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Consider the set of all spanning trees (not just minimum ones) of a weighted, connected, undirected graph G = (V,E). Recall that adding an edge e to a spanning tree T creates an unique cycle, and subsequently removing any other edge e0 6= e from this cycle gives back a different spanning tree T0. We will say that T and T0 differ by a single edge swap (e, e0) and that they are neighbors. (a) Show that it is possible to move from any spanning tree T to any other spanning tree T0 by performing a series of edge-swaps, that is, by moving from neighbor to neighbor. At most how many edge-swaps are needed? (b) Show that if T0 is an MST, then it is possible to choose these swaps so that the costs of the spanning trees encountered along the way are non-increasing. In other words, if the sequence of spanning trees encountered is T = T0 ! T1 ! T2 ! Tk = T0; then cost(Ti+1)  cost(Ti) for all i < k.

Consider the set of all spanning trees (not just minimum ones) of a weighted, connected, undirected
graph G = (V,E). Recall that adding an edge e to a spanning tree T creates an unique cycle, and
subsequently removing any other edge e0 6= e from this cycle gives back a different spanning tree T0.
We will say that T and T0 differ by a single edge swap (e, e0) and that they are neighbors.
(a) Show that it is possible to move from any spanning tree T to any other spanning tree T0 by
performing a series of edge-swaps, that is, by moving from neighbor to neighbor. At most how
many edge-swaps are needed?
(b) Show that if T0 is an MST, then it is possible to choose these swaps so that the costs of the
spanning trees encountered along the way are non-increasing. In other words, if the sequence of
spanning trees encountered is T = T0 ! T1 ! T2 ! Tk = T0; then cost(Ti+1)  cost(Ti) for all
i < k. (c) Consider the following local search algorithm which is given as input an undirected graph G. Let T be any spanning tree of G while there is an edge-swap (e, e0) which reduces cost(T): T T + e − e0 return T Show that this procedure always returns a minimum spanning tree. At most how many iterations does it take?

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