297 research outputs found
Sparse Fault-Tolerant BFS Trees
This paper addresses the problem of designing a sparse {\em fault-tolerant}
BFS tree, or {\em FT-BFS tree} for short, namely, a sparse subgraph of the
given network such that subsequent to the failure of a single edge or
vertex, the surviving part of still contains a BFS spanning tree for
(the surviving part of) . Our main results are as follows. We present an
algorithm that for every -vertex graph and source node constructs a
(single edge failure) FT-BFS tree rooted at with O(n \cdot
\min\{\Depth(s), \sqrt{n}\}) edges, where \Depth(s) is the depth of the BFS
tree rooted at . This result is complemented by a matching lower bound,
showing that there exist -vertex graphs with a source node for which any
edge (or vertex) FT-BFS tree rooted at has edges. We then
consider {\em fault-tolerant multi-source BFS trees}, or {\em FT-MBFS trees}
for short, aiming to provide (following a failure) a BFS tree rooted at each
source for some subset of sources . Again, tight bounds
are provided, showing that there exists a poly-time algorithm that for every
-vertex graph and source set of size constructs a
(single failure) FT-MBFS tree from each source , with
edges, and on the other hand there exist
-vertex graphs with source sets of cardinality , on
which any FT-MBFS tree from has edges.
Finally, we propose an approximation algorithm for constructing
FT-BFS and FT-MBFS structures. The latter is complemented by a hardness result
stating that there exists no approximation algorithm for these
problems under standard complexity assumptions
Towards Resistance Sparsifiers
We study resistance sparsification of graphs, in which the goal is to find a
sparse subgraph (with reweighted edges) that approximately preserves the
effective resistances between every pair of nodes. We show that every dense
regular expander admits a -resistance sparsifier of size , and conjecture this bound holds for all graphs on nodes. In
comparison, spectral sparsification is a strictly stronger notion and requires
edges even on the complete graph.
Our approach leads to the following structural question on graphs: Does every
dense regular expander contain a sparse regular expander as a subgraph? Our
main technical contribution, which may of independent interest, is a positive
answer to this question in a certain setting of parameters. Combining this with
a recent result of von Luxburg, Radl, and Hein~(JMLR, 2014) leads to the
aforementioned resistance sparsifiers
Edge-disjoint spanners in Cartesian products of graphs
AbstractA spanning subgraph S=(V,E′) of a connected graph G=(V,E) is an (x+c)-spanner if for any pair of vertices u and v, dS(u,v)⩽dG(u,v)+c where dG and dS are the usual distance functions in G and S, respectively. The parameter c is called the delay of the spanner. We study edge-disjoint spanners in graphs, focusing on graphs formed as Cartesian products. Our approach is to construct sets of edge-disjoint spanners in a product based on sets of edge-disjoint spanners and colorings of the component graphs. We present several results on general products and then narrow our focus to hypercubes
Improved Deterministic Distributed Construction of Spanners
Graph spanners are fundamental graph structures with a wide range of applications in distributed networks. We consider a standard synchronous message passing model where in each round O(log n) bits can be transmitted over every edge (the CONGEST model).
The state of the art of deterministic distributed spanner constructions suffers from large messages. The only exception is the work of Derbel et al., which computes an optimal-sized (2k-1)-spanner but uses O(n^(1-1/k)) rounds.
In this paper, we significantly improve this bound. We present a deterministic distributed algorithm that given an unweighted n-vertex graph G = (V,E) and a parameter k > 2, constructs a (2k-1)-spanner with O(k n^(1+1/k)) edges within O(2^k n^(1/2 - 1/k)) rounds for every even k. For odd k, the number of rounds is O(2^k n^(1/2 - 1/(2k))). For the weighted case, we provide the first deterministic construction of a 3-spanner with O(n^(3/2)) edges that uses O(log n)-size messages and ~O(1) rounds. If the vertices have IDs in [1,Theta(n)], the spanner is computed in only 2 rounds
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