20 research outputs found
Tight Bounds on the Round Complexity of the Distributed Maximum Coverage Problem
We study the maximum -set coverage problem in the following distributed
setting. A collection of sets over a universe is
partitioned across machines and the goal is to find sets whose union
covers the most number of elements. The computation proceeds in synchronous
rounds. In each round, all machines simultaneously send a message to a central
coordinator who then communicates back to all machines a summary to guide the
computation for the next round. At the end, the coordinator outputs the answer.
The main measures of efficiency in this setting are the approximation ratio of
the returned solution, the communication cost of each machine, and the number
of rounds of computation.
Our main result is an asymptotically tight bound on the tradeoff between
these measures for the distributed maximum coverage problem. We first show that
any -round protocol for this problem either incurs a communication cost of or only achieves an approximation factor of
. This implies that any protocol that simultaneously achieves
good approximation ratio ( approximation) and good communication cost
( communication per machine), essentially requires
logarithmic (in ) number of rounds. We complement our lower bound result by
showing that there exist an -round protocol that achieves an
-approximation (essentially best possible) with a communication
cost of as well as an -round protocol that achieves a
-approximation with only communication per each
machine (essentially best possible).
We further use our results in this distributed setting to obtain new bounds
for the maximum coverage problem in two other main models of computation for
massive datasets, namely, the dynamic streaming model and the MapReduce model
The Teleportation Design Pattern for Hardware Transactional Memory
We identify a design pattern for concurrent data structures, called teleportation, that uses best- effort hardware transactional memory to speed up certain kinds of legacy concurrent data struc- tures. Teleportation unifies and explains several existing data structure designs, and it serves as the basis for novel approaches to reducing the memory traffic associated with fine-grained locking, and with hazard pointer management for memory reclamation
On the Tree Conjecture for the Network Creation Game
Selfish Network Creation focuses on modeling real world networks from a game-theoretic point of view. One of the classic models by Fabrikant et al.[PODC\u2703] is the network creation game, where agents correspond to nodes in a network which buy incident edges for the price of alpha per edge to minimize their total distance to all other nodes. The model is well-studied but still has intriguing open problems. The most famous conjectures state that the price of anarchy is constant for all alpha and that for alpha >= n all equilibrium networks are trees.
We introduce a novel technique for analyzing stable networks for high edge-price alpha and employ it to improve on the best known bounds for both conjectures. In particular we show that for alpha > 4n-13 all equilibrium networks must be trees, which implies a constant price of anarchy for this range of alpha. Moreover, we also improve the constant upper bound on the price of anarchy for equilibrium trees
Distributed algorithms for low stretch spanning trees
Given an undirected graph with integer edge lengths, we study the problem of approximating the distances in the graph by a spanning tree based on the notion of stretch. Our main contribution is a distributed algorithm in the CONGEST model of computation that constructs a random spanning tree with the guarantee that the expected stretch of every edge is O(log3 n), where n is the number of nodes in the graph. If the graph is unweighted, then this algorithm can be implemented to run in O(D) rounds, where D is the hop-diameter of the graph, thus being asymptotically optimal. In the weighted case, the run-time of our algorithm matches the currently best known bound for exact distance computations, i.e., Õ(min{√nD, √nD1/4 + n3/5 + D}). We stress that this is the first distributed construction of spanning trees leading to poly-logarithmic expected stretch with non-trivial running time
The Fagnano Triangle Patrolling Problem
We investigate a combinatorial optimization problem that involves patrolling
the edges of an acute triangle using a unit-speed agent. The goal is to
minimize the maximum (1-gap) idle time of any edge, which is defined as the
time gap between consecutive visits to that edge. This problem has roots in a
centuries-old optimization problem posed by Fagnano in 1775, who sought to
determine the inscribed triangle of an acute triangle with the minimum
perimeter. It is well-known that the orthic triangle, giving rise to a periodic
and cyclic trajectory obeying the laws of geometric optics, is the optimal
solution to Fagnano's problem. Such trajectories are known as Fagnano orbits,
or more generally as billiard trajectories. We demonstrate that the orthic
triangle is also an optimal solution to the patrolling problem.
Our main contributions pertain to new connections between billiard
trajectories and optimal patrolling schedules in combinatorial optimization. In
particular, as an artifact of our arguments, we introduce a novel 2-gap
patrolling problem that seeks to minimize the visitation time of objects every
three visits. We prove that there exist infinitely many well-structured
billiard-type optimal trajectories for this problem, including the orthic
trajectory, which has the special property of minimizing the visitation time
gap between any two consecutively visited edges. Complementary to that, we also
examine the cost of dynamic, sub-optimal trajectories to the 1-gap patrolling
optimization problem. These trajectories result from a greedy algorithm and can
be implemented by a computationally primitive mobile agent