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If you're required to solve this "one-shot" problem, you're not allowed to perform any pre-processing, and you don't have any additional assumption, then a running time of $O(m)$ is the best you can hope for. Indeed, consider an edge $e$ such that the fundamental cut $C_e$ induced by $e$ contains $\Theta(m)$ edges. All edges in $C_e$ have distinct ...

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Let $n=|V|$ and $m=|E|$. Intuitively you want the to return the union of the edges in 1) a maximal spanning forest $F$ of the graph induced by the edges of weight $1$, with 2) a maximal spanning forest $F'$ of the graph obtained by identifying the edges of each tree in $F$ into a single vertex (where each edge in $F'$ actually represents an edge of $G$). ...

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It looks like you misread or misunderstand the standard set notation "$E'\subseteq E$". That notation just means $E'$ is a subset of $E$, i.e. every element of $E'$ is also an element of $E$. We do not speak of "an improper subset", most of the time if not ever. Why? Suppose $A$ and $B$ are two sets. $A$ is a subset of $B$ iff every ...

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BFS (or Breadth-First Search) from its name implies that it will search for the destination along the breadth. That means it looks at all the current possible routes ie nodes connected to the current one and divides itself into all of those. Basically, at every junction, it divides itself and continues down that path until it encounters the destination node. ...

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