Results 1  10
of
16
Biconnectivity Approximations and Graph Carvings
, 1994
"... A spanning tree in a graph is the smallest connected spanning subgraph. Given a graph, how does one find the smallest (i.e., least number of edges) 2connected spanning subgraph (connectivity refers to both edge and vertex connectivity, if not specified) ? Unfortunately, the problem is known to be ..."
Abstract

Cited by 84 (3 self)
 Add to MetaCart
A spanning tree in a graph is the smallest connected spanning subgraph. Given a graph, how does one find the smallest (i.e., least number of edges) 2connected spanning subgraph (connectivity refers to both edge and vertex connectivity, if not specified) ? Unfortunately, the problem is known to be NP hard. We consider the problem of finding a better approximation to the smallest 2connected subgraph, by an efficient algorithm. For 2edge connectivity our algorithm guarantees a solution that is no more than 3 2 times the optimal. For 2vertex connectivity our algorithm guarantees a solution that is no more than 5 3 times the optimal. The previous best approximation factor is 2 for each of these problems. The new algorithms (and their analyses) depend upon a structure called a carving of a graph, which is of independent interest. We show that approximating the optimal solution to within an additive constant is NP hard as well. We also consider the case where the graph has edge weigh...
Balanced Matroids
"... We introduce the notion of "balance", and say that a matroid is balanced if the matroid and all its minors satisfy the property that, for a randomly chosen basis, the presence of an element can only make any other element less likely. We establish strong expansion properties for the base ..."
Abstract

Cited by 76 (3 self)
 Add to MetaCart
We introduce the notion of "balance", and say that a matroid is balanced if the matroid and all its minors satisfy the property that, for a randomly chosen basis, the presence of an element can only make any other element less likely. We establish strong expansion properties for the basesexchange graph of balanced matroids; consequently, the set of bases of a balanced matroid can be sampled and approximately counted using rapidly mixing Markov chains. Thus, the general problem of approximately counting bases (known to be #Pcomplete) is reduced to that of showing balance. Specific classes for which balance is known to hold include graphic and regular matroids.
Marked Ancestor Problems
, 1998
"... Consider a rooted tree whose nodes can be marked or unmarked. Given a node, we want to find its nearest marked ancestor. This generalises the wellknown predecessor problem, where the tree is a path. ..."
Abstract

Cited by 52 (7 self)
 Add to MetaCart
Consider a rooted tree whose nodes can be marked or unmarked. Given a node, we want to find its nearest marked ancestor. This generalises the wellknown predecessor problem, where the tree is a path.
DynFO: A Parallel, Dynamic Complexity Class
 Journal of Computer and System Sciences
, 1994
"... Traditionally, computational complexity has considered only static problems. Classical Complexity Classes such as NC, P, and NP are defined in terms of the complexity of checking  upon presentation of an entire input  whether the input satisfies a certain property. For many applications of compu ..."
Abstract

Cited by 50 (4 self)
 Add to MetaCart
Traditionally, computational complexity has considered only static problems. Classical Complexity Classes such as NC, P, and NP are defined in terms of the complexity of checking  upon presentation of an entire input  whether the input satisfies a certain property. For many applications of computers it is more appropriate to model the process as a dynamic one. There is a fairly large object being worked on over a period of time. The object is repeatedly modified by users and computations are performed. We develop a theory of Dynamic Complexity. We study the new complexity class, Dynamic FirstOrder Logic (DynFO). This is the set of properties that can be maintained and queried in firstorder logic, i.e. relational calculus, on a relational database. We show that many interesting properties are in DynFO including multiplication, graph connectivity, bipartiteness, and the computation of minimum spanning trees. Note that none of these problems is in static FO, and this f...
Minimizing Diameters of Dynamic Trees
 In Proc. 24th International Colloquium on Automata, Languages, and Programming (ICALP
, 1997
"... . In this paper we consider an online problem related to minimizing the diameter of a dynamic tree T . A new edge f is added, and our task is to delete the edge e of the induced cycle so as to minimize the diameter of the resulting tree T [ffgnfeg. Starting with a tree with n nodes, we show how e ..."
Abstract

Cited by 28 (12 self)
 Add to MetaCart
. In this paper we consider an online problem related to minimizing the diameter of a dynamic tree T . A new edge f is added, and our task is to delete the edge e of the induced cycle so as to minimize the diameter of the resulting tree T [ffgnfeg. Starting with a tree with n nodes, we show how each such best swap can be found in worstcase O(log 2 n) time. The problem was raised by Italiano and Ramaswami at ICALP'94 together with a related problem for edge deletions. Italiano and Ramaswami solved both problems in O(n) time per operation. 1 Introduction The diameter of a tree is the length of a longest simple path in the tree and such a path is called a diameter path. The unique midpoint on all diameter paths is called the center, hence the center is the point whose maximal distance to any node is as small as possible. In 1973 Handler [4] showed how one in linear time can compute the diameter (and center) of a tree. However, as pointed out by Rauch [8], too little work has been...
On Certificates and Lookahead in Dynamic Graph Problems
, 1996
"... Recent work in dynamic graph algorithms has led to efficient algorithms for dynamic undirected graph problems such as connectivity. However, no efficient deterministic algorithms are known for the dynamic versions of fundamental directed graph problems like strong connectivity and transitive closur ..."
Abstract

Cited by 17 (3 self)
 Add to MetaCart
Recent work in dynamic graph algorithms has led to efficient algorithms for dynamic undirected graph problems such as connectivity. However, no efficient deterministic algorithms are known for the dynamic versions of fundamental directed graph problems like strong connectivity and transitive closure, as well as some undirected graph problems such as maximum matchings and cuts. We provide some explanation for this lack of success by presenting quadratic lower bounds on the certificate complexity of the seemingly difficult problems, in contrast to the known linear certificate complexity for the problems which have efficient dynamic algorithms. A direct outcome of our lower bounds is the demonstration that a generic technique for designing efficient dynamic graph algorithms, viz., sparsification, will not apply to the difficult problems. More generally, it is our belief that the boundary between tractable and intractable dynamic graph problems can be demarcated in terms of certificate co...
Topology BTrees and Their Applications
"... . The wellknown Btree data structure provides a mechanism for dynamically maintaining balanced binary trees in external memory. We present an externalmemory dynamic data structure for maintaining arbitrary binary trees. Our data structure, which we call the topology Btree, is an externalmemory ..."
Abstract

Cited by 15 (0 self)
 Add to MetaCart
. The wellknown Btree data structure provides a mechanism for dynamically maintaining balanced binary trees in external memory. We present an externalmemory dynamic data structure for maintaining arbitrary binary trees. Our data structure, which we call the topology Btree, is an externalmemory analogue to the internalmemory topology tree data structure of Frederickson. It allows for dynamic expression evaluation and updates as well as various tree searching and evaluation queries. We show how to apply this data structure to a number of externalmemory dynamic problems, including approximate nearestneighbor searching and closestpair maintenance. 1 Introduction The Btree [8, 12, 14, 15] data structure is a very efficient and powerful way for maintaining balanced binary trees in external memory [1, 11, 13, 18, 19, 21, 22, 2]. Indeed, in his wellknown survey paper [8], Comer calls Btrees "ubiquitous," for they are found in a host of different applications. Nevertheless, there ar...
Finding All the Best Swaps of a Minimum Diameter Spanning Tree Under Transient Edge Failures
 Journal of Graph Algorithms and Applications
, 1998
"... Abstract. In network communication systems, frequently messages are routed along a minimum diameter spanning tree (MDST) of the network, to minimize the maximum delay in delivering a message. When a transient edge failure occurs, it is important to choose a temporary replacement edge which minimizes ..."
Abstract

Cited by 14 (6 self)
 Add to MetaCart
Abstract. In network communication systems, frequently messages are routed along a minimum diameter spanning tree (MDST) of the network, to minimize the maximum delay in delivering a message. When a transient edge failure occurs, it is important to choose a temporary replacement edge which minimizes the diameter of the new spanning tree. Such an optimal replacement is called the best swap. As a natural extension, the allbestswaps (ABS) problem is the problem of finding the best swap for every edge of the MDST. Given a weighted graph G =(V, E), where V  = n and E  = m,wesolvetheABSprobleminO(n √ m)time and O(m + n) space, thus improving previous bounds for m = o(n 2). 1
Maintaining information in fullydynamic trees with top trees
 ACM Transactions on Algorithms
, 2003
"... We introduce top trees as a design of a new simpler interface for data structures maintaining information in a fullydynamic forest. We demonstrate how easy and versatile they are to use on a host of different applications. For example, we show how to maintain the diameter, center, and median of eac ..."
Abstract

Cited by 12 (0 self)
 Add to MetaCart
We introduce top trees as a design of a new simpler interface for data structures maintaining information in a fullydynamic forest. We demonstrate how easy and versatile they are to use on a host of different applications. For example, we show how to maintain the diameter, center, and median of each tree in the forest. The forest can be updated by insertion and deletion of edges and by changes to vertex and edge weights. Each update is supported in O(log n) time, where n is the size of the tree(s) involved in the update. Also, we show how to support nearest common ancestor queries and level ancestor queries with respect to arbitrary roots in O(log n) time. Finally, with marked and unmarked vertices, we show how to compute distances to a nearest marked vertex. The later has applications to approximate nearest marked vertex in general graphs, and thereby to static optimization problems over shortest path metrics. Technically speaking, top trees are easily implemented either with Frederickson’s topology trees [Ambivalent Data Structures for Dynamic 2EdgeConnectivity and k Smallest Spanning Trees, SIAM J. Comput. 26 (2) pp. 484–538, 1997] or with Sleator and Tarjan’s dynamic
Using Sparsification for Parametric Minimum Spanning Tree Problems
 Nordic J. Computing
, 1996
"... Two applications of sparsification to parametric computing are given. The first is a fast algorithm for enumerating all distinct minimum spanning trees in a graph whose edge weights vary linearly with a parameter. The second is an asymptotically optimal algorithm for the minimum ratio spanning t ..."
Abstract

Cited by 7 (2 self)
 Add to MetaCart
Two applications of sparsification to parametric computing are given. The first is a fast algorithm for enumerating all distinct minimum spanning trees in a graph whose edge weights vary linearly with a parameter. The second is an asymptotically optimal algorithm for the minimum ratio spanning tree problem, as well as other search problems, on dense graphs. 1 Introduction In the parametric minimum spanning tree problem, one is given an nnode, medge undirected graph G where each edge e has a linear weight function w e (#)=a e +#b e . Let Z(#) denote the weight of the minimum spanning tree relative to the weights w e (#). It can be shown that Z(#) is a piecewise linear concave function of # [Gus80]; the points at which the slope of Z changes are called breakpoints. We shall present two results regarding parametric minimum spanning trees. First, we show that Z(#) can be constructed in O(min{nm log n, TMST (2n, n) # Department of Computer Science, Iowa State University, Ames, IA...