ISOM MIS 215 Module 4 – Recursion

ISOM Where are we? 2 Intro to Java, Course Java lang. basics Arrays Introduction NewbieProgrammersDevelopersProfessionalsDesigners MIS215 Binary Search Search Techniques Sorting Techniques Bubblesort Basic Algorithms Fast Sorting algos (quicksort, mergesort) Hashtables Graphs, Trees Linked Lists Stacks, Queues List StructuresAdvanced structures

ISOM Today’s buzzwords Recursion  A data structure where items can be added to the top and removed from the top  A LIFO (Last In, First Out) Structure Recursion  A programming style where a method directly or indirectly calls itself.  Typically leads to simpler and more elegant looking implementations, although not necessarily more efficient Recursive structures  A data structure including a reference to itself Base Case/Condition  The case that forms the basis from which other cases can be built

ISOM Recursion: What is it? It is a problem solving technique  divide and conquer It is a programming technique  let a function call itself

ISOM Towers of Hanoi: A Classical Example 12 3

ISOM Towers of Hanoi: Recursive Function int Move(int count, int start, int finish, int temp); Pre: There are at least count disks on the tower start. The top disk (if any) on each of towers temp and finish is larger than any of the top count disks on tower start. Post: The top count disks on start have been moved to finish; temp has been returned to its starting position.

ISOM An Example with Two Disks: Trace of the Function Move (2,1,3,2) Move (1,1,2,3) Move (1,2,3,1) Move (0,1,3,2) Move (0,3,1,2) Move (0,2,1,3) Move (0,3,2,1) Outer Call First instruction printed Trivial recursive Call End of first recursive call Second instruction period Second recursive call Trivial recursive call Third instruction printed Trivial recursive call End of second recursive call End of outer Call First recursive call Trivial recursive call Move disk 1 from 1 to 2 Move disk 1 from 2 to 3 Move disk 2 from 1 to 3

ISOM Find the key step Find a stopping rule Outline your algorithm Check termination Draw a recursion tree Designing Recursive Algorithms

ISOM Examples Factorial Fibonacci Sequence Tower’s of Hanoi Linear Search in LinkList? Binary Search in Array? Any problem that can be defined in terms of a smaller version of itself!

ISOM Building A recursion Say, I want you to write a recursion for finding power(x, n) i.e, find the n th power of x (x n ) What is x 0 ? Can you write x 2 in terms of x? Can you write x 3 in terms of x 2 ? Can you write x 25 in terms of x 24 ? So, can you write x n in terms of x n-1 ?

ISOM So, to find recursion Find the base case Try out a few small examples from the base case … building on top of another Now try to generalize

ISOM Recursively define factorial fact(n) Base case: Recursion:

ISOM Recursively define gcd(x, y) Base case: Recursion:

ISOM Recursively define find method in a linked list find(Node n, int target) Base case: Recursion:

ISOM Recursively define binary search find(int lb, int ub, int target) Base case: Recursion: