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

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By Alexander Shen

That publication does not include natural algorithms thought (like Kormen's or Skiena's book), yet difficulties (and so much of them are with solutions). each bankruptcy begins with the straightforward challenge, via a few discussions of attainable ideas, and after an increasing number of tough projects ends with lovely difficult problems.
The booklet (in my opinion) is de facto very useful (well, it includes a few concept, yet now not very formal) and is important in case you are getting ready to the programming contests or Google/Microsoft-like interviews.

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

Example text

B [n] whose elements mark the start of the "non-scanned" portions of arrays a [1] . . . a [ n ] . 26. Our solution of the preceding problem requires mn2 operations. , not more than Cnm operations for some C). [Hint. We have to break the symmetry and choose one of the rows as a "principal" row. 27. (Binary search) An array x [ l ] _< ... _< x [ n ] of integers and an integer a are given. Determine if a is present in x, that is, if there exists an i in 1 . n such that x [ i ] = a. ) Solution.

N operations (for some constant C). When is it possible to increase the s-th term of a bit string with k ls without changing the preceding terms? If x [s] is changed from 0 to 1, we should replace 1 by 0 somewhere to keep the total number of is fixed. Therefore, it is necessary to have is on the right of x [ s ] . Conclusion: If we want to find the next bit string with k ls, we need x [s] to be the rightmost 0 that has some ls on the right. In this case we have x [ s + l ] =1 (otherwise, x [s] is not the rightmost one).

Suppose we want to multiply two polynomials of degree 2k. Represent these polynomials as A ( x ) x k + B(x) and C(x) x k + D(x) where A, B, C, D are polynomials of degree k. The product in question is equal to A ( x ) C ( x ) x 2k + ( A ( x ) D ( x ) + B(x)C(x)) x k + B(x)D(x). The natural way to compute AC, A D + BC, BD requires four multiplications of degree k polynomials. 17. Two arrays x : a r r a y [ 1 . k] of i n t e g e r and y: a r r a y [ 1 . 1] of i n t e g e r are sorted (x [1] < ...