Marcel Roeloffzen

Hanabi is NP-complete, Even for Cheaters who Look at Their Cards

This paper studies a cooperative card game called Hanabi from an algorithmic combinatorial game theory viewpoint. The aim of the game is to play cards from 1 to n in increasing order (this has to be done independently in c different colors). Cards are drawn from a deck one by one. Drawn cards are either immediately played, discarded or stored for future use (overall each player can store up to h cards). The main feature of the game is that players know the cards their partners hold (but not theirs. This information must be shared through hints). We introduce a simplified mathematical model of a single-player version of the game, and show several complexity results: the game is intractable in a general setting even if we forego with the hidden information aspect of the game. On the positive side, the game can be solved in linear time for some interesting restricted cases (i.e., for small values of h and c).

Improved Time-Space Trade-Offs for Computing Voronoi Diagrams

Let P be a planar n-point set in general position. For k between 1 and n-1, the Voronoi diagram of order k is obtained by subdividing the plane into regions such that points in the same cell have the same set of nearest k neighbors in P. The (nearest point) Voronoi diagram (NVD) and the farthest point Voronoi diagram (FVD) are the particular cases of k=1 and k=n-1, respectively. It is known that the family of all higher-order Voronoi diagrams of order 1 to K for P can be computed in total time O(n K^2 + n log n) using O(K^2(n-K)) space. Also NVD and FVD can be computed in O(n log n) time using O(n) space. For s in {1, ..., n}, an s-workspace algorithm has random access to a read-only array with the sites of P in arbitrary order. Additionally, the algorithm may use O(s) words of Theta(log n) bits each for reading and writing intermediate data. The output can be written only once and cannot be accessed afterwards. We describe a deterministic s-workspace algorithm for computing an NVD and also an FVD for P that runs in O((n^2/s) log s) time. Moreover, we generalize our s-workspace algorithm for computing the family of all higher-order Voronoi diagrams of P up to order K in O(sqrt(s)) in total time O( (n^2 K^6 / s) log^(1+epsilon)(K) (log s / log K)^(O(1)) ) for any fixed epsilon > 0. Previously, for Voronoi diagrams, the only known s-workspace algorithm was to find an NVD for P in expected time O((n^2/s) log s + n log s log^*s). Unlike the previous algorithm, our new method is very simple and does not rely on advanced data structures or random sampling techniques.

Time-Space Trade-offs for Triangulating a Simple Polygon

An s-workspace algorithm is an algorithm that has read-only access to the values of the input, write-only access to the output, and only uses O(s) additional words of space. We give a randomized s-workspace algorithm for triangulating a simple polygon P of n vertices, for any s up to n. The algorithm runs in O(n^2/s+n(log s)log^5(n/s)) expected time using O(s) variables, for any s up to n. In particular, the algorithm runs in O(n^2/s) expected time for most values of s.

Line segment covering of cells in arrangements

Abstract Given a collection L of line segments, we consider its arrangement and study the problem of covering all cells with line segments of L . That is, we want to find a minimum-size set L ′ of line segments such that every cell in the arrangement has a line from L ′ defining its boundary. We show that the problem is NP-hard, even when all segments are axis-aligned. In fact, the problem is still NP-hard when we only need to cover rectangular cells of the arrangement. For the latter problem we also show that it is fixed parameter tractable with respect to the size of the optimal solution. Finally we provide a linear time algorithm for the case where cells of the arrangement are created by recursively subdividing a rectangle using horizontal and vertical cutting segments.

Time–space trade-offs for triangulations and Voronoi diagrams

Let

Time-Space Trade-offs for Triangulations and Voronoi Diagrams

S

Dynamic Graph Coloring

be a planar

Balanced Line Separators of Unit Disk Graphs

n

Routing in Polygonal Domains.

-point set. A triangulation for

Time-space trade-offs for triangulating a simple polygon

S