Suppose you are given as input a rooted tree t. a labeling of the vertices with labels from the set {1, 2, 3} is called valid if the label of every vertex is different from the label of its parent. the cost of a labeling is the number of vertices with smaller labels than their parents. design an efficient algorithm to find the minimum cost of a valid labeling of t.
prove that your algorithm is correct and analyze its running time.