Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Entrywise Approximate Laplacian Solving

Jingbang Chen, Mehrdad Ghadiri, Hoai-An Nguyen, Richard Peng, Junzhao Yang

TL;DR

Algorithm and analyses for weighted directed graphs under floating-point arithmetic are presented for weighted directed graphs under floating-point arithmetic and the previous best running times are improved in terms of the number of bit operations.

Abstract

We study the escape probability problem in random walks over graphs. Given vertices, $s,t,$ and $p$, the problem asks for the probability that a random walk starting at $s$ will hit $t$ before hitting $p$. Such probabilities can be exponentially small even for unweighted undirected graphs with polynomial mixing time. Therefore current approaches, which are mostly based on fixed-point arithmetic, require $n$ bits of precision in the worst case. We present algorithms and analyses for weighted directed graphs under floating-point arithmetic and improve the previous best running times in terms of the number of bit operations. We believe our techniques and analysis could have a broader impact on the computation of random walks on graphs both in theory and in practice.

Entrywise Approximate Laplacian Solving

TL;DR

Algorithm and analyses for weighted directed graphs under floating-point arithmetic are presented for weighted directed graphs under floating-point arithmetic and the previous best running times are improved in terms of the number of bit operations.

Abstract

We study the escape probability problem in random walks over graphs. Given vertices, and , the problem asks for the probability that a random walk starting at will hit before hitting . Such probabilities can be exponentially small even for unweighted undirected graphs with polynomial mixing time. Therefore current approaches, which are mostly based on fixed-point arithmetic, require bits of precision in the worst case. We present algorithms and analyses for weighted directed graphs under floating-point arithmetic and improve the previous best running times in terms of the number of bit operations. We believe our techniques and analysis could have a broader impact on the computation of random walks on graphs both in theory and in practice.
Paper Structure (16 sections, 14 theorems, 65 equations, 2 figures)

This paper contains 16 sections, 14 theorems, 65 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Discussion and Related Work
  3. Near-linear-time Laplacian solvers.
  4. Fast-matrix-multiplication.
  5. Shifted and $p$-adic numbers.
  6. Gaussian elimination.
  7. Bit complexity.
  8. Preliminaries
  9. Schur complement.
  10. Floating-point numbers.
  11. Matrix Associated with Escape Probability
  12. Schur Complement View
  13. Repeated Squaring
  14. Floating Point Edge Weights
  15. Case 3:
  16. ...and 1 more sections

Key Result

Theorem 1.1

Given a weighted directed graph $G=(V,E)$ with $n$ vertices and nonnegative edge weights that are given with $L$ bits in floating-point, and $t,p\in V$, there is an algorithm that computes the escape probability for all starting vertices $s\in V$ within an $e^{\epsilon}$ multiplicative factor with $

Figures (2)

  • Figure 1: Exponentially small escape probability.
  • Figure 2: Psuedocode for recursive inversion with varying precision

Theorems & Definitions (29)

  • Theorem 1.1
  • Theorem 1.2
  • Example 1.1
  • Example 1.2
  • Definition 2.1: Escape Probability
  • Definition 2.2
  • Definition 2.3: RDDL
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • ...and 19 more