Sign in
Get StudyX it's free
Sign in
Open in AppQUESTION
Text
Image
Professor Borden proposes a new divide-and-conquer algorithm for computing minimum spanning trees, which goes as follows. Given a graph \( G=(V, E) \), partition the set \( V \) of vertices into two sets \( V_{1} \) and \( V_{2} \) such that \( \left|V_{1}\right| \) and \( \left|V_{2}\right| \) differ>by at most 1. Let \( E_{1} \) be the set of edges that are incident only on vertices in \( V_{1} \), and let \( E_{2} \) be the set of edges that are incident only on vertices in \( V_{2} \). Recursively solve a minimum-spanning-tree problem on each of the two subgraphs \( G_{1}=\left(V_{1}, E_{1}\right) \) and \( G_{2}=\left(V_{2}, E_{2}\right) \). Finally, select the minimum-weight edge in \( E \) that crosses the cut \( \left(V_{1}, V_{2}\right) \), and use this edge to unite the resulting two minimum spanning trees into a single spanning tree.
Either argue that the algorithm correctly computes a minimum spanning tree of \( G \), or provide an example for which the algorithm fails.
Professor Borden proposes a new divide-and-conquer algorithm for computing minimum spanning trees, which goes as follows. Given a graph $G=(V, E)$, partition the set $V$ of vertices into two sets $V_{1}$ and $V_{2}$ such that $\left|V_{1}\right|$ and $\left|V_{2}\right|$ differ
by at most 1 . Let $E_{1}$ be the set of edges that are incident only on vertices in $V_{1}$, and let $E_{2}$ be the set of edges that are incident only on vertices in $V_{2}$. Recursively solve a minimum-spanning-tree problem on each of the two subgraphs $G_{1}=\left(V_{1}, E_{1}\right)$ and $G_{2}=\left(V_{2}, E_{2}\right)$. Finally, select the minimum-weight edge in $E$ that crosses the cut $\left(V_{1}, V_{2}\right)$, and use this edge to unite the resulting two minimum spanning trees into a single spanning tree. Either argue that the algorithm correctly computes a minimum spanning tree of $G$, or provide an example for which the algorithm fails.
Either argue that the algorithm correctly computes a minimum spanning tree of \( G \), or provide an example for which the algorithm fails.
Professor Borden proposes a new divide-and-conquer algorithm for computing minimum spanning trees, which goes as follows. Given a graph $G=(V, E)$, partition the set $V$ of vertices into two sets $V_{1}$ and $V_{2}$ such that $\left|V_{1}\right|$ and $\left|V_{2}\right|$ differ
by at most 1 . Let $E_{1}$ be the set of edges that are incident only on vertices in $V_{1}$, and let $E_{2}$ be the set of edges that are incident only on vertices in $V_{2}$. Recursively solve a minimum-spanning-tree problem on each of the two subgraphs $G_{1}=\left(V_{1}, E_{1}\right)$ and $G_{2}=\left(V_{2}, E_{2}\right)$. Finally, select the minimum-weight edge in $E$ that crosses the cut $\left(V_{1}, V_{2}\right)$, and use this edge to unite the resulting two minimum spanning trees into a single spanning tree. Either argue that the algorithm correctly computes a minimum spanning tree of $G$, or provide an example for which the algorithm fails.
Public Answer
WINEUX The First Answerer
