By Alexander Shen
That publication does not include natural algorithms concept (like Kormen's or Skiena's book), yet difficulties (and such a lot of them are with solutions). each bankruptcy begins with the straightforward challenge, by means of a few discussions of attainable ideas, and after a growing number of hard projects ends with beautiful tough problems.
The ebook (in my opinion) is de facto very functional (well, it comprises a few concept, yet no longer very formal) and is valuable when you are getting ready to the programming contests or Google/Microsoft-like interviews.
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Additional info for Algorithms and Programming: Problems and Solutions (Modern Birkhäuser Classics)
Baur, V. Strassen) Assume that a "straight-line" program (containing only assignment statements) computes the value of some polynomial P ( x l . . . Xn) given the variables xl . . . Xn. We assume that the right-hand sides of the assignment statements are expressions that contain only addition, multiplication, constants, variables x l , . . , xn and the variables that appear on the left-hand side of previous assignment statements. Prove that there exists a program of the same type that computes all n derivatives 8 P / a x l .
2. Generate all sequences of length n composed of the numbers 1 . k in such an order that neighboring sequences differ only in one place, and the numbers at this place differ by 1. Solution. Consider a rectangular chess board of width n and height k. Place a piece in each column of the chess board. The position is represented by a sequence of n integers (each between i and k); the i-th number represents a position of the piece in the i-th column. At each piece we draw a small arrow that points up or down.
Let us prove by induction on n that all sequences of length n composed of the numbers 1 . k will appear. 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.
Algorithms and Programming: Problems and Solutions (Modern Birkhäuser Classics) by Alexander Shen