Pipage Rounding, Pessimistic Estimators and Matrix Concentration

Pipage rounding is a dependent random sampling technique that has several interesting properties and diverse applications. One property that has been particularly useful is negative correlation of the resulting vector. Unfortunately negative correlation has its limitations, and there are some further desirable properties that do not seem to follow from existing techniques. In particular, recent concentration results for sums of independent random matrices are not known to extend to a negatively dependent setting. We introduce a simple but useful technique called concavity of pessimistic estimators. This technique allows us to show concentration of submodular functions and conc

Structural Routability of n-Pairs Information Networks

Information does not generally behave like a conservative fluid flow in communication networks with multiple sources and sinks. However, it is often conceptually and practically useful to be able to associate separate data streams with each source-sink pair, with only routing and no coding performed at the network nodes. This raises the question of whether there is a nontrivial class of network topologies for which achievability is always equivalent to routability, for any combination of source signals and positive channel capacities. This chapter considers possibly cyclic, directed, errorless networks with n source-sink pairs and mutually independent source signals. The concept of downward dominance is introduced and it is shown that, if the network topology is downward dominated, then the achievability of a given combination of source signals and channel capacities implies the existence of a feasible multicommodity flow.Comment: The final publication is available at link.springer.com http://link.springer.com/chapter/10.1007/978-3-319-02150-8_

Pipage Rounding, Pessimistic Estimators and Matrix Concentration

Pipage rounding is a dependent random sampling technique that has several interesting properties and diverse applications. One property that has been particularly useful is negative correlation of the resulting vector. Unfortunately negative correlation has its limitations, and there are some further desirable properties that do not seem to follow from existing techniques. In particular, recent concentration results for sums of independent random matrices are not known to extend to a negatively dependent setting. We introduce a simple but useful technique called concavity of pessimistic estimators. This technique allows us to show concentration of submodular functions and conc

The Serializability of Network Codes

Network coding theory studies the transmission of information in networks whose vertices may perform nontrivial encoding and decoding operations on data as it passes through the network. A solution to a network coding problem is a specification of a coding function on each edge of the network. This specification is subject to constraints that ensure the existence of a protocol by which the messages on each vertex’s outgoing edges can be computed from the data it received on its incoming edges. In directed acyclic graphs it is clear how to verify these causality constraints, but in graphs with cycles this becomes more subtle because of the possibility of cyclic dependencies among the coding functions. Sometimes the system of coding functions is serializable — meaning that the cyclic dependencies (if any) can be “unraveled ” by a protocol in which a vertex sends a few bits of its outgoing messages, waits to receive more information, then send a few more bits, and so on — but in other cases, there is no way to eliminate a cyclic dependency by an appropriate sequencing of partial messages. How can we decide whether a given system of coding functions is serializable? When it is not serializable, how much extra information must be transmitted in order to permit a serialization? Our work addresses both of these questions. We show that the first one is decidable in polynomial time, whereas the second one is NP-hard, and in fact it is logarithmically inapproximable.

Tiara: A self-stabilizing deterministic skip list

Abstract. We present Tiara — a self-stabilizing peer-to-peer network maintenance algorithm. Tiara is truly deterministic which allows it to achieve exact performance bounds. Tiara allows logarithmic searches and topology updates. It is based on a novel sparse 0-1 skip list. We rigorously prove the algorithm correct in the shared register model. We then describe its extension to a ring and incorporation of crash tolerance.

ART: sub-logarithmic decentralized range query processing with probabilistic guarantees

We focus on range query processing on large-scale, typically distributed infrastructures, such as clouds of thousands of nodes of shared-datacenters, of p2p distributed overlays, etc. In such distributed environments, efficient range query processing is the key for managing the distributed data sets per se, and for monitoring the infrastructure’s resources. We wish to develop an architecture that can support range queries in such large-scale decentralized environments and can scale in terms of the number of nodes as well as in terms of the data items stored. Of course, in the last few years there have been a number of solutions (mostly from researchers in the p2p domain) for designing such large-scale systems. However, these are inadequate for our purposes, since at the envisaged scales the classic logarithmic complexity (for point queries) is still too expensive while for range queries it is even more disappointing. In this paper we go one step further and achieve a sub-logarithmic complexity. We contribute the ART (Autonomous Range Tree) structure, which outperforms the most popular decentralized structures, including Chord (and some of its successors), BATON (and its successor) and Skip-Graphs. We contribute theoretical analysis, backed up by detailed experimental results, showing that the communication cost of query and update operations is O(log2blogN) hops, where the base b is a double-exponentially power of two and N is the total number of nodes. Moreover, ART is a fully dynamic and fault-tolerant structure, which supports the join/leave node operations in O(loglogN) expected w.h.p. number of hops. Our experimental performance studies include a detailed performance comparison which showcases the improved performance, scalability, and robustness of ART