32. Which of the following are false? i. The worst-case running time and expected running time are equal to within constant factors for any randomized algorithm. ii. Sorting 6 elements with a comparison sort requires at least 10 comparisons in the worst case. iii. In Blum, Floyd, Pratt, Rivest, and Tarjan [1973] worst-case linear-time order statistics algorithm, instead of dividing the n elements into groups of 5, dividing them into groups of 3 gives the same linear time complexity. iv. If a dynamic programming problem satisfies the optimal substructure property, then a locally optimal solution is a globally optimal. (A)i (B)i,iii,iv (C)i,ii；,iv (D)ii,iv (E)iii,iv

28. Which one of the following statements is not correct? (A) If an edge is contained in some minimum spanning tree, then it is a light edge crossing some cut of the graph. (Definition: An edge is a light edge crossing a cut if its weight is the minimum of any edge crossing the cut.) (B) If a graph has a unique minimum spanning tree then, for every cut of the graph, there is a unique light edge crossing the cut. (C) A graph has a unique minimum spanning tree if, for every cut of the graph, there is a unique light edge crossing the cut. (D) Let e be a maximum-weight edge on some cycle of the graph G = (V, E), then there is a minimum spanning tree of G' = (V, E-{e}) that is also a minimum spanning tree of G = (V, E). (E) Let (u, ) be a minimum-weight edge in a graph G. Then (u, v) belongs to some minimum spanning tree of G.

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