Graph traversals associated with iterative methods for solving systems of linear equations

TL;DR

This paper explores graph traversal algorithms linked to iterative methods for solving linear systems, highlighting their differences from traditional searches and their efficiency in visiting graph vertices.

Contribution

It introduces a class of traversal algorithms associated with iterative linear solvers, such as Gauss-Seidel, and analyzes their iteration complexity compared to BFS.

Findings

Traversal algorithms can be more efficient than BFS for large problem instances.

These algorithms are not equivalent to standard graph searches like DFS or BFS.

Fewer iterations are often needed for large, complex graphs.

Abstract

To solve many problems on graphs, graph traversals are used, the usual variants of which are the depth-first search and the breadth-first search. Implementing a graph traversal we consequently reach all vertices of the graph that belong to a connected component. The breadth-first search is the usual choice when constructing efficient algorithms for finding connected components of a graph. Methods of simple iteration for solving systems of linear equations with modified graph adjacency matrices and with the properly specified right-hand side can be considered as graph traversal algorithms. These traversal algorithms, generally speaking, turn out to be non-equivalent neither to the depth-first search nor the breadth-first search. The example of such a traversal algorithm is the one associated with the Gauss-Seidel method. For an arbitrary connected graph, to visit all its vertices, the algorithm requires not more iterations than that is required for BFS. For a large number of instances of the problem, fewer iterations will be required.