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Optimal Vertex Fault-Tolerant Spanners in Polynomial Time

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Abstract

Recent work has pinned down the existentially optimal size bounds for vertex fault-tolerant spanners: for any positive integer $k$, every $n$-node graph has a $(2k-1)$-spanner on $O(f^{1-1/k} n^{1+1/k})$ edges resilient to $f$ vertex faults, and there are examples of input graphs on which this bound cannot be improved. However, these proofs work by analyzing the output spanner of a certain exponential-time greedy algorithm. In this work, we give the first algorithm that produces vertex fault tolerant spanners of optimal size and which runs in polynomial time. Specifically, we give a randomized algorithm which takes $\widetilde{O}\left( f^{1-1/k} n^{2+1/k} + mf^2\right)$ time. We also derandomize our algorithm to give a deterministic algorithm with similar bounds. This reflects an exponential improvement in runtime over [Bodwin-Patel PODC '19], the only previously known algorithm for constructing optimal vertex fault-tolerant spanners.
arXiv:2007.08401v1 [cs.DS] 16 Jul 2020
Optimal Vertex Fault-Tolerant Spanners in Polynomial Time
Greg Bodwin
University of Michigan
greg.bodwin@gmail.com
Michael Dinitz
Johns Hopkins University
mdinitz@cs.jhu.edu
Caleb Robelle
UMBC
carobel1@umbc.edu
Abstract
Recent work has pinned down the existentially optimal size bounds for vertex fault-tolerant
spanners: for any positive integer k, every n-node graph has a (2k1)-spanner on O(f11/k n1+1/k)
edges resilient to fvertex faults, and there are examples of input graphs on which this bound
cannot be improved. However, these proofs work by analyzing the output spanner of a certain
exponential-time greedy algorithm. In this work, we give the first algorithm that produces vertex
fault tolerant spanners of optimal size and which runs in polynomial time. Specifically, we give
a randomized algorithm which takes e
Of11/kn2+1/k +mf 2time. We also derandomize our
algorithm to give a deterministic algorithm with similar bounds. This reflects an exponential
improvement in runtime over [Bodwin-Patel PODC ’19], the only previously known algorithm
for constructing optimal vertex fault-tolerant spanners.
Supported in part by NSF awards CCF-1717349, DMS-183932 and CCF-1909756.
Supported in part by NSF award CCF-1909111.
1 Introduction
Let G= (V, E ) be a graph, possibly with edge lengths w:ER0. A t-spanner of G, for t1,
is a subgraph H= (V, E) that preserves all pairwise distances within a factor of t, i.e.,
dH(u, v)t·dG(u, v) (1)
for all u, v V(where dXdenotes the shortest-path distance in a graph X). Since His a subgraph
of Git is also true that dG(u, v)dH(u, v), and so distances in Hare the same as in Gup to a
factor of t. The distance preservation factor tis called the stretch of the spanner. Spanners were
introduced by Peleg and Ullman [PU89a] and Peleg and Sch¨affer [PS89], and have a wide range
of applications in routing [PU89b], synchronizers [AP90], broadcasting [ABP91, Pel00], distance
oracles [TZ05], graph sparsifiers [KP12], preconditioning of linear systems [EEST08], etc.
The most common objective in spanners research is to achieve the best possible existential
size-stretch trade-off, and to do this with algorithms that are as fast as possible. Most notably,
a landmark result of Alth¨ofer et al. [ADD+93] analyzed the following simple and natural greedy
algorithm: given an n-node graph Gand an integer k1, consider the edges of Gin non-
decreasing order of their weight and add an edge {u, v}to the current spanner Hif and only
if dH(u, v)>(2k1)w(u, v). They proved that this algorithm produces (2k1)-spanners of
existentially optimal size : the spanner produced has size O(n1+1/k ), and (assuming the well-known
Erd˝os girth conjecture [Erd64]) there are graphs in which every (2k1) spanner (and in fact every
2k-spanner) has at least Ω(n1+1/k ) edges.
1.1 Fault Tolerance
A crucial aspect of real-life systems that is not captured by the standard notion of spanners is
the possibility of failure. If some edges (e.g., communication links) or vertices (e.g., computer
processors) fail, what remains of the spanner might not still approximate the distances of what
remains of the original graph. This motivates the notion of fault tolerant spanners:
Definition 1. A subgraph His an f-vertex fault tolerant (f-VFT) t-spanner of Gif
dH\F(u, v)t·dG\F(u, v) (2)
for all u, v Vand FV\ {u, v}with |F| ≤ f.
In other words, an f-VFT spanner contains a spanner of G\Ffor every set of |F| ≤ fnodes
that could fail. The definition for edge fault tolerance (EFT) is equivalent, with the only change
being that FErather than FV\ {u, v}.
Fault tolerant spanners were originally introduced in the geometric setting (where the vertices
are points in Rdand the initial graph Gis the complete graph with Euclidean distances) by
Levcopoulos, Narasimhan, and Smid [LNS98a] and have since been studied extensively in that
setting [LNS98b, Luk99, CZ04, NS07]. Chechik, Langberg, Peleg and Roditty [CLPR10] were the
first to study fault-tolerant spanners in general graphs, giving a construction of an f-VFT (2k1)-
spanner of size approximately O(f2kf+1 ·n1+1/k log11/k n) and an f-EFT (2k1)-spanner of size
O(f·n1+1/k). So they showed that introducing tolerance to fedge faults costs us an extra factor
of fin the size of the spanner, while introducing tolerance to fvertex faults costs us a factor of
f2kf+1 in the size (compared to the size of a non-fault tolerant spanner of the same stretch).
Since [CLPR10], there has been a line of work focused on improving these bounds, particularly
for vertex faults (see Table 1). The first improvement was by [DK11], who improved the bound for
1
Spanner Size Runtime Greedy? Citation
e
OkO(f)·n1+1/ke
OkO(f)·n3+1/k[CLPR10]
e
Of21/k ·n1+1/ke
Of22/k ·mn1+1/k[DK11]
Oexp(k)f11/k ·n1+1/kOexp(k)·mnO(f)X[BDPW18]
Of11/k ·n1+1/kOmnO(f)X[BP19]
Ok2f11/k ·n1+1/ke
Of21/k ·mn1+1/k(X) [DR20]
Of11/k ·n1+1/ke
Of11/kn2+1/k +mf 2(X)(this paper)
Table 1: Prior work on f-VFT (2k1)-spanners of weighted input graphs on nnodes and medges.
Size bounds in red are existentially optimal, and runtimes in blue are polynomial. The (X) entries
indicate a greedy algorithm with slack, as discussed below. With e
Owe hide factors of log n(or k,
since we may assume klog n).
vertex faults to O(f21
kn1+ 1
klog n) via a black-box reduction to non-fault tolerant spanners. Fol-
lowing this, the area turned towards analyses of the FT-greedy algorithm, the obvious extension of
the greedy algorithm of [ADD+93] to the fault tolerant setting: look at the edges (u, v) of the input
graph Gin order of nondecreasing weight, and add (u, v) to the spanner Hiff currently there exists
a set of |F| ≤ ffaults such that (2) fails. This algorithm was first analyzed by [BDPW18] who ob-
tained a size bound of O(exp(k)f11/kn1+1/k). They also proved a lower bound of Ω(f11/kn1+1/k )
for f-VFT (2k1)-spanners: assuming the girth conjecture of Erd¨os [Erd64], there are graphs
which require that many edges for any f-VFT (2k1)-spanner. An improved analysis of the
FT-greedy algorithm was then given by [BP19], who remove the exp(k) factor and so proved that
this algorithm gives existentially optimal VFT spanners.
While the FT-greedy algorithm inherits some of the nice properties of the non-faulty greedy
algorithm (such as simplicity, easy-to-prove correctness, and existential optimality), it unfortunately
has serious issues in runtime. The edge test in the FT greedy algorithm, i.e., whether or not there
exists a fault set under which (2) holds, is an NP-hard problem known as length-bounded
cut [BEH+06], and hence the algorithm inherently runs in exponential time. Addressing this, a
greedy algorithm with slack was recently proposed in [DR20]. This algorithm is an adaptation of
the FT-greedy algorithm which replaces the exponential-time edge test with a different subroutine
test(u, v), which accepts every edge (u, v) where there exist |F| ≤ ffaults under which (2) fails,
and possibly some other edges too. This slack maintains correctness and allows one to escape NP-
hardness, but it introduces the challenge of bounding the number of additional edges added. The
approach in [DR20] is to design an O(k)-approximation algorithm for length -bounded cut and
use this in an efficiently computable test subroutine. This gives a polynomial runtime, but pays
the approximation ratio of O(k) in spanner size over optimal. So the result in [DR20] takes an
important step forward (polynomial time) but also a step back (non-optimal size, by a factor of
O(k)).
It thus remains an important open problem to design a polynomial time algorithm which obtains
truly optimal size. We note that kfactors are often considered particularly important for spanners,
since the regime k= Θ(log n) yields the sparsest possible spanners and hence arises commonly in
algorithmic applications (see, e.g., [BBG+20] for a recent example), and here an extra factor of k
2
in the size of the spanner is significant. Accordingly, for spanners and many related objects there
has been significant effort expended to remove unnecessary factors of k. For example, Chechik and
Wulff-Nilsen recently received the best paper award at SODA ’16 for removing a factor of kfrom
the “lightness” bound for spanners [CWN16], and removing the final factor of kfrom the size of a
“distance oracle” (a data structure related to spanners) appeared in STOC ’15 [Che15]. It seems
to often be the case that initial algorithms pay a factor of k, which can later be removed through
smarter algorithms and analyses. Our results fit into this tradition, addressing the remaining open
question: can we get truly optimal-size fault tolerant spanners in polynomial time?
1.2 Our Results and Techniques
We answer this in the affirmative, giving both randomized and deterministic algorithms for con-
structing optimal-size fault-tolerant spanners. More formally, we prove the following theorems.
Theorem 1.1. There is a randomized algorithm which runs in expected time
e
Of11/kn2+1/k +mf 2
which with high probability returns an f-VFT (2k1)-spanner with O(f11/kn1+1/k)edges.
Theorem 1.2. There is a deterministic algorithm which constructs an f-VFT (2k1) spanner
with at most O(f11/kn1+1/k )edges in time
˜
Of41/kn2+1/k +mf 5.
If f=poly(n)(i.e., fncfor some constant c > 0) then the running time improves to
˜
Of11/kn2+1/k +mf 2,
where the polynomial exponent cappears in both the spanner size and the running time but is hidden
by the O(·)notation.
So if fis subpolynomial in nthen our deterministic algorithm is slower than our randomized
algorithm by about f3, while if fis polynomial in nthen we get determinism essentially for free
(although the hidden polylogarithmic factors are larger in the deterministic case).
To put these results in context, note that this is an exponential improvement in running time
over [BP19], the only previous algorithm to give optimal-size fault tolerant spanners. And un-
like [DR20,DK11] it gives spanners with existentially optimal size, saving an O(k) factor over [DR20]
and an O(flog n) factor over [DK11]. It is also polynomially faster than both [DR20,DK11]. We
note that not only is k= Θ(log n) (and superconstant kmore generally) a particularly interesting
regime (as discussed), large values of fare also particularly interesting. If we only ever think of f
as small then the dependence on fin the size of the spanner does not matter much, but of course
we are interested in protecting against as many faults as possible! So our results are strongest
(compared to previous work) precisely in one of the most interesting regimes for fault-tolerant
spanners: k= Θ(log n) and fpolynomially large in n.
Additional Properties. Our algorithms and techniques have a few other properties that we
briefly mention here, but which will not be a focus in the paper. First, a corollary of our techniques
and analysis is that we actually speed up the running time of the non-fault tolerant greedy algorithm
3
from O(mn1+1/k) to O(kn2+1/k). Second, our algorithms and bounds continue to hold for edge
fault tolerance, but for simplicity we will only discuss the VFT case. These size bounds are also
not known to be optimal for edge fault tolerance (except for spanners of stretch 3) since the known
lower bounds are weaker, making our results more interesting in the VFT setting. See Section 6
for more discussion of edge fault tolerance. Finally, since our algorithms are slack-greedy, they are
unconditionally optimal: even if the Erd˝os girth conjecture is false, our algorithms still produce
spanners of optimal size (whatever that is).
1.2.1 Our First Algorithm
Our first algorithm is a surprisingly simple randomized algorithm that, while not as efficient as the
algorithm we will use to prove Theorem 1.1, achieves our main goal: it has polynomial running time
and produces spanners of optimal size. It also illustrates the main ideas that our more advanced
algorithms (faster and/or deterministic) will utilize.
Theorem 1.3. There is a randomized algorithm which, given an undirected weighted n-node graph
and positive integers fand k, runs in polynomial time and with high probability returns an f-VFT
(2k1)-spanner with O(f11/k n1+1/k)edges.
The main new ingredient is the following simple test(u, v) subroutine. To test an edge (u, v) we
first randomly sample Θ(log n) induced subgraphs of the current spanner, each of which is obtained
by including uand vand then including each other node with probability 1/(2f). Then we test
(1) in each of the sampled subgraphs, and we add (u, v) to the spanner iff a large enough fraction
of these subgraphs violate (1). For correctness, one observes that if there exists any set of |F|=f
vertex deletions under which dH\F(u, v) is large, then with high probability a large fraction of the
subgraphs will delete all of F. Hence d(u, v) will be large in these subgraphs, and we will correctly
include (u, v) in the spanner.
The more interesting part of the argument is bounding the size of the output spanner. This
relates to the blocking set technique introduced in [BP19] and also used in [DR20]. This technique
uses the following observation: if the spanner His sufficiently dense, and one samples a random
induced subgraph on n/f nodes, then that subgraph will still be dense enough that it must have
some 2k-cycles. This statement is even somewhat robust, in that one will not be able to destroy
all 2k-cycles by deleting only a constant fraction of remaining edges in the subgraph. Thus one
can certify sparsity of the spanner by arguing that it is in fact possible to sample a subgraph,
remove a constant fraction of the surviving edges, and destroy all 2k-cycles in the process.
In [BP19, DR20], one roughly predefines a small set of edges in the spanner (called the “blocking
set”) that intersects all 2k-cycles; the strategy is then to sample a subgraph and remove the parts
of the blocking set that survive. This approach works, but it is limited: if we use a slack FT-greedy
algorithm (as in [DR20]) then the size of the blocking set increases with the slack, giving spanners
that no longer have optimal size.
Our main idea is to bypass blocking sets by more closely tying together the algorithm and the
analysis. The analysis of the blocking set technique uses random subgraph sampling, but this does
not appear in the actual algorithm of [DR20] or [BP19]. Our new algorithm, by explicitly sampling
subgraphs as part of the test(u,v) subroutine, is in some sense doing algorithmically exactly the
minimum needed for the analysis to work. As is shown more formally in Section 3, if an edge
passes our new test(u,v) then by construction the probability that it will have to be removed in
the analysis in order to obtain high-girth is at most some constant less than 1. So getting our
sampled subgraph to be high-girth in the analysis requires removing only a constant fraction of the
remaining edges. This is in contrast to the ideas behind blocking sets, where the blocking set is
4
predefined by the algorithm and so an edge either has to be removed from the analysis subgraph
(if it is part of the blocking set) or does not (if it isn’t). Hence another way to think of our idea
is that we are moving from a global analysis of which edges need to be removed (blocking sets,
where edges are either in the set or not) to a local analysis (where each edge has only a constant
probability of being removed).
Details of this algorithm and analysis are given in Section 3.
1.2.2 Our Faster Randomized Algorithm
As discussed, obtaining fast algorithms for spanners – not just any polynomial time – is a long-
standing and important line of research. Once we are able to achieve polynomial time, we naturally
want to minimize this time. Our next algorithm (which achieves the running time bound of Theo-
rem 1.1) optimizes the runtime in two ways:
1. It is costly to randomly sample subgraphs anew in each round of the algorithm. A more
efficient approach is to randomly sample vertex sets once at the very beginning, use these to
perform the test in each round, and incrementally maintain the sampled subgraphs as edges
are added to the spanner. We show that this approach works as long as we sample Θ(f3log n)
total subgraphs in the beginning.
2. Since we incrementally maintain a fixed set of subgraphs, to test (1) on each subgraph we
can use an appropriate incremental data structure rather than computing from scratch each
time. The obvious way of doing this requires using an incremental dynamic distance oracle /
APSP algorithm, but unfortunately known constructions are not sufficient for our purposes.
However, in the setting of unweighted input graphs one can easily reduce to reachability
(rather than distance) on a (k+ 1)-layered version of the subgraph, and we can use an
observation from [DR20] to make this reduction work for weighted input graphs as well
thanks to the fact that our algorithm is a slack version of the greedy algorithm. We can then
use a classical data structure for incremental reachability by Italiano [Ita86].
Together, these improvements give the runtime listed in Theorem 1.1. Details are given in
Section 4.
1.2.3 Our Deterministic Algorithm
Our second improvement is to regain determinism. Both the exponential time algorithm of [BDPW18]
and the polynomial time algorithm of [DR20] are deterministic, while the core ideas of our previous
two algorithms seem to require randomization (particularly our first, non-optimized algorithm).
But by constructing set systems with specific properties through the use of almost-universal hash
functions, we are able to derandomize the algorithm of Theorem 1.1.
The main idea is to leverage the fact that our fast randomized algorithm samples vertex sets
only once at the very beginning. By examining the proof of Theorem 1.1 we can determine what
properties we need these sets to have. Informally, we need that for every {u, v} ∈ Ethere are
not many sets containing both uand v, and that for every {u, v} ∈ Eand FV\ {u, v}with
|F| ≤ f, a constant fraction of the sets which contain both uand vdo not contain any vertex in
F(note that this guarantee has to hold simultaneously for all possible fault sets). So we just need
to give a deterministic construction of such a set system. We show how to do this by building an
(almost-)universal hash family from Vto [Θ(f)], and for each hash function in the family creating
Θ(f2) sets based on pairs of hash values. Interestingly, we are able to tolerate relatively large
5
values of “almost”: our spanner construction still works even if the universal hashing guarantee is
violated by large constants. Most of the literature on hashing, on the other hand, is optimized for
the case of only a (1 + ǫ) violation. By taking advantage of our ability to withstand weaker hashing
guarantees, we can design an extremely small hash family based on Message Authentication Codes
from cryptography, which is a standard and classical construction but to the best of our knowledge
has not previously been used in the context of hashing or derandomization. This allows us to obtain
running time that is essentially the same as our fast randomized algorithm when fis polynomial
in n.
These ideas give the deterministic runtime in Theorem 1.2. Details are given in Section 5.
2 Preliminaries and Notation
We will use ˜
O(·) to suppress polylogarithmic (in n) factors. For any integer a1, let [a] =
{1,2,...,a}. Given an edge-weighted graph G= (V, E, w), let dG(u, v) denote the shortest-path
distance from uto vin Gaccording to the weight function wand let d
G(u, v) denote the unweighted
distance (minimum number of hops) from uto vin G.
Many of our algorithms are randomized, and so we make claims that hold with high probability.
Formally, this means that they hold with probability at least 1 1/n.1
We will use the following Chernoff bounds (see [DP09]).
Theorem 2.1. Let X=Pn
i=1 Xi, where Xi(i[n]) are independently distributed in [0,1]. Then:
For 0< ǫ < 1:Pr[X < (1 ǫ)E[X]] exp((ǫ2/2) E[X]) and Pr[X > (1 + ǫ)E[X]]
exp((ǫ2/3) E[X]).
If t > 2eE[X], then Pr[X > t]<2t.
We will use the following structural lemma about fault-tolerant spanners, which was given
explicitly in [DR20] but appeared implicitly in essentially all previous papers on fault-tolerant
spanners. It essentially says that we only have to worry about spanning edges (not all pairs of
nodes), and only edges for which the shortest path between the endpoints is the edge itself.
Lemma 2.2. Let G= (V, E)be a graph with weight function wand let Hbe a subgraph of G.
Then His an f-VFT t-spanner of Gif and only if dH\F(u, v)t·w(u, v)for all FVwith
|F| ≤ fand u, v V\Fsuch that {u, v} ∈ E.
3 Optimal Fault-Tolerant Spanners in Polynomial Time
In this section we resolve the main open question left by [DR20,BP19,BDPW18] by proving The-
orem 1.3: we give a polynomial time algorithm which constructs optimal-size vertex fault tolerant
spanners. As discussed in Section 1.2.1, the algorithm itself is quite simple: we just use the greedy
algorithm but test whether to include an edge by sampling subgraphs and checking whether the
distance between the two endpoints is too large. This algorithm is given formally as Algorithm 1.
The following definitions will be useful in our analysis. Let Hbe the final spanner, and let H
be an induced subgraph of Hobtained by including each node, independently, with probability
1By changing the constants in the algorithm/analysis, all high probability claims we make can be made to hold
with probability at least 1 1/ncfor any constant c. We choose c= 1 only for simplicity.
6
Algorithm 1 Basic f-VFT (2k1)-Spanner Algorithm
Input: Graph G= (V, E ) on nnodes, edge weights w:ER+, integers k1 and f1.
1: H(V, )
2: for all e={u, v} ∈ Ein nondecreasing weight order do
3: Sample α=clog nsubgraphs {c
Hi
eH}i[α], where each is an induced subgraph on a vertex
set obtained by including uand v, and then each other node independently with probability
1/(2f).
4: Let c
Pebe the fraction of these subgraphs in which dc
Hi
e(u, v)>(2k1) ·w(u, v).
5: if c
Pe1/4then
6: Add eto H
7: return H
1/(2f) (note that His only an analytical tool, not part of the algorithm). Let He(H
e) denote the
subgraph of H(H) containing only the edges considered strictly before ein the algorithm. Let
Pe=(u,v):= Pr dH
e(u, v)>(2k1) ·w(u, v)|u, v V(H)
where the probability is over the random construction of H. So c
Pein the algorithm is an experi-
mental estimate of Pe, and we can bound its accuracy as follows:
Lemma 3.1. With high probability, for every edge eE, we have c
PePe±1/8.
Proof. We will prove the lower bound c
PePe1/8; the upper bound is essentially identical. The
random variable αc
Peis the sum of αrandom variables nc
Pi
eo, where
c
Pi
e:=
0 if dc
Hi
e(u, v)(2k1) ·w(u, v)
1 if dc
Hi
e(u, v)>(2k1) ·w(u, v).
Thus we may apply Chernoff bounds (Theorem 2.1), giving:
Pr hαc
Pe< α(Pe1/8)i= Pr hαc
Pe<Ehαc
Pe1/8iiPr αc
Pe<7
8Ehαc
Pei
e(1/8)2
2αE[c
Pe]eα/128
If we set α128 ·3 ln n, then the probability is at most 1/n3. So, by a union bound over the
mn2edges in the input graph, the probability that the lower bound fails for any edge is at most
1/n, proving the lemma.
We are now ready to prove the properties of Algorithm 1.
Lemma 3.2. With high probability, Algorithm 1 returns an f-VFT (2k1)-spanner.
Proof. Let e= (u, v) be an edge considered by the algorithm, and suppose there exists a fault set
FV\ {u, v}with |F| ≤ fsuch that
dHe\F(u, v)>(2k1) ·w(e).
In the event that H
econtains u, v but it does not contain any node in F, we thus also have
dH
e(u, v)>(2k1) ·w(e).
7
Thus Peis at least the probability that none of the nodes in Fsurvive in H
e, which we may bound:
Pe(1 p)|F|11
2ff
1/2.
By Lemma 3.1, with high probability we have c
PePe1/81/21/8>1/4, and so we add e
to Hin the algorithm. So for any edge enot added to the spanner, no such fault set Fexists. It
then follows from Lemma 2.2 that His an f-VFT (2k1)-spanner.
Lemma 3.3. With high probability, |E(H)| ≤ Of11/kn1+1/k.
Proof. Recall that His a induced subgraph of Hobtained by including every vertex independently
with probability 1/(2f), and hence E[|V(H)|] = n/(2f). For each edge (u, v)E(H), we say that
(u, v) is bad if dH
e(u, v)(2k1) ·w(u, v), and otherwise (if dH
e(u, v)>(2k1) ·w(u, v)) we
say that (u, v) is good. Let H′′ Hbe obtained by deleting all bad edges. We will now bound its
expected number of edges E[|E(H′′)|], conditioned on the high probability event from Lemma 3.1
holding, in two different ways:
For any cycle Cin Hwith at most 2kedges, notice that the edge (u, v)Cconsidered last
by the algorithm is bad, since there is a u vpath around the cycle consisting of at most
2k1 edges, each of weight at most w(u, v). Thus (u, v) is removed in H′′. It follows that
H′′ has no cycles with at most 2kedges; the folklore Moore Bounds then imply that
E(H′′)=OV(H′′)1+1/k.
Since each of the nnodes in Hare included in H′′ independently with probability 1/(2f), we
have
EE(H′′)=OEhV(H′′)1+1/ki=OEV(H′′)1+1/k=O n
f1+1/k!.(3)
In particular, the reason we can pull the exponent outside the expectation in the second step
is due to the following computation:
EhV(H′′)1+1/k iEV(H′′)1+1/k EhV(H′′)2iEV(H′′ )2
=V ar V(H′′)=O(n/f )
and hence the difference between these two terms may be hidden in the O.
For an edge e= (u, v)E(H), the probability that esurvives in H′′ may be decomposed as
the probability that it survives in H, times the probability that it survives in H′′ given that
it survives in H. This gives:
Pr (u, v)E(H′′)= Pr u, v V(H)·Pr (u, v) is good |u, v E(H)
= Θ(1/f2)·Pe.
Since (u, v) was added to H, we have c
Pe1/4, and so (since we condition on Lemma 3.1)
we have Pe1/8. Hence the probability of (u, v)E(H′′ ) is Θ(1/f2). By linearity of
expectations, we then have
|E(H′′)|= Θ |E(H)|
f2.(4)
8
Combining (3) and (4), we have
|E(H)|
f2=EE(H′′)=O n
f1+1/k!
and so, comparing the left- and right-hand sides and rearranging, we get
|E(H)|=On1+1/kf11/k .
Lemma 3.4. Algorithm 1 runs in polynomial time.
Proof. We first need to sort the edges by weight, which takes at most O(mlog n) time. Then for each
edge eE, we must sample Θ(log n) independent subgraphs and then run a single-source shortest
path computation in each. Sampling such a subgraph takes O(n+|E(H)|)O(f11/kn1+1/k ) time
(using the bound on |E(H)|from Lemma 3.3). The running time of the shortest path computation
on the subgraph is at most O(|E(H)|log n),2which is at most O(f11/kn1+1/k log n). Since we
repeat O(log n) times per round, and we have mtotal rounds in the algorithm, the total runtime
is O(mf11/kn1+1/k log2n).
4 An Even Faster Randomized Algorithm
Algorithm 2 is a bit more complicated than Algorithm 1, but it is significantly faster: in terms of
runtime, it essentially turns the O(m) factor in Lemma 3.4 into an O(k2n) factor. In Algorithm 1,
up to a logarithmic factor the runtime per sampled subgraph is dominated by the time required to
compute the subgraph rather than the time to measure distances on the subgraph, since sampling
the subgraph requires time linear in |E(H)|while distances are computed in the subgraph itself,
which has fewer edges than H. This leads to essentially O(|E(H)| · mlog n) total runtime, since we
sample O(mlog n) subgraphs in total: O(log n) subgraphs in each round, and mtotal rounds. In
our new algorithm, we improve this by (1) pre-sampling Θ(f3log n) subgraphs and using them in
each round, and (2) measuring relevant distances on these subgraphs using a certain incremental
dynamic algorithm, rather than recomputing from scratch. See Section 1.2.2 for a more detailed
overview. We now state the algorithm formally.
Definition 2. Let G= (V , E) be an undirected graph, and let k1 be an integer. The layered
graph G2kis the directed graph with vertex set V×[2k] and edges
((u, i),(v, i + 1)) for all i[2k1],{u, v} ∈ E, and
((u, i),(u, i + 1)) for all i[2k1], u V.
Notice that the unweighted distance between uand vin Gis at most 2k1 if and only if there
is a path from (u, 1) to (v, 2k) in G2k.
Theorem 4.1 ( [Ita86]).There is a data structure which takes O(n2)time to initialize on an empty
n-node graph, and which can then support directed edge insertions in O(n)time (amortized) and
reachability queries (answering “is there currently a u vpath in the graph?) in O(1) time.
2This estimate is conservative; except in a small range of parameters the subgraph is much smaller than Hand
thus the running time of the shortest path computation is dominated by the time needed to sample the subgraph in
the first place.
9
Algorithm 2 Faster f-VFT (2k1)-Spanner Algorithm
Input: Graph G= (V, E ), edge weights w:ER+, integers k1 and f1.
{preprocessing phase}
1: H(V, )
2: for i= 1 to α=cf 3log ndo
3: Create Viby including every vertex of Vindependently with probability 1/(2f)
4: Let Hi= (Vi,)
5: Create the layered graph H2k
i, and initialize the data structure of Theorem 4.1 for H2k
i
6: for all e={u, v} ∈ Edo Let Le={i|u, v Vi}
{main greedy algorithm}
7: for all e={u, v} ∈ Ein nondecreasing weight order do
8: for all HiLedo
9: Query whether there is a (u, 1) (v, 2k) path in H2k
i
10: Let Pebe the fraction of subgraphs HiLewhere the query returns NO
11: Let 0 < τ < 1 be an absolute constant that we choose in the analysis
12: if Peτthen
13: Add eto H
14: for all iLedo
15: Add eto Hi
16: Update the connectivity data structure for H2k
iby inserting the edges ((u, j),(v, j + 1))
and ((v, j),(u, j + 1)) in H2k
ifor all j[2k1]
17: return H
Like for Algorithm 1, we let Hbe the final spanner, and now we let Hbe a uniform random
subgraph among those selected in the preprocessing phase. Let He(H
e) denote the subgraph of H
(H) containing only the edges considered strictly before ein the algorithm. We note that we do
not have separate analogous definitions of c
Pe, Pethis time: the relevant probability Peis computed
exactly by the algorithm. We start our analysis with the following technical lemma:
Lemma 4.2. With high probability over the choice of random subgraphs in the preprocessing phase,
for every e= (u, v)Eand |F| ⊆ V\ {u, v}with |F| ≤ f, we have:
1. |Le|=O(flog n)
2. |{iLe|FVi=∅}| = Ω(flog n)
Proof. For the first part, by linearity of expectations the expected number of sets Lethat contain
both uand vis exactly α/(2f)2= (c/4)flog n. Applying Chernoff bounds (Theorem 2.1), we have
(for sufficiently large c):
Pr [|Le|> ce ·flog n] = Pr [|Le|>4eE[|Le|]] <2ce·flog n<1/(2n3)
A union bound over the mn2edges in the graph implies that |Le|=O(flog n) for all e,
simultaneously, with probability at least 1 1/(2n).
For the second part of the lemma, for any e= (u, v)Eand FV\ {u, v}with |F| ≤ f, and
10
for any i[α], we compute:
Pr[iLeand FVi=] = Pr[u, v Vi]·Pr[FVi6=]
=1
4f211
2f|F|
1
4f211
2ff
1
8f2.
Thus by linearity of expectations,
E[|{iLe|FVi=∅}|]α
8f2=c
8flog n.
Again by Chernoff bounds, we have
Pr |{iLe|FVi=∅}| <1
2c
8flog nexp 1
8c
8flog n
1
2n4fassuming sufficiently large c.
Taking a union bound over all nfpossible choices of Fand all mn2edges, we have
|{iLe|FVi=∅}| = Ω(flog n)
for all choices of e, F , simultaneously, with probability 11/(2n). Hence, by an intersection
bound, the two parts of the lemma hold jointly with high probability.
We are now ready to prove the properties of Algorithm 2:
Lemma 4.3. With high probability, Algorithm 2 returns an f-VFT (2k1)-spanner.
Proof. Let e= (u, v). As with our proof of correctness of Algorithm 1 (Lemma 3.2), by Lemma 2.2
we just need to show that when the algorithm considers e, if there is a fault set Ffor which
dHe\F(u, v)>(2k1) ·w(e),
then the algorithm adds eto H. Notice that this implies
d
He\F(u, v)>2k1,
since by construction the weight of every edge in Heis no larger than w(e) (recall from Section 2
that ddenotes the unweighted distance). By Lemma 4.2, with high probability a constant fraction
of the iLehave FVi=and thus we have d(u, v)>2k1 in the corresponding subgraphs.
Thus Peis at least an absolute constant; by setting τless than this constant, we will add eto
H.
Lemma 4.4. With high probability, |E(H)| ≤ Of11/kn1+1/k.
Proof. This loosely follows the proof of Lemma 3.3. For each edge (u, v)E(H), let us say:
(u, v) is bad if d
H
e(u, v)2k1
(u, v) is good if d
H
e(u, v)>2k1.
Let H′′ Hbe obtained by deleting all bad edges. We now again bound E[|E(H′′ )|] in two ways:
11
By essentially the same argument as in Lemma 3.3, H′′ has no cycles on 2kedges, and thus
EE(H′′)=OEhV(H′′ )1+1/ki=OEV(H′′)1+1/k
where we may pull the exponent outside the expectation by the same argument as in Lemma
3.3. When we choose random subgraphs in the preprocessing phase, the total number of nodes
added to all subgraphs can be viewed as the sum of independent binary random variables, and
the expectation is Θ(nf 2log n). Thus, by Chernoff bounds, with high probability over the
choice of random subgraphs in the preprocessing phase, we do indeed have Θ(nf2log n) total
nodes in our subgraphs. Conditioned on this high probability event, since we have exactly
α= Θ(f3log n) subgraphs, the average sampled subgraph Hihas |V(Hi)|= Θ(n/f ). We
then have E[|V(H′′)|] = Θ(n/f), and so with high probability
EE(H′′)=O n
f1+1/k!
(where the expectation is only over the choice of Hamong the subgraphs sampled in the
preprocessing phase).
By Lemma 4.2, with high probability over the choice of random subgraphs, we have |Le|=
Θ(flog n) for all edges e. In this event, whenever we add an edge e= (u, v) to the spanner
we also add the edge to Θ(flog n) out of the α= Θ(f3log n) subgraphs. Moreover, by
construction, in at least a constant τfraction of these subgraphs Hi, our (u, 1) (v, 2k) path
query returns NO. It follows that the current unweighted u vdistance in these subgraphs
is >2k1, and hence eis a good edge in Θ(τ·flog n) = Θ(flog n) subgraphs. So the total
number of good edges among the Θ(f3log n) subgraphs is Θ(|E(H)|flog n) We thus have
E|E(H′′)|= Θ |E(H)|
f2.
Combining these, we have
|E(H)|
f2=E|E(H′′)|=O n
f1+1/k!,
and again the lemma follows by comparing the left- and right-hand sides and rearranging.
Lemma 4.5. The expected running time of Algorithm 2 is at most
Ok2f11/kn2+1/k log n+mf 2log n.
Proof. We first analyze the preprocessing phase. Note that without loss of generality, m
f11/kn1+1/k f n or else we are already finished (we can simply return the input graph).
We can create the Visets in time O(nf 3log n) = O(mf2log n) by flipping α=O(f3log n)
weighted coins for each vertex. Once we do this, we may assume that every vertex vhas a sorted
list Lvof the values of ifor which vVi. In expectation, each of the layered graphs H2k
ihas
O(kn/f ) vertices, so we can then create the layered graphs in expected time O((kn/f)f3log n) =
O(knf 2log n) = O(kf11/kn2+1/k log n) (by linearity of expectations). Note that this includes
the initial edges of the form ((u, i),(u, i + 1)) in each of the layered graphs. Initializing the data
12
structure of Theorem 4.1 for each of the layered graphs takes time O((kn/f)2) in expectation
(since the number of nodes is binomial random variable and E[X2]O(E[X]2) for any bino-
mial random variable X), and thus the total expected time to initialize these data structures is
O((kn/f )2f3log n) = O(k2f n2log n)O(k2f11/k n2+1/k log n). We also need to insert all of the
initial edges of the form ((u, i),(u, i + 1)) into these data structures, which takes expected time
O(f3log n·(kn/f )·(kn/f)) = O(k2fn2log n)O(k2f11/kn2+1/k log n) (again using the square of
a binomial random variable). Hence we can construct and initialize the vertex sets and the needed
data structures in expected time at most
O(mf2log n) + O(kf 11/k n2+1/k log n) + O(k2f11/kn2+1/k log n) + O(k2f11/kn2+1/k log n)
=O(mf2log n) + O(k2f11/kn2+1/k log n).
To create the Lesets we need to be a little more careful, since doing it naively (looping through
each edge and each i[α] and checking if both endpoints are in Vi) would take O(mf 3log n) time.
But we can speed this up: since every vertex vhas Lvin a sorted list, for each edge e={u, v}we
can just do a single pass through Luand Lvto compute Le=LuLv. Thus this can be done in
O(|Lu|+|Lv|) time, and since |Lv| ≤ O(f2log n) with high probability for every vV, this takes
time O(f2log n) per edge and thus O(mf 2log n) total.
Putting all of this together, we get that the preprocessing takes time O(k2f11/kn2+1/k log n+
mf2log n).
We now analyze the main loop. For every e={u, v} ∈ E, the algorithm performs a connec-
tivity query in O(flog n) of the layered subgraphs. By Theorem 4.1, this takes O(mf log n) total
time. When we decide to add an edge eto the spanner (which happens at most O(f11/k n1+1/k)
times by Lemma 4.4), we have to do O(k) insertions into each of the O(flog n) layered graphs
in Le. The amortized cost of each insertion is O(kn/f) by Theorem 4.1 (since the number of
nodes in each layered graph is O(kn/f) by Lemma 4.2), and hence the total time of all insertions
is O(f11/kn1+1/k (kn/f )kf log n) = O(k2f11/kn2+1/k log n). This is asymptotically larger than
mlog nsince m < n2, and hence the running time of the main loop is O(k2f11/kn2+1/k log n).
5 Deterministic Algorithm
We now design a deterministic algorithm by derandomizing Algorithm 2. Recall that Algorithm 2
uses randomization in the preprocessing phase to create Θ(f3log n) vertex sets. We will derandom-
ize this by deterministically creating sets with the same properties by using appropriately chosen
hash functions. This results in a deterministic algorithm whose running time depends on the size
of the hash family that we use. If we use universal or pairwise-independent hash functions, then
we end up paying additional factors of nin the running time. We can improve this by using
almost-universal hash functions, since our analysis is robust to changes in constants. Standard
constructions then give the same dependence on nas in Algorithm 2, but polynomially worse de-
pendence on f. But in the most important regime where fis polynomial in n, we can use ideas from
message authentication codes in cryptography to design a hash family which is significantly more
efficient, allowing us to get running time that is essentially identical to the randomized algorithm!
5.1 Set System
Intuitively, we want sets which “act like” the Θ(f3log n) random sets of Algorithm 2. So they
should each have size about n/f, there shouldn’t be too many sets in each Le(each edge shouldn’t
be in too many of the subsets), and for every fault set Fa constant fraction of the sets in Leshould
13
not intersect F. We will proceed somewhat similarly to the approach of [Par19], who also needed
a set system where a reasonable fraction of the sets both had subsets of some given size and did
not intersect with subsets of a given size. But trying to apply [Par19] as a black box, or even using
the construction from [Par19], gives highly suboptimal bounds: the number of sets we would need
would be exponential in f. So we need to change the construction. We will also use a different
hash family, at least for the regime where fis polynomial in n, in order to get even faster running
time.
5.1.1 Almost Universal Hashing
Parter [Par19] began with “almost-pairwise independent” hash families. We will use a slightly
different (but related) definition of “almost-universal” hash families, which is a weaker requirement
but is sufficient for our needs and will allow us to design faster algorithms than if we required
almost-pairwise independence.
Definition 3. A family H={h:UR}is δ-almost universal if:
1. For all x1, x2Uwith x16=x2: PrhH[h(x1) = h(x2)] δ
|R|, and
2. For all yRand h∈ H,|{xU:h(x) = y}| ≤ O(|U|/|R|).
3. Each h∈ H can be described with ˜
O(1) bits and can be evaluated in ˜
O(1) time.
Setting δ= 1 recovers the standard definition of universal hash families. The second and third
parts of this definition are not always part of the standard definition of universality, but easily
follow from most standard constructions (and, in particular, from the constructions that we will
use). The third property implies that in time ˜
O(|H||U|) we can compute all hash functions from
the family on all elements of the domain.
We will use two different constructions of δ-almost universal hash families, one which works for
all regimes and one which only gives meaningful bounds when the size of the range is polynomial
in the size of the domain. The first construction we will use is the following:
Theorem 5.1. For every δ > 1, there is a δ-almost universal hash family Hwith O(poly(1/(δ
1))|R|4log2|U|)functions.
To the best of our knowledge, this theorem does not appear explicitly anywhere in the literature,
since most papers just bound the number of random bits as O(log |R|+ log log |U|). Since the
constant is unspecified, this is not enough to prove Theorem 5.1. But Theorem 5.1 can easily be
derived from Theorem 2 of [AGHP92] by using N=|U|log |R|bits to define a hash function and
setting k= 2 log |R|and ǫ= (δ1)/|R|2.
If |R|is close to |U|, we can use a different construction based on ideas from the cryptography
literature, and in particular from Message Authentication Codes (MACs). This construction and
analysis is essentially standard (see [KR09,dB93,Tay94,BJKS93]), but to the best of our knowledge
has not been explicitly phrased as a hash function before in the literature. We will actually use a
slightly weaker (and thus more efficient) version of the standard construction since we only need
almost-universality, not almost-pairwise independence. We give the proof for completeness.
Theorem 5.2. There is a llog |U|
log |R|m-almost universal hash family Hwith O(|R|)functions.
Proof. Without loss of generality, let U={0,1}uand let R={0,1}rwith udivisible by r, and
we interpret Uas F2uand Ras F2r. A single hash function is defined by a single element aF2r.
14
Given xU, we split xinto u/r chunks x0, x1, x(u/r)1, each of which has rbits and so is an
element of F2r. This defines a polynomial Mx(a) = P(u/r)1
i=0 xiaiof degree (u/r)1. Given an
element aF2r, we define a hash function ha(x) = Mx(a).
Let H={ha:aF2r}. Clearly |H| = 2r=|R|. It is also not hard to see that His a u/r-almost
universal family. To see this, let x, y Uwith x6=y. Then ha(x) = ha(y) if and only if Mx(a) =
My(a), which is equivalent to Mx(a)My(a) = 0. Clearly Mx(a)My(a) = P(u/r)1
i=0 (xiyi)ai
is a non-zero polynomial (in a) of degree at most u/r, so there are at most u/r roots and thus the
probability that we choose an awhich satisfies this is at most (u/r)/2r= (u/r)/|R|.
Clearly we can compute these functions quickly enough, so the third property of Definition 3
holds. The second property of Definition 3 also holds, since if we divide the possible xUinto
equivalence classes by everything except their lowest-order chunk x0(so each class has 2relements
and there are 2urclasses), then for every fixed aand every equivalence class there is exactly one
element from the class which gets hashed to every possible value.
Since log |U|/log |R|is a constant if Rand Uare polynomially related, Theorem 5.2 gives an
O(1)-almost universal family in the case of |R| ≥ poly(|U|). We will only use this theorem in that
regime.
5.1.2 Creating Our Sets
We will use δ-almost universal hash families to create the subsets in the preprocessing stage of
Algorithm 2 rather than creating these sets randomly. More formally, rather than create O(f3log n)
sets independently, we will use the following construction.
Let Hbe a δ-almost universal hash family with domain Vand range [4δf].3We will create
|H|4δf
2= Θ(|H|f2) sets as follows: for every hHand y, z [4δf] with y6=z, we let
Vh,{y,z}={vV:h(v)∈ {y, z}}
In order to keep our previous notation, we will let α=|H|4δf
2and will arbitrarily number
these sets and refer to them as V1,...,Vα. And as before, we will let Le={i[α] : eVi}be
the sets which contain both endpoints of e. We first need the equivalent of Lemma 4.2, but now
all bounds are deterministic.
Lemma 5.3. If His δ-universal hash family for some constant δ > 1then there exist constants
c1, c2, c3>0so that for sufficiently large nthe sets V1,...,Vαhave the following properties:
1. |Vi| ≤ c1n/f for all i[α],
2. |Le| ≤ c2δ|H| for all eE, and
3. |{iLe:FVi=∅}| ≥ c3|H| for all eEand FVwith |F| ≤ fand Fe=.
Proof. The second property of Definition 3 implies that each set has size at most O(n/(4δf )) =
O(n/f), so the first property is clearly true. For the second property, let e={u, v} ∈ E, and let
h∈ H. If h(u) = h(v) then eis contained in 4δf 1 of the 4δf
2subsets defined by h, and if
3This might seem strange in conjunction with Theorem 5.2, since then the range is a function of δbut δis also
a function of the range. We show how to set the parameters appropriately when we actually instantiate this hash
family in the proof of Theorem 1.2.
15
h(u)6=h(v) then eis contained in 1 of the subsets defined by h. By the first part of the definition
of almost universal, Prh∼H[h(u) = h(v)] δ
4δf =1
4f. Thus
|Le| ≤ X
h∈H
1 + X
h∈H:h(u)=h(v)
(4δf 1) |H| +δ|H| = (1 + δ)|H|.
For the third property, fix e={u, v} ∈ Eand FV\ {u, v}with |F| ≤ f. For every xF,
let Xxu be the indicator random variable for the event that h(x) = h(u), and similarly let Xxv
be the indicator random variable for h(x) = h(v). Note that by the definition of almost-universal,
both of these random variables have expectation at most δ/(4δf) = 1/(4f). Let ZFbe the
set of elements of Fthat hash to the same value as uor v(so Zis a random subset). Then
|Z| ≤ PxF(Xxu +Xxv), and thus
E[|Z|]E"X
xF
(Xxu +Xxv)#=X
xF
(E[Xxu] + E[Xxv ]) X
xF
2
4f1
2
So by Markov’s inequality, the probability that |Z| ≥ 1 is at most 1/2. Thus in at least half of
the hash functions from H, nothing from Fhas the same hash value as uor v. If we have such an
h,and if h(u)6=h(v), then this means that FVh,{h(u),h(v)}=. As discussed earlier, the number
of hash functions in which h(u) = h(v) is at most |H|/(4f). Thus the number of sets which contain
both uand vbut do not contain any element of Fis at least
|H| 1
21
4f·13
8|H|
as claimed.
5.2 The Algorithm and Analysis
Our deterministic algorithm is the same as Algorithm 2, except:
1. Instead of sampling O(f3log n) sets, we use the above construction to create O(|H|f2) sets
which obey Lemma 5.3,
2. We set the threshold τto c3/c2, i.e., we add eto the spanner if Pec3/c2(where c2and c3
are the constants from Lemma 5.3).
We now prove correctness and size bounds in essentially the same way as in the randomized
algorithm. They just hold deterministically rather than with high probability.
Lemma 5.4. This algorithm returns an f-VFT (2k1)-spanner.
Proof. Let e={u, v}. As with our proofs of correctness of Algorithms 1 and 2, by Lemma 2.2 we
just need to show that if when the algorithm considers ethere is a fault set Fwhich is bad for e
(i.e., |F| ≤ fand dH\F(u, v)>(2k1)w(e)), then the algorithm adds eto H.
Note that for any i[α] where eVibut FVi=, we have dHi(u, v)>(2k1)w(e) (since
removing Fis enough to make the distance too large) and hence there is no (2k1)-hop path from
uto vin Hi(since we consider the edges in nondecreasing weight order). This implies that our
reachability query from (u, 1) to (v, 2k) in H2k
iwill return NO. Thus by Lemma 5.3, Pec3|H|
c2|H| =τ,
and so the algorithm will add eto the spanner. Lemma 2.2 then implies the theorem.
16
We can now prove our size bound.
Lemma 5.5. |E(H)| ≤ O(f11/kn1+1/k).
Proof. We use randomness in our analysis in the same way it was used in the proof of Lemma 4.3,
but note that the algorithm itself is deterministic.
Choose an i[α] uniformly at random, and let H=Hi. Then for each (2k)-cycle in H,
remove the heaviest edge to get H′′.
By part 1 of Lemma 5.3, we know that |V(H′′)| ≤ O(n/f). Since H′′ has no (2k)-cycles, this
implies that
|E(H′′)| ≤ O(n/f )1+1/k .(5)
Fix some e={u, v} ∈ E(H). Then the third property of Lemma 5.3 (with F=) implies
that Pr[eH]Ω(|H|/(|H|f2)) = Ω(1/f2). Conditioned on this, efails to survive to H′′ only
if it was the heaviest edge on some (2k)-cycle in H. Since the algorithm considers the edges in
nondecreasing weight order, eis the heaviest edge on some (2k)-cycle if and only if when it was
added by the algorithm the hop-distance between uand vin Hwas at most 2k1.
The probability of this happening (conditioned on ebeing in H) is by definition equal to 1Pe,
and so the probability that edoes survive to H′′ (conditioned on being in H) is precisely Pe. Since
ewas added by the algorithm we know that Pec3/c2, and thus the probability that esurvives
to H′′ (conditioned on being in H) is at least c3/c2.
Hence
E[|E(H′′)|]≥ |E(H)| · Ω(1/f 2)·(c3/c2)Ω(|E(H)|/f 2) (6)
Combining (5) and (6) implies that
|E(H)| ≤ O(f2·(n/f)1+1/k) = O(f11/kn1+1/k )
as claimed.
We now analyze the running time. Clearly it depends on |H|, but for now we will leave this as
a parameter.
Lemma 5.6. The running time of the deterministic algorithm is at most
˜
Of1/kn2+1/k |H| +m|H|f.
Proof. We proceed as in Lemma 4.5 by first analyzing the preprocessing, and assuming without
loss of generality that mf11/kn1+1/k f n.
Our set system has O(|H|f2) sets. For each hash function in H, we can create the fbuckets in
˜
O(n) time (by evaluating the hash function on each vertex in ˜
O(1) time). Then we can build each
of the Θ(f2) sets for that function in time O(n/f ), so the time to create all of the Θ(|H|f2) sets is
˜
O(|H|fn). Creating the layered graphs takes additional O(|H|f2(n/f)k) = O(|H|nfk) = O(mk|H|)
time. Initializing all O(|H|f2) data structures from Theorem 4.1 and inserting the initial edges
takes time O(|H|f2·(kn/f)2) = O(k2n2|H|) = ˜
O(f1/kn2+1/k |H|). As in the analysis of the fast
randomized algorithm (Lemma 4.5), while creating the sets we can record for each vertex a sorted
list of which sets it is in, and then can create each Leset by simple set intersection. Since each
vertex is in O(|H|f) sets, this takes time O(m|H|f).
Thus our total preprocessing time is ˜
O(f1/kn2+1/k |H| +m|H|f).
We now analyze the main greedy loop. For every e={u, v} ∈ E, the algorithm performs
a connectivity query in |Le| ≤ O(|H|) of the layered subgraphs. By Theorem 4.1, this takes
17
O(m|H|) total time. When we decide to add an edge eto the spanner (which happens at most
O(f11/kn1+1/k ) times by Lemma 5.5), we have to do O(k) insertions into each of the O(|Le|) =
O(|H|) layered graphs in Le. The amortized cost of each insertion is O(kn/f ) by Theorem 4.1 (since
the number of nodes in each layered graph is O(kn/f) by Lemma 5.3), and hence the total time
of all insertions is O(f11/kn1+1/k(kn/f)k|H|) = O(k2f1/kn2+1/k |H|). This is asymptotically
larger than m|H| since m < n2and f < n, and hence the running time of the main loop is
O(k2f1/kn2+1/k |H|).
This now finally allows us to prove Theorem 1.2, which we restate here for clarity.
Theorem 1.2. There is a deterministic algorithm which constructs an f-VFT (2k1) spanner
with at most O(f11/kn1+1/k )edges in time ˜
O(f41/kn2+1/k +mf 5), and if f=poly(n)(i.e.,
fncfor some constant c > 0) then the running time improves to ˜
O(f11/kn2+1/k +mf 2)
Proof. The fault tolerance and size bounds are from Lemmas 5.4 and 5.5. Using Lemma 5.6
with the almost universal construction of Theorem 5.1 (with δ= 2) gives a running time of
˜
O(k2f1/kn2+1/k f4+mf 5) = ˜
O(f41/kn2+1/k +mf 5).
On the other hand, if fncfor some constant c > 0 then we want to use the almost universal
construction of Theorem 5.2. But as discussed earlier, we have to be a little careful since if we
use this construction then the range is a function of δbut δis also a function of the range. So we
need to show that we can set δso that it is a constant and the range is [4δf ] (since that is the
construction we used). We first set δ=log n
log f1/c and set the range of the hash family to be 4δf.
Then if we use Theorem 5.2 with this range, the theorem implies that this is a δ=log n
log(4δf )-almost
universal family. But δis clearly at most δ, and hence it is also a δ-almost universal family. So
we have, as required, a δ-almost universal family with range [4δf ] where δ1/c is a constant.
When we use this family, Lemma 5.6 implies that the running time is ˜
O(k2f1/kn2+1/k f+mf 2) =
˜
O(f11/kn2+1/k +mf 2).
6 Conclusion and Open Questions
In this paper we gave the first polynomial-time algorithm to construct optimal-size vertex fault-
tolerant spanners. Our algorithm, after being optimized for running time, is also significantly
faster than the previous best (non-optimal) polynomial time algorithm. We also derandomized
our algorithm to get a deterministic algorithm, which is always polynomial time and, in the most
interesting regime of fbeing polynomial in n, is just as fast (ignoring polylogarithmic factors) as
our randomized algorithm.
There are still a number of tantalizing open problems involving fault tolerant spanners. Algo-
rithmically, while we significantly optimized the running time to make it surprisingly efficient, even
in the regime f=O(1) it is still not as fast the fastest algorithms for non-FT spanners (e.g., [BS07]
which has running time ˜
O(m)). These fast non-FT spanner algorithms are not the greedy algo-
rithm, which seems to be unable to achieve such an efficient running time. Is it possible to compute
optimal-size fault-tolerant spanners in time ˜
O(m)? Similarly, there has been significant work on
computing spanners in other models of computation, most notably in distributed and parallel mod-
els (see [BS07, DGP07, DGP07, PY18, BDG+20] for a small sampling of such results). The greedy
algorithm is typically difficult to parallelize or implement efficiently distributedly (particularly in
the presence of congestion), so there is the obvious question of computing optimal-size fault tolerant
spanners efficiently in these models.
18
Next, we note that while all of our constructions and bounds work as well in the case of edge
faults, they are not known to be optimal. The best-known lower bound on the size of an f-edge
fault tolerant (2k1) spanner, proved in [BDPW18], is only Ω f1
2(11/k)n1+1/krather than
f11/kn1+1/k as for vertex faults (for k3; for k= 2 the lower bounds are both Ω f1/2n3/2,
and hence the size bounds achieved here are optimal). Closing this gap for general k, by either
giving improved upper bounds or improved lower bounds (or both), is probably the most important
open question about fault-tolerant spanners.
Finally, there are some additional nice properties of the non-faulty greedy algorithm, and it
would be interesting to determine whether these have desirable analogs for FT (slack-)greedy al-
gorithms as well. In particular: the non-FT greedy algorithm gives optimal spanners for several
important graph classes like Euclidean graphs and doubling metrics [LS19], it produces optimal
spanners as measured by lightness [FS20], and there is experimental evidence that it performs
particularly well on graphs encountered in practice [FG05].
Acknowledgements
We would like to thank Xin Li for many helpful discussions about derandomization, and in particular
for pointing us towards message authentication codes.
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