Instance-Optimality in PageRank Computation
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866911684963074048 |
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| author | Thorup, Mikkel Wang, Hanzhi |
| author_facet | Thorup, Mikkel Wang, Hanzhi |
| contents | We study the problem of estimating a vertex's PageRank within a constant relative error, with constant probability. We prove that an adaptive variant of the simple classic bidirectional algorithm is instance-optimal up to a polylogarithmic factor for all directed graphs of order $n$ whose maximum in- and out-degrees are at most a constant fraction of $n$. In other words, there is no correct algorithm that can be faster than our algorithm on any such graph by more than a polylogarithmic factor. We further extend the instance-optimality to all graphs in which at most a polylogarithmic number of vertices have unbounded degrees. This covers all sparse graphs with $\tilde{O}(n)$ edges. In addition, we provide a counterexample showing that the bidirectional algorithm is not instance-optimal for graphs whose degrees are mostly equal to $n$. We also consider weighted graphs and multigraphs. We show that the bidirectional algorithm is instance-optimal on \emph{all} multigraphs, but for weighted simple graphs, we have almost the same limitations as for unweighted simple graphs. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_16087 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Instance-Optimality in PageRank Computation Thorup, Mikkel Wang, Hanzhi Data Structures and Algorithms We study the problem of estimating a vertex's PageRank within a constant relative error, with constant probability. We prove that an adaptive variant of the simple classic bidirectional algorithm is instance-optimal up to a polylogarithmic factor for all directed graphs of order $n$ whose maximum in- and out-degrees are at most a constant fraction of $n$. In other words, there is no correct algorithm that can be faster than our algorithm on any such graph by more than a polylogarithmic factor. We further extend the instance-optimality to all graphs in which at most a polylogarithmic number of vertices have unbounded degrees. This covers all sparse graphs with $\tilde{O}(n)$ edges. In addition, we provide a counterexample showing that the bidirectional algorithm is not instance-optimal for graphs whose degrees are mostly equal to $n$. We also consider weighted graphs and multigraphs. We show that the bidirectional algorithm is instance-optimal on \emph{all} multigraphs, but for weighted simple graphs, we have almost the same limitations as for unweighted simple graphs. |
| title | Instance-Optimality in PageRank Computation |
| topic | Data Structures and Algorithms |
| url | https://arxiv.org/abs/2512.16087 |