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Let us start by observing that after the $k$-th iteration in the main loop, the Bellman-Ford algorithm has computed minimal weight paths (or the weight of such a path if we do not store the predecessors) of length at most $k$ from the starting vertex $s$ to every other vertex of our graph $G$ (if such paths exists). To prove this, we can use induction: ...


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When solving most optimization problems on paths (e. g. shortest, longest path) a cycle is either useless or makes the optimum not exist (allowing infinitely long or short paths). Thus there's generally no point in allowing cycles.


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