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Bottom-Up Strategy

Let $D^{(i)} \;=\; W^i$ be the distance matrix after considering paths of length $\leq$ i.

$D^{(1)} \;=\; W$
$D^{(2)} \;=\; D^{(1)} \cdot W \;=\; W^2$
...
$D^{(n-1)} \;=\; D^{(n-2)} \cdot W \;=\; W^{n-1}$

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