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By Jeff Erickson

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N]) i=1 Again, the base case is OptCost( f [1 .. 0]) = 0; the best way to organize no keys, which we will plan to search zero times, is by storing them in the empty tree! 6 Lecture 3: Backtracking [Fa’10] Algorithms This recursive definition can be translated mechanically into a recursive algorithm, whose running time T (n) satisfies the recurrence n T (n) = Θ(n) + T (k − 1) + T (n − k) . k=1 n The Θ(n) term comes from computing the total number of searches i=1 f [i]. Yeah, that’s one ugly recurrence, but it’s actually easier to solve than it looks.

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. 4 Lecture 3: Backtracking [Fa’10] Algorithms LIS(A[1 .. n]): return LISBIGGER(−∞, A[1 .. n]) LISBIGGER(prev, A[1 .. n]): if n = 0 return 0 else max ← LISBIGGER(prev, A[2 .. n]) if A[1] > pr ev L ← 1 + LISBIGGER(A[1], A[2 ..

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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