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By Jeff Erickson

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Whenever the algorithm needs to decide between multiple alternatives to the next component of the solution, it simply tries all possible options recursively. 1 n Queens The prototypical backtracking problem is the classical n Queens Problem, first proposed by German chess enthusiast Max Bezzel in 1848 for the standard 8 × 8 board, and both solved and generalized to larger boards by Franz Nauck in 1850. The problem is to place n queens on an n × n chessboard, so that no two queens can attack each other.

What is the worst-case running time of the resulting sorting algorithm? ) 15. Suppose we have n points scattered inside a two-dimensional box. A kd-tree recursively subdivides the points as follows. First we split the box into two smaller boxes with a vertical line, then we split each of those boxes with horizontal lines, and so on, always alternating between horizontal and vertical splits. Each time we split a box, the splitting line partitions the rest of the interior points as evenly as possible by passing through a median point inside the box (not on its boundary).

If A[1] ≤ x, the LIS of A[1 .. n] with elements larger than x is the LIS of A[2 .. n] with elements larger than x. Otherwise, the LIS of A[1 .. n] with elements larger than x is either the LIS of A[2 .. n] with elements larger than x or A[1] followed by the LIS of A[2 .. n] with elements larger than A[1], whichever is longer. The longest increasing subsequence without restrictions can now be redefined as the longest increasing subsequence with elements larger than −∞. Rewriting this recursive definition into pseudocode gives us the following recursive algorithm.

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