Sparse Fault-Tolerant BFS Structures

Abstract

A fault-tolerant structure for a network is required for continued functioning following the failure of some of the network’s edges or vertices. This article considers breadth-first search (BFS) spanning trees and addresses the problem of designing a sparse fault-tolerant BFS structure (FT-BFS structure), namely, a sparse subgraph T of the given network G such that subsequent to the failure of a single edge or vertex, the surviving part T ′ of T still contains a BFS spanning tree for (the surviving part of) G . For a source node s , a target node t , and an edge e ∈ G , the shortest s − t path P s , t , e that does not go through e is known as a replacement path . Thus, our FT-BFS structure contains the collection of all replacement paths P s , t , e for every t ∈ V ( G ) and every failed edge e ∈ E ( G ). Our main results are as follows. We present an algorithm that for every n -vertex graph G and source node s constructs a (single edge failure) FT-BFS structure rooted at s with O ( n ċ min {Depth( s ), √n{) edges, where Depth( s ) is the depth of the BFS tree rooted at s . This result is complemented by a matching lower bound, showing that there exist n -vertex graphs with a source node s for which any edge (or vertex) FT-BFS structure rooted at s has Ω( n 3/2 ) edges. We then consider fault-tolerant multi-source BFS structures (FT-MBFS structures), aiming to provide (following a failure) a BFS tree rooted at each source s ∈ S for some subset of sources S ⊆ V . Again, tight bounds are provided, showing that there exists a poly-time algorithm that for every n -vertex graph and source set S ⊆ V of size σ constructs a (single failure) FT-MBFS structure T *( S ) from each source s i ∈ S , with O (√σ ċ n <sup;>3/2</sup;>) edges, and, on the other hand, there exist n -vertex graphs with source sets S ⊆ V of cardinality σ, on which any FT-MBFS structure from S has Ω(√σ ċ n 3/2 ) edges. Finally, we propose an O (log  n ) approximation algorithm for constructing FT-BFS and FT-MBFS structures. The latter is complemented by a hardness result stating that there exists no Ω(log  n ) approximation algorithm for these problems under standard complexity assumptions. In comparison with previous constructions, our algorithm is deterministic and may improve the number of edges by a factor of up to √ n for some instances. All our algorithms can be extended to deal with one vertex failure as well, with the same performance.