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Engineering Data Reduction for Nested Dissection

Lara Ost, Christian Schulz, Darren Strash

TL;DR

The paper tackles computing low-fill, high-quality node orderings for sparse linear systems by introducing a practical data-reduction framework that precedes nested dissection. It combines four exact and two inexact reduction rules—including simplicial, indistinguishable, twin, path compression, degree-2 elimination, and triangle contraction—to shrink the input graph to a smaller, equivalent instance before ordering, then lifts the result back to the original graph. This approach improves both the quality of the elimination orders and the running time, outperforming state-of-the-art methods and achieving substantial speedups in road-network benchmarks. By integrating theory from minimum triangulation with engineering-focused reductions, the method provides a scalable, effective preprocessing step for order generation in large sparse systems.

Abstract

Many applications rely on time-intensive matrix operations, such as factorization, which can be sped up significantly for large sparse matrices by interpreting the matrix as a sparse graph and computing a node ordering that minimizes the so-called fill-in. In this paper, we engineer new data reduction rules for the minimum fill-in problem, which significantly reduce the size of the graph while producing an equivalent (or near-equivalent) instance. By applying both new and existing data reduction rules exhaustively before nested dissection, we obtain improved quality and at the same time large improvements in running time on a variety of instances. Our overall algorithm outperforms the state-of-the-art significantly: it not only yields better elimination orders, but it does so significantly faster than previously possible. For example, on road networks, where nested dissection algorithms are typically used as a preprocessing step for shortest path computations, our algorithms are on average six times faster than Metis while computing orderings with less fill-in.

Engineering Data Reduction for Nested Dissection

TL;DR

The paper tackles computing low-fill, high-quality node orderings for sparse linear systems by introducing a practical data-reduction framework that precedes nested dissection. It combines four exact and two inexact reduction rules—including simplicial, indistinguishable, twin, path compression, degree-2 elimination, and triangle contraction—to shrink the input graph to a smaller, equivalent instance before ordering, then lifts the result back to the original graph. This approach improves both the quality of the elimination orders and the running time, outperforming state-of-the-art methods and achieving substantial speedups in road-network benchmarks. By integrating theory from minimum triangulation with engineering-focused reductions, the method provides a scalable, effective preprocessing step for order generation in large sparse systems.

Abstract

Many applications rely on time-intensive matrix operations, such as factorization, which can be sped up significantly for large sparse matrices by interpreting the matrix as a sparse graph and computing a node ordering that minimizes the so-called fill-in. In this paper, we engineer new data reduction rules for the minimum fill-in problem, which significantly reduce the size of the graph while producing an equivalent (or near-equivalent) instance. By applying both new and existing data reduction rules exhaustively before nested dissection, we obtain improved quality and at the same time large improvements in running time on a variety of instances. Our overall algorithm outperforms the state-of-the-art significantly: it not only yields better elimination orders, but it does so significantly faster than previously possible. For example, on road networks, where nested dissection algorithms are typically used as a preprocessing step for shortest path computations, our algorithms are on average six times faster than Metis while computing orderings with less fill-in.

Paper Structure

This paper contains 9 sections, 9 theorems, 1 equation, 2 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Related Work
  4. Advanced Node Ordering
  5. Data Reduction Rules
  6. The Simplicial Node Reduction
  7. The Indistinguishable Node Reduction
  8. The Twin Reduction
  9. Path Compression

Key Result

Theorem 1

Let $G = (V, E)$ be a graph with a simplicial node $x$. The ordering $x \Sigma(G_x)$ is a minimum fill-in ordering of $G$.

Figures (2)

  • Figure 1: Examples for simplicial nodes, indistinguishable nodes and twins. The neighborhood of $s$ is a clique, so $s$ is simplicial. Nodes $i_1$ and $i_2$ are indistinguishable, since they are neighbors and adjacent to all unlabeled nodes, i.e., $\Gamma[i_1] = \Gamma[i_2]$. Nodes $t_1$ and $t_2$ are twins, since they are both adjacent to all unlabeled nodes, but not to each other. $\Gamma(t_1) = \Gamma(t_2)$.
  • Figure 2: Examples for the three cases in the proof of Theorem \ref{['thm:pathcompression']}. Red nodes are nodes in $P$, black nodes are in $\Gamma(P)$. Dashed edges lead to some other nodes in the graph.

Theorems & Definitions (9)

  • Theorem 1
  • Lemma 2
  • Theorem 3
  • Theorem 4
  • Lemma 5: Corollary 1 from rose72
  • Lemma 6: Proposition 2 from rose72
  • Lemma 7
  • Lemma 8
  • Theorem 9