Optimal fault-tolerant routings with small routing tables for k-connected graphs

Abstract We study the problem of designing fault-tolerant routings with small routing tables for a k -connected network of n processors in the surviving route graph model. The surviving route graph R ( G , ρ )/ F for a graph G , a routing ρ and a set of faults F is a directed graph consisting of nonfaulty nodes of G with a directed edge from a node x to a node y iff there are no faults on the route from x to y . The diameter of the surviving route graph could be one of the fault-tolerance measures for the graph G and the routing ρ and it is denoted by D ( R ( G , ρ )/ F ). We want to reduce the total number of routes defined in the routing, and the maximum of the number of routes defined for a node (called route degree) as least as possible. In this paper, we show that we can construct a routing λ for every n -node k -connected graph such that n ⩾2 k 2 , in which the route degree is O (k n ) , the total number of routes is O( k 2 n ) and D ( R ( G , λ )/ F )⩽3 for any fault set F (|F| . In particular, in the case that k =2 we can construct a routing λ ′ for every biconnected graph in which the route degree is O ( n ) , the total number of routes is O( n ) and D ( R ( G , λ ′)/{ f })⩽3 for any fault f . We also show that we can construct a routing ρ 1 for every n -node biconnected graph, in which the total number of routes is O( n ) and D ( R ( G , ρ 1 )/{ f })⩽2 for any fault f , and a routing ρ 2 (using ρ 1 ) for every n -node biconnected graph, in which the route degree is O ( n ) , the total number of routes is O (n n ) and D ( R ( G , ρ 2 )/{ f })⩽2 for any fault f .