The edge e is in the set (S_{(u,langle i rangle )}), if and only if there is a path (P in mathcal {S}_{s,(u,langle i rangle )}) such that (e in P).
The central notion in evolving graph theory is the restriction imposed upon paths to traverse arcs strictly in non-decreasing order of arc schedule times, implying that there are no paths in (mathcal{G} ) going to the "past".
Also note that for any two different paths (mu, nu in mathcal {P}_{mathrm {min}} v,H)), we have (mu ^* nu = 0 ) in (L_mathbb {K}(E)).
That is because any path that contains an edge in (mathcal {E} backslash mathcal {E}_M) is not a shortest path.
We will show that (tin[0,1]mapsto C_{varphi_{t}}) is a continuous path in (mathcal{C}(A_{alpha}^{2}(Pi^))).
For a given (x_{0} in mathcal {I}), let (c_{x_{0}}) denote the execution path the program takes for processing x 0. Intuitively, we would assign a high-quality rating to the new input x 0 if it drives the targeted program to a previously undiscovered execution path, i.e., if (c_{x_{0}}) differs significantly from all previously explored execution paths ({c_{x} | x in mathcal {I}'}).
Also, for all (s,t in V), let begin{aligned} mathcal {S}_{s,t} := arg min { mathbf {t}(P) : P in mathcal {P}_{s,t} } end{aligned}when (mathcal {P}_{s,t}) is a set of all paths from s to t and (arg min) returns a set of paths such that all members of the set minimize the value of (mathbf {t}(P)).
Note that any path ending in (v_0) can be written uniquely as (gamma mu _0^k) for some (gamma in mathcal {P}) and (k in mathbb {N}_0).
Since there exists a path from source S i to destination D j, i≠j, i,j∈{1,…,K}, we can find a node (mathsf {V}^{ast } in mathcal {V}) such that (mathsf {V}^{ast } in mathcal {G}_{ij}) and (mathsf {V}^{ast } in mathcal {G}_{jj}), see Fig. 18.
