$5\mathrm{.}22.$ You are given a graph $G=(V,E)$ with positive edge Note, you are given the tree $T$ and the edge $e=(y,z)$ whose weight is changed; you are told it's

old weight $w(e)$ and it's new weight widehat$(w)(e)($which you type in latex by Iwidehat $\{w\}(e)$ surrounded by

double dollar signs$).$

In each case specify if the tree might change. And if it might change then give an algorithm to find

the weight of the possibly new MST $($just return the weight or the MST$,$ whatever's easier$).$ You can

use the algorithms from class $($DFS$,$ Explore, BFS$,$ Dijkstra's, SCC$,$ Topological Sort$)$ as black$-$box

algorithms if you'd like. Explain your algorithm in words. Note the running time of your algorithm

in terms of $n$ and $m.$

Part $($a$)$: $e!$inT and widehat$(w)(e)>w(e)$ :weights, and a minimum spanning tree$5\mathrm{.}22.$ You are given a graph $G=(V,E)$ with positive edge weights, and a minimum spanning tree

$T=(V,E^{\text{'}})$ with respect to these weights; you may assume $G$ and $T$ are given as adjacency lists.

Now suppose the weight of a particular edge einE is modified from $w(e)$ to a new value hat$(w)(e).$ You

wish to quickly update the minimum spanning tree $T$ to reflect this change, without recomputing

the entire tree from scratch. There are four cases. In each case give a linear$-$time algorithm for

updating the tree.

$($a$)e!in{E}^{\text{'}}$ and hat$(w)(e)>w(e).$

$($b$)e!in{E}^{\text{'}}$ and hat$(w)(e)$

$T=(V,E^{\text{'}})$ with respect to these weights; you may assume $G$ and $T$ are given as adjacency lists.

Now suppose the weight of a particular edge einE is modified from $w(e)$ to a new value hat$(w)(e).$ You

wish to quickly update the minimum spanning tree $T$ to reflect this change, without recomputing

the entire tree from scratch. There are four cases. In each case give a linear$-$time algorithm for

updating the tree.

$($a$)e!in{E}^{\text{'}}$ and hat$(w)(e)>w(e).$

$($b$)e!in{E}^{\text{'}}$ and $ein{E}^{\text{'}}ein{E}^{\text{'}}$hat$(w)(e)>w(e)hat(w)(e).$

$(d)ein{E}^{\text{'}}$ and hat$(w)(e)>w(e).$hat$(w)(e).$

$(c)ein{E}^{\text{'}}$ and hat$(w)(e).$

$(d)ein{E}^{\text{'}}$ and hat$(w)(e)>w(e)\mathrm{.}5\mathrm{.}22.$ You are given a graph $G=(V,E)$ with positive edge weights, and a minimum spanning tree

$T=(V,E^{\text{'}})$ with respect to these weights; you may assume $G$ and $T$ are given as adjacency lists.

Now suppose the weight of a particular edge einE is modified from $w(e)$ to a new value hat$(w)(e).$ You

wish to quickly update the minimum spanning tree $T$ to reflect this change, without recomputing

the entire tree from scratch. There are four cases. In each case give a linear$-$time algorithm for

updating the tree.

$($a$)e!in{E}^{\text{'}}$ and hat$(w)(e)>w(e).$

$($b$)e!in{E}^{\text{'}}$ and $ein{E}^{\text{'}}ein{E}^{\text{'}}$hat$(w)(e)>w(e)hat(w)(e).$

$(d)ein{E}^{\text{'}}$ and hat$(w)(e)>w(e).$hat$(w)(e).$

$(c)ein{E}^{\text{'}}$ and hat$(w)(e).$

$(d)ein{E}^{\text{'}}$ and hat$(w)(e)>w(e).$Note, you are given the tree $T$ and the edge $e=(y,z)$ whose weight is changed; you are told it's

old weight $w(e)$ and it's new weight widehat$(w)(e)($which you type in latex by Iwidehat $\{w\}(e)$ surrounded by

double dollar signs$).$

In each case specify if the tree might change. And if it might change then give an algorithm to find

the weight of the possibly new MST $($just return the weight or the MST$,$ whatever's easier$).$ You can

use the algorithms from class $($DFS$,$ Explore, BFS$,$ Dijkstra's, SCC$,$ Topological Sort$)$ as black$-$box

algorithms if you'd like. Explain your algorithm in words. Note the running time of your algorithm

in terms of $n$ and $m.$

Part $($a$)$: $e!$inT and widehat$(w)(e)>w(e)$ :