Results 1  10
of
31
Dynamic Generators of Topologically Embedded Graphs
, 2003
"... We provide a data structure for maintaining an embedding of a graph on a surface (represented combinatorially by a permutation of edges around each vertex) and computing generators of the fundamental group of the surface, in amortized time O(logn + logg(loglogg) 3) per update on a surface of genus g ..."
Abstract

Cited by 42 (1 self)
 Add to MetaCart
We provide a data structure for maintaining an embedding of a graph on a surface (represented combinatorially by a permutation of edges around each vertex) and computing generators of the fundamental group of the surface, in amortized time O(logn + logg(loglogg) 3) per update on a surface of genus g; we can also test orientability of the surface in the same time, and maintain the minimum and maximum spanning tree of the graph in time O(log n + log 4 g) per update. Our data structure allows edge insertion and deletion as well as the dual operations; these operations may implicitly change the genus of the embedding surface. We apply similar ideas to improve the constant factor in a separator theorem for lowgenus graphs, and to find in linear time a treedecomposition of lowgenus lowdiameter graphs.
Logarithmic lower bounds in the cellprobe model
 SIAM Journal on Computing
"... Abstract. We develop a new technique for proving cellprobe lower bounds on dynamic data structures. This enables us to prove Ω(lg n) bounds, breaking a longstanding barrier of Ω(lg n/lg lg n). We can also prove the first Ω(lgB n) lower bound in the external memory model, without assumptions on the ..."
Abstract

Cited by 34 (4 self)
 Add to MetaCart
Abstract. We develop a new technique for proving cellprobe lower bounds on dynamic data structures. This enables us to prove Ω(lg n) bounds, breaking a longstanding barrier of Ω(lg n/lg lg n). We can also prove the first Ω(lgB n) lower bound in the external memory model, without assumptions on the data structure. We use our technique to prove better bounds for the partialsums problem, dynamic connectivity and (by reductions) other dynamic graph problems. Our proofs are surprisingly simple and clean. The bounds we obtain are often optimal, and lead to a nearly complete understanding of the problems. We also present new matching upper bounds for the partialsums problem. Key words. cellprobe complexity, lower bounds, data structures, dynamic graph problems, partialsums problem AMS subject classification. 68Q17
Cell probe complexity  a survey
 In 19th Conference on the Foundations of Software Technology and Theoretical Computer Science (FSTTCS), 1999. Advances in Data Structures Workshop
"... The cell probe model is a general, combinatorial model of data structures. We give a survey of known results about the cell probe complexity of static and dynamic data structure problems, with an emphasis on techniques for proving lower bounds. 1 ..."
Abstract

Cited by 29 (0 self)
 Add to MetaCart
The cell probe model is a general, combinatorial model of data structures. We give a survey of known results about the cell probe complexity of static and dynamic data structure problems, with an emphasis on techniques for proving lower bounds. 1
A Fully Dynamic Algorithm for Recognizing and Representing Proper Interval Graphs
 SIAM J. COMPUT
, 1999
"... In this paper we study the problem of recognizing and representing dynamically changing proper interval graphs. The input to the problem consists of a series of modifications to be performed on a graph, where a modification can be a deletion or an addition of a vertex or an edge. The objective is to ..."
Abstract

Cited by 25 (1 self)
 Add to MetaCart
In this paper we study the problem of recognizing and representing dynamically changing proper interval graphs. The input to the problem consists of a series of modifications to be performed on a graph, where a modification can be a deletion or an addition of a vertex or an edge. The objective is to maintain a representation of the graph as long as it remains a proper interval graph, and to detect when it ceases to be so. The representation should enable one to efficiently construct a realization of the graph by an inclusionfree family of intervals. This problem has important applications in physical mapping of DNA. We give a nearoptimal fully dynamic algorithm for this problem. It operates in time O(log n) per edge insertion or deletion. We prove a close lower bound of\Omega\Gamma/24 n=(log log n + log b)) amortized time per operation, in the cell probe model with wordsize b. We also construct optimal incremental and decremental algorithms for the problem, which handle each edge operation in O(1) time.
Efficient and simple generation of random simple connected graphs with prescribed degree sequence
 in The Eleventh International Computing and Combinatorics Conference, Aug. 2005, kumming
, 2005
"... degree sequence ..."
SingleStrip Triangulation of Manifolds with Arbitrary Topology
, 2004
"... Triangle strips have been widely used for efficient rendering. It is NPcomplete to test whether a given triangulated model can be represented as a single triangle strip, so many heuristics have been proposed to partition models into few long strips. In this paper, we present a new algorithm for cre ..."
Abstract

Cited by 16 (5 self)
 Add to MetaCart
Triangle strips have been widely used for efficient rendering. It is NPcomplete to test whether a given triangulated model can be represented as a single triangle strip, so many heuristics have been proposed to partition models into few long strips. In this paper, we present a new algorithm for creating a single triangle loop or strip from a triangulated model. Our method applies a dual graph matching algorithm to partition the mesh into cycles, and then merges pairs of cycles by splitting adjacent triangles when necessary. New vertices are introduced at midpoints of edges and the new triangles thus formed are coplanar with their parent triangles, hence the visual fidelity of the geometry is not changed. We prove that the increase in the number of triangles due to this splitting is 50 % in the worst case, however for all models we tested the increase was less than 2%. We also prove tight bounds on the number of triangles needed for a singlestrip representation of a model with holes on its boundary. Our strips can be used not only for efficient rendering, but also for other applications including the generation of space filling curves on a manifold of any arbitrary topology.
Lower bounds for dynamic connectivity
 STOC
, 2004
"... We prove an Ω(lg n) cellprobe lower bound on maintaining connectivity in dynamic graphs, as well as a more general tradeoff between updates and queries. Our bound holds even if the graph is formed by disjoint paths, and thus also applies to trees and plane graphs. The bound is known to be tight fo ..."
Abstract

Cited by 15 (0 self)
 Add to MetaCart
We prove an Ω(lg n) cellprobe lower bound on maintaining connectivity in dynamic graphs, as well as a more general tradeoff between updates and queries. Our bound holds even if the graph is formed by disjoint paths, and thus also applies to trees and plane graphs. The bound is known to be tight for these restricted cases, proving optimality of these data structures (e.g., Sleator and Tarjan’s dynamic trees). Our tradeoff is known to be tight for trees, and the best two data structures for dynamic connectivity in general graphs are points on our tradeoff curve. In this sense these two data structures are optimal, and this tightness serves as strong evidence that our lower bounds are the best possible. From a more theoretical perspective, our result is the first logarithmic cellprobe lower bound for any problem in the natural class of dynamic language membership problems, breaking the long standing record of Ω(lg n / lg lg n). In this sense, our result is the first datastructure lower bound that is “truly ” logarithmic, i.e., logarithmic in the problem size counted in bits. Obtaining such a bound is listed as one of three major challenges for future research by Miltersen [13] (the other two challenges remain unsolved). Our techniques form a general framework for proving cellprobe lower bounds on dynamic data structures. We show how our framework also applies to the partialsums problem to obtain a nearly complete understanding of the problem in cellprobe and algebraic models, solving several previously posed open problems.
Compact Forbiddenset Routing
 Proceedings of STACS’07, Aachen, 2007, volume 4393 of LNCS
, 2007
"... We study labelling schemes for Xconstrained path problems. Given a graph (V, E) and X ⊆ V, a path is Xconstrained if all intermediate vertices avoid X. We study the problem of assigning labels J(x) to vertices so that given {J(x) : x ∈ X} for any X ⊆ V, we can route on the shortest Xconstrained p ..."
Abstract

Cited by 15 (6 self)
 Add to MetaCart
We study labelling schemes for Xconstrained path problems. Given a graph (V, E) and X ⊆ V, a path is Xconstrained if all intermediate vertices avoid X. We study the problem of assigning labels J(x) to vertices so that given {J(x) : x ∈ X} for any X ⊆ V, we can route on the shortest Xconstrained path between x, y ∈ X. This problem is motivated by Internet routing, where the presence of routing policies means that shortestpath routing is not appropriate. For graphs of tree width k, we give a routing scheme using routing tables of size O(k 2 log 2 n). We introduce mclique width, generalizing clique width, to show that graphs of mclique width k also have a routing scheme using size O(k 2 log 2 n) tables.
Dynamic Subgraph Connectivity with Geometric Applications
 Proc. 34th ACM Sympos. Theory Comput
, 2002
"... Inspired by dynamic connectivity applications in computational geometry, we consider a problem we call dynamic subgraph connectivity : design a data structure for an undirected graph G = (V, E) and a subset of vertices S # V , to support insertions and deletions in S and connectivity queries (are ..."
Abstract

Cited by 12 (3 self)
 Add to MetaCart
Inspired by dynamic connectivity applications in computational geometry, we consider a problem we call dynamic subgraph connectivity : design a data structure for an undirected graph G = (V, E) and a subset of vertices S # V , to support insertions and deletions in S and connectivity queries (are two vertices connected?) in the subgraph induced by S. We develop the first sublinear, fully dynamic method for this problem for general sparse graphs, using an elegant combination of several simple ideas. Our method requires linear space, # O(E 4#/(3#+3) ) = O(E 0.94 ) amortized update time, and # O(E 1/3 ) query time, where # is the matrix multiplication exponent and # O hides polylogarithmic factors.
Lombardi Drawings of Graphs
"... We introduce the notion of Lombardi graph drawings, named after the American abstract artist Mark Lombardi. In these drawings, edges are represented as circular arcs rather than as line segments or polylines, and the vertices have perfect angular resolution: the edges are equally spaced around each ..."
Abstract

Cited by 9 (6 self)
 Add to MetaCart
We introduce the notion of Lombardi graph drawings, named after the American abstract artist Mark Lombardi. In these drawings, edges are represented as circular arcs rather than as line segments or polylines, and the vertices have perfect angular resolution: the edges are equally spaced around each vertex. We describe algorithms for finding Lombardi drawings of regular graphs, graphs of bounded degeneracy, and certain families of planar graphs.