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65
Lower Bounds for Fully Dynamic Connectivity Problems in Graphs
, 1998
"... We prove lower bounds on the complexity of maintaining fully dynamic kedge or kvertex connectivity in plane graphs and in (k − 1)vertex connected graphs. We show an amortized lower bound of �(log n/k(log log n + log b)) per edge insertion, deletion, or query operation in the cell probe model, whe ..."
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Cited by 32 (5 self)
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We prove lower bounds on the complexity of maintaining fully dynamic kedge or kvertex connectivity in plane graphs and in (k − 1)vertex connected graphs. We show an amortized lower bound of �(log n/k(log log n + log b)) per edge insertion, deletion, or query operation in the cell probe model, where b is the word size of the machine and n is the number of vertices in G. We also show an amortized lower bound of �(log n/(log log n + log b)) per operation for fully dynamic planarity testing in embedded graphs. These are the first lower bounds for fully dynamic connectivity problems.
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 ..."
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Cited by 28 (0 self)
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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
ONLINE PLANARITY TESTING
, 1996
"... The online planaritytesting problem consists of performing the following operations on a planar graph G: (i) testing if a new edge can be added to G so that the resulting graph is itself planar; (ii) adding vertices and edges such that planarity is preserved. An efficient technique for online plan ..."
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Cited by 24 (2 self)
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The online planaritytesting problem consists of performing the following operations on a planar graph G: (i) testing if a new edge can be added to G so that the resulting graph is itself planar; (ii) adding vertices and edges such that planarity is preserved. An efficient technique for online planarity testing of a graph is presented that uses O(n) space and supports tests and insertions of vertices and edges in O(log n) time, where n is the current number of vertices of G. The bounds for tests and vertex insertions are worstcase and the bound for edge insertions is amortized. We also present other applications of this technique to dynamic algorithms for planar graphs.
Searching Constant Width Mazes Captures the AC° Hierarchy
 In Proceedings of the 15th Annual Symposium on Theoretical Aspects of Computer Science
, 1997
"... We show that searching a width /' maze is complete for II, i.e., for the /"th level of the AC hierarchy. Equivalently, stconnectivity for width /' grid graphs is complete for II. As an application, we show that there is a data structure solving dynamic stconnectivity for constan ..."
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Cited by 24 (5 self)
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We show that searching a width /' maze is complete for II, i.e., for the /"th level of the AC hierarchy. Equivalently, stconnectivity for width /' grid graphs is complete for II. As an application, we show that there is a data structure solving dynamic stconnectivity for constant width grid graphs with time bound O (log log n) per operation on a random access machine. The dynamic algorithm is derived from the parallel one in an indirect way using algebraic tools.
Offline Algorithms for Dynamic Minimum Spanning Tree Problems
, 1994
"... We describe an efficient algorithm for maintaining a minimum spanning tree (MST) in a graph subject to a sequence of edge weight modifications. The sequence of minimum spanning trees is computed offline, after the sequence of modifications is known. The algorithm performs O(log n) work per modificat ..."
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Cited by 17 (9 self)
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We describe an efficient algorithm for maintaining a minimum spanning tree (MST) in a graph subject to a sequence of edge weight modifications. The sequence of minimum spanning trees is computed offline, after the sequence of modifications is known. The algorithm performs O(log n) work per modification, where n is the number of vertices in the graph. We use our techniques to solve the offline geometric MST problem for a planar point set subject to insertions and deletions; our algorithm for this problem performs O(log 2 n) work per modification. No previous dynamic geometric MST algorithm was known.
Finding the k Smallest Spanning Trees
, 1992
"... We give improved solutions for the problem of generating the k smallest spanning trees in a graph and in the plane. Our algorithm for general graphs takes time O(m log #(m, n)+k 2 ); for planar graphs this bound can be improved to O(n + k 2 ). We also show that the k best spanning trees for a set of ..."
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Cited by 17 (2 self)
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We give improved solutions for the problem of generating the k smallest spanning trees in a graph and in the plane. Our algorithm for general graphs takes time O(m log #(m, n)+k 2 ); for planar graphs this bound can be improved to O(n + k 2 ). We also show that the k best spanning trees for a set of points in the plane can be computed in time O(min(k 2 n + n log n, k 2 + kn log(n/k))). The k best orthogonal spanning trees in the plane can be found in time O(n log n + kn log log(n/k)+k 2 ).
A Linear Algorithm for Analysis of Minimum Spanning and Shortest Path Trees of Planar Graphs
 Algorithmica
, 1992
"... We give a linear time and space algorithm for analyzing trees in planar graphs. The algorithm can be used to analyze the sensitivity of a minimum spanning tree to changes in edge costs, to find its replacement edges, and to verify its minimality. It can also be used to analyze the sensitivity of a s ..."
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Cited by 16 (0 self)
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We give a linear time and space algorithm for analyzing trees in planar graphs. The algorithm can be used to analyze the sensitivity of a minimum spanning tree to changes in edge costs, to find its replacement edges, and to verify its minimality. It can also be used to analyze the sensitivity of a singlesource shortest path tree to changes in edge costs, and to analyze the sensitivity of a minimum cost network flow. The algorithm is simple and practical. It uses the properties of a planar embedding, combined with a heapordered queue data structure. Let G = (V; E) be a planar graph, either directed or undirected, with n vertices and m = O(n) edges. Each edge e 2 E has a realvalued cost cost(e). A minimum spanning tree of a connected, undirected planar graph G is a spanning tree of minimum total edge cost. If G is directed and r is a vertex from which all other vertices are reachable, then a shortest path tree from r is a spanning tree that contains a minimumcost path from r to every...
Kinetic Connectivity of Rectangles
 In Proc. 15th Annu. ACM Sympos. Comput. Geom
, 1999
"... We develop a kinetic data structure (KDS) for maintaining the connectivity of a set of axisaligned rectangles moving in the plane. In the kinetic framework, each rectangle is assumed to travel along a lowdegree algebraic path, specified by a flight planif the flight plan changes, the data struc ..."
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Cited by 15 (1 self)
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We develop a kinetic data structure (KDS) for maintaining the connectivity of a set of axisaligned rectangles moving in the plane. In the kinetic framework, each rectangle is assumed to travel along a lowdegree algebraic path, specified by a flight planif the flight plan changes, the data structure is informed about it. The connectivity of rectangles changes only at discrete moments, given by the times when the order of rectangles along either axis changes. Our main result is a kinetic data structure of size O(n log n) that requires O(log 2 n) amortized time for each update, and answers connectivity queries in worstcase time O(log n= log log n). 1 Introduction Connectivity is the most basic of graph properties, with many applications to realworld problems. Applications of connectivity range from electrical connectivity in integrated circuits to network connectivity in communication networks. In this paper, we explore the problem of maintaining the connectivity of n axisalig...
Experimental Analysis of Dynamic Minimum Spanning Tree Algorithms (Extended Abstract)
, 1997
"... ) Giuseppe Amato Giuseppe Cattaneo y Giuseppe F. Italiano z Abstract We conduct an extensive empirical study on the performance of several algorithms for maintaining the minimum spanning tree of a dynamic graph. In particular, we implemented and tested Frederickson's algorithms, and spa ..."
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Cited by 15 (2 self)
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) Giuseppe Amato Giuseppe Cattaneo y Giuseppe F. Italiano z Abstract We conduct an extensive empirical study on the performance of several algorithms for maintaining the minimum spanning tree of a dynamic graph. In particular, we implemented and tested Frederickson's algorithms, and sparsification on top of Frederickson's algorithms, and compared them to other dynamic algorithms. Moreover, we propose a variant of a dynamic algorithm by Frederickson, which was in our experience always faster than the other implementations derived from the papers. In our experiments, we considered both random and nonrandom inputs, with nonrandom inputs trying to enforce bad update patterns on the algorithms. For random inputs, a simple adaptation of a partially dynamic data structure on Kruskal's algorithm was the fastest implementation. For nonrandom inputs, sparsification yielded the fastest algorithm. In both cases, the performance of our variant of the algorithm of Frederickson was clos...
Dynamic Trees as Search Trees via Euler Tours, Applied to the Network Simplex Algorithm
 Mathematical Programming
, 1997
"... The dynamic tree is an abstract data type that allows the maintenance of a collection of trees subject to joining by adding edges (linking) and splitting by deleting edges (cutting), while at the same time allowing reporting of certain combinations of vertex or edge values. For many applications of ..."
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Cited by 14 (1 self)
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The dynamic tree is an abstract data type that allows the maintenance of a collection of trees subject to joining by adding edges (linking) and splitting by deleting edges (cutting), while at the same time allowing reporting of certain combinations of vertex or edge values. For many applications of dynamic trees, values must be combined along paths. For other applications, values must be combined over entire trees. For the latter situation, we show that an idea used originally in parallel graph algorithms, to represent trees by Euler tours, leads to a simple implementation with a time of O(log n) per tree operation, where n is the number of tree vertices. We apply this representation to the implementation of two versions of the network simplex algorithm, resulting in a time of O(log n) per pivot, where n is the number of vertices in the problem network.