Recursion

recursion =

In programming languages, recursion happens when a method calls itself

Rules of recursion

1)

2)

Example: binary search

Algorithm: binarySearch(dictionary, word)

Example: choosing k out of n things

How many ways are there to choose k out of n items (e.g., 3 out of 5 items)?

Example:

Write a recursive method that determines if a string represents a non-negative integer. Assume '+' has been stripped off (if necessary).

A string is a non-negative integer if

• it is a single character and that character is a digit, or
• it has more than 1 character and its first character is a digit and the rest of the string is a non-negative integer.

Example:

Write a recursive method that determines if a chain of linked nodes containing integers is strictly increasing.

• a sequence of integers n1, n2, n3, ... is strictly increasing if n1 < n2 < n3 < ...
• a chain of linked nodes containing integers is strictly increasing if
  • it is a single number, or
  • head < head→next and the chain starting with head→next is strictly increasing

Example: palindromes

Write a recursive method that determines if string is a palindrome.

Examples of palindromes:

• deed
• eye
• racecar
• radar
• Step on, no pets
• Never odd or even
• A nut for a jar of tuna
• Do offer a ref food
• Ed, I saw Harpo Marx ram Oprah W aside
• No, it is open on one position
• Are we not drawn onward, we few, drawn onward to new era?

Assume: input string is not null, has been stripped of all spaces and punctuation, and is in all lower-case

Useful string methods:

• char charAt(int index)
• int length()
• String substring(int begin, int last_plus_one)

Challenge:

Write a recursive method that counts the number of even values in a chain of linked nodes containing integer values.

First complete the English description for the base cases and recursive case:

• The number of even values in an empty chain is ...
• The number of even values in a chain with 1 node is ...
• The number of even values in a chan with > 1 node is ...

Then complete the recursive countEven method:

public static int countEven(Listnode<Integer> head) {

Analyzing complexity for recursive methods

Two options:

1)

2)

Using recurrence relations

Step 1: write recurrence relations

Step 2: guess a solution

Step 3: verify solution

Step 4: apply big-O notation to guessed solution

Analyzing recursive method: strictlyIncreasing

Analyzing recursive method: isPalindrome

Challenge: analyze the complexity of countEven using recurrence relations

Example: the Towers of Hanoi