Introduction to Algorithms
3rd Edition
ISBN: 9780262033848
Author: Thomas H. Cormen, Ronald L. Rivest, Charles E. Leiserson, Clifford Stein
Publisher: MIT Press
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Chapter 5.3, Problem 1E
Program Plan Intro
To rewrite the procedure RANDOMIZE-IN-PLACE so that the corresponding loop invariants holds to a nonempty subarray prior to first equation.
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Appendix A
10-Fold Cross Validation for Parameter Selection
Cross Validation is the standard method for evaluation in empirical machine learning. It can also be used
for parameter selection if we make sure to use the training set only.
To select parameter A of algorithm A(X) over an enumerated range d E [A1,..., A] using dataset D, we do
the following:
1. Split the data D into 10 disjoint folds.
2. For each value of A e (A1,..., Ar]:
(a) For i = 1 to 10
Train A(A) on all folds but ith fold
Test on ith fold and record the error on fold i
(b) Compute the average performance of A on the 10 folds.
3. Pick the value of A with the best average performance
Now, in the above, D only includes the training data and the parameter A is chosen without the knowledge
of the test data. We then re-train on the entire train set D using the chosen A and evaluate the result on
the test set.
how to use a histrogram to estimate the the size of selection of the form σA<=γ(r)?
In order to determine the full histogram for all matchings of a given size, we need to generate every single possible matching in a unique way. This is where the inductive
description from the introduction becomes useful, as it provides a way to do so recursively: We can generate all arc diagrams with n arcs from all arc diagrams with
(n - 1) arcs by adding one arc to each of them in precisely 2n 1 ways. To this end, we take an arc diagram with (n - 1) arcs, insert one new point at the left end and
one more point somewhere to the right of it (2n − 1 options), and then match the newly inserted points to obtain the additional arc. The only "problem" is that we need
to relabel some points in doing so.
-
-
Inserting a point to the left implies that the indices of the other points all have to be increased by one. Moreover, if we insert another point at some position m, then all
the indices with values m and larger again have to be increased by one.
.
=
1, 2, 3. This implies that all indices…
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Introduction to Algorithms
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