A Reverse Hex Solver

We present Solrex,an automated solver for the game of Reverse Hex.Reverse Hex, also known as Rex, or Misere Hex, is the variant of the game of Hex in which the player who joins her two sides loses the game. Solrex performs a mini-max search of the state space using Scalable Parallel Depth First Proof Number Search, enhanced by the pruning of inferior moves and the early detection of certain winning strategies. Solrex is implemented on the same code base as the Hex program Solver, and can solve arbitrary positions on board sizes up to 6x6, with the hardest position taking less than four hours on four threads.Comment: Presented at Computers and Games 2016 Leiden, International Conference on Computers and Games. Springer International Publishing, 201

Dynamic Data Structures for Parameterized String Problems

We revisit classic string problems considered in the area of parameterized complexity, and study them through the lens of dynamic data structures. That is, instead of asking for a static algorithm that solves the given instance efficiently, our goal is to design a data structure that efficiently maintains a solution, or reports a lack thereof, upon updates in the instance. We first consider the Closest String problem, for which we design randomized dynamic data structures with amortized update times $d^{\mathcal{O}(d)}$ and $|\Sigma|^{\mathcal{O}(d)}$, respectively, where $\Sigma$ is the alphabet and $d$ is the assumed bound on the maximum distance. These are obtained by combining known static approaches to Closest String with color-coding. Next, we note that from a result of Frandsen et al.~[J. ACM'97] one can easily infer a meta-theorem that provides dynamic data structures for parameterized string problems with worst-case update time of the form $\mathcal{O}(\log \log n)$, where $k$ is the parameter in question and $n$ is the length of the string. We showcase the utility of this meta-theorem by giving such data structures for problems Disjoint Factors and Edit Distance. We also give explicit data structures for these problems, with worst-case update times $\mathcal{O}(k2^{k}\log \log n)$ and $\mathcal{O}(k^2\log \log n)$, respectively. Finally, we discuss how a lower bound methodology introduced by Amarilli et al.~[ICALP'21] can be used to show that obtaining update time $\mathcal{O}(f(k))$ for Disjoint Factors and Edit Distance is unlikely already for a constant value of the parameter $k$.Comment: 28 page

Scalable Parallel DFPN Search

Abstract. We present Scalable Parallel Depth-First Proof Number Search, a new shared-memory parallel version of depth-first proof number search. Based on the serial DFPN 1+ε method of Pawlewicz and Lew, SPDFPN searches effectively even as the transposition table becomes almost full, and so can solve large prob-lems. To assign jobs to threads, SPDFPN uses proof and disproof numbers and two parameters. SPDFPN uses no domain-specific knowledge or heuristics, so it can be used in any domain. Our experiments show that SPDFPN scales well and performs well on hard problems. We tested SPDFPN on problems from the game of Hex. On a 24-core machine and a 4.2-hour single-thread task, parallel efficiency ranges from 0.8 on 4 threads to 0.74 on 16 threads. SPDFPN solved all previously intractable 9×9 Hex open-ing moves; the hardest opening took 111 days. Also, in 63 days, it solved one 10×10 Hex opening move. This is the first time a computer or human has solved a 10×10 Hex opening move.