4.4. The Three Laws of Recursion¶
Like robots in Asimov’s stories, all recursive algorithms must obey three important laws:
A recursive algorithm must have a base case.
A recursive algorithm must change its state and move toward the base case.
A recursive algorithm must call itself recursively.
Let’s look at each one of these laws in more detail and see how it was
used in the list_sum algorithm. First, a base case is the condition
that allows the algorithm to stop recursing. A base case is typically a
problem that is small enough to solve directly. In the list_sum
algorithm the base case is a list of length 1.
To obey the second law, we must arrange for a change of state that moves
the algorithm toward the base case. A change of state means that some
data that the algorithm is using is modified. Usually the data that
represents our problem gets smaller in some way. In the list_sum
algorithm our primary data structure is a list, so we must focus our
state-changing efforts on the list. Since the base case is a list of
length 1, a natural progression toward the base case is to shorten the
list. This is exactly what happens on line 5 of ActiveCode 4.3.2 when we call list_sum with a shorter list.
The final law is that the algorithm must call itself. This is the very definition of recursion. Recursion is a confusing concept to many beginning programmers. As a novice programmer, you have learned that functions are good because you can take a large problem and break it up into smaller problems. The smaller problems can be solved by writing a function to solve each problem. When we talk about recursion it may seem that we are talking ourselves in circles. We have a problem to solve with a function, but that function solves the problem by calling itself! But the logic is not circular at all; the logic of recursion is an elegant expression of solving a problem by breaking it down into a smaller and easier problems.
In the remainder of this chapter we will look at more examples of recursion. In each case we will focus on designing a solution to a problem by using the three laws of recursion.
Self Check
- 6
- There are only five numbers on the list, the number of recursive calls will not be greater than the size of the list.
- 5
- The initial call to list_sum is not a recursive call.
- 4
- the first recursive call passes the list [4, 6, 8, 10], the second [6, 8, 10] and so on until [10].
- 3
- This would not be enough calls to cover all the numbers on the list
Q-1: How many recursive calls are made when computing the sum of the list [2, 4, 6, 8, 10]?
- n == 0
- Although this would work there are better and slightly more efficient choices. since fact(1) and fact(0) are the same.
- n == 1
- A good choice, but what happens if you call fact(0)?
- n >= 0
- This basecase would be true for all numbers greater than zero so fact of any positive number would be 1.
- n <= 1
- Good, this is the most efficient, and even keeps your program from crashing if you try to compute the factorial of a negative number.
Q-2: Suppose you are going to write a recusive function to calculate the factorial of a number. fact(n) returns n * n-1 * n-2 * … Where the factorial of zero is defined to be 1. What would be the most appropriate base case?