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3.2 AN EXAMPLE OF SIMPLE REGRESSION An experiment was conducted to find the relationship between two procedures for assessing the acid content of a chemical. The two procedures measure somewhat different but related quantities. The organic acid content of a sample of chemical can be determined by a method of extraction and weighing, which is expensive, but a relatively cheap titration method is available for determining the acid number. It was hoped that, by using regression, the cheap method could be used instead of the expensive method to measure organic acid content. Both procedures were used on 20 samples of chemical. The data are displayed in Table 3.1. Using these data we would like to obtain an equation that expresses the organic acid content measurement as an approximate function of the acid number measurement. For notation, let

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The unionlfind algorithm, as described so far, is quite acceptable for most cases It is very simple and linear on average for a sequence of M instructions However the worst case is still unappealing The reason is that a sequence of u n i o n operations occurring in some particular application (such as the NCA problem) is not obviously random (in fact, for certain trees, it is far from random) Hence we have to seek a better bound for the worst case of a sequence of M operations Seemingly, no more improvements to the union algorithm are possible because the worst case is achievable when identical trees are merged The only way to speed up the algorithm, without reworking the data structure entirely, then is to do something clever with the f i n d operation That something clever is path compression Clearly, after we perform a f i n d on x, changing x's parent to the root would make sense In that way, a second f i n d on x or any item in x's subtree becomes easier There is no need to stop there, however We might as well change the parents for all the nodes on the access path In path compression e v e n node on the path from x to the root has its parent changed to the root Figure 2419 shows the effect of path compression after f i n d ( 14) on the generic worst tree shown in Figure 2417 With an extra two parent changes, nodes 12 and 13 are now one position closer to the root and nodes 14 and 15 are now two positions closer The fast future accesses on the nodes pay (we hope) for the extra work to do the path compression Note that subsequent unions push the nodes deeper When u n i o n s are done arbitrarily, path compression is a good idea because of the abundance of deep nodes; they are brought near the root by path compression It has been proved that when path compression is done in this case, a sequence of M operations requires at most O(M log N ) time, so path compression by itself guarantees logarithmic amortized cost for the f i n d operation.

U = Q2(1 - x g }.

Path compression is perfectly compatible with union-by-size Thus both routines can be implemented at the same time However, path compression is not entirely compatible with union-by-height because path compression can change the heights of the trees We do not know how to recompute them efficiently, so we do not attempt to do so Then the heights stored for each tree become estimated heights, called ranks, which is not a problem The resulting algorithm, union-by-rank, is thus obtained from union-by-height when compression is performed As we show in Section 246, the combination of a smart union rule and path compression gives an almost linear guarantee on the running time for a sequence of M operations

Path compression and a Smart root guarantee essentially constant amortized cost per operation 0% a long sequence can be executed in almost linear time)

The data points (x"y,), ... ,(X 20 'Y20) are plotted in Figure 3.1. The relationship between X and Y appears to be approximately linear.

For instance,

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