Lecture 12

Recursion

MCS 275 Spring 2021
Emily Dumas

Lecture 12: Recursion

Course bulletins:

  • Quiz 4 due Tuesday at Noon CST
  • Please complete anonymous feedback survey (link in Blackboard announcement)

Recursion

In computer science, recursion refers to a method of solving a problem that depends on solving smaller versions of the same problem.

Usually, recursion involves a function calling itself.

Strategies using recursion

  • Divide and conquer: A problem can be split into pieces; solutions for the pieces can be combined to the full solution.
    • e.g. Mergesort
  • Decrease and conquer: Reduce a problem for a given input (e.g. n) to the answer for a slightly smaller input (e.g. n-1) and a bit of extra work.
    • e.g. Factorial

Iteration

Recursive solutions are often contrasted with iterative solutions.

  • Iterative: Loops and local variables keep track of all state (work to be done, work completed, next ...)
  • Recursive: Arguments keep track of current state; call stack stores work in progress; return values send back results.

Recursive solutions can always be converted to iterative ones, often at the cost of more complex code.

Stop condition

A function that always calls itself will never finish!

Recursion must include some kind of stop condition---a case in which the function can directly return an answer instead of calling itself.

Today's examples

  • Factorial
  • Fibonacci numbers
  • Paper folding sequence

Factorial

The classic first example of recursion, computing $$n! = n \times (n-1) \times \cdots \times 2 \times 1.$$

The argument to the function decreases with each subsequent call, so it eventually reaches the stop condition ($n \leq 1$).

Fibonacci

The Fibonacci numbers are defined by $$F_0=0,\; F_1=1,\; \text{and }F_n = F_{n-1} + F_{n-2}$$

So the sequence begins $0,1,1,2,3,5,8,13,...$

The definition immediately suggests a recursive implementation.

Paper folding sequence

Paper folding sequence

Paper folding sequence

Let $\oplus$ denote concatenation of binary sequences, so $0110 \oplus 11 = 011011$.

If $A$ is a binary sequence, let $\bar{A}$ denote the sequence with $0$ and $1$ switched, e.g. $\overline{11101} = 00010$

Finally, let $A^r$ denote the sequence in opposite order, e.g. $10010^r = 01001$.

$$PFS(n) = PFS(n-1) \oplus 1 \oplus \overline{PFS(n-1)^r}$$

References

Revision history

  • 2021-02-08 Fixed value of $F_0$
  • 2021-02-08 Initial publication