 Tentative distance

A tentative distance $d(v)$ is the cheapest cost so far recorded by a shortest path problem algorithm. For the single-source convention, initialize $d(s)=0$ and $d(v)=\infty$ elsewhere. Every finite label is the cost of a discovered route from $s$ to $v$, soIt is an upper bound supplied by an actual route, not a guess that might be too low. The loop invariant follows directly from edge relaxation: extending a discovered route to $u$ by $u\to v$ supplies another discovered route. The usual label initialization and this safety property are stated and proved in the authored MIT notes.^[ref]

In the trace under edge relaxation, $d(t)=10$ first records the direct directed path $s\to t$. Later $d(t)=8$ records $s\to a\to t$, and finally $d(t)=3$ records $s\to b\to a\to t$. Each label is a usable route cost, but the first two are not shortest. An infinite tentative distance means no route has yet been discovered; only a completed search establishes unreachability. The isolated vertex $z$ in that example remains at $\infty$.

In a directed graph, the discovered route is formally a directed walk, which may repeat vertices; in the pictured acyclic graph it is always a directed path. Negative-cost directed cycles can keep supplying cheaper directed walks without a finite optimum. In the all-to-one convention, the same meaning applies to discovered routes from $i$ to $r$. Do not confuse either version with the lower bounds supplied by feasible potentials in the Bellman inequalities: one records something achievable, the other rules out anything cheaper.