Algorithms and Programming: Problems and Solutions (Modern by Alexander Shen

By Alexander Shen

That booklet does not include natural algorithms thought (like Kormen's or Skiena's book), yet difficulties (and such a lot of them are with solutions). each bankruptcy begins with the easy challenge, by way of a few discussions of attainable suggestions, and after progressively more not easy initiatives ends with lovely tough problems.
The ebook (in my opinion) is absolutely very useful (well, it comprises a few conception, yet no longer very formal) and is necessary if you happen to are getting ready to the programming contests or Google/Microsoft-like interviews.

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Extra info for Algorithms and Programming: Problems and Solutions (Modern Birkhäuser Classics)

Example text

The case n = l is evident, so assume that n > 1. Divide all moves into two categories. The first category is formed by moves where the last (rightmost) piece is moving. The second category is formed by moves where the moving piece is not the last one. In this case the rightmost piece is near the border and is turned over. Therefore, each move of the second category is followed by k - 1 moves of the first category; during this period the rightmost piece visits all the cells. Let us forget now about the rightmost piece.

Solution. When a new coefficient is added, the polynomial changes from P (x) to Q (x) = x P (x) + c. The derivative QI (x) is equal to x P ' (x) + P (x). Therefore we can easily compute Q(x) and Q'(x) if we know x, c, P ( x ) and P ' ( x ) . 9 This solution has a unexpected feature: we do not need to know in advance the degree of the polynomial. If we add this requirement and ask to compute the value of the derivative only (not mentioning the polynomial itself), we get a rather confusing problem.

2 n - 1 in binary notation. For example, for n = 3 we have: 000 001 010 011 100 101 110 111 Each number is transformed according to the following rule: each digit (except the first one) is replaced by its sum (modulo 2) with the preceding (untransformed) digit. In other words, the number with binary digits al, a2 . . , an is transformed into the number with binary digits al, al -q- a2, a2 -q- a3 . . . an-1 -b an (addition modulo 2). For n = 3, we get the following list: 000 001 011 010 110 111 101 100 It is easy to check that the transformation described (which can be applied to any sequence of n binary digits, giving another sequence of the same length) is invertible.

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