We show how to visualize a graph, G=(V,E), as a layered drawing, layer-respecting arc diagram, or layer-respecting linear cylindric drawing with a minimum number of edge crossings, where layer-respecting means that layers appear in order on a single line and vertices are grouped by their layers. Even though this problem is NP-hard for general arc diagrams, we show how to create such diagrams with fixed-parameter tractable linear-time algorithms, where the parameter that allows this is the width of a layered graph. Such a layered graph can be obtained from a breadth-first search (BFS), in which case the width is upper bounded by a graph width parameter called the BFS width.

The Erd\H{o}s-Anning theorem states that every point set in the Euclidean plane with integer distances must be either collinear or finite. More strongly, for any (non-degenerate) triangle of diameter $\delta$, at most $O(\delta^2)$ points can have integer distances from all three triangle vertices. We prove the same results for any strictly convex distance function on the plane, and analogous results for every two-dimensional complete Riemannian manifold of bounded genus and for geodesic distance on the boundary of every three-dimensional Euclidean convex set. As a consequence, we resolve a 1983 question of Richard Guy on the equilateral dimension of Riemannian manifolds. Our proofs are based on the properties of additively weighted Voronoi diagrams of these distances.

Motivated by worst-case algorithmic time bounds for solving sudoku, we prove that a majority of filled-in n^2× n^2 sudoku grids require all but a logarithmic fraction of cells to be filled by clues. For 9× 9 and 16× 16 sudoku, we construct grids that require 18 clues and 80 clues.

We study the problem of aggregating polygons by covering them with disjoint representative regions, thereby inducing a clustering of the polygons. Our objective is to minimize a weighted sum of the total area and the total perimeter of the regions. This problem has applications in cartographic map generalization and urban analytics. Here, the polygons represent building footprints and the clusters may represent urban areas. Previous approaches forced the boundaries of the regions to come from a fixed subdivision of the plane, which allows the optimal solution (restricted in this way) to be found from a minimum cut in a dual graph. It is natural to ask whether the problem can still be solved efficiently if this restriction is removed, allowing output regions to be bounded by arbitrary curves. We provide a positive answer in the form of a polynomial-time algorithm. Additionally, we fully characterize the optimal solutions by showing that their boundaries are composed of input polygon edges and circular arcs of constant radius. Since some applications favor straight edges, we also study two problem variants in which the output regions must be polygons, but are not restricted to have boundaries from a fixed subdivision. In the first variant, region vertices must lie on the boundaries of the input polygons. The second variant requires them to be vertices of the input polygons. We show that both variants can be approximated up to a constant factor in polynomial time by altering an optimal solution for the unrestricted problem. Our experimental evaluation on real-world building footprints demonstrates that these approximate solutions are visually similar to the optimal unrestricted ones and achieve near-optimal objective values.

We construct a convex polygon for which the minimum-weight Steiner triangulation requires an interior Steiner point. This provides a counterexample to a 1994 conjecture of Eppstein that minimum-weight Steiner triangulation of convex polygons needs only Steiner points on the boundary of the polygon.

The Erdős-Anning theorem states that any point set for which all distances are integers, in a Euclidean space of any dimension, must be either finite or collinear. We prove the same result in hyperbolic space of any dimension. A quantitative form of our result also extends for the first time to Euclidean spaces of dimension greater than two: if a set of points with integer distances in 𝔼^D or ℍ^D has a subset of D+1 points in general position whose diameter is d, then the whole set has size O(D(d+1)^D). To prove these results we formulate a lemma that, if the graph of external tangencies of a system of spheres in Euclidean or hyperbolic space contains a K_a,b subgraph for a,b≥ 3, then the sets of spheres on each side of this biclique have centers that lie on a hyperplane. This lemma also implies that, in multilateration (determining a position from differences of distances to known landmarks), D+1 non-coplanar landmarks always suffice to limit the position to two possibilities.

Let 𝒜 be the subdivision of ℝ^d induced by m convex polyhedra having n facets in total. We prove that 𝒜 has combinatorial complexity O(m^⌈ d/2 ⌉ n^⌊ d/2 ⌋) and that this bound is tight. The bound is mentioned several times in the literature, but no proof for arbitrary dimension has been published before.

In this paper, we introduce zip-tries, which are simple, dynamic, memory-efficient data structures for strings. Zip-tries support search and update operations for k-length strings in 𝒪(k+log n) time in the standard RAM model or in 𝒪(k/α+log n) time in the word RAM model, where α is the length of the longest string that can fit in a memory word, and n is the number of strings in the trie. Importantly, we show how zip-tries can achieve this while only requiring 𝒪(loglogn + loglogk/α) bits of metadata per node w.h.p., which is an exponential improvement over previous results for long strings. Despite being considerably simpler and more memory efficient, we show how zip-tries perform competitively with state-of-the-art data structures on large datasets of long strings. Furthermore, we provide a simple, general framework for parallelizing string comparison operations in linked data structures, which we apply to zip-tries to obtain parallel zip-tries. Parallel zip-tries are able to achieve good search and update performance in parallel, performing such operations in 𝒪(logn) span. We also apply our techniques to an existing external-memory string data structure, the string B-tree, obtaining a parallel string B-tree which performs search operations using 𝒪(log_Bn) I/O span and 𝒪(k/αB + log_Bn) I/O work in the parallel external memory (PEM) model. The parallel string B-tree can perform prefix searches using only 𝒪(logn/loglogn) span under the practical PRAM model. For the case of long strings that share short common prefixes, we provide LCP-aware variants of all our algorithms that should be quite efficient in practice, which we justify empirically.

We show that the max-min-angle polygon in a planar point set can be found in time O(nlog n) and a max-min-solid-angle convex polyhedron in a three-dimensional point set can be found in time O(n^2). We also study the maxmin-angle polygonal curve in 3d, which we show to be 𝖭𝖯-hard to find if repetitions are forbidden but can be found in near-cubic time if repeated vertices or line segments are allowed, by reducing the problem to finding a bottleneck cycle in a graph. We formalize a class of problems on which a decremental greedy algorithm can be guaranteed to find an optimal solution, generalizing our max-min-angle and bottleneck cycle algorithms, together with a known algorithm for graph degeneracy.

We study entropy-bounded computational geometry, that is, geometric algorithms whose running times depend on a given measure of the input entropy. Specifically, we introduce a measure that we call range-partition entropy, which unifies and subsumes previous definitions of entropy used for sorting problems and structural entropy used in computational geometry. We provide simple algorithms for several problems, including 2D maxima, 2D and 3D convex hulls, and some visibility problems, and we show that they have running times depending on the range-partition entropy.

A string graph is the intersection graph of curves in the plane. Kratochvíl previously showed the existence of infinitely many obstacles: graphs that are not string graphs but for which any edge contraction or vertex deletion produces a string graph. Kratochvíl's obstacles contain arbitrarily large cliques, so they have girth three and unbounded degree. We extend this line of working by studying obstacles among graphs of restricted girth and/or degree. We construct an infinite family of obstacles of girth four; in addition, our construction is K_2,3-subgraph-free and near-planar (planar plus one edge). Furthermore, we prove that there is a subcubic obstacle of girth three, and that there are no subcubic obstacles of high girth. We characterize the subcubic string graphs as having a matching whose contraction yields a planar graph, and based on this characterization we find a linear-time algorithm for recognizing subcubic string graphs of bounded treewidth.

The subdivided double construction on 4-regular graphs was used by Potočnik and Wilson to explore semi-symmetric (edge-transitive but not vertex-transitive) graphs, and can be used to construct every semi-symmetric 4-regular graph that contains a pair of twin vertices. We show that (regardless of symmetry) subdivided doubles have another curious property: they have exponentially many Hamiltonian cycles each of which is complementary to another Hamiltonian cycle.

The famous Tower of Hanoi puzzle involves moving n discs of distinct sizes from one of p≥ 3 pegs (traditionally p=3) to another of the pegs, subject to the constraints that only one disc may be moved at a time, and no disc can ever be placed on a disc smaller than itself. Much is known about the Hanoi graph H_p^n, whose p^n vertices represent the configurations of the puzzle, and whose edges represent the pairs of configurations separated by a single legal move. In a previous paper, the present authors presented nearly tight asymptotic bounds of O((p-2)^n) and Ω(n^(1-p)/2(p-2)^n) on the treewidth of this graph for fixed p ≥ 3. In this paper we show that the upper bound is tight, by giving a matching lower bound of Ω((p-2)^n) for the expansion of H_p^n.

Edge crossings in geometric graphs are sometimes undesirable as they could lead to unwanted situations such as collisions in motion planning and inconsistency in VLSI layout. Short geometric structures such as shortest perfect matchings, shortest spanning trees, shortest spanning paths, and shortest spanning cycles on a given point set are inherently noncrossing. However, the longest such structures need not be noncrossing. In fact, it is intuitive to expect many edge crossings in various geometric graphs that are longest. Recently, Álvarez-Rebollar, Cravioto-Lagos, Marín, Solé-Pi, and Urrutia (Graphs and Combinatorics, 2024) constructed a set of points for which the longest perfect matching is noncrossing. They raised several challenging questions in this direction. In particular, they asked whether the longest spanning path, on every finite set of points in the plane, must have a pair of crossing edges. They also conjectured that the longest spanning cycle must have a pair of crossing edges. In this paper, we give a negative answer to the question and also refute the conjecture. We present a framework for constructing arbitrarily large point sets for which the longest perfect matchings, the longest spanning paths, and the longest spanning cycles are noncrossing.

We provide the first approximation quality guarantees for the Cuthull-McKee heuristic for reordering symmetric matrices to have low bandwidth, and we provide an algorithm for reconstructing bounded-bandwidth graphs from distance oracles with near-linear query complexity. To prove these results we introduce a new width parameter, BFS width, and we prove polylogarithmic upper and lower bounds on the BFS width of graphs of bounded bandwidth. Unlike other width parameters, such as bandwidth, pathwidth, and treewidth, BFS width can easily be computed in polynomial time. Bounded BFS width implies bounded bandwidth, pathwidth, and treewidth, which in turn imply fixed-parameter tractable algorithms for many problems that are NP-hard for general graphs. In addition to their applications to matrix ordering, we also provide applications of BFS width to graph reconstruction, to reconstruct graphs from distance queries, and graph drawing, to construct arc diagrams of small height.

Much prior work has been done on designing computational geometry algorithms that handle input degeneracies, data imprecision, and arithmetic round-off errors. We take a new approach, inspired by the noisy sorting literature, and study computational geometry algorithms subject to noisy Boolean primitive operations in which, e.g., the comparison "is point q above line L?" returns the wrong answer with some fixed probability. We propose a novel technique called path-guided pushdown random walks that generalizes the results of noisy sorting. We apply this technique to solve point-location, plane-sweep, convex hulls in 2D and 3D, dynamic 2D convex hulls, and Delaunay triangulations for noisy primitives in optimal time with high probability.

Graph embedding, especially as a subgraph of a grid, is an old topic in VLSI design and graph drawing. In this paper, we investigate related questions concerning the complexity of embedding a graph G in a host graph that is the strong product of a path P with a graph H that satisfies some properties, such as having small treewidth, pathwidth or tree depth. We show that this is NP-hard, even under numerous restrictions on both G and H. In particular, computing the row pathwidth and the row treedepth is NP-hard even for a tree of small pathwidth, while computing the row treewidth is NP-hard even for series-parallel graphs.

We prove that, for every plane graph G and every smooth convex curve C not on a single line, there exists a straight-line drawing of G for which every face is crossed by C.

The shortest path passing on the surface of a polyhedron is called a geodesic path. A geodesic path of a polyhedron has a property that it becomes a single line segment on a development. A geodesic path is the shortest path and it mostly passes a small number of faces. We, however, consider a problem “is there a case that a geodesic path passes all faces of a polyhedron?” For this problem the answer is “yes”: we found that a regular tetrahedron has such a geodesic path. The next question is “what polyhedra have such geodesic paths?” We define a face-guard geodesic path (FGG path, for short) as a geodesic path connecting two points on a polyhedron and passing through all its faces, call a polyhedron that has an FGG path an FGG polyhedron, and try to characterize FGG polyhedra. For this new problem, we prove that there exists an FGG n-hedron for any integer n≥ 4 , all tetrahedra and all triangular prisms with one exception are FGG polyhedra, and all cuboids and all regular polyhedra except regular tetrahedra are not FGG polyhedra.