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- W3097720716 abstract "Over the last two decades, a significant line of work in theoretical algorithms has been progress in solving linear systems of the form $mathbf{L}mathbf{p} = mathbf{b}$, where $mathbf{L}$ is the Laplacian matrix of a weighted graph with weights $w(i,j)>0$ on the edges. The solution $mathbf{p}$ of the linear system can be interpreted as the potentials of an electrical flow. Kelner, Orrechia, Sidford, and Zhu cite{KOSZ13} give a combinatorial, near-linear time algorithm that maintains the Kirchoff Current Law, and gradually enforces the Kirchoff Potential Law. Here we consider a dual version of the algorithm that maintains the Kirchoff Potential Law, and gradually enforces the Kirchoff Current Law. We prove that this dual algorithm also runs in a near-linear number of iterations. Each iteration requires updating all potentials on one side of a fundamental cut of a spanning tree by a fixed amount. If this update step can be performed in polylogarithmic time, we can also obtain a near-linear time algorithm to solve $mathbf{L}mathbf{p} = mathbf{b}$. However, if we abstract this update step as a natural data structure problem, we show that we can use the data structure to solve a problem that has been conjectured to be difficult for dynamic algorithms, the online vector-matrix-vector problem cite{HKNS15}. The conjecture implies that the data structure does not have an $O(n^{1-epsilon})$ time algorithm for any $epsilon > 0$. Thus our dual algorithm cannot be near-linear time algorithm for solving $mathbf{L}mathbf{p} = mathbf{b}$ unless we are able to take advantage of the structure of the particular update steps that our algorithm uses." @default.
- W3097720716 created "2020-11-09" @default.
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- W3097720716 date "2020-10-30" @default.
- W3097720716 modified "2023-10-07" @default.
- W3097720716 title "A Combinatorial Cut-Based Algorithm for Solving Laplacian Linear Systems." @default.
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