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Efficient Computation of the Characteristic Polynomial of a Tree and Related Tasks

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Abstract

An O(nlog2 n) algorithm is presented to compute all coefficients of the characteristic polynomial of a tree on n vertices improving on the previously best quadratic time. With the same running time, the algorithm can be generalized in two directions. The algorithm is a counting algorithm for matchings, and the same ideas can be used to count other objects. For example, one can count the number of independent sets of all possible sizes simultaneously with the same running time. These counting algorithms not only work for trees, but can be extended to arbitrary graphs of bounded tree-width.

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Authors and Affiliations

  1. Department of Computer Science and Engineering, Pennsylvania State University, University Park, PA, 16802, USA

    Martin Fürer

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  1. Martin Fürer
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Correspondence to Martin Fürer.

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Research supported in part by NSF Grants CCF-0728921 and CCF-0964655. Part of this research has been done while visiting the Laboratory of Algorithms (ALGO) at EPFL Lausanne and the Institute for Mathematics at the University of Zürich.

A preliminary version of this paper appeared in LNCS 5757, Proc. 17th ESA 2009, pp. 11–22.

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Fürer, M. Efficient Computation of the Characteristic Polynomial of a Tree and Related Tasks. Algorithmica 68, 626–642 (2014). https://doi.org/10.1007/s00453-012-9688-5

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  • DOI: https://doi.org/10.1007/s00453-012-9688-5

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