COMP-202. Recursion. COMP Recursion, 2013 Jörg Kienzle and others

Transcription

1 COMP-202 Recursion

2 Recursion Recursive Definitions Run-time Stacks Recursive Programming Recursion vs. Iteration Indirect Recursion Lecture Outline 2

3 Recursive Definitions (1) A recursive definition is one which uses the word or concept being defined in the definition itself Consider the following list of numbers: 24, 88, 40, 37 Such a list can be defined as A LIST is either a number or a number comma LIST That is, a LIST is defined to be a single number, or a number followed by a comma followed by a LIST The concept of a LIST is used to define itself 3

4 Recursive Definitions (2) The recursive part of the LIST definition is used several times, terminating with the non-recursive part: number comma LIST 24, 88, 40, 37 number comma LIST 88, 40, 37 number comma LIST 40, 37 number 37 4

5 Infinite Recursions All recursive definitions have to have a nonrecursive part If they didn't, there would be no way to terminate the recursive path Such a definition would cause infinite recursion This problem is similar to an infinite loop The non-recursive part is often called the base case 5

6 Recursive Definition Example (1) n!, for any positive integer n, is defined to be the product of all integers between 1 and n inclusive This definition can be expressed recursively as: 1! = 1 n! = n * (n-1)! The concept of the factorial is defined in terms of another factorial Eventually, the base case of 1! is reached 6

7 Recursive Definition Example (2) 5! 5 * 4! 4 * 3! 3 * 2! 2 * 1!

8 Recursive Programming A method in Java can invoke itself; if set up that way, it is called a recursive method The code of a recursive method must be structured to handle both the base case and the recursive case Each call to the method sets up a new execution environment, with new parameters and local variables As always, when the method completes, control returns to the method that invoked it (which may be an earlier invocation of itself) 8

9 An Executing Program in Memory Dynamic Data Run-time Heap: where objects are created Run-time Stack: memory needed for method calls, e.g. parameters, local variables, return values Program Code Static Data Your compiled code All non-local data: class definitions, static variables, constants 9

10 Method Call Frame At every call to a method a frame is added to the top of the stack. This is referred to as a push. Method Call Frame When the method terminates the frame is removed from the top of the stack. This is referred to as a pop. Stacks function much like a stack of plates. You put plates on the top and you remove them from the top. Return Address Return Value Parameters Local Variables 10

11 Recursive Programming Example (1) Consider the problem: Compute the sum of all the numbers between 1 and any positive integer n sum(5) =

12 Recursive Programming Example (2) public class RecursionTest { public int sum(int n) { int result = 0; if (n == 1) // base case! result = 1; else! // recursive part! result = n + sum(n-1); return result; } } public static void main(string [] args) { System.out.println(sum(3)); } 12

13 Recursive Programming Example (3) public int sum(int n) { int result = 0; if (n == 1) // base case result = 1; else! // recursive part result = n + sum(n-1); return result; } main: println(sum(3)) Method Call Frame result = 3 + sum(2) ra main rv 6 n 3 result 06 result = 2 + sum(1) ra sum(3) rv 3 n 2 result 03 result = 1 ra sum(2) rv 1 n 1 result 01 13

14 Recursive vs. Iterative public int sum_recursive(int n) { int result = 0; if (n == 1) // base case result = 1; else! // recursive part result = n + sum_recursive(n-1); return result; } public int sum_iterative(int n) { int result = 0; for (int i = 1; i <= n; i++) result += i; return result; } 14

15 Recursion: When to Use It (1) Note that just because we can use recursion to solve a problem, doesn't mean we should (there is a lot of overhead: method calls, variable declarations, etc.) For instance, we usually would not use recursion to solve the sum of 1 to N problem, because the iterative version is easier to understand, and also faster However, for some problems, recursion provides an elegant solution, often cleaner than an iterative version You must carefully decide whether recursion is the correct technique for any problem 15

16 Palindrome Testing public class PalindromeTesters { } public static boolean iterativetester (String str { int left = 0; int right = str.length() - 1; while (left < right && str.charat(left) == str.charat(right)){ left++; right--; } return (left >= right); } public static boolean recursivetester (String str)! { boolean result = false; if (str.length() <= 1) result = true; else result = (str.charat(0) == str.charat(str.length() - 1)) && recursivetester(str.substring(1,str.length()-2)); return result; } 16

17 Recursion: When to Use It (2) Notice that we have many ways to repeatedly perform some statements: do while while for Recursion They all do the same thing, so the choice should be based on some benefit: Easier to program Resource usage: runs faster, uses less memory Ideally you want to optimize on both criteria 17

18 Designing for Recursion A recursive algorithm always looks like this: Base Case The part of the loop that has the stop condition. It also returns the default (simplest case) result Incrementing Part The part of the program that moves us on to the next data value. Incrementing variable Reading data Moving to a new data item in a structure (like array) Recursion Part The part of the program that initiates the recursion, i.e. makes the recursive call Note that the Incrementing and Recursion Parts are often together in the same statement (but not always) 18

19 Indirect Recursion A method invoking itself is considered to be direct recursion A method could invoke another method, which invokes another, etc., until eventually the original method is invoked again For example, method m1 could invoke m2, which invokes m3, which in turn invokes m1 again This is called indirect recursion, and requires all the same care as direct recursion It is often more difficult to trace and debug 19

20 Exercise: Towers of Hanoi (1) In the great temple at Benares beneath the dome that marks the centre of the world, rests a brass plate in which are fixed three diamond needles, each a cubit high and as thick as the body of a bee. On one of these needles, at the creation, God placed sixty-four discs of pure gold, the largest disk resting on the brass plate, and the others getting smaller and smaller up to the top one. This is the tower of Bramah. Day and night unceasingly the priest transfer the discs from one diamond needle to another according to the fixed and immutable laws of Bramah, which require that the priest on duty must not move more than one disc at a time and that he must place this disc on a needle so that there is no smaller disc below it. When the sixty-four discs shall have been thus transferred from the needle which at creation God placed them, to one of the other needles, tower, temple, and Brahmins alike will crumble into dust and with a thunderclap the world will vanish. 20

21 Towers of Hanoi (2) Three rods (a, b, c), n disks, initially stacked in ascending order of size on rod a Transfer all n disks to c, obeying the following rules Only one disk can be moved at a time Each move consists of taking the uppermost disk off one rod and sliding it onto another rod, on top of the disks already present No disk may be placed on top of a smaller disk Write a program that prints instructions for the monks Example output for 2 disks: Move disk from a to b Move disk from a to c Move disk from b to c 21