7,042 research outputs found
Randomized Proof-Labeling Schemes
International audienceA proof-labeling scheme, introduced by Korman, Kutten and Peleg [PODC 2005], is a mechanism enabling to certify the legality of a network configuration with respect to a boolean predicate. Such a mechanism finds applications in many frameworks, including the design of fault-tolerant distributed algorithms. In a proof-labeling scheme, the verification phase consists of exchanging labels between neighbors. The size of these labels depends on the network predicate to be checked. There are predicates requiring large labels, of poly-logarithmic size (e.g., MST), or even polynomial size (e.g., Symmetry). In this paper, we introduce the notion of randomized proof-labeling schemes. By reduction from deterministic schemes, we show that randomization enables the amount of communication to be exponentially reduced. As a consequence, we show that checking any network predicate can be done with probability of correctness as close to one as desired by exchanging just a logarithmic number of bits between neighbors. Moreover, we design a novel space lower bound technique that applies to both deterministic and randomized proof-labeling schemes. Using this technique, we establish several tight bounds on the verification complexity of classical distributed computing problems, such as MST construction, and of classical predicates such as acyclicity, connectivity, and cycle length
Randomized proof-labeling schemes
International audienceProof-labeling schemes, introduced by Korman et al. (Distrib Comput 22(4):215–233, 2010. https://doi.org/10.1007/s00446-010-0095-3), are a mechanism to certify that a network configuration satisfies a given boolean predicate. Such mechanisms find applications in many contexts, e.g., the design of fault-tolerant distributed algorithms. In a proof-labeling scheme, predicate verification consists of neighbors exchanging labels, whose contents depends on the predicate. In this paper, we introduce the notion of randomized proof-labeling schemes where messages are randomized and correctness is probabilistic. We show that randomization reduces verification complexity exponentially while guaranteeing probability of correctness arbitrarily close to one. We also present a novel message-size lower bound technique that applies to deterministic as well as randomized proof-labeling schemes. Using this technique, we establish several tight bounds on the verification complexity of MST, acyclicity, connectivity, and longest cycle size
Trade-Offs in Distributed Interactive Proofs
The study of interactive proofs in the context of distributed network computing is a novel topic, recently introduced by Kol, Oshman, and Saxena [PODC 2018]. In the spirit of sequential interactive proofs theory, we study the power of distributed interactive proofs. This is achieved via a series of results establishing trade-offs between various parameters impacting the power of interactive proofs, including the number of interactions, the certificate size, the communication complexity, and the form of randomness used. Our results also connect distributed interactive proofs with the established field of distributed verification. In general, our results contribute to providing structure to the landscape of distributed interactive proofs
Universal Communication, Universal Graphs, and Graph Labeling
We introduce a communication model called universal SMP, in which Alice and Bob receive a function f belonging to a family ?, and inputs x and y. Alice and Bob use shared randomness to send a message to a third party who cannot see f, x, y, or the shared randomness, and must decide f(x,y). Our main application of universal SMP is to relate communication complexity to graph labeling, where the goal is to give a short label to each vertex in a graph, so that adjacency or other functions of two vertices x and y can be determined from the labels ?(x), ?(y). We give a universal SMP protocol using O(k^2) bits of communication for deciding whether two vertices have distance at most k in distributive lattices (generalizing the k-Hamming Distance problem in communication complexity), and explain how this implies a O(k^2 log n) labeling scheme for deciding dist(x,y) ? k on distributive lattices with size n; in contrast, we show that a universal SMP protocol for determining dist(x,y) ? 2 in modular lattices (a superset of distributive lattices) has super-constant ?(n^{1/4}) communication cost. On the other hand, we demonstrate that many graph families known to have efficient adjacency labeling schemes, such as trees, low-arboricity graphs, and planar graphs, admit constant-cost communication protocols for adjacency. Trees also have an O(k) protocol for deciding dist(x,y) ? k and planar graphs have an O(1) protocol for dist(x,y) ? 2, which implies a new O(log n) labeling scheme for the same problem on planar graphs
Dynamic and Multi-functional Labeling Schemes
We investigate labeling schemes supporting adjacency, ancestry, sibling, and
connectivity queries in forests. In the course of more than 20 years, the
existence of labeling schemes supporting each of these
functions was proven, with the most recent being ancestry [Fraigniaud and
Korman, STOC '10]. Several multi-functional labeling schemes also enjoy lower
or upper bounds of or
respectively. Notably an upper bound of for
adjacency+siblings and a lower bound of for each of the
functions siblings, ancestry, and connectivity [Alstrup et al., SODA '03]. We
improve the constants hidden in the -notation. In particular we show a lower bound for connectivity+ancestry and
connectivity+siblings, as well as an upper bound of for connectivity+adjacency+siblings by altering existing
methods.
In the context of dynamic labeling schemes it is known that ancestry requires
bits [Cohen, et al. PODS '02]. In contrast, we show upper and lower
bounds on the label size for adjacency, siblings, and connectivity of
bits, and to support all three functions. There exist efficient
adjacency labeling schemes for planar, bounded treewidth, bounded arboricity
and interval graphs. In a dynamic setting, we show a lower bound of
for each of those families.Comment: 17 pages, 5 figure
Survey of Distributed Decision
We survey the recent distributed computing literature on checking whether a
given distributed system configuration satisfies a given boolean predicate,
i.e., whether the configuration is legal or illegal w.r.t. that predicate. We
consider classical distributed computing environments, including mostly
synchronous fault-free network computing (LOCAL and CONGEST models), but also
asynchronous crash-prone shared-memory computing (WAIT-FREE model), and mobile
computing (FSYNC model)
Near-Optimal Induced Universal Graphs for Bounded Degree Graphs
A graph is an induced universal graph for a family of graphs if every
graph in is a vertex-induced subgraph of . For the family of all
undirected graphs on vertices Alstrup, Kaplan, Thorup, and Zwick [STOC
2015] give an induced universal graph with vertices,
matching a lower bound by Moon [Proc. Glasgow Math. Assoc. 1965].
Let . Improving asymptotically on previous results by
Butler [Graphs and Combinatorics 2009] and Esperet, Arnaud and Ochem [IPL
2008], we give an induced universal graph with vertices for the family of graphs with vertices of maximum degree
. For constant , Butler gives a lower bound of
. For an odd constant , Esperet et al.
and Alon and Capalbo [SODA 2008] give a graph with
vertices. Using their techniques for any
(including constant) even values of gives asymptotically worse bounds than
we present.
For large , i.e. when , the previous best
upper bound was due to Adjiashvili and
Rotbart [ICALP 2014]. We give upper and lower bounds showing that the size is
. Hence the optimal size is
and our construction is within a factor of
from this. The previous results were
larger by at least a factor of .
As a part of the above, proving a conjecture by Esperet et al., we construct
an induced universal graph with vertices for the family of graphs with
max degree . In addition, we give results for acyclic graphs with max degree
and cycle graphs. Our results imply the first labeling schemes that for any
are at most bits from optimal
Pseudo-random graphs and bit probe schemes with one-sided error
We study probabilistic bit-probe schemes for the membership problem. Given a
set A of at most n elements from the universe of size m we organize such a
structure that queries of type "Is x in A?" can be answered very quickly.
H.Buhrman, P.B.Miltersen, J.Radhakrishnan, and S.Venkatesh proposed a bit-probe
scheme based on expanders. Their scheme needs space of bits, and
requires to read only one randomly chosen bit from the memory to answer a
query. The answer is correct with high probability with two-sided errors. In
this paper we show that for the same problem there exists a bit-probe scheme
with one-sided error that needs space of O(n\log^2 m+\poly(\log m)) bits. The
difference with the model of Buhrman, Miltersen, Radhakrishnan, and Venkatesh
is that we consider a bit-probe scheme with an auxiliary word. This means that
in our scheme the memory is split into two parts of different size: the main
storage of bits and a short word of bits that is
pre-computed once for the stored set A and `cached'. To answer a query "Is x in
A?" we allow to read the whole cached word and only one bit from the main
storage. For some reasonable values of parameters our space bound is better
than what can be achieved by any scheme without cached data.Comment: 19 page
Non-malleable codes for space-bounded tampering
Non-malleable codes—introduced by Dziembowski, Pietrzak and Wichs at ICS 2010—are key-less coding schemes in which mauling attempts to an encoding of a given message, w.r.t. some class of tampering adversaries, result in a decoded value that is either identical or unrelated to the original message. Such codes are very useful for protecting arbitrary cryptographic primitives against tampering attacks against the memory. Clearly, non-malleability is hopeless if the class of tampering adversaries includes the decoding and encoding algorithm. To circumvent this obstacle, the majority of past research focused on designing non-malleable codes for various tampering classes, albeit assuming that the adversary is unable to decode. Nonetheless, in many concrete settings, this assumption is not realistic
On the Cryptographic Hardness of Local Search
We show new hardness results for the class of Polynomial Local Search problems (PLS):
- Hardness of PLS based on a falsifiable assumption on bilinear groups introduced by Kalai, Paneth, and Yang (STOC 2019), and the Exponential Time Hypothesis for randomized algorithms. Previous standard model constructions relied on non-falsifiable and non-standard assumptions.
- Hardness of PLS relative to random oracles. The construction is essentially different than previous constructions, and in particular is unconditionally secure. The construction also demonstrates the hardness of parallelizing local search.
The core observation behind the results is that the unique proofs property of incrementally-verifiable computations previously used to demonstrate hardness in PLS can be traded with a simple incremental completeness property
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