As our first example of a self-healing structure, we consider the special case of a simple line segment. As we recall from the previous section, starting with two distantly located endpoints, we wish to organize the intermediate processors such that a set of labelled processors lies on the shortest path between the two endpoints. The shortest path metric is determined by a gradient field which has been initiated by one of the endpoints.

In this example, we can use a number of properties of the line segment algorithm to help us maintain its connectivity. In order to be more precise about the problem we are trying to solve, we propose the following definitions:

Two identically labelled processors *a* and *b* are **pairwise
connected** *C*_{p} if a message can be transmitted from one endpoint to
the other only through pairs of connected elements. i.e. for

Processors labelled with `A-material` are responsible for
detecting the failure of a structure to maintain invariants after they
have been generated by the `A-to-B-segment` growing point. In
this case, the invariant to be maintained is the continuity of the
materials which connect point `a` to point `b`. The
natural candidates to inform the structure that a connection has been
broken are the materials themsleves.

Our algorithms will initially run on processors we know to be pairwise connected.