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By Herbert S. Wilf

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100000, 10000, 1000, 100, 10, 1. Then, to represent an integer, we can specify how many copies of each power of 10 we would like to have. If we write 237, for example, then that means that we want 2 100s, 3 10s, and 7 1s. In general, if we write out the string of digits that represents a number in the decimal system, as dm dm−1 · · · d1 d0 , then the number that is being represented by that string of digits is: n= m X di 10i . i=0 Now let’s try the binary system. Instead of using 10s we’re going to use 2s.

But it doesn’t. It calls something that’s just slightly different from itself in order to get its job done, and that won’t work. Observe the exact purpose of Quicksort, as described above. We are given an array of length n, and we want to sort it, all of it. Now look 54 2. Recursive Algorithms at the two ‘recursive calls,’ which really aren’t quite recursive. The first one of them sorts the array to the left of xi . That is indeed a recursive call, because we can just change the ‘n’ to ‘i − 1’ and call Quicksort.

Indeed, if not, then we arrived at v 0 one more time than we departed from it, each time using a new edge, and finding no edges remaining at the end. Thus, there was an odd number of edges of G incident with v 0 , a contradiction. Hence, we are indeed back at our starting point when the walk terminates. Let W denote the sequence of edges along which we have so far walked. If W includes all edges of G, then we have found an Euler tour and we are finished. Else there are edges of G that are not in W .

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