Fun with Fonts: Algorithmic Typography

Over the past decade, we have designed six typefaces based on mathematical theorems and open problems, specifically computational geometry. These typefaces expose the general public in a unique way to intriguing results and hard problems in hinged dissections, geometric tours, origami design, computer-aided glass design, physical simulation, and protein folding. In particular, most of these typefaces include puzzle fonts, where reading the intended message requires solving a series of puzzles which illustrate the challenge of the underlying algorithmic problem.Comment: 14 pages, 12 figures. Revised paper with new glass cane font. Original version in Proceedings of the 7th International Conference on Fun with Algorithm

Bidimensionality of Geometric Intersection Graphs

Let B be a finite collection of geometric (not necessarily convex) bodies in the plane. Clearly, this class of geometric objects naturally generalizes the class of disks, lines, ellipsoids, and even convex polygons. We consider geometric intersection graphs GB where each body of the collection B is represented by a vertex, and two vertices of GB are adjacent if the intersection of the corresponding bodies is non-empty. For such graph classes and under natural restrictions on their maximum degree or subgraph exclusion, we prove that the relation between their treewidth and the maximum size of a grid minor is linear. These combinatorial results vastly extend the applicability of all the meta-algorithmic results of the bidimensionality theory to geometrically defined graph classes

A Pseudopolynomial Algorithm for Alexandrov's Theorem

Alexandrov's Theorem states that every metric with the global topology and local geometry required of a convex polyhedron is in fact the intrinsic metric of a unique convex polyhedron. Recent work by Bobenko and Izmestiev describes a differential equation whose solution leads to the polyhedron corresponding to a given metric. We describe an algorithm based on this differential equation to compute the polyhedron to arbitrary precision given the metric, and prove a pseudopolynomial bound on its running time. Along the way, we develop pseudopolynomial algorithms for computing shortest paths and weighted Delaunay triangulations on a polyhedral surface, even when the surface edges are not shortest paths.Comment: 25 pages; new Delaunay triangulation algorithm, minor other changes; an abbreviated v2 was at WADS 200

Contraction Bidimensionality: the Accurate Picture

We provide new combinatorial theorems on the structure of graphs that are contained as contractions in graphs of large treewidth. As a consequence of our combinatorial results we unify and significantly simplify contraction bidimensionality theory -- the meta algorithmic framework to design efficient parameterized and approximation algorithms for contraction closed parameters

Greedy Selfish Network Creation

We introduce and analyze greedy equilibria (GE) for the well-known model of selfish network creation by Fabrikant et al.[PODC'03]. GE are interesting for two reasons: (1) they model outcomes found by agents which prefer smooth adaptations over radical strategy-changes, (2) GE are outcomes found by agents which do not have enough computational resources to play optimally. In the model of Fabrikant et al. agents correspond to Internet Service Providers which buy network links to improve their quality of network usage. It is known that computing a best response in this model is NP-hard. Hence, poly-time agents are likely not to play optimally. But how good are networks created by such agents? We answer this question for very simple agents. Quite surprisingly, naive greedy play suffices to create remarkably stable networks. Specifically, we show that in the SUM version, where agents attempt to minimize their average distance to all other agents, GE capture Nash equilibria (NE) on trees and that any GE is in 3-approximate NE on general networks. For the latter we also provide a lower bound of 3/2 on the approximation ratio. For the MAX version, where agents attempt to minimize their maximum distance, we show that any GE-star is in 2-approximate NE and any GE-tree having larger diameter is in 6/5-approximate NE. Both bounds are tight. We contrast these positive results by providing a linear lower bound on the approximation ratio for the MAX version on general networks in GE. This result implies a locality gap of $\Omega(n)$ for the metric min-max facility location problem, where n is the number of clients.Comment: 28 pages, 8 figures. An extended abstract of this work was accepted at WINE'1

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 2O(k·log k). Our approach combines tools from topological graph theory and analytic combinatorics.Postprint (updated version

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

Combining Binary Search Trees

We present a general transformation for combining a constant number of binary search tree data structures (BSTs) into a single BST whose running time is within a constant factor of the minimum of any “well-behaved” bound on the running time of the given BSTs, for any online access sequence. (A BST has a well-behaved bound with f(n) overhead if it spends at most O(f(n)) time per access and its bound satisfies a weak sense of closure under subsequences.) In particular, we obtain a BST data structure that is O(loglogn) competitive, satisfies the working set bound (and thus satisfies the static finger bound and the static optimality bound), satisfies the dynamic finger bound, satisfies the unified bound with an additive O(loglogn) factor, and performs each access in worst-case O(logn) time

Signal Transmission Across Tile Assemblies: 3D Static Tiles Simulate Active Self-Assembly by 2D Signal-Passing Tiles

The 2-Handed Assembly Model (2HAM) is a tile-based self-assembly model in which, typically beginning from single tiles, arbitrarily large aggregations of static tiles combine in pairs to form structures. The Signal-passing Tile Assembly Model (STAM) is an extension of the 2HAM in which the tiles are dynamically changing components which are able to alter their binding domains as they bind together. For our first result, we demonstrate useful techniques and transformations for converting an arbitrarily complex STAM$^+$ tile set into an STAM$^+$ tile set where every tile has a constant, low amount of complexity, in terms of the number and types of ``signals'' they can send, with a trade off in scale factor. Using these simplifications, we prove that for each temperature $\tau>1$ there exists a 3D tile set in the 2HAM which is intrinsically universal for the class of all 2D STAM$^+$ systems at temperature $\tau$ (where the STAM$^+$ does not make use of the STAM's power of glue deactivation and assembly breaking, as the tile components of the 2HAM are static and unable to change or break bonds). This means that there is a single tile set $U$ in the 3D 2HAM which can, for an arbitrarily complex STAM$^+$ system $S$, be configured with a single input configuration which causes $U$ to exactly simulate $S$ at a scale factor dependent upon $S$. Furthermore, this simulation uses only two planes of the third dimension. This implies that there exists a 3D tile set at temperature $2$ in the 2HAM which is intrinsically universal for the class of all 2D STAM$^+$ systems at temperature $1$. Moreover, we show that for each temperature $\tau>1$ there exists an STAM$^+$ tile set which is intrinsically universal for the class of all 2D STAM$^+$ systems at temperature $\tau$, including the case where $\tau = 1$.Comment: A condensed version of this paper will appear in a special issue of Natural Computing for papers from DNA 19. This full version contains proofs not seen in the published versio

(Total) Vector Domination for Graphs with Bounded Branchwidth

Given a graph $G=(V,E)$ of order $n$ and an $n$-dimensional non-negative vector $d=(d(1),d(2),\ldots,d(n))$, called demand vector, the vector domination (resp., total vector domination) is the problem of finding a minimum $S\subseteq V$ such that every vertex $v$ in $V\setminus S$ (resp., in $V$) has at least $d(v)$ neighbors in $S$. The (total) vector domination is a generalization of many dominating set type problems, e.g., the dominating set problem, the $k$-tuple dominating set problem (this $k$ is different from the solution size), and so on, and its approximability and inapproximability have been studied under this general framework. In this paper, we show that a (total) vector domination of graphs with bounded branchwidth can be solved in polynomial time. This implies that the problem is polynomially solvable also for graphs with bounded treewidth. Consequently, the (total) vector domination problem for a planar graph is subexponential fixed-parameter tractable with respectto $k$, where $k$ is the size of solution.Comment: 16 page