$(30$ pts$.)$ We apply the GRACE algorithm to perform $R|><mi\; id="EditorElement\_100519">|</mi><mi\; id="EditorElement\_100521">S</mi>$ on a small shared$-$

nothing system with four processing nodes $($PNs$).R$ has $4,000$ pages and $S$ has

$8,000$ pages. Each relation is evenly divided among the four PNs$.$ Thus, each PN

has $3,000$ pages of tuples. The Hashing Phase results in data skew as follows:

$50\%$ of the data in the first $8$ bucket pairs: $\frac{R_0}{S_0}-\frac{R_7}{S_7}$

$20\%$ of the data in the second $8$ bucket pairs: $\frac{R_8}{S_8}-\frac{R_{15}}{S_{15}}$

$15\%$ of the data in the third $8$ bucket pairs: $\frac{R_{16}}{S_{16}}-\frac{R_{23}}{S_{23}}$

$15\%$ of the data in the fourth $8$ bucket pairs: $\frac{R_{24}}{S_{24}}-\frac{R_{31}}{S_{31}}$

For each of the three parallel phases, estimate the read cost, the write cost, and

the total computation cost. Show and explain the derivation of your

mathematical analysis.