Jim Lambers
Math 6A
Spring Quarter 2003-04
Lecture 24 Notes
These notes correspond to Section 6.3 in the text.
Divide-and-Conquer Algorithms and Recurrence Relations
In the previous lectures, we have studied recurrence relations in which a term anof a sequence is
determined by preceding terms an−1, an−2, . . . , an−kwhere kis the degree of the recurrence relation.
These recurrence relations arise from problems in which the solution for an input of size n, where
the notion of size of the input depends on the problem, can be obtained from solutions of slightly
smaller instances of the same problem, such as those of size n−1, n −2, and so on.
However, in many cases, a problem with input of size ncan be solved by reducing the problem
to aproblems with input of size n/b, for some b > 1, and then constructing the solution of the
original problem from the asolutions that are obtained. This approach to problem-solving is known
as divide-and-conquer.
The number of operations performed to solve a problem in this manner, which we denote by
f(n), satisfies a recurrence relation of the form
f(n) = af(n/b) + g(n),
where g(n) denotes the number of operations needed to construct the solution of the original
problem of size nfrom the asolutions of problems of size n/b.
The following theorem provides essential insight into the complexity of certain classes of divide-
and-conquer algorithms. We use the “big-Oh” notation to measure complexity. A function f(n)
is said to be O(g(n)) if there exists a constant csuch that f(n)≤cg(n) for all n≥n0for some
integer n0.
Theorem (Master Theorem) Let fbe an increasing function that satisfies the recurrence relation
f(n) = af(n/b) + cnd,
whenever n=bkfor some positive integer k,a≥1,bis an integer greater than 1, c > 0and d≥0.
Then
f(n) =
O(nd)if a < bd
O(ndlog n)if a=bd
O(nlogba)if a > bd
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