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We present the rst sublinear-time algorithms for computing order statistics in the Farey sequence and for the related problem of ranking. Our algorithms achieve a running times of nearly O(n 2/3 ), which is a signicant improvement over the previous algorithms taking time O(n).We also initiate the study of a more general problem: counting primitive lattice points inside planar shapes. For rational polygons containing the origin, we obtain a running time proportional to D 6/7 , where D is the diameter of the polygon.

Abstract. Various efficient game problem solvers are based on PNSearch. Especially depth-first versions of PN-Search like DF-PN or PDS -contrary to other known techniques -are able to solve really hard problems. However, the performance of DF-PN and PDS decreases dramatically when the search space significantly exceeds available memory. A simple trick to overcome this problem is presented. Experiments on Atari Go and Lines of Action show great practical value of the proposed enhancement.

In the Bin Packing problem one is given n items with weights w 1 , . . . , w n and m bins with capacities c 1 , . . . , c m . The goal is to find a partition of the items into sets S 1 , . . . , S m such that w(S j ) c j for every bin j, where w(X) denotes i∈X w i .Björklund, Husfeldt and Koivisto (SICOMP 2009) presented an O (2 n ) time algorithm for Bin Packing. In this paper, we show that for every m ∈ N there exists a constant σ m > 0 such that an instance of Bin Packing with m bins can be solved in O(2 (1−σ m )n ) randomized time. Before our work, such improved algorithms were not known even for m equals 4.A key step in our approach is the following new result in Littlewood-Offord theory on the additive combinatorics of subset sums: For every δ > 0 there exists an ε > 0 such that if |{X ⊆ {1, . . . , n} : w(X) = v}| 2 (1−ε)n for some v then |{w(X) : X ⊆ {1, . . . , n}}| 2 δn .

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 problems. 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 opening 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.