Design of fault-tolerant on-board networks with variable switch sizes

An (n, k, r)-network is a triple N = (G, in, out) where G = (V,E) is a graph and in, out are nonnegative integer functions defined on V called the input and output functions, such that for any v ∈ V , in(v)+out(v)+deg(v) ≤ 2r where deg(v) is the degree of v in the graph G. The total number of inputs is in(V ) = ∑ v∈V in(v) = n, and the total number of outputs is out(V ) = ∑ v∈V out(v) = n+ k. An (n, k, r)-network is valid, if for any faulty output function out′ (that is such that 0 ≤ out′(v) ≤ out(v) for any v ∈ V , and out′(V ) = n), there are n edge-disjoint paths in G such that each vertex v ∈ V is the initial vertex of in(v) paths and the terminal vertex of out′(v) paths. We investigate the design problem of determining the minimum number N (n, k, r) of vertices in a valid (n, k, r)-network and of constructing minimum (n, k, r)-networks, or at least valid (n, k, r)networks with a number of vertices close to the optimal value. We first give some upper bounds on N (n, k, r). We show N (n, k, r) ≤ ⌈ k+2 2r−2 ⌉ ⌈ n 2 ⌉ . When r ≥ k/2, we prove a better upper bound: N (n, k, r) ≤ r−2+k/2 r2−2r+k/2n+O(1). Next, we establish some lower bounds. We show that if k ≥ r, then N (n, k, r) ≥ 3n+k 2r . We improve this bound when k ≥ 2r: N (n, k, r) ≥ 3n+ 2k/3− r/2 2r − 2 + 3r b k r c . Finally, we determineN (n, k, r) up to additive constants for k ≤ 6.