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Offline algorithms for dynamic minimum spanning tree problems
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 (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
A Randomized Algorithm for Single-Source Shortest Path on Undirected Real-Weighted Graphs
In undirected graphs with real non-negative weights, we give a new randomized
algorithm for the single-source shortest path (SSSP) problem with running time
in the comparison-addition model. This is
the first algorithm to break the time bound for real-weighted
sparse graphs by Dijkstra's algorithm with Fibonacci heaps. Previous undirected
non-negative SSSP algorithms give time bound of in comparison-addition model, where is the
inverse-Ackermann function and is the ratio of the maximum-to-minimum edge
weight [Pettie & Ramachandran 2005], and linear time for integer edge weights
in RAM model [Thorup 1999]. Note that there is a proposed complexity lower
bound of for hierarchy-based
algorithms for undirected real-weighted SSSP [Pettie & Ramachandran 2005], but
our algorithm does not obey the properties required for that lower bound. As a
non-hierarchy-based approach, our algorithm shows great advantage with much
simpler structure, and is much easier to implement.Comment: 17 page
Augmenting graphs to minimize the diameter
We study the problem of augmenting a weighted graph by inserting edges of
bounded total cost while minimizing the diameter of the augmented graph. Our
main result is an FPT 4-approximation algorithm for the problem.Comment: 15 pages, 3 figure
Approximate Data Structures with Applications
In this paper we introduce the notion of approximate
data structures, in which a small amount of error is
tolerated in the output. Approximate data structures
trade error of approximation for faster operation, leading to theoretical and practical speedups for a wide variety of algorithms. We give approximate variants of the van Emde Boas data structure, which support the same dynamic operations as the standard van Emde Boas data structure [28, 201, except that answers to queries are approximate. The variants support all operations in constant time provided the error of approximation is l/polylog(n), and in O(loglog n) time provided the error
is l/polynomial(n), for n elements in the data structure.
We consider the tolerance of prototypical algorithms to approximate data structures. We study in particular Prim’s minimumspanning tree algorithm, Dijkstra’s single-source shortest paths algorithm, and an on-line variant of Graham’s convex hull algorithm. To obtain output which approximates the desired output
with the error of approximation tending to zero, Prim’s algorithm requires only linear time, Dijkstra’s algorithm requires O(mloglogn) time, and the on-line variant of Graham’s algorithm requires constant amortized time per operation
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