Let G be a directed graph with n vertices and non-negative weights in its directed edges, embedded on a surface of genus g, and let f be an arbitrary face of G. We describe a randomized algorithm to preprocess the graph in O(g n log n) time with high probability, so that the shortest-path distance from any vertex on the boundary of f to any other vertex in G can be retrieved in O(log n) time. Our result directly generalizes the O(n log n)-time algorithm of Klein [Multiple-source shortest paths in planar graphs. In Proc. 16th Ann. ACM-SIAM Symp. Discrete Algorithms, 2005] for multiple-source shortest paths in planar graphs. Intuitively, our preprocessing algorithm maintains a shortest-path tree as its source point moves continuously around the boundary of f . As an application of our algorithm, we describe algorithms to compute a shortest non-contractible or non-separating cycle in embedded, undirected graphs in O(g 2 n log n) time with high probability. Our high-probability time bounds hold in the worst-case for generic edge weights, or with an additional O(log n) factor for arbitrary edge weights.
We describe the first algorithms to compute minimum cuts in surface-embedded graphs in near-linear time. Given an undirected graph embedded on an orientable surface of genus g, with two specified vertices s and t, our algorithm computes a minimum (s, t)-cut in g O(g) n log n time. Except for the special case of planar graphs, for which O(n log n)-time algorithms have been known for more than 20 years, the best previous time bounds for finding minimum cuts in embedded graphs follow from algorithms for general sparse graphs. A slight generalization of our minimum-cut algorithm computes a minimum-cost subgraph in every 2 -homology class. We also prove that finding a minimum-cost subgraph homologous to a single input cycle is NP-hard.
A Roman dominating function of a graph G is a labeling f : V (G) → {0, 1, 2} such that every vertex with label 0 has a neighbor with label 2. The Roman domination number γ R (G) of G is the minimum of v∈V (G) f (v) over such functions. Let G be a connected n-vertex graph. We prove that γ R (G) ≤ 4n/5, and we characterize the graphs achieving equality. We obtain sharp upper and lower bounds for γ R (G)+γ R (G) and γ R (G)γ R (G), improving known results for domination number. We prove that γ R (G) ≤ 8n/11 when δ(G) ≥ 2 and n ≥ 9, and this is sharp. IntroductionAccording to [6], Constantine the Great (Emperor of Rome) issued a decree in the 4th century A.D. for the defense of his cities. He decreed that any city without a legion stationed to secure it must neighbor another city having two stationed legions. If the first were attacked, then the second could deploy a legion to protect it without becoming vulnerable itself.The objective, of course, is to minimize the total number of legions needed. The problem generalizes to arbitrary graphs. A Roman dominating function (RDF) on a graph G is a vertex labeling f : V (G) → {0, 1, 2} such that every vertex with label 0 has a neighbor with label 2. For an RDF f , let V i (f ) = {v ∈ V (G) : f (v) = i}. In the context of a fixed RDF, we suppress the argument and simply write V 0 , V 1 , and V 2 . Since this partition determines f , we can equivalently write f = (V 0 , V 1 , V 2 ). The weight w(f ) of an RDF f is v∈V (G) f (v), which equals |V 1 | + 2|V 2 |. The Roman domination number γ R (G) is the minimum weight of an RDF of G. Thus, γ R (G) is the minimum number of legions needed to protect cities whose adjacency graph is G.
We describe the first algorithm to compute maximum flows in surface-embedded graphs in near-linear time. Specifically, given a graph embedded on a surface of genus g, with two specified vertices s and t and integer edge capacities that sum to C, our algorithm computes a maximum (s, t)-flow in O(g 8 n log 2 n log 2 C) time. We also present a combinatorial algorithm that takes g O(g) n 3/2 arithmetic operations. Except for the special case of planar graphs, for which an O(n log n)-time algorithm has been known for 20 years, the best previous time bounds for maximum flows in surface-embedded graphs follow from algorithms for general sparse graphs. For graphs of any fixed genus, our algorithms improve these time bounds by roughly a factor of n. Our key insight is to optimize the homology class of the flow, rather than directly optimizing the flow itself; two flows are in the same homology class if their difference is a weighted sum of directed facial cycles. A dual formulation of our algorithm computes the minimum-cost circulation in a given (real or integer) homology class.
Given a distribution of pebbles on the vertices of a graph G, a pebbling move takes two pebbles from one vertex and puts one on a neighboring vertex. The pebbling number Π(G) is the least k such that for every distribution of k pebbles and every vertex r, a pebble can be moved to r. The optimal pebbling number Π OP T (G) is the least k such that some distribution of k pebbles permits reaching each vertex.Using new tools (such as the "Squishing" and "Smoothing" Lemmas), we give short proofs of prior results on these parameters for paths, cycles, trees, and hypercubes, a new linear-time algorithm for computing Π(G) on trees, and new results on Π OP T
Thinning is a commonly used approach for computing skeleton descriptors. Traditional thinning algorithms often have a simple, iterative structure, yet producing skeletons that are overly sensitive to boundary perturbations. We present a novel thinning algorithm, operating on objects represented as cell complexes, that preserves the simplicity of typical thinning algorithms but generates skeletons that more robustly capture global shape features. Our key insight is formulating a skeleton significance measure, called medial persistence, which identify skeleton geometry at various dimensions (e.g., curves or surfaces) that represent object parts with different anisotropic elongations (e.g., tubes or plates). The measure is generally defined in any dimensions, and can be easily computed using a single thinning pass. Guided by medial persistence, our algorithm produces a family of topology and shape preserving skeletons whose shape and composition can be flexible controlled by desired level of medial persistence.
While playing a fundamental role in shape understanding, the medial axis is known to be sensitive to small boundary perturbations. Methods for pruning the medial axis are usually guided by some measure of significance. The majority of significance measures over the medial axes of 3D shapes are locally defined and hence unable to capture the scale of features. We introduce a global significance measure that generalizes in 3D the classical Erosion Thickness (ET) measure over the medial axes of 2D shapes. We give precise definition of ET in 3D, analyze its properties, and present an efficient approximation algorithm with bounded error on a piece-wise linear medial axis. Experiments showed that ET outperforms local measures in differentiating small boundary noise from prominent shape features, and it is significantly faster to compute than existing global measures. We demonstrate the utility of ET in extracting clean, shape-revealing and topology-preserving skeletons of 3D shapes.
Let M be an orientable surface without boundary. A cycle on M is splitting if it has no self-intersections and it partitions M into two components, neither homeomorphic to a disk. In other words, splitting cycles are simple, separating, and non-contractible. We prove that finding the shortest splitting cycle on a combinatorial surface is NP-hard but fixed-parameter tractable with respect to the surface genus. Specifically, we describe an algorithm to compute the shortest splitting cycle in g O(g) n 2 log n time.
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