We present a new iterative scheme for PageRank computation. The algorithm is applied to the linear system formulation of the problem, using inner-outer stationary iterations. It is simple, can be easily implemented and parallelized, and requires minimal storage overhead. Our convergence analysis shows that the algorithm is effective for a crude inner tolerance and is not sensitive to the choice of the parameters involved. The same idea can be used as a preconditioning technique for nonstationary schemes. Numerical examples featuring matrices of dimensions exceeding 100,000,000 in sequential and parallel environments demonstrate the merits of our technique. Our code is available online for viewing and testing, along with several large scale examples.

Abstruct-We consider the problem of power controlled minimum frame length scheduling for TbMA wireless networks, Given a set of one-hop transmission requests, our objective is to schedule them in a minimum number of time slots, so that each slot schedule is free of' self-interferences and meets desired SINR constraints. Additionally, the transmit power vector corresponding to each slot schedule should be minimal. We consider two different versions of the problem, a per-slot version and a per-frame version, and develop mixed integer linear programming models which can be used for solving the problems optimally. In addition, vie propose a heuristic algorithm for the per-slot version. I. INTRODUCTlONIn this paper, we consider the problem of power controlled adaptive frame length scheduling in TDMA wireless networks. To the best of our knowledge, the issue of joint scheduling and power control was first addressed by Tamer and Ephremides in 11, 21. Given a set of one-hop transmission requests constituting a request list, they suggest a two-phase algorithm which essentially decouples the scheduling and power control objectives. The scheduling objective, which is used to remove "selfinterferences" from the request list, is achieved by executing a centralized algorithm at the scheduler. After successful execution of this phase, the authors show that the power control problem in TDMA or hybrid TDMMCDMA ad hoc networks is similar to the power control problem in cellular systems. Consequently. algorithms developed for the latter can be used for ad hoc networks.In [ l , 21, the authors assume that the frame length comprises a fixed number of time slots, which is determined heuristically. Our work! on the other hand, focusses on the adaptive frame length case wherein, given a request list, the objective is to schedule them in a minimum number of time slots, so that each slot schedule is free of self-interferences and meets desired SINR constraints. Additionally, the transmit power vector corresponding to each slot schedule should be minimal, Ideally. the optimization should be carried out on a per-frame basis. However, this approach requires an excessive number of variables prohibiting optimal offline analysis even for moderately sized request lists. Consequently, for the most part, we focus on a (suboptimal) per-slot optimization approach which is much more lractable than a per-frame approach. Another aspect of our work. which differs from [ 1-21, is our consideration of sectored anlennas as opposed to omnidirectional antennas.The minimum frame length scheduling objective in our work has also been researched in the context of link scheduling in spatial-TDMA (STDMA) networks 13. 43. The work in 131 provides a comparison of graph-based versus SINR based link scheduling policies while [41 proposes optimal link and node scheduling algorithms using mixed integer linear programming (MILP) techniques. However, transmitter power is assumed to be fixed in these papers. Our work can therefore he seen as an extension of the link schedu...