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- Björklund, Andreas, et al.
(author)
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Approximate counting of K-paths : Deterministic and in polynomial space
- 2019
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In: 46th International Colloquium on Automata, Languages, and Programming, ICALP 2019. - 9783959771092 ; 132
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Conference paper (peer-reviewed)abstract
- A few years ago, Alon et al. [ISMB 2008] gave a simple randomized O((2e)km∊−2)-time exponential-space algorithm to approximately compute the number of paths on k vertices in a graph G up to a multiplicative error of 1 ± ∊. Shortly afterwards, Alon and Gutner [IWPEC 2009, TALG 2010] gave a deterministic exponential-space algorithm with running time (2e)k+O(log3 k)m log n whenever ∊−1 = kO(1). Recently, Brand et al. [STOC 2018] provided a speed-up at the cost of reintroducing randomization. Specifically, they gave a randomized O(4km∊−2)-time exponential-space algorithm. In this article, we revisit the algorithm by Alon and Gutner. We modify the foundation of their work, and with a novel twist, obtain the following results. We present a deterministic 4k+O(√k(log2 k+log2 ∊−1))m log n-time polynomial-space algorithm. This matches the running time of the best known deterministic polynomial-space algorithm for deciding whether a given graph G has a path on k vertices. Additionally, we present a randomized 4k+O(log k(log k+log ∊−1))m log n-time polynomial-space algorithm. While Brand et al. make non-trivial use of exterior algebra, our algorithm is very simple; we only make elementary use of the probabilistic method. Thus, the algorithm by Brand et al. runs in time 4k+o(k)m whenever ∊−1 = 2o(k), while our deterministic and randomized algorithms run in time 4k+o(k)m log n whenever ∊−1 = 2o(k 4 ) and 1 ∊−1 = 2o(log k k ), respectively. Prior to our work, no 2O(k)nO(1)-time polynomial-space algorithm was known. Additionally, our approach is embeddable in the classic framework of divide-and-color, hence it immediately extends to approximate counting of graphs of bounded treewidth; in comparison, Brand et al. note that their approach is limited to graphs of bounded pathwidth.
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| 2. |
- Lokshtanov, Daniel, et al.
(author)
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Approximate Counting of k-Paths : Simpler, Deterministic, and in Polynomial Space
- 2021
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In: ACM Transactions on Algorithms. - : Association for Computing Machinery (ACM). - 1549-6325 .- 1549-6333. ; 17:3
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Journal article (peer-reviewed)abstract
- Recently, Brand et al. [STOC 2018] gave a randomized mathcal O(4kmϵ-2-time exponential-space algorithm to approximately compute the number of paths on k vertices in a graph G up to a multiplicative error of 1 ± ϵ based on exterior algebra. Prior to our work, this has been the state-of-the-art. In this article, we revisit the algorithm by Alon and Gutner [IWPEC 2009, TALG 2010], and obtain the following results: •We present a deterministic 4k+ O(sk(log k+log2ϵ-1))m-time polynomial-space algorithm. This matches the running time of the best known deterministic polynomial-space algorithm for deciding whether a given graph G has a path on k vertices. •Additionally, we present a randomized 4k+mathcal O(logk(logk+logϵ-1))m-time polynomial-space algorithm. Our algorithm is simple - we only make elementary use of the probabilistic method. Here, n and m are the number of vertices and the number of edges, respectively. Additionally, our approach extends to approximate counting of other patterns of small size (such as q-dimensional p-matchings).
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