Given a directed multigraph G= (V, E), with | V| = n nodes and | E| = m edges, and an integer z, we are asked to assess whether the number # ET(G) of node-distinct Eulerian trails of G is at least z; two trails are called node-distinct if their node sequences are different. This problem has been formalized by Bernardini et al.&nbsp;[ALENEX 2020] as it is the core computational problem in several string processing applications. It can be solved in O(nω) arithmetic operations by applying the well-known BEST theorem, where ω&lt; 2.373 denotes the matrix multiplication exponent. The algorithmic challenge is: Can we solve this problem faster for certain values of m and z? Namely, we want to design a combinatorial algorithm for assessing whether # ET(G) ≥ z, which does not resort to the BEST theorem and has a predictably bounded cost as a function of m and z. We address this challenge here by providing a combinatorial algorithm requiring O(m· min { z, # ET(G) }) time.

### Beyond the BEST Theorem: Fast Assessment of Eulerian Trails

#### Abstract

Given a directed multigraph G= (V, E), with | V| = n nodes and | E| = m edges, and an integer z, we are asked to assess whether the number # ET(G) of node-distinct Eulerian trails of G is at least z; two trails are called node-distinct if their node sequences are different. This problem has been formalized by Bernardini et al. [ALENEX 2020] as it is the core computational problem in several string processing applications. It can be solved in O(nω) arithmetic operations by applying the well-known BEST theorem, where ω< 2.373 denotes the matrix multiplication exponent. The algorithmic challenge is: Can we solve this problem faster for certain values of m and z? Namely, we want to design a combinatorial algorithm for assessing whether # ET(G) ≥ z, which does not resort to the BEST theorem and has a predictably bounded cost as a function of m and z. We address this challenge here by providing a combinatorial algorithm requiring O(m· min { z, # ET(G) }) time.
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2021
978-3-030-86592-4
978-3-030-86593-1
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Utilizza questo identificativo per citare o creare un link a questo documento: `https://hdl.handle.net/11568/1108265`
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