Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userTight Bounds for Set Disjointness in the Message Passing Model
203
0
0.0
(
0
)
Added by
Rotem Oshman
Publication date
2013
fields
Informatics Engineering
and research's language is
English
Ask ChatGPT about the research
No Arabic abstract
In a multiparty message-passing model of communication, there are $k$ players. Each player has a private input, and they communicate by sending messages to one another over private channels. While this model has been used extensively in distributed computing and in multiparty computation, lower bounds on communication complexity in this model and related models have been somewhat scarce. In recent work cite{phillips12,woodruff12,woodruff13}, strong lower bounds of the form $Omega(n cdot k)$ were obtained for several functions in the message-passing model; however, a lower bound on the classical Set Disjointness problem remained elusive. In this paper, we prove tight lower bounds of the form $Omega(n cdot k)$ for the Set Disjointness problem in the message passing model. Our bounds are obtained by developing information complexity tools in the message-passing model, and then proving an information complexity lower bound for Set Disjointness. As a corollary, we show a tight lower bound for the task allocation problem cite{DruckerKuhnOshman} via a reduction from Set Disjointness.
rate research
Read More
Vizings celebrated theorem asserts that any graph of maximum degree $Delta$ admits an edge coloring using at most $Delta+1$ colors. In contrast, Bar-Noy, Naor and Motwani showed over a quarter century that the trivial greedy algorithm, which uses $2Delta-1$ colors, is optimal among online algorithms. Their lower bound has a caveat, however: it only applies to low-degree graphs, with $Delta=O(log n)$, and they conjectured the existence of online algorithms using $Delta(1+o(1))$ colors for $Delta=omega(log n)$. Progress towards resolving this conjecture was only made under stochastic arrivals (Aggarwal et al., FOCS03 and Bahmani et al., SODA10). We resolve the above conjecture for emph{adversarial} vertex arrivals in bipartite graphs, for which we present a $(1+o(1))Delta$-edge-coloring algorithm for $Delta=omega(log n)$ known a priori. Surprisingly, if $Delta$ is not known ahead of time, we show that no $big(frac{e}{e-1} - Omega(1) big) Delta$-edge-coloring algorithm exists. We then provide an optimal, $big(frac{e}{e-1}+o(1)big)Delta$-edge-coloring algorithm for unknown $Delta=omega(log n)$. Key to our results, and of possible independent interest, is a novel fractional relaxation for edge coloring, for which we present optimal fractional online algorithms and a near-lossless online rounding scheme, yielding our optimal randomized algorithms.
The following online bin packing problem is considered: Items with integer sizes are given and variable sized bins arrive online. A bin must be used if there is still an item remaining which fits in it when the bin arrives. The goal is to minimize the total size of all the bins used. Previously, a lower bound of 5/4 on the competitive ratio of this problem was achieved using jobs of size S and 2S-1. For these item sizes and maximum bin size 4S-3, we obtain asymptotically matching upper and lower bounds, which vary depending on the ratio of the number of small jobs to the number of large jobs.
We consider the following online optimization problem. We are given a graph $G$ and each vertex of the graph is assigned to one of $ell$ servers, where servers have capacity $k$ and we assume that the graph has $ell cdot k$ vertices. Initially, $G$ does not contain any edges and then the edges of $G$ are revealed one-by-one. The goal is to design an online algorithm $operatorname{ONL}$, which always places the connected components induced by the revealed edges on the same server and never exceeds the server capacities by more than $varepsilon k$ for constant $varepsilon>0$. Whenever $operatorname{ONL}$ learns about a new edge, the algorithm is allowed to move vertices from one server to another. Its objective is to minimize the number of vertex moves. More specifically, $operatorname{ONL}$ should minimize the competitive ratio: the total cost $operatorname{ONL}$ incurs compared to an optimal offline algorithm $operatorname{OPT}$. Our main contribution is a polynomial-time randomized algorithm, that is asymptotically optimal: we derive an upper bound of $O(log ell + log k)$ on its competitive ratio and show that no randomized online algorithm can achieve a competitive ratio of less than $Omega(log ell + log k)$. We also settle the open problem of the achievable competitive ratio by deterministic online algorithms, by deriving a competitive ratio of $Theta(ell lg k)$; to this end, we present an improved lower bound as well as a deterministic polynomial-time online algorithm. Our algorithms rely on a novel technique which combines efficient integer programming with a combinatorial approach for maintaining ILP solutions. We believe this technique is of independent interest and will find further applications in the future.
We consider the file maintenance problem (also called the online labeling problem) in which n integer items from the set {1,...,r} are to be stored in an array of size m >= n. The items are presented sequentially in an arbitrary order, and must be stored in the array in sorted order (but not necessarily in consecutive locations in the array). Each new item must be stored in the array before the next item is received. If r<=m then we can simply store item j in location j but if r>m then we may have to shift the location of stored items to make space for a newly arrived item. The algorithm is charged each time an item is stored in the array, or moved to a new location. The goal is to minimize the total number of such moves done by the algorithm. This problem is non-trivial when n=<m<r. In the case that m=Cn for some C>1, algorithms for this problem with cost O(log(n)^2) per item have been given [IKR81, Wil92, BCD+02]. When m=n, algorithms with cost O(log(n)^3) per item were given [Zha93, BS07]. In this paper we prove lower bounds that show that these algorithms are optimal, up to constant factors. Previously, the only lower bound known for this range of parameters was a lower bound of Omega(log(n)^2) for the restricted class of smooth algorithms [DSZ05a, Zha93]. We also provide an algorithm for the sparse case: If the number of items is polylogarithmic in the array size then the problem can be solved in amortized constant time per item.
We consider the point-to-point message passing model of communication in which there are $k$ processors with individual private inputs, each $n$-bit long. Each processor is located at the node of an underlying undirected graph and has access to private random coins. An edge of the graph is a private channel of communication between its endpoints. The processors have to compute a given function of all their inputs by communicating along these channels. While this model has been widely used in distributed computing, strong lower bounds on the amount of communication needed to compute simple functions have just begun to appear. In this work, we prove a tight lower bound of $Omega(kn)$ on the communication needed for computing the Tribes function, when the underlying graph is a star of $k+1$ nodes that has $k$ leaves with inputs and a center with no input. Lower bound on this topology easily implies comparable bounds for others. Our lower bounds are obtained by building upon the recent information theoretic techniques of Braverman et.al (FOCS13) and combining it with the earlier work of Jayram, Kumar and Sivakumar (STOC03). This approach yields information complexity bounds that is of independent interest.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
