Added section on path bottlenck (it was accidentally removed) to the …

…report
frangente · Feb 13, 2024 · 0b5c512 · 0b5c512
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diff --git a/report/report.pdf b/report/report.pdf
diff --git a/report/src/sections/method.tex b/report/src/sections/method.tex
@@ -138,3 +138,5 @@ \subsection{Planning}
     \caption{An example of path bottleneck. The tiles in yellow belong to the path bottleneck from the start position to the end tile.}
     \label{fig:path-bottleneck}
 \end{figure}
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+The recomputed paths are then stored in a cache to be reused in the future. However, given the highly dynamic nature of the environment, an invalidation mechanism has been implemented to remove from the cache the paths no longer valid. To this end, the concept of path bottleneck is introduced. When moving from the current position to the destination tile, the set of all tiles that the agent must necessarily traverse to reach the destination in the shortest time is called the path bottleneck (see Figure~\ref{fig:path-bottleneck}). In other words, the path bottleneck is equal to the intersection of all the shortest paths from the current position to the destination. Given such definition, it is easy to see that the path recomputation is necessary only when the path bottleneck is no longer free. Thus, as soon as the path bottleneck becomes free again, the recomputed path can be discarded and the Seidel's cache can be used again.