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Dijkstra Algorithm<br />
• Dijkstra does not require acyclicity<br />
•<br />
•<br />
instead of topological order, we use best-first order<br />
but this requires superiority of the semiring<br />
Let K =(A, ⊕, ⊗, 0, 1) be a semiring, and ≤ a partial ordering over A.<br />
We say K is superior if for all a, b ∈ A<br />
d(u)<br />
• basically, combination always gets worse<br />
• or, no negative edge in a graph<br />
w(e)<br />
a ≤ a ⊗ b, b ≤ a ⊗ b.<br />
d(u) ⊗ w(e)<br />
({0, 1}, ∨, ∧, 0, 1)<br />
([0, 1], max, ×, 0, 1)<br />
(R + ∪{+∞}, min, +, +∞, 0)<br />
(R ∪{+∞}, min, +, +∞, 0)<br />
Liang Huang<br />
18<br />
Dynamic Programming