🤖 AI Summary
This paper addresses the fundamental connection between solving linear systems and graph traversal. Method: We propose and formalize a novel graph traversal paradigm grounded in iterative linear solvers—specifically Gauss–Seidel—and systematically construct modified adjacency matrices and tailored right-hand-side vectors to map iterative updates onto vertex visitation sequences. Contribution/Results: We prove that this paradigm is neither depth-first nor breadth-first search, yet guarantees deterministic, complete coverage of any connected graph; its iteration count is strictly bounded above by BFS depth and empirically subsumes BFS on most graph instances. This work establishes the first rigorous equivalence framework linking linear iterative methods with graph traversal, revealing that Gauss–Seidel and related solvers inherently implement efficient, non-standard graph traversal algorithms. The result extends both the theoretical foundations and practical applicability of graph algorithm design.
📝 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.