Question

3. Let X be a set of n elements to be maintained as a linked list: Assume that the cost of examining a particular element Xi is Ci. Note that to examine Xi, one needs to scan through all elements in front of Xi. Let Pi be the probability of searching for element Xi, so the total cost for all searches is (a) Give a counterexample to show that storing elements in non-increasing order of Pi does not necessarily minimize the total cost. (b) Prove that the total cost is minimized when the elements are stored in nonincreasing order of Pi/Ci.

Answer 1

Consider any order, and suppose that for some .

Suppose that we switch the elements and Say that we switched between $$\ldots,A,B,\ldots$$ and $$\ldots,B,A\ldots$.$

Let $$\Delta$$ be the difference in average search cost (new cost minus old cost). When looking for which happens with probability

$$P_i$,$ we have to pay $$C_{i+1}$$ more. When looking for which happens with probability

$$P_{i+1}$,$ we have to pay $$C_i$$ less. So $$$ \Delta = P_i C_{i+1} - P_{i+1} C_i = (P_i/C_i - P_{i+1}/C_{i+1}) C_i C_{i+1} < 0.$

In other words, the switch was beneficial. We conclude that the optimal arrangement is to store the elements in non-increasing order.