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Fast APSP

Note that $D^{(m)} \;=\; D^{(n-1)}$ for all m $\geq$ n-1.

$D^{(1)} \;=\; W$
$D^{(2)} \;=\; W^2 \;=\; W.W$
$D^{(4)} \;=\; W^4 \;=\; W^2.W^2$
$D^{(8)} \;=\; W^8 \;=\; W^4.W^4$

Continue until $D^{2^{\lceil lg(n-1) \rceil}}$ since $2^{\lceil lg(n-1) \rceil} \;\geq\; n-1$ .

This requires only $\lceil lg(n-1) \rceil$ matrix products (calls to Extend).

This is called ``repeated squaring''.

Fast-APSP(W)
$\;\;\;\;\;$ n = rows(W)
$\;\;\;\;\;$ D⁽¹⁾ = W
$\;\;\;\;\;$ m = 1
$\;\;\;\;\;$ while n-1 > m $\;\;\;\;\;$ $\;\;\;\;\;$ $\;\;\;\;\;$ $\;\;\;\;\;$ $\;\;\;\;\;$ $\;\;\;\;\;$ $\;\;\;\;\;$ $\;\;\;\;\;$ ; $\lg n$
$\;\;\;\;\;$ $\;\;\;\;\;$ D^(2m) = Extend-Shortest-Paths(D^(m), D^(m)) $\;\;\;\;\;$ $\;\;\;\;\;$ ; n³
$\;\;\;\;\;$ $\;\;\;\;\;$ m = 2m
$\;\;\;\;\;$ return D^(m)

Running time: $\Theta(n^3 \lg n)$

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