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By Even S.

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First let’s choose our unit of cost to be one comparison of two numbers, and then we will choose a different unit of cost, namely one interchange of position (‘swap’) of two numbers. 31 Chapter 2: Recursive Algorithms How many paired comparisons does the algorithm make? Reference to procedure slowsort shows that it makes one comparison for each value of j = r + 1, . . , n in the inner loop. This means that the total number of comparisons is n−1 n f1 (n) = 1 r=1 j=r+1 n−1 = (n − r) r=1 = (n − 1)n/2.

5(b) of this construction, we show in Fig. 5(a) a map of a distant planet, and in Fig. 5(b) the graph that results from the construction that we have just described. By a ‘planar graph’ we mean a graph G that can be drawn in the plane in such a way that two edges never cross (except that two edges at the same vertex have that vertex in common). The graph that results from changing a map of countries into a graph as described above is always a planar graph. In Fig. 6(a) we show a planar graph G.

Every vertex of G − {e} other than v or w keeps the same color that it has in the coloring of G/{e}. Vertex v and vertex w each receive the color that vertex vw has in the coloring of G/{e}. Now we have a K-coloring of the vertices of G − {e}. It is a proper coloring because if f is any edge of G − {e} then the two endpoints of f have different colors. Indeed, this is obviously true if neither endpoint of f is v or w because the coloring of G/{e} was a proper one. There remains only the case where one endpoint of f is, say, v and the other one is some vertex x other than v or w.

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