ 2
dist
2
p
0
q
Hence, the graph G
θ
k 1
S is a k
1
1 2sin θ 2
-VFTS for S. Clearly,it contains O C kn
edges, where C d θ
O d
. It can be constructed in O C nlog
d
1
n
knloglogn
time using the algorithm of Levcopoulos et al [12]. They compute for each point p S
and each cone c C the set N
k
c
p in order to determine so-called strong approximated
neighbors.
Corollary 1. Let S be a set of n points in IR
d
, t
1 a real constant, and k an integer,
1 k n 2. Then there is a k t -VFTS for S with O kn edges. Such a spanner can
be constructed in O nlog
d
1
n
knloglogn time.
Proof. We set θ such that t
1
1 2sin θ 2
and 0 θ π 3 and construct G
θ
k 1
S . If
t 1 then the constant factors hidden in the O-calculus are
d
t 1
O d
.
3 k-edge fault tolerant t-spanners
Levcopoulos et al [12] claimed that any k t -VFTS is also a k t -EFTS. We give our
own proof of this fact. The proof is simple and holds also for directed spanners.
Theorem 2. Let S be a set of n points in IR
d
, t 1 a real constant, and k an integer,
1 k n 2. Then every (directed) k t -VFTS for S is also a (directed) k t -EFTS
for S.
Proof. Let G S E be a (directed) k t -VFTS for S. Let E E be a set of at most
k edges. Consider two arbitrary points p q S. Let P be the shortest (directed) pq-
path in K
S
E . Such a path exists, since the set of pq-paths in K
S
E is not empty. It
contains, for example, at least one of the n 2 paths in K
S
of two edges P
s
p s q, for
s S p q , or the immediate path P
0
p q, because at least one of them is distinct
from E .
We have to show that there is a (directed) pq-path P in G E such that the length of
P is at most t times the length of P . The edges e in P that are contained in G will also
be contained in P. Consider an edge u v ( u v in the directed case) in P which is not
contained in G. We show that this edge can besubstituted by a uv-path P
uv
in G E such
that length P
uv
t dist
2
u v : For each edge e E (for each e E v u in the
directed case) we fix one of its endpoints p
e
such that p
e
S u v . Let S
uv
:
p
e
:
e E (S
uv
: p
e
: e E v u in the directed case). Note that S
uv
E k.
Since G is a (directed)
k t -VFTS for S, there is a (directed) uv-path P
uv
in G S
uv
such
that P
uv
does not contain any edge ofE
and length P
uv
t dist
2
u v . The desired pq-
path P is composed of the edges of P G and the uv-paths for the edges u v P G
( u v P G in the directed case). Clearly, length P t length P .
This, together with Corollary 1, leads to
Corollary 2. Let S be a set of n points in IR
d
, t 1 a real constant, and k an integer,
1 k n 2. Then there is a (directed) k t -EFTS for S with O kn edges. Such a
spanner can be constructed in O nlog
d
1
n
knloglogn time.
The proof of Theorem 2 implies also the following for directed graphs.
Theorem 3. Let S be a set of n points in IR
d
, t
1 a real constant, and k an integer,
1 k n 2. Let G V E be a directed k t -VFTS for S. Let E E be a set of at
most k edges and let E : v u : u v E . Then for each two points p q S the
graph G E E contains a pq-path P such that the length of P is at most t times the
length of the shortest pq-path in K
S
E E .
4 A k-vertex fault tolerant t-spanners with degree O k
2
We now turn to the problem of constructing fault tolerant spanners with bounded de-
gree. We proceed similar to the method in [1] which constructs a spanner with constant
degree. However, we must take much more care, because of the fault tolerant property
and the goal of keeping the number of edges small. We have shown that for any real
constant t 1 we can construct a directed k t -VFTS/EFTS for S whose outdegree is
O k . In this section we give a method to construct a k t -VFTS whose degree is O k
2
from a directed k t
1
3
-VFTS whose outdegree is O k .
In order toshowthis construction we need the notion of k-vertex fault tolerant single
sink spanner. This is a generalization of single sink spanners introduced in [1]. Let V be
a set of m points in IR
d
, v V,
ˆ
t 1 a real constant, and k an integer, 1 k m 2. A
directed graph G V E is a k-vertex fault tolerant v-single sink
ˆ
t-spanner, or k
ˆ
t
v -
VFTssS for V if for each u V v and each V V v u , V k, there is an
ˆ
t-spanner path in G
V from u to v.
Now let V be a set of m points in IR
d
, v V a fixed point, 1 i m 1 an integer,
θ an angle, 0 θ π 3, and C a θ-frame. We define a directed graph
ˆ
G
v
θ i
V V E
whose edges are directedstraight line segments between points ofV as follows. First we
partition the set V in clusters such that each cluster contains at most i points. Then we
build a tree-like structure based on these clusters. For the clustering we use the cones
of C. Now we describe this procedure more precisely.
First we create a cluster cl v containing the unique point v. For each cluster that
we create, we choose a point as the representative of the cluster. The representative of
cl v is v. The clustering of the setV v is recursive. The recursion stops if V v
is the empty set. Otherwise, we do the following. For each cone c C letV
c
v
be the set
of points of V v contained in c. If a point is contained in more than one cone then
assign the point only to one of them. If one cone, say c, contains more than m 2 points,
then partition the points of V
c
v
arbitrarily into two sets V
1
c v
and V
2
c v
both having at
most m 2 points. For each nonempty set V
c
v
, c C (or in the case if V
c
v
had to be
partitioned, for each V
1
c
v
and V
2
c
v
), let N
i
c
v V
c
s
be the set of the min i V
c
v
-
nearest neighbors of v in V
c
v
w.r.t the distance dist
c
. The points contained in the same
N
i
c
v define a new cluster cl N
i
c
v . Note that in this way we obtain at most C 1
new clusters. We say that these clusters are the children of cl v and cl v is the
parent of these clusters. For each new cluster cl N
i
c
v we choose a representative
u
c
N
i
c
v such that dist
c
v u
c
max dist
c
v u : u N
i
c
v . Then, for each set
V
c
v
, c C (and V
1
c
v
, V
2
c
v
if exist), we recursively cluster V
c
v
N
i
c
v using the
cones around u
c
.
After the clustering is done, for each clustercl cl v we add an edge in
ˆ
G
v
θ i
V
from each point u cl to each point w of the parent cluster of cl. Figure 1 shows an
example for
ˆ
G
v
θ i
V . The dotted lines represent the boundaries of the cones at the
representatives of the clusters.
v
Fig.1. The directed graph
ˆ
G
v θ 3
V for a point set V in IR
2
.
Lemma 2. Let V be a set of m points in IR
d
, v V a fixed point and 1 k m 2
an integer number. Let 0 θ π 3 be an angle and C be a θ-frame. Then the graph
ˆ
G
v
θ k 1
V is a k
1
1 2sin θ 2
2
v -VFTssS for V. Its degree is bounded by O C k
and it can be computed in O C mlogm km time.
Proof. For each point u V let cl u denote the cluster containing it. The outdegree of
each point u V v in
ˆ
G
v
θ k 1
V is bounded by k 1, because each point u has only
edges to the points contained in the parent cluster of cl u and the number of points in
each cluster is bounded by k 1. (Each internal cluster i.e , a cluster which is different
from cl v and has at least one child – contains exactly k 1 points). Since each cluster
has at most C 1 children, the indegree of the points is bounded by C 1 k 1 .
The bound for the construction time follows from the fact that the recursion has depth
O logm .
Now we prove the fault tolerant single sink spanner property. Consider an arbitrary
point u V v . Let P
0
: u
0
u
l
, u
0
u and u
l
v, be the unique path from u to
v in
ˆ
G
v
θ k 1
V such that each internal vertex u
i
, 1 i l, is the representative of a
cluster. Note that l O logm . The length of P
0
is at most
1
1 2sin θ 2
dist
2
u v . If the
edge u v
ˆ
G
v
θ k 1
V , this claim holds trivially, otherwise, it follows by applying
Lemma 1 iteratively for the triples u
i
1
u
i
u
i
1
, i
1 l 1, in the same way as in
the proof of Theorem 1.
Now let V V u v be a set of at most k points. We show that there is a uv-
path P in
ˆ
G
v
θ k 1
V V such that length P
1
1 2sin θ 2
length P
0
. This will imply
the desired stretch factor
1
1 2sin θ 2
2
. Let P be the path constructed as follows. Let
v
0
:
u, i : 0 and let P contain the single point v
0
. If v
i
v then stop. Otherwise, let
v
i
1
be an arbitrary point with
v
i
v
i
1
ˆ
G
v
θ k 1
V V . Add the vertex v
i
1
to P and
repeat the procedure with v
i
1
.
v=v =u
l l
v
u
2
2
v
1
u
1
u=u =v
0 0
Fig.2. The paths P
0
: u
0
u
l
and P : v
0
v
l
. The dotted lines show the cone boundaries.
The above algorithm is well defined in each step. To see this, consider the ith it-
eration. If the cluster cl v is the parent of cl v
i
then the algorithm chooses v as v
i
1
and terminates. Otherwise, the parent of cl v
i
contains k 1 points and, hence, at least
one point disjoint from V . The algorithm chooses such a point as v
i
1
. Clearly, the
algorithm terminates after l
O logm steps and constructs a uv-path P v
0
v
l
(Figure 2) with
length
P
0 i l
dist
2
v
i
v
i
1
0
i l
dist
2
v
i
u
i
1
dist
2
u
i
1
v
i
1
(1)
0 i l
dist
2
v
i
u
i
1
dist
2
u
i
v
i
dist
2
u
l
v
l
0
dist
2
u
0
v
0
0
0 i l
1
1 2sin θ 2
dist
2
u
i
u
i 1
(2)
1
1 2sin θ 2
length P
0
(1) holds because of the triangle inequality and (2) follows by applying Lemma 1 for
the triples u
i
1
v
i
u
i
, i 0 l 1. Hence, the claimed stretch factor of
ˆ
G
v
θ k 1
V
follows.
Theorem 4. Let S be a set of n points in IR
d
, t 1 a real constant, and k an integer,
1 k n 2. Then there is a k t -VFTS G for S whose degree is bounded by O k
2
.
The total number of edges in G is O k
2
n and G can be constructed in O nlog
d
1
n
knlogn k
2
n time.
Proof. Let G
0
be a directed k t
1 3
-VFTS for S whose outdegree is O k , for exmple,
let G
0
be the k 1 th order θ-graph G
θ
k 1
S with t
1 3
1
1 2sin θ 2
. For each point
p
S let N
in
p : q S : q p G
0
. Let G be the directed graph with vertex set S
which is created such that for each p S we construct the graph
ˆ
G
p
θ k 1
N
in
p p
and we add the edges of
ˆ
G
p
θ k 1
N
in
p p to G.
We can bound the degree of G as follows. For each q S, the graph G contains the
edges of
ˆ
G
q
θ k 1
N
in
q q . In this VFTssS each vertex p has an in- and outdegree
O k . Now for each p S, we have to count the graphs
ˆ
G
q
θ k 1
N
in
q q , q S,
that contain p. Clearly, the number of such graphs is equal to one plus the outdegree of
p in G
0
, which is O k . Therefore, the degree of each p S in G is O k
2
.
Now we show that G is a k t -VFTS for S. Let S S, S k. Consider two
arbitrary points p q S S . Since G
0
is a k t
1 3
-VFTS for S, there is an t
1 3
-spanner
path P
0
in G
0
S between p andq. Furthermore, foreach edge u v P
0
, there is a t
2
3
-
spanner path P
uv
in G
S , because G contains all edges of the graph
ˆ
G
v
θ k 1
N
in
v
v which is, by Lemma 2, a k t
2 3
v -VFTssS for N
in
v v . Therefore, the path
P :
u v P
0
P
uv
is contained in G
S and P is an t-spanner path between p and q.
5 Fault tolerant spanners with Steiner points
In this section we show a very simple method which constructs for an arbitrary set S of
n points in IR
d
, t 1, and k IN, a k t -EFTSS and k t -VFTSS for S with O kn
edges and kn Steiner points. Then we prove the surprising fact that these upper bounds
on the number of edges and on the number Steiner points in a k t -EFTSS are optimal
up to constant factors.
Theorem 5. Let S
IR
d
be a set of n points, k IN, and let t 1 a real constant. Then
there is a k t -EFTSSand k t -VFTSS G for S with kn Steinerpoints and O kn edges.
Proof. Assume that the Euclidean distance between the closest pair of S is one. Other-
wise, we scale S accordingly. Let ε be a real number such that 0 ε t 1 3. Let
t t 2ε and let G S E be a t -spanner for S with O n edges. G can be com-
puted, for example, using the method described in [4] or in [14]. We construct from G
a k t -EFTSS/VFTSS G for S in the following way. Let o IR
d
be a fixed point and
let D : x IR
d
: dist
2
o x ε be the sphere with radius ε whose center is o. Let
s
1
s
k
be k distinct points on D. (In the case if d 1, let s
1
s
k
be k distinct points
such that 0 dist
2
o p
i
ε, 1 i k.) For each point p S translate the sphere D
and the points s
1
s
k
on D such that p becomes the center of the sphere. Let p
1
p
k
denote the translated points around p. We construct the graph G V E such that
V : p p
1
p
k
: p S and
E : p q : p q E p p
i
: p S 1 i k
p
i
q
i
: p q E 1 i k
1
2
3
2
1
3
p
p
p
p
p
q
q
q
q
q
Fig.3. Example for the graphs G and G for k 3 d 2.
Clearly, the graph G has kn Steiner points and O kn edges. It is obvious that G
is a k-EFTSS and k-VFTSS for S, because for each pair of points p q S and for
each t -spanner path P p p
1
p
l
1
q in G between p and q, there are k 1 edge
disjoint and up to the endpoints vertex disjoint pq-paths P
0
p p
1
p
l
1
q and P
i
p p
i
p
i
1
p
i
l
1
q
i
q, 1 i k, in G whose length is at most
length P 2ε t dist
2
p q 2ε t dist
2
p q
Figure 3 shows an example.
Now we prove a lower bound on the number of edges and Steiner points which
shows that the above upper bound is optimal up to a constant factor.
Theorem 6. For each k IN, n 2, andt 1, there exists a set S IR
d
of n pointssuch
that each k t -EFTSS for S contains at least kn Steiner points and kn edges.
Proof. We give an example for a set S of n points in the plane for which we show that
each k t -EFTSS for S contains kn Steiner points and kn edges. For two points
p q IR
d
let
el p q : x IR
d
: dist
2
p x dist
2
q x t dist
2
p q
If p q are two points in S and G is a k t -EFTSS for S then each t-spanner path be-
tween p and q must be contained entirely in el p q . Clearly, a path which contains
a point v outside el p q has a length at least dist
2
p v dist
2
q v which is greater
than t dist
2
p q . For p q S let G
pq
be the smallest subgraph of G which contains
all t-spanner paths between p and q. Since G is a k t -EFTSS, G
pq
must be k-edge
connected. Otherwise, we could separate p from q in G
pq
by deletion of a set E of k
edges, and therefore, we would not have any t-spanner path in G E . Since the graph
G
pq
is k-edge connected, Menger’s Theorem implies that it contains at least k 1 edge
disjoint pq-paths. Hence, G
pq
and, therefore, el p q contains at least k vertices
different from p and q and at least 2k 1 edges of G.
Now we show how to place the points of S in order to get the desired lower bounds.
We construct the set S of n points in the plane hierarchically bottom-up. For simplic-
ity of the description we assume that n is a power of two. Let l be a horizontal line
and let o be any fixed point of l. We place the points of S on l. We put p
1
S to o
and p
2
S right from p
1
such that dist
2
p
1
p
2
1. Let ε 0 be a fixed real num-
ber. We translate el p
1
p
2
with the points p
1
and p
2
right on l, by a Euclidean dis-
tance t ε. This translation guarantees that el p
1
p
2
and the translated ellipsoid are
distinct. Let p
3
and p
4
denote the translated points p
1
and p
2
, respectively. In gen-
eral, in the ith step, 1
i logn, we translate the ellipsoid el p
1
p
2
i
with the points
p
1
p
2
i
right on l, by a Euclidean distance t dist
2
p
1
p
2
i
ε. Denote the translated
points by p
1
2
i
p
2
i
1
(Figure 4). Then the ellipsoids el
p
1
p
2
i
and el p
2
i
1
p
2
i
1
are distinct. We say that the ellipsoid el p
1
p
2
i 1
is the parent of el p
1
p
2
i
and
el p
1
2
i
p
2
i
1
. Furthermore, we call the two children of an ellipsoid siblings of one
another. We denote by parent el and sib el the parent and the sibling of an
ellipsoid el , respectively.
Fig.4. Example for a set S of n points for which each k t -EFTSS contains at least n 2 Steiner
points and 3k 3 n 2 k 1 edges.
Now we count the Steiner points and the edges in an arbitrary k t -EFTSS G for
the set S. Consider a pair of points p
2j
1
p
2j
S, 1 j n 2. For this pair, there are
at least k 1 edge disjoint paths in G contained entirely in el p
2j
1
p
2j
. Since, for
j j the ellipsoids el p
2j
1
p
2j
and el p
2j
1
p
2j
are disjoint, each el p
2j
1
p
2j
contains in the interior at least k Steiner points and 2k 1 edges. Furthermore, p
2j
1
and p
2j
must be k 1 -edge connected with the points of sib el p
2j
1
,p
2j
. There-
fore, we have at least k 1 edges contained entirely in parent el p
2j
1
,p
2j
that have
exactly one endpoint in el p
2j
1
,p
2j
. We can repeat these arguments at each level of
the hierarchy of the ellipsoids. Then we obtain that the number of edges in G is at least
2k 1 n 2 k 1 n 2 1 3k 2 n 2 k 1 and the number of Steinerpoints
is at least kn 2.
In the case if n is not a power of two, we place 2
i
points, where 2
i
1
n 2
i
, in
the same way as described above. Then we remove the points p
n
1
p
2
i
. Using the
above arguments we obtain that for this point set, each k t -EFTSS contains at least
k n 2 Steiner points and 3k 2 n 2 k 1 edges. This proves the claim of the
theorem.
6 Conclusion and open problems
Some interesting problems remain to be solved. Is it possible to construct a k t -VFTS
whose degree is bounded by O k ? Levcopoulos et al. [12] studied fault tolerant span-
ners with low weight. Let w MST be the weight of the minimum spanning tree of S.
In [12] it is proven that for each S a k t -VFTS can be constructed whose weight is
O c
k 1
w MST for some constant c. Can this upper bound be improved? In [12] it is
also proven that k
2
w MST is a lower bound on the weight. Is it possible to con-
struct a k t -VFTS with lower weight using Steiner points? Finally, we do not know
any results for fault tolerant spanners with low diameter.
Acknowledgment:I would like to thank Artur Czumaj, Matthias Fischer,and Silvia
G¨otz for their helpful comments and suggestions.
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... A subgraph H = (V, E, w) is an f -vertex-fault-tolerant t-spanner of a weighted graph G = (V, E, w), if for every set F ⊂ V of at most f vertices it holds that ∀u, v ∉ F , d H∖F (u, v) ≤ t ⋅ d G∖F (u, v). It is known that every n-point set in R d , or more generally in a space of doubling dimension d, admits an f -vertex-fault-tolerant (1 + )-spanner with −O(d) ⋅ f ⋅ n edges [LNS02,Luk99,Sol14]. For general graphs, after a long line of works [CLPR10, DK11, BDPW18, BP19, DR20, BDR20], it was shown that every n-vertex graph admits an efficiently constructible f -vertex-fault-tolerant (2k−1)-spanner with O(f 1− 1 k ⋅ n 1+ 1 k ) edges, which is optimal assuming the Erdös' Girth Conjecture [Erd64]. ...
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A highly desirable property of networks is robustness to failures. Consider a metric space $(X,d_X)$, a graph $H$ over $X$ is a $\vartheta$-reliable $t$-spanner if, for every set of failed vertices $B\subset X$, there is a superset $B^+\supseteq B$ such that the induced subgraph $H[X\setminus B]$ preserves all the distances between points in $X\setminus B^+$ up to a stretch factor $t$, while the expected size of $B^+$ is as most $(1+\vartheta)|B|$. Such a spanner could withstand a catastrophe: failure of even $90\%$ of the network. Buchin, Har-Peled, and Ol{\'{a}}h [2019,2020], constructed very sparse reliable spanners with stretch $1+\epsilon$ for Euclidean space using locality-sensitive orderings. Har-Peled and Ol{\'{a}}h [2020] constructed reliable spanners for various non-Euclidean metric spaces using sparse covers. However, this second approach has an inherent dependency on the aspect ratio (a.k.a. spread) and gives sub-optimal stretch and sparsity parameters. Our contribution is twofold: 1) We construct a locality-sensitive ordering for doubling metrics with a small number of orderings. As a corollary, we obtain reliable spanners for doubling metrics matching the sparsity parameters of known reliable spanners for Euclidean space. 2) We introduce new types of locality-sensitive orderings suitable for non-Euclidean metrics and construct such orderings for various metric families. We then construct reliable spanners from the newly introduced locality-sensitive orderings via reliable 2-hop spanners for paths. The number of edges in our spanner has no dependency on the spread.
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Given a set S of n points, a weight function w to associate a non-negative weight to each point in S, a positive integer \(k \ge 1\), and a real number \(\epsilon > 0\), we devise the following algorithms to compute a k-vertex fault-tolerant spanner network G(S, E) for the metric space induced by the weighted points in S: (1) When the points in S are located in a simple polygon, we present an algorithm to compute G with multiplicative stretch \(\sqrt{10}+\epsilon \), and the number of edges in G is \(O(k n (\lg {n})^2)\). (2) When the points in S are located in the free space of a polygonal domain \(\mathcal{P}\), we present an algorithm to compute G of size \(O(\sqrt{h} k n(\lg {n})^2)\) and its multiplicative stretch is \(6+\epsilon \). Here, h is the number of simple polygonal holes in \(\mathcal{P}\).
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- A topological method is given for obtaining lower bounds for the height of algebraic computation trees, and algebraic decision trees. Using this method we are able to generalize, and present in a uniform and easy way, almost all the known nonlinear lower bounds for algebraic computations. Applying the method to decision trees we extend all the apparently known lower bounds for linear decision trees to bounded degree algebraic decision trees, thus answering the open questions raised by Steele and Yao [20]. We also show how this new method can be used to establish lower bounds on the complexity of constructions with ruler and compass in plane Euclidean geometry. 1
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Conference Paper
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