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 non-negative 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∈Vin(v)=n, and the total number of outputs is out(V)=∑v∈Vout(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+22r−2⌉⌈n2⌉. When r≥k/2, we prove a better upper bound: N(n,k,r)≤r−2+k/2r2−2r+k/2n+O(1). Next, we establish some lower bounds. We show that if k≥r, then N(n,k,r)≥3n+k2r. We improve this bound when k≥2r: N(n,k,r)≥3n+2k/3−r/22r−2+3r⌊kr⌋. Finally, we determine N(n,k,r) up to additive constants for k≤6.