Online Control Message Aggregation in Chain Networks

Abstract

In the Control Message Aggregation (CMA) problem, control packets are generated over time at the nodes of a tree T and need to be transmitted to the root of T. To optimize the overall cost, these transmissions can be delayed and different packets can be aggregated, that is a single transmission can include all packets from a subtree rooted at the root of T. The cost of this transmission is then equal to the total edge length of this subtree, independently of the number of packets that are sent. A sequence of transmissions that transmits all packets is called a schedule. The objective is to compute a schedule with minimum cost, where the cost of a schedule is the sum of all the transmission costs and delay costs of all packets. The problem is known to be $\mathbb{NP}$-hard, even for trees of depth 2. In the online scenario, it is an open problem whether a constant-competitive algorithm exists.

We address the special case of the problem when T is a chain network. For the online case, we present a 5-competitive algorithm and give a lower bound of 2 + φ ≈ 3.618, improving the previous bounds of 8 and 2, respectively. Furthermore, for the offline version, we give a polynomial-time algorithm that computes the optimum solution.

Research partially supported by NSF grants CCF-1217314 and OISE-1157129, MNiSW grant no. N N206 368839, 2010-2013, EU ERC project 259515 PAAl, CE-ITI (project P202/12/G061 of GA ČR), and grant IAA100190902 of GA AV ČR.