The problem

Let $G=(V,E)$ be a graph. $k = O\left(\log(|V|)\right)$ distinct vertices are picked randomly from $V$. We call the set of chosen $k$ vertices $T$.

We define the graph metric $d$ for every two vertices $x,y$ to be $d(x,y)$: The length of shortest path between $x$ and $y$.

For every $a,b \in T$, $d(a,b)$ is known. This is called the "global knowledge". Also for every $x \in V, a \in T$, the distance $d(x,a)$ is known. This is called the "local knowledge" for the vertex $x$.

For a vertex $x \in V$, The address of $x$, $A(x)$, is a set of shortest paths from $x$ to all the vertices in $T$.

A person should travel from $x$ to $y$. This is done by hopping from one vertex to one of its neighbours.

Information given:

• $A(y)$
• The global knowledge
• While staying at the vertex $z$, the local knowledge with respect to $z$ is known.

Find an algorithm to travel from $x$ to $y$. (It might be a probabilistic one). The following properties are required:

• The total length of the journey from $x$ to $y$ should not be more than $\log(|V|)\cdot d(x,y)$
• Assume that use the proposed algorithm to send $R$ people between random pairs $x,y \in V$. Let $J(x)$ be the amount of journeys passing through $x$. Then $J(x) \leq J(y)\cdot\log(|V|)$ with high probability when $R \rightarrow \infty$.

My attempts

I tried a few things, though I didn't have much progress. If you have any kind of idea, please post it.

Naive solution

The naive solution to go from $x$ to $y$ would be to use an intermediate vertex $t \in T$: First we travel from $x$ to $t$, and then as we know $A(y)$, we could follow a shortest path from $t$ to $y$.

This generally works, however the load on the vertices of $T$ will be too high, which means that the second requirement from the algorithm will not be satisfied.

Metric information

Using the basic properties of metrics, we can conclude the following: $\max_{a \in T} \left|d(a,y) - d(x,a)\right| \leq d(x,y) \leq \min_{a \in T}\left(d(x,a) + d(a,y)\right)$

This could give us some kind of approximation to how far we are from $y$ at any point of the journey, but this approximation is usually not so good.

Continuity

When hopping from a vertex $z$ to his neighbour $w$, one can prove that the distance to each of the vertices inside $T$ will change only by $+1$, $-1$ or $0$. I didn't manage to get further with this idea, but it is kind of interesting.

Embedding

If we manage to embed the graph in an euclidean space (Maybe of high dimension) so that all distances are preserved, we could use euclidean distance and have some better understanding of the geometry of the graph. Just an idea. I never managed to do anything of this sort.