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Combinatorics

By Herbert S. Wilf

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N is equally likely to occur. The reason for this is that we chose the splitter originally by choosing a random array entry. Since all orderings of the array entries are equally likely, the one that we happened to have chosen was just as likely to have been the largest entry as to have been the smallest, or the 17th -from-largest, or whatever. Since each value of i is equally likely, each i has probability 1/n of being chosen as the residence of the splitter. If the splitting element lives in the ith array position, the two recursive calls to Quicksort will be on two subarrays, one of which has length i − 1 and the other of which has length n − i.

2. Let G be a path of n vertices. What is the size of the largest independent set of vertices in V (G)? 3. Let G be a connected graph in which every vertex has degree 2. What must such a graph consist of? Prove. 4. Let G be a connected graph in which every vertex has degree ≤ 2. What must such a graph look like? 5. Let G be a not-necessarily-connected graph in which every vertex has degree ≤ 2. What must such a graph look like? What is the size of the largest independent set of vertices in such a graph?

Again, the complexity of the algorithm will be exponential, but as a small consolation we note that no polynomial time algorithm for this problem is known. Choose an edge e of the graph, and let its endpoints be v and w. Now delete the edge e from the graph, and let the resulting graph be called G − {e}. Then we will distinguish two kinds of proper colorings of G−{e}: those in which vertices v and w have the same color and those in which v and w have different colors. Obviously the number of proper colorings of G − {e} in K colors is the sum of the numbers of colorings of each of these two kinds.

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Algorithms and Complexity by Herbert S. Wilf


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