Results 1  10
of
11
An optimal minimum spanning tree algorithm
 J. ACM
, 2000
"... Abstract. We establish that the algorithmic complexity of the minimum spanning tree problem is equal to its decisiontree complexity. Specifically, we present a deterministic algorithm to find a minimum spanning tree of a graph with n vertices and m edges that runs in time O(T ∗ (m, n)) where T ∗ is ..."
Abstract

Cited by 58 (11 self)
 Add to MetaCart
Abstract. We establish that the algorithmic complexity of the minimum spanning tree problem is equal to its decisiontree complexity. Specifically, we present a deterministic algorithm to find a minimum spanning tree of a graph with n vertices and m edges that runs in time O(T ∗ (m, n)) where T ∗ is the minimum number of edgeweight comparisons needed to determine the solution. The algorithm is quite simple and can be implemented on a pointer machine. Although our time bound is optimal, the exact function describing it is not known at present. The current best bounds known for T ∗ are T ∗ (m, n) = �(m) and T ∗ (m, n) = O(m · α(m, n)), where α is a certain natural inverse of Ackermann’s function. Even under the assumption that T ∗ is superlinear, we show that if the input graph is selected from Gn,m, our algorithm runs in linear time with high probability, regardless of n, m, or the permutation of edge weights. The analysis uses a new martingale for Gn,m similar to the edgeexposure martingale for Gn,p.
A shortest path algorithm for realweighted undirected graphs
 in 13th ACMSIAM Symp. on Discrete Algs
, 1985
"... Abstract. We present a new scheme for computing shortest paths on realweighted undirected graphs in the fundamental comparisonaddition model. In an efficient preprocessing phase our algorithm creates a linearsize structure that facilitates singlesource shortest path computations in O(m log α) ti ..."
Abstract

Cited by 16 (4 self)
 Add to MetaCart
Abstract. We present a new scheme for computing shortest paths on realweighted undirected graphs in the fundamental comparisonaddition model. In an efficient preprocessing phase our algorithm creates a linearsize structure that facilitates singlesource shortest path computations in O(m log α) time, where α = α(m, n) is the very slowly growing inverseAckermann function, m the number of edges, and n the number of vertices. As special cases our algorithm implies new bounds on both the allpairs and singlesource shortest paths problems. We solve the allpairs problem in O(mnlog α(m, n)) time and, if the ratio between the maximum and minimum edge lengths is bounded by n (log n)O(1) , we can solve the singlesource problem in O(m + nlog log n) time. Both these results are theoretical improvements over Dijkstra’s algorithm, which was the previous best for real weighted undirected graphs. Our algorithm takes the hierarchybased approach invented by Thorup. Key words. singlesource shortest paths, allpairs shortest paths, undirected graphs, Dijkstra’s
Optimal resilient dynamic dictionaries
 IN PROCEEDINGS OF 15TH EUROPEAN SYMPOSIUM ON ALGORITHMS (ESA
, 2007
"... We investigate the problem of computing in the presence of faults that may arbitrarily (i.e., adversarially) corrupt memory locations. In the faulty memory model, any memory cell can get corrupted at any time, and corrupted cells cannot be distinguished from uncorrupted ones. An upper bound δ on the ..."
Abstract

Cited by 12 (8 self)
 Add to MetaCart
(Show Context)
We investigate the problem of computing in the presence of faults that may arbitrarily (i.e., adversarially) corrupt memory locations. In the faulty memory model, any memory cell can get corrupted at any time, and corrupted cells cannot be distinguished from uncorrupted ones. An upper bound δ on the number of corruptions and O(1) reliable memory cells are provided. In this model, we focus on the design of resilient dictionaries, i.e., dictionaries which are able to operate correctly (at least) on the set of uncorrupted keys. We first present a simple resilient dynamic search tree, based on random sampling, with O(log n+δ) expected amortized cost per operation, and O(n) space complexity. We then propose an optimal deterministic static dictionary supporting searches in Θ(log n+δ) time in the worst case, and we show how to use it in a dynamic setting in order to support updates in O(log n + δ) amortized time. Our dynamic dictionary also supports range queries in O(log n+δ+t) worst case time, where t is the size of the output. Finally, we show that every resilient search tree (with some reasonable properties) must take Ω(log n + δ) worstcase time per search.
Sensitivity Analysis of Minimum Spanning Trees in SubInverseAckermann Time
, 2015
"... We present a deterministic algorithm for computing the sensitivity of a minimum spanning tree (MST) or shortest path tree in O(m logα(m,n)) time, where α is the inverseAckermann function. This improves upon a long standing bound of O(mα(m,n)) established by Tarjan. Our algorithms are based on an e ..."
Abstract

Cited by 8 (4 self)
 Add to MetaCart
We present a deterministic algorithm for computing the sensitivity of a minimum spanning tree (MST) or shortest path tree in O(m logα(m,n)) time, where α is the inverseAckermann function. This improves upon a long standing bound of O(mα(m,n)) established by Tarjan. Our algorithms are based on an efficient splitfindmin data structure, which maintains a collection of sequences of weighted elements that may be split into smaller subsequences. As far as we are aware, our splitfindmin algorithm is the first with superlinear but subinverseAckermann complexity. We also give a reduction from MST sensitivity to the MST problem itself. Together with the randomized linear time MST algorithm of Karger, Klein, and Tarjan, this gives another randomized linear time MST sensitivity algorithm.
An InverseAckermann Type Lower Bound for Online Minimum Spanning Tree Verification
 Combinatorica
"... Given a spanning tree T of some graph G, the problem of minimum spanning tree verication is to decide whether T = MST(G). A celebrated result of Komlos shows that this problem can be solved in linear time. Somewhat unexpectedly, MST verication turns out to be useful in actually computing minimum spa ..."
Abstract

Cited by 5 (3 self)
 Add to MetaCart
(Show Context)
Given a spanning tree T of some graph G, the problem of minimum spanning tree verication is to decide whether T = MST(G). A celebrated result of Komlos shows that this problem can be solved in linear time. Somewhat unexpectedly, MST verication turns out to be useful in actually computing minimum spanning trees from scratch. It is this application that has led some to wonder whether a more flexible version of MST Verification could be used to derive a faster deterministic minimum spanning tree algorithm.
Randomized Minimum Spanning Tree Algorithms Using Exponentially Fewer Random Bits
"... For many fundamental problems there exist randomized algorithms that are asymptotically optimal and are superior to the best known deterministic algorithm. Among these are the minimum spanning tree (MST) problem, the MST sensitivity analysis problem, the parallel connected components and parallel mi ..."
Abstract

Cited by 5 (1 self)
 Add to MetaCart
For many fundamental problems there exist randomized algorithms that are asymptotically optimal and are superior to the best known deterministic algorithm. Among these are the minimum spanning tree (MST) problem, the MST sensitivity analysis problem, the parallel connected components and parallel minimum spanning tree problems, and the local sorting and set maxima problems. (For the first two problems there are provably optimal deterministic algorithms with unknown, and possibly superlinear running times.) One downside of the randomized methods for solving these problems is that they use a number of random bits linear in the size of the input. In this paper we develop some general methods for reducing exponentially the consumption of random bits in comparison based algorithms. In some cases we are able to reduce the number of random bits from linear to nearly constant without affecting the expected running time. Most of our results are obtained by adjusting or reorganizing existing randomized algorithms to work well with a pairwise or O(1)wise independent sampler. The prominent exception — and the main focus of this paper — is a lineartime randomized minimum spanning tree algorithm that is not derived from the well known KargerKleinTarjan algorithm. In many ways it resembles more closely the deterministic minimum spanning tree algorithms based on Soft Heaps. Further,
An InverseAckermann Style Lower Bound for Online Minimum Spanning Tree Verification
 Combinatorica
"... 1 Introduction The minimum spanning tree (MST) problem has seen a flurry of activity lately, driven largely by the success of a new approach to the problem. The recent MST algorithms [20, 8, 29, 28], despite their superficial differences, are all based on the idea of progressively improving an appro ..."
Abstract

Cited by 3 (2 self)
 Add to MetaCart
(Show Context)
1 Introduction The minimum spanning tree (MST) problem has seen a flurry of activity lately, driven largely by the success of a new approach to the problem. The recent MST algorithms [20, 8, 29, 28], despite their superficial differences, are all based on the idea of progressively improving an approximately minimum solution, until the actual minimum spanning tree is found. It is still likely that this progressive improvement approach will bear fruit. However, the current
Data Structures: Sequence Problems, Range Queries and Fault Tolerance
, 2010
"... The focus of this dissertation is on algorithms, in particular data structures that give provably efficient solutions for sequence analysis problems, range queries, and fault tolerant computing. The work presented in this dissertation is divided into three parts. In Part I we consider algorithms for ..."
Abstract

Cited by 1 (0 self)
 Add to MetaCart
The focus of this dissertation is on algorithms, in particular data structures that give provably efficient solutions for sequence analysis problems, range queries, and fault tolerant computing. The work presented in this dissertation is divided into three parts. In Part I we consider algorithms for a range of sequence analysis problems that have risen from applications in pattern matching, bioinformatics, and data mining. On a high level, each problem is defined by a function and some constraints and the job at hand is to locate subsequences that score high with this function and are not invalidated by the constraints. Many variants and similar problems have been proposed leading to several different approaches and algorithms. We consider problems where the function is the sum of the elements in the sequence and the constraints only bound the length of the subsequences considered. We give optimal algorithms for several variants of the problem based on a simple idea and classic algorithms and data structures. In Part II we consider range query data structures. This a category of
On Cartesian Trees and Range Minimum Queries
"... We present new results on Cartesian trees with applications in range minimum queries and bottleneck edge queries. We introduce a cacheoblivious Cartesian tree for solving the range minimum query problem, a Cartesian tree of a tree for the bottleneck edge query problem on trees and undirected graphs, ..."
Abstract
 Add to MetaCart
(Show Context)
We present new results on Cartesian trees with applications in range minimum queries and bottleneck edge queries. We introduce a cacheoblivious Cartesian tree for solving the range minimum query problem, a Cartesian tree of a tree for the bottleneck edge query problem on trees and undirected graphs, and a proof that no Cartesian tree exists for the twodimensional version of the range minimum query problem. 1