Dynamic Programming for Graphs on Surfaces

We provide a framework for the design and analysis of dynamic programming algorithms for surface-embedded graphs on n vertices and branchwidth at most k. Our technique applies to general families of problems where standard dynamic programming runs in 2^{O(k log k)} n steps. Our approach combines tools from topological graph theory and analytic combinatorics. In particular, we introduce a new type of branch decomposition called "surface cut decomposition", generalizing sphere cut decompositions of planar graphs introduced by Seymour and Thomas, which has nice combinatorial properties. Namely, the number of partial solutions that can be arranged on a surface cut decomposition can be upper-bounded by the number of non-crossing partitions on surfaces with boundary. It follows that partial solutions can be represented by a single-exponential (in the branchwidth k) number of configurations. This proves that, when applied on surface cut decompositions, dynamic programming runs in 2^{O(k)} n steps. That way, we considerably extend the class of problems that can be solved in running times with a single-exponential dependence on branchwidth and unify/improve most previous results in this direction.Comment: 28 pages, 3 figure

Optimal randomized incremental construction for guaranteed logarithmic planar point location

Given a planar map of $n$ segments in which we wish to efficiently locate points, we present the first randomized incremental construction of the well-known trapezoidal-map search-structure that only requires expected $O(n \log n)$ preprocessing time while deterministically guaranteeing worst-case linear storage space and worst-case logarithmic query time. This settles a long standing open problem; the best previously known construction time of such a structure, which is based on a directed acyclic graph, so-called the history DAG, and with the above worst-case space and query-time guarantees, was expected $O(n \log^2 n)$. The result is based on a deeper understanding of the structure of the history DAG, its depth in relation to the length of its longest search path, as well as its correspondence to the trapezoidal search tree. Our results immediately extend to planar maps induced by finite collections of pairwise interior disjoint well-behaved curves.Comment: The article significantly extends the theoretical aspects of the work presented in http://arxiv.org/abs/1205.543

A dedicated algorithm for calculating ground states for the triangular random bond Ising model

In the presented article we present an algorithm for the computation of ground state spin configurations for the 2d random bond Ising model on planar triangular lattice graphs. Therefore, it is explained how the respective ground state problem can be mapped to an auxiliary minimum-weight perfect matching problem, solvable in polynomial time. Consequently, the ground state properties as well as minimum-energy domain wall (MEDW) excitations for very large 2d systems, e.g. lattice graphs with up to N=384x384 spins, can be analyzed very fast. Here, we investigate the critical behavior of the corresponding T=0 ferromagnet to spin-glass transition, signaled by a breakdown of the magnetization, using finite-size scaling analyses of the magnetization and MEDW excitation energy and we contrast our numerical results with previous simulations and presumably exact results.Comment: 5 pages, 5 figure

Improved Implementation of Point Location in General Two-Dimensional Subdivisions

We present a major revamp of the point-location data structure for general two-dimensional subdivisions via randomized incremental construction, implemented in CGAL, the Computational Geometry Algorithms Library. We can now guarantee that the constructed directed acyclic graph G is of linear size and provides logarithmic query time. Via the construction of the Voronoi diagram for a given point set S of size n, this also enables nearest-neighbor queries in guaranteed O(log n) time. Another major innovation is the support of general unbounded subdivisions as well as subdivisions of two-dimensional parametric surfaces such as spheres, tori, cylinders. The implementation is exact, complete, and general, i.e., it can also handle non-linear subdivisions. Like the previous version, the data structure supports modifications of the subdivision, such as insertions and deletions of edges, after the initial preprocessing. A major challenge is to retain the expected O(n log n) preprocessing time while providing the above (deterministic) space and query-time guarantees. We describe an efficient preprocessing algorithm, which explicitly verifies the length L of the longest query path in O(n log n) time. However, instead of using L, our implementation is based on the depth D of G. Although we prove that the worst case ratio of D and L is Theta(n/log n), we conjecture, based on our experimental results, that this solution achieves expected O(n log n) preprocessing time.Comment: 21 page

Optimal Online Edge Coloring of Planar Graphs with Advice

Using the framework of advice complexity, we study the amount of knowledge about the future that an online algorithm needs to color the edges of a graph optimally, i.e., using as few colors as possible. For graphs of maximum degree $\Delta$, it follows from Vizing's Theorem that $O(m\log \Delta)$ bits of advice suffice to achieve optimality, where $m$ is the number of edges. We show that for graphs of bounded degeneracy (a class of graphs including e.g. trees and planar graphs), only $O(m)$ bits of advice are needed to compute an optimal solution online, independently of how large $\Delta$ is. On the other hand, we show that $\Omega (m)$ bits of advice are necessary just to achieve a competitive ratio better than that of the best deterministic online algorithm without advice. Furthermore, we consider algorithms which use a fixed number of advice bits per edge (our algorithm for graphs of bounded degeneracy belongs to this class of algorithms). We show that for bipartite graphs, any such algorithm must use at least $\Omega(m\log \Delta)$ bits of advice to achieve optimality.Comment: CIAC 201