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algorithms(bf-to-astar): mention time complexity in BF

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Signed-off-by: Matej Focko <me@mfocko.xyz>
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Matej Focko 2024-01-01 21:32:04 +01:00
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@ -387,6 +387,27 @@ possibly look it up, if necessary.
Also the correctness of the BF relies on the conclusion we've made when fixing Also the correctness of the BF relies on the conclusion we've made when fixing
the infinite-loop on our naïve BF solution. the infinite-loop on our naïve BF solution.
## Time complexity
Let's have a short look at the time complexities of the presented algorithms:
1. naïve approach: given that there are no negative loops, we are bound by the
worst-case ordering of the relaxations which results in
$$
\mathcal{O}(\vert V \vert \cdot \vert E \vert)
$$
2. our naïve approach with the fixed count of iterations instead of the
`do-while` loop results in the same worst-case time complexity:
$$
\mathcal{O}(\vert V \vert \cdot \vert E \vert)
$$
3. and finally the well-known Bellman-Ford's algorithm time complexity:
$$
\mathcal{O}(\vert V \vert \cdot \vert E \vert)
$$
## Small refactor ## Small refactor
Since we are literally copy-pasting the body of the loops just for the sake of Since we are literally copy-pasting the body of the loops just for the sake of