Solving Laplacian Systems in Logarithmic Space
نویسنده
چکیده
We investigate the space complexity of solving linear systems of equations. While all known deterministic or randomized algorithms solving a square system of n linear equations in n variables require Ω(log 2 n) space, Ta-Shma (STOC 2013) recently showed that on a quantum computer an approximate solution can be computed in logarithmic space, giving the first explicit computational task for which quantum computation seems to outperform classical computation with respect to space complexity. In this paper we show that for systems of linear equations in the Laplacian matrix of graphs, the same logarithmic space complexity can actually be achieved by a classical (i.e., non-quantum) algorithm. More precisely, given a system of linear equations Lx = b, where L is the (normalized) Laplacian matrix of a graph on n vertices and b is a unit-norm vector, our algorithm outputs a vector˜x such that˜x − x ≤ 1/poly(n) and uses only O(log n) space if the underlying graph has polynomially bounded weights. We also show how to estimate, again in logarithmic space, the smallest non-zero eigenvalue of L.
منابع مشابه
CAS WAVELET METHOD FOR THE NUMERICAL SOLUTION OF BOUNDARY INTEGRAL EQUATIONS WITH LOGARITHMIC SINGULAR KERNELS
In this paper, we present a computational method for solving boundary integral equations with loga-rithmic singular kernels which occur as reformulations of a boundary value problem for the Laplacian equation. Themethod is based on the use of the Galerkin method with CAS wavelets constructed on the unit interval as basis.This approach utilizes the non-uniform Gauss-Legendre quadrature rule for ...
متن کاملProbabilistic Logarithmic-Space Algorithms for Laplacian Solvers
A recent series of breakthroughs initiated by Spielman and Teng culminated in the construction of nearly linear time Laplacian solvers, approximating the solution of a linear system Lx = b, where L is the normalized Laplacian of an undirected graph. In this paper we study the space complexity of the problem. Surprisingly we are able to show a probabilistic, logspace algorithm solving the proble...
متن کاملMultiplicity of Positive Solutions of laplacian systems with sign-changing weight functions
In this paper, we study the multiplicity of positive solutions for the Laplacian systems with sign-changing weight functions. Using the decomposition of the Nehari manifold, we prove that an elliptic system has at least two positive solutions.
متن کاملThe fibering map approach to a quasilinear degenerate p(x)-Laplacian equation
By considering a degenerate $p(x)-$Laplacian equation, a generalized compact embedding in weighted variable exponent Sobolev space is presented. Multiplicity of positive solutions are discussed by applying fibering map approach for the corresponding Nehari manifold.
متن کاملFault Detection and Isolation of Multi-Agent Systems via Complex Laplacian
This paper studies the problem of fault detection and isolation (FDI) for multi-agent systems (MAS) via complex Laplacian subject to actuator faults. A planar formation of point agents in the plane using simple and linear interaction rules related to complex Laplacian is achieved. The communication network is a directed, and yet connected graph with a fixed topology. The loss of symmetry in the...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
برای دانلود متن کامل این مقاله و بیش از 32 میلیون مقاله دیگر ابتدا ثبت نام کنید
ثبت ناماگر عضو سایت هستید لطفا وارد حساب کاربری خود شوید
ورودعنوان ژورنال:
- CoRR
دوره abs/1608.01426 شماره
صفحات -
تاریخ انتشار 2016