Arboricity, h-Index, and Dynamic Algorithms

In this paper we present a modification of a technique by Chiba and Nishizeki [Chiba and Nishizeki: Arboricity and Subgraph Listing Algorithms, SIAM J. Comput. 14(1), pp. 210--223 (1985)]. Based on it, we design a data structure suitable for dynamic graph algorithms. We employ the data structure to formulate new algorithms for several problems, including counting subgraphs of four vertices, recognition of diamond-free graphs, cop-win graphs and strongly chordal graphs, among others. We improve the time complexity for graphs with low arboricity or h-index.Comment: 19 pages, no figure

Degree Constrained Triangulation of Annular Regions and Point Sites

Generating constrained triangulations of point sites distributed in the plane is a significant problem in computational geometry. We present theoretical and experimental investigation results for generating triangulations for polygons and point sites that address node degree constraints. We characterize point sites that have almost all vertices of odd degree. We present experimental results on the node degree distribution of Delaunay triangulations of point sites generated randomly. Additionally, we present a heuristic algorithm for triangulating a given normal annular region with an increment of even degree nodes

Recognition of affine-equivalent polyhedra by their natural developments

The classical Cauchy rigidity theorem for convex polytopes reads that if two convex polytopes have isometric developments then they are congruent. In other words, we can decide whether two polyhedra are isometric or not by using their developments only. In this article, we study a similar problem about whether it is possible, using only the developments of two convex polyhedra of Euclidean 3-space, to understand that these polyhedra are (or are not) affine-equivalent.Comment: 23 pages, 4 figures, in Russia

In-Memory Storage for Labeled Tree-Structured Data

In this thesis, we design in-memory data structures for labeled and weights trees, so that various types of path queries or operations can be supported with efficient query time. We assume the word RAM model with word size w, which permits random accesses to w-bit memory cells. Our data structures are space-efficient and many of them are even succinct. These succinct data structures occupy space close to the information theoretic lower bounds of the input trees within lower order terms. First, we study the problems of supporting various path queries over weighted trees. A path counting query asks for the number of nodes on a query path whose weights lie within a query range, while a path reporting query requires to report these nodes. A path median query asks for the median weight on a path between two given nodes, and a path selection query returns the k-th smallest weight. We design succinct data structures to support path counting queries in O(lg σ/ lg lg n + 1) time, path reporting queries in O((occ + 1)(lg σ/ lg lg n + 1)) time, and path median and path selection queries in O(lg σ/ lg lg σ) time, where n is the size of the input tree, the weights of nodes are drawn from [1..σ] and occ is the size of the output. Our results not only greatly improve the best known data structures [31, 75, 65], but also match the lower bounds for path counting, median and selection queries [86, 87, 71] when σ = Ω(n/polylog(n)). Second, we study the problem of representing labeled ordinal trees succinctly. Our new representations support a much broader collection of operations than previous work. In our approach, labels of nodes are stored in a preorder label sequence, which can be compressed using any succinct representation of strings that supports access, rank and select operations. Thus, we present a framework for succinct representations of labeled ordinal trees that is able to handle large alphabets. This answers an open problem presented by Geary et al. [54], which asks for representations of labeled ordinal trees that remain space-efficient for large alphabets. We further extend our work and present the first succinct representations for dynamic labeled ordinal trees that support several label-based operations including finding the level ancestor with a given label. Third, we study the problems of supporting path minimum and semigroup path sum queries. In the path minimum problem, we preprocess a tree on n weighted nodes, such that given an arbitrary path, the node with the smallest weight along this path can be located. We design novel succinct indices for this problem under the indexing model, for which weights of nodes are read-only and can be accessed with ranks of nodes in the preorder traversal sequence of the input tree. One of our index structures supports queries in O(α(m,n)) time, and occupies O(m) bits of space in addition to the space required for the input tree, where m is an integer greater than or equal to n and α(m, n) is the inverse-Ackermann function. Following the same approach, we also develop succinct data structures for semigroup path sum queries, for which a query asks for the sum of weights along a given query path. Then, using the succinct indices for path minimum queries, we achieve three different time-space tradeoffs for path reporting queries. Finally, we study the problems of supporting various path queries in dynamic settings. We propose the first non-trivial linear-space solution that supports path reporting in O((lgn/lglgn)^2 +occlgn/lglgn)) query time, where n is the size of the input tree and occ is the output size, and the insertion and deletion of a node of an arbitrary degree in O(lg^{2+ε} n) amortized time, for any constant ε ∈ (0, 1). Obvious solutions based on directly dynamizing solutions to the static version of this problem all require Ω((lg n/ lg lg n)^2) time for each node reported. We also design data structures that support path counting and path reporting queries in O((lg n/ lg lg n)^2) time, and insertions and deletions in O((lg n/ lg lg n)^2) amortized time. This matches the best known results for dynamic two-dimensional range counting [62] and range selection [63], which can be viewed as special cases of path counting and path selection