A set of identical, mobile agents is deployed in a weighted network. Each agent has a battery--a power source allowing it to move along network edges. An agent uses its battery proportionally to the distance traveled. We consider two tasks: convergecast, in which at the beginning, each agent has some initial piece of information, and information of all agents has to be collected by some agent; and broadcast in which information of one specified agent has to be made available to all other agents. In both tasks, the agents exchange the currently possessed information when they meet. The objective of this paper is to investigate what is the minimal value of power, initially available to all agents, so that convergecast or broadcast can be achieved. We study this question in the centralized and the distributed settings. In the centralized setting, there is a central monitor that schedules the moves of all agents. In the distributed setting every agent has to perform an algorithm being unaware of the network. In the centralized setting, we give a linear-time algorithm to compute the optimal battery power and the strategy using it, both for convergecast and for broadcast, when agents are on the line. We also show that finding the optimal battery power for convergecast or for broadcast is NP-hard for the class of trees. On the other hand, we give a polynomial algorithm that finds a 2-approximation for convergecast and a 4-approximation for broadcast, for arbitrary graphs.In the distributed setting, we give a 2-competitive algorithm for convergecast in trees and a 4-competitive algorithm for broadcast in trees. The competitive ratio of 2 is proved to be the best for the problem of convergecast, even if we only consider line networks. Indeed, we show that there is no ( @math 2-∈)-competitive algorithm for convergecast or for broadcast in the class of lines, for any @math ∈>0.
 Computing energy efficient broadcast trees is one of the most prominent operations in wireless networks. For stations embedded in the Euclidean plane, the best analytic result known to date is a 6.33-approximation algorithm based on computing an Euclidean minimum spanning tree. We improve the analysis of this algorithm and show that its approximation ratio is 6, which matches a previously known lower bound for this algorithm. We consider the problem of selecting threshold times to transition a device to low-power sleep states during an idle period. The two-state case in which there is a single active and a single sleep state is a continuous version of the ski-rental problem. We consider a generalized version in which there is more than one sleep state, each with its own power consumption rate and transition costs. We give an algorithm that, given a system, produces a deterministic strategy whose competitive ratio is arbitrarily close to optimal. We also give an algorithm to produce the optimal online strategy given a system and a probability distribution that generates the length of the idle period. We also give a simple algorithm that achieves a competitive ratio of 3 + 2 spl radic 2 spl ap 5.828 for any system. We consider offline scheduling algorithms that incorporate speed scaling to address the bicriteria problem of minimizing energy consumption and a scheduling metric. For makespan, we give a linear-time algorithm to compute all non-dominated solutions for the general uniprocessor problem and a fast arbitrarily-good approximation for multiprocessor problems when every job requires the same amount of work. We also show that the multiprocessor problem becomes NP-hard when jobs can require different amounts of work. For total flow, we show that the optimal flow corresponding to a particular energy budget cannot be exactly computed on a machine supporting exact real arithmetic, including the extraction of roots. This hardness result holds even when scheduling equal-work jobs on a uniprocessor. We do, however, extend previous work by to give an arbitrarily-good approximation for scheduling equal-work jobs on a multiprocessor. The energy usage of computer systems is becoming an important consideration, especially for battery-operated systems. Various methods for reducing energy consumption have been investigated, both at the circuit level and at the operating systems level. In this paper, we propose a simple model of job scheduling aimed at capturing some key aspects of energy minimization. In this model, each job is to be executed between its arrival time and deadline by a single processor with variable speed, under the assumption that energy usage per unit time, P, is a convex function, of the processor speed s. We give an off-line algorithm that computes, for any set of jobs, a minimum-energy schedule. We then consider some on-line algorithms and their competitive performance for the power function P(s)=s sup p where p spl ges 2. It is shown that one natural heuristic, called the Average Rate heuristic, uses at most a constant times the minimum energy required. The analysis involves bounding the largest eigenvalue in matrices of a special type. A cost aware metric for wireless networks based on remaining battery power at nodes was proposed for shortest-cost routing algorithms, assuming constant transmission power. Power-aware metrics, where transmission power depends on distance between nodes and corresponding shortest power algorithms were also proposed. We define a power-cost metric based on the combination of both node's lifetime and distance-based power metrics. We investigate some properties of power adjusted transmissions and show that, if additional nodes can be placed at desired locations between two nodes at distance d, the transmission power can be made linear in d as opposed to d sup spl alpha dependence for spl alpha spl ges 2. This provides basis for power, cost, and power-cost localized routing algorithms where nodes make routing decisions solely on the basis, of location of their neighbors and destination. The power-aware routing algorithm attempts to minimize the total power needed to route a message between a source and a destination. The cost-aware routing algorithm is aimed at extending the battery's worst-case lifetime at each node. The combined power-cost localized routing algorithm attempts to minimize the total power needed and to avoid nodes with a short battery's remaining lifetime. We prove that the proposed localized power, cost, and power-cost efficient routing algorithms are loop-free and show their efficiency by experiments. Competitive analysis of algorithms.- Self-organizing data structures.- Competitive analysis of paging.- Metrical task systems, the server problem and the work function algorithm.- Distributed paging.- Competitive analysis of distributed algorithms.- On-line packing and covering problems.- On-line load balancing.- On-line scheduling.- On-line searching and navigation.- On-line network routing.- On-line network optimization problems.- Coloring graphs on-line.- On-Line Algorithms in Machine Learning.- Competitive solutions for on-line financial problems.- On the performance of competitive algorithms in practice.- Competitive odds and ends. This article examines two different mechanisms for saving power in battery-operated embedded systems. The first strategy is that the system can be placed in a sleep state if it is idle. However, a fixed amount of energy is required to bring the system back into an active state in which it can resume work. The second way in which power savings can be achieved is by varying the speed at which jobs are run. We utilize a power consumption curve P(s) which indicates the power consumption level given a particular speed. We assume that P(s) is convex, nondecreasing, and nonnegative for s ≥ 0. The problem is to schedule arriving jobs in a way that minimizes total energy use and so that each job is completed after its release time and before its deadline. We assume that all jobs can be preempted and resumed at no cost. Although each problem has been considered separately, this is the first theoretical analysis of systems that can use both mechanisms. We give an offline algorithm that is within a factor of 2 of the optimal algorithm. We also give an online algorithm with a constant competitive ratio.