Collective communications are ubiquitous in parallel applications. We present two new algorithms for performing a reduction. The operation associated with our reduction needs to be associative and commutative. The two algorithms are developed under two different communication models (unidirectional and bidirectional). Both algorithms use a greedy scheduling scheme. For a unidirectional, fully connected network, we prove that our greedy algorithm is optimal when some realistic assumptions are respected. Previous algorithms fit the same assumptions and are only appropriate for some given configurations. Our algorithm is optimal for all configurations. We note that there are some configuration where our greedy algorithm significantly outperform any existing algorithms. This result represents a contribution to the state-of-the art. For a bidirectional, fully connected network, we present a different greedy algorithm. We verify by experimental simulations that our algorithm matches the time complexity of an optimal broadcast (with addition of the computation). Beside reversing an optimal broadcast algorithm, the greedy algorithm is the first known reduction algorithm to experimentally attain this time complexity. Simulations show that this greedy algorithm performs well in practice, outperforming any state-of-the-art reduction algorithms. Positive experiments on a parallel distributed machine are also presented.
Optimizing the reduction operation is closely related to optimization of the broadcast operation. Any broadcast algorithm can be reversed to perform a reduction. Bar- @cite_10 and Tr "a ff and Ripke @cite_1 both provide algorithms that produce an optimal broadcast schedule for a bidirectional system. Here the messages are split into segments and the segments are broadcast in rounds. In both cases the optimality is in the sense that the algorithm meets the lower bound on the number of communication rounds. For the theoretical case when @math , reversing an optimal broadcast will provide an optimal reduction. However, for @math this is no longer valid as an optimal schedule will most like take into account the computation.
@cite_1 @cite_10
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In this paper, we investigate the transmission completion time minimization problem in an additive white Gaussian noise (AWGN) broadcast channel, where the transmitter is able to harvest energy from the nature, using a rechargeable battery. The harvested energy is modeled to arrive at the transmitter during the course of transmissions. The transmitter has a fixed number of packets to be delivered to each receiver. The objective is to minimize the time by which all of the packets are delivered to their respective destinations. To this end, we optimize the transmit powers and transmission rates in a deterministic setting. We first analyze the structural properties of the optimal transmission policy in a two-user broadcast channel via the dual problem of maximizing the departure region by a fixed time T. We prove that the optimal total transmit power sequence has the same structure as the optimal single-user transmit power sequence in . In addition, the total power is split optimally based on a cut-off power level; if the total transmit power is lower than this cut-off level, all transmit power is allocated to the stronger user; otherwise, all transmit power above this level is allocated to the weaker user. We then extend our analysis to an M-user broadcast channel. We show that the optimal total power sequence has the same structure as the two-user case and optimally splitting the total power among M users involves M-1 cut-off power levels. Using this structure, we propose an algorithm that finds the globally optimal policy. Our algorithm is based on reducing the broadcast channel problem to a single-user problem as much as possible. Finally, we illustrate the optimal policy and compare its performance with several suboptimal policies under different settings. We address the minimum-energy broadcast problem under the assumption that nodes beyond the nominal range of a transmitter can collect the energy of unreliably received overheard signals. As a message is forwarded through the network, a node will have multiple opportunities to reliably receive the message by collecting energy during each retransmission. We refer to this cooperative strategy as accumulative broadcast. We seek to employ accumulative broadcast in a large scale loosely synchronized, low-power network. Therefore, we focus on distributed network layer approaches for accumulative broadcast in which loosely synchronized nodes use only local information. To further simplify the system architecture, we assume that nodes forward only reliably decoded messages. Under these assumptions, we formulate the minimum-energy accumulative broadcast problem. We present a solution employing two subproblems. First, we identify the ordering in which nodes should transmit. Second, we determine the optimum power levels for that ordering. While the second subproblem can be solved by means of linear programming, the ordering subproblem is found to be NP-complete. We devise a heuristic algorithm to find a good ordering. Simulation results show the performance of the algorithm to be close to optimum and a significant improvement over the well known BIP algorithm for constructing energy-efficient broadcast trees. We then formulate a distributed version of the accumulative broadcast algorithm that uses only local information at the nodes and has performance close to its centralized counterpart. This paper studies the resource allocation algorithm design for secure information and renewable green energy transfer to mobile receivers in distributed antenna communication systems. In particular, distributed remote radio heads (RRHs antennas) are connected to a central processor (CP) via capacity-limited backhaul links to facilitate joint transmission. The RRHs and the CP are equipped with renewable energy harvesters and share their energies via a lossy micropower grid for improving the efficiency in conveying information and green energy to mobile receivers via radio frequency signals. The considered resource allocation algorithm design is formulated as a mixed nonconvex and combinatorial optimization problem taking into account the limited backhaul capacity and the quality-of-service requirements for simultaneous wireless information and power transfer (SWIPT). We aim at minimizing the total network transmit power when only imperfect channel state information of the wireless energy harvesting receivers, which have to be powered by the wireless network, is available at the CP. In light of the intractability of the problem, we reformulate it as an optimization problem with binary selection, which facilitates the design of an iterative resource allocation algorithm to solve the problem optimally using the generalized Bender's decomposition (GBD). Furthermore, a suboptimal algorithm is proposed to strike a balance between computational complexity and system performance. Simulation results illustrate that the proposed GBD-based algorithm obtains the global optimal solution and the suboptimal algorithm achieves a close-to-optimal performance. In addition, the distributed antenna network for SWIPT with renewable energy sharing is shown to require a lower transmit power compared with a traditional system with multiple colocated antennas. We consider the minimization of the transmission completion time with a battery limited energy harvesting transmitter in an M-user AWGN broadcast channel where the transmitter is able to harvest energy from the nature, using a finite storage capacity rechargeable battery. The harvested energy is modeled to arrive (be harvested) at the transmitter during the course of transmissions at arbitrary time instants. The transmitter has fixed number of packets for each receiver. Due to the finite battery capacity, energy may overflow without being utilized for data transmission. We derive the optimal offline transmission policy that minimizes the time by which all of the data packets are delivered to their respective destinations. We analyze the structural properties of the optimal transmission policy using a dual problem. We find the optimal total transmit power sequence by a directional water-filling algorithm. We prove that there exist M-1 cut-off power levels such that user i is allocated the power between the i-1st and the ith cut-off power levels subject to the availability of the allocated total power level. Based on these properties, we propose an algorithm that gives the globally optimal offline policy. The proposed algorithm uses directional water-filling repetitively. Finally, we illustrate the optimal policy and compare its performance with several suboptimal policies under different settings.