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Using this idea, we can develop a dynamic-programming algorithm for nding optimal join orders Dynamic programming algorithms store results of computations and reuse them, a procedure that can reduce execution time greatly A recursive procedure implementing the dynamic programming algorithm appears in Figure 145 The procedure stores the evaluation plans it computes in an associative array bestplan, which is indexed by sets of relations Each element of the associative array contains two components: the cost of the best plan of S, and the plan itself The value of bestplan[S]cost is assumed to be initialized to if bestplan[S] has not yet been computed The procedure rst checks if the best plan for computing the join of the given set of relations S has been computed already (and stored in the associative array bestplan); if so it returns the already computed plan Otherwise, the procedure tries every way of dividing S into two disjoint subsets For each division, the procedure recursively nds the best plans for each of the two subsets, and then computes the cost of the overall plan by using that division The procedure picks the cheapest plan from among all the alternatives for dividing S into two sets The cheapest plan and its cost are stored in the array bestplan, and returned by the procedure The time complexity of the procedure can be shown to be O(3n ) (see Exercise 1411) Actually, the order in which tuples are generated by the join of a set of relations is also important for nding the best overall join order, since it can affect the cost of further joins (for instance, if merge join is used) A particular sort order of the tuples is said to be an interesting sort order if it could be useful for a later operation For instance, generating the result of r1 1 r2 1 r3 sorted on the attributes common with r4 or r5 may be useful, but generating it sorted on the attributes common to only r1 and r2 is not useful Using merge join for computing r1 1 r2 1 r3 may be costlier than using some other join technique, but may provide an output sorted in an interesting sort order Hence, it is not suf cient to nd the best join order for each subset of the set of n given relations Instead, we have to nd the best join order for each subset, for.

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each interesting sort order of the join result for that subset The number of subsets of n relations is 2n The number of interesting sort orders is generally not large Thus, about 2n join expressions need to be stored The dynamic-programming algorithm for nding the best join order can be easily extended to handle sort orders The cost of the extended algorithm depends on the number of interesting orders for each subset of relations; since this number has been found to be small in practice, the cost remains at O(3n ) With n = 10, this number is around 59000, which is much better than the 176 billion different join orders More important, the storage required is much less than before, since we need to store only one join order for each interesting sort order of each of 1024 subsets of r1 , , r10 Although both numbers still increase rapidly with n, commonly occurring joins usually have less than 10 relations, and can be handled easily We can use several techniques to reduce further the cost of searching through a large number of plans For instance, when examining the plans for an expression, we can terminate after we examine only a part of the expression, if we determine that the cheapest plan for that part is already costlier than the cheapest evaluation plan for a full expression examined earlier Similarly, suppose that we determine that the cheapest way of evaluating a subexpression is costlier than the cheapest evaluation plan for a full expression examined earlier Then, no full expression involving that subexpression needs to be examined We can further reduce the number of evaluation plans that need to be considered fully by rst making a heuristic guess of a good plan, and estimating that plan s cost Then, only a few competing plans will require a full analysis of cost These optimizations can reduce the overhead of query optimization signi cantly.

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