33 research outputs found

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### On the Computational Power of Radio Channels

Radio networks can be a challenging platform for which to develop distributed algorithms, because the network nodes must contend for a shared channel. In some cases, though, the shared medium is an advantage rather than a disadvantage: for example, many radio network algorithms cleverly use the shared channel to approximate the degree of a node, or estimate the contention. In this paper we ask how far the inherent power of a shared radio channel goes, and whether it can efficiently compute "classicaly hard" functions such as Majority, Approximate Sum, and Parity.
Using techniques from circuit complexity, we show that in many cases, the answer is "no". We show that simple radio channels, such as the beeping model or the channel with collision-detection, can be approximated by a low-degree polynomial, which makes them subject to known lower bounds on functions such as Parity and Majority; we obtain round lower bounds of the form Omega(n^{delta}) on these functions, for delta in (0,1). Next, we use the technique of random restrictions, used to prove AC^0 lower bounds, to prove a tight lower bound of Omega(1/epsilon^2) on computing a (1 +/- epsilon)-approximation to the sum of the nodes\u27 inputs. Our techniques are general, and apply to many types of radio channels studied in the literature

### Rounds vs Communication Tradeoffs for Maximal Independent Sets

We consider the problem of finding a maximal independent set (MIS) in the
shared blackboard communication model with vertex-partitioned inputs. There are
$n$ players corresponding to vertices of an undirected graph, and each player
sees the edges incident on its vertex -- this way, each edge is known by both
its endpoints and is thus shared by two players. The players communicate in
simultaneous rounds by posting their messages on a shared blackboard visible to
all players, with the goal of computing an MIS of the graph. While the MIS
problem is well studied in other distributed models, and while shared
blackboard is, perhaps, the simplest broadcast model, lower bounds for our
problem were only known against one-round protocols.
We present a lower bound on the round-communication tradeoff for computing an
MIS in this model. Specifically, we show that when $r$ rounds of interaction
are allowed, at least one player needs to communicate $\Omega(n^{1/20^{r+1}})$
bits. In particular, with logarithmic bandwidth, finding an MIS requires
$\Omega(\log\log{n})$ rounds. This lower bound can be compared with the
algorithm of Ghaffari, Gouleakis, Konrad, Mitrovi\'c, and Rubinfeld [PODC 2018]
that solves MIS in $O(\log\log{n})$ rounds but with a logarithmic bandwidth for
an average player. Additionally, our lower bound further extends to the closely
related problem of maximal bipartite matching.
To prove our results, we devise a new round elimination framework, which we
call partial-input embedding, that may also be useful in future work for
proving round-sensitive lower bounds in the presence of edge-sharing between
players.
Finally, we discuss several implications of our results to multi-round
(adaptive) distributed sketching algorithms, broadcast congested clique, and to
the welfare maximization problem in two-sided matching markets.Comment: Full version of the paper in FOCS 2022, 44 page

### Optimal Multi-Pass Lower Bounds for MST in Dynamic Streams

The seminal work of Ahn, Guha, and McGregor in 2012 introduced the graph
sketching technique and used it to present the first streaming algorithms for
various graph problems over dynamic streams with both insertions and deletions
of edges. This includes algorithms for cut sparsification, spanners, matchings,
and minimum spanning trees (MSTs). These results have since been improved or
generalized in various directions, leading to a vastly rich host of efficient
algorithms for processing dynamic graph streams.
A curious omission from the list of improvements has been the MST problem.
The best algorithm for this problem remains the original AGM algorithm that for
every integer $p \geq 1$, uses $n^{1+O(1/p)}$ space in $p$ passes on $n$-vertex
graphs, and thus achieves the desired semi-streaming space of $\tilde{O}(n)$ at
a relatively high cost of $O(\frac{\log{n}}{\log\log{n}})$ passes. On the other
hand, no lower bounds beyond a folklore one-pass lower bound is known for this
problem.
We provide a simple explanation for this lack of improvements: The AGM
algorithm for MSTs is optimal for the entire range of its number of passes! We
prove that even for the simplest decision version of the problem -- deciding
whether the weight of MSTs is at least a given threshold or not -- any $p$-pass
dynamic streaming algorithm requires $n^{1+\Omega(1/p)}$ space. This implies
that semi-streaming algorithms do need $\Omega(\frac{\log{n}}{\log\log{n}})$
passes.
Our result relies on proving new multi-round communication complexity lower
bounds for a variant of the universal relation problem that has been
instrumental in proving prior lower bounds for single-pass dynamic streaming
algorithms. The proof also involves proving new composition theorems in
communication complexity, including majority lemmas and multi-party XOR lemmas,
via information complexity approaches

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### Interactive Compression for Multi-Party Protocol

The field of compression studies the question of how many bits of communication are necessary to convey a given piece of data. For one-way communication between a sender and a receiver, the seminal work of Shannon and Huffman showed that the communication required is characterized by the entropy of the data; in recent years, there has been a great amount of interest in extending this line of research to interactive communication, where instead of a sender and a receiver we have two parties communication back-and-forth. In this paper we initiate the study of interactive compression for distributed multi-player protocols. We consider the classical shared blackboard model, where players take turns speaking, and each player\u27s message is immediately seen by all the other players. We show that in the shared blackboard model with k players, one can compress protocols down to ~O(Ik), where I is the information content of the protocol and k is the number of players. We complement this result with an almost matching lower bound of ~Omega(Ik), which shows that a nearly-linear dependence on the number of players cannot be avoided

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### A Candidate for a Strong Separation of Information and Communication

The weak interactive compression conjecture asserts that any two-party communication protocol with communication complexity C and information complexity I can be compressed to a protocol with communication complexity poly(I)polylog(C).
We describe a communication problem that is a candidate for refuting that conjecture. Specifically, while we show that the problem can be solved by a protocol with communication complexity C and information complexity I=polylog(C), the problem seems to be hard for protocols with communication complexity poly(I)polylog(C)=polylog(C)

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### Computation over the Noisy Broadcast Channel with Malicious Parties

We study the n-party noisy broadcast channel with a constant fraction of malicious parties. Specifically, we assume that each non-malicious party holds an input bit, and communicates with the others in order to learn the input bits of all non-malicious parties. In each communication round, one of the parties broadcasts a bit to all other parties, and the bit received by each party is flipped with a fixed constant probability (independently for each recipient). How many rounds are needed?
Assuming there are no malicious parties, Gallager gave an (n log log n)-round protocol for the above problem, which was later shown to be optimal. This protocol, however, inherently breaks down in the presence of malicious parties.
We present a novel n ⋅ ̃(√{log n})-round protocol, that solves this problem even when almost half of the parties are malicious. Our protocol uses a new type of error correcting code, which we call a locality sensitive code and which may be of independent interest. Roughly speaking, these codes map "close" messages to "close" codewords, while messages that are not close are mapped to codewords that are very far apart.
We view our result as a first step towards a theory of property preserving interactive coding, i.e., interactive codes that preserve useful properties of the protocol being encoded. In our case, the naive protocol over the noiseless broadcast channel, where all the parties broadcast their input bit and output all the bits received, works even in the presence of malicious parties. Our simulation of this protocol, unlike Gallager’s, preserves this property of the original protocol

### Protecting Single-Hop Radio Networks from Message Drops

Single-hop radio networks (SHRN) are a well studied abstraction of communication over a wireless channel. In this model, in every round, each of the n participating parties may decide to broadcast a message to all the others, potentially causing collisions. We consider the SHRN model in the presence of stochastic message drops (i.e., erasures), where in every round, the message received by each party is erased (replaced by ?) with some small constant probability, independently.
Our main result is a constant rate coding scheme, allowing one to run protocols designed to work over the (noiseless) SHRN model over the SHRN model with erasures. Our scheme converts any protocol ? of length at most exponential in n over the SHRN model to a protocol ?\u27 that is resilient to constant fraction of erasures and has length linear in the length of ?.
We mention that for the special case where the protocol ? is non-adaptive, i.e., the order of communication is fixed in advance, such a scheme was known. Nevertheless, adaptivity is widely used and is known to hugely boost the power of wireless channels, which makes handling the general case of adaptive protocols ? both important and more challenging. Indeed, to the best of our knowledge, our result is the first constant rate scheme that converts adaptive protocols to noise resilient ones in any multi-party model

### Near-Optimal Two-Pass Streaming Algorithm for Sampling Random Walks over Directed Graphs

For a directed graph G with n vertices and a start vertex u_start, we wish to (approximately) sample an L-step random walk over G starting from u_start with minimum space using an algorithm that only makes few passes over the edges of the graph. This problem found many applications, for instance, in approximating the PageRank of a webpage. If only a single pass is allowed, the space complexity of this problem was shown to be ??(n ? L). Prior to our work, a better space complexity was only known with O?(?L) passes.
We essentially settle the space complexity of this random walk simulation problem for two-pass streaming algorithms, showing that it is ??(n ? ?L), by giving almost matching upper and lower bounds. Our lower bound argument extends to every constant number of passes p, and shows that any p-pass algorithm for this problem uses ??(n ? L^{1/p}) space. In addition, we show a similar ??(n ? ?L) bound on the space complexity of any algorithm (with any number of passes) for the related problem of sampling an L-step random walk from every vertex in the graph