By Herbert S. Wilf

ISBN-10: 1568811780

ISBN-13: 9781568811789

**Read or Download Algorithms and Complexity (Second edition) PDF**

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**Extra resources for Algorithms and Complexity (Second edition)**

**Example text**

N−1 is uniquely representable. Now consider the integer n. Define d = n mod b. Then d is one of the b permissible digits. By induction, the number n0 = (n − d)/b is uniquely representable, say: n−d = d0 + d1 b + d2 b2 + . . b Then clearly n n−d b b = d + d0 b + d1 b2 + d2 b3 + . . = d+ is a representation of n that uses only the allowed digits. Finally, suppose that n has some other representation in this form also. Then we would have: n = a0 + a 1 b + a 2 b 2 + . . = c 0 + c1 b + c2 b 2 + .

10). Most counting problems on graphs are much easier for labeled than for unlabeled graphs. Consider the following question: How many graphs are there that have exactly n vertices? Suppose first¡ that we mean labeled graphs. A graph of n vertices has ¢ a maximum of n2 edges. To construct a graph, we would decide which ¡ ¢ of these possible edges would be used. We can make each of these n2 decisions independently, and for every way of deciding where to put the edges, we would get a diﬀerent graph.

This is a graph of 5 vertices and 5 edges. A nice way to present a graph to an audience is to draw a picture of it, instead of just listing the pairs of vertices that are its edges. To draw a picture of a graph, we would first make a point for each vertex, and then we would draw an arc between two vertices v and w if and only if (v, w) is an edge of the graph that we are talking about. The graph 40 1. 3. 3(a). 3(b). They’re both the same graph. Only the pictures are diﬀerent, but the pictures aren’t “really” the graph; the graph is the vertex list and the edge list.

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