In this paper, we quantify the amount of "practical" information (i.e. views obtained from the neighbors, colors attributed to the nodes and links) to obtain "theoretical" information (i.e. the local topology of the network up to distance k) in anonymous networks. In more details, we show that a coloring at distance 2k + 1 is necessary and sufficient to obtain the local topology at distance A; that includes outgoing links. This bound drops to 2k when outgoing links are not needed. A second contribution of this paper deals with color bootstrapping (from which local topology can be obtained using the aforementioned mechanisms). On the negative side, we show that (ⅰ) with a distributed daemon, it is impossible to achieve deterministic color bootstrap, even if the whole network topology can be instantaneously obtained, and (ⅱ) with a central daemon, it is impossible to achieve distance m when instantaneous topology knowledge is limited to m — 1. On the positive side, we show that (ⅰ) under the k-central daemon, deterministic self-stabilizing bootstrap of colors up to distance k is possible provided that k-local topology can be instantaneously obtained, and (ⅱ) under the distributed daemon, probabilistic self-stabilizing bootstrap is possible for any range.
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