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I have this problem : given undirected graph we can build a tree that approximates the distances from given vertex ,r, and the total weight,
i.e. for every vertex x, $d_G(r,x) \le d_T(r,x) \le 3d_G(r,x)$ and : $w(T) \le 3MST(G)$ could any one help me to understand how to build this tree and if there are any material that would help ...

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You are looking for a Tree Spanner. In general it is NP-Hard to find such a tree for weighted and unweighted graphs, but it can be found in polynomial time in few special cases. You can find more information about it and more related results in this paper:

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