More on Recursion

More techniques

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Binary search algorithm

• Binary searching for a key in an array is similar to looking for a word in dictionary

• Take the midpoint of the dictionary and see if the key is in the lower half or upper half

• Pick the half the contains the key and take the midpoint of the half and see which quarter the key is in

• Repeat the above step until the key is matched or the dictionary cannot be halved any more

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Use helper method

• Given int[] list and int key• You are asked to return

– the position of key in the array if matched or– -1 if not matched

• What additional information do you need to apply the algorithm above?

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/** Use binary search to find the key in the list */ public static int recursiveBinarySearch(int[] list, int key) { int low = 0; int high = list.length - 1; return recursiveBinarySearch(list, key, low, high); // use helper method }// recursiveBinarySearch method

/** Use binary search to find the key in the list between list[low] and list[high] */ public static int recursiveBinarySearch(int[] list, int key, int low, int high) { if (low > high) // The list has been exhausted without a match return -1;

int mid = (low + high) / 2; if (key < list[mid]) return recursiveBinarySearch(list, key, low, mid - 1); else if (key == list[mid]) return mid; else return recursiveBinarySearch(list, key, mid + 1, high); }// recursiveBinarySearch method

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What is the difference between:

public static int recursiveBinarySearch(int[] list, int key){ . . . }

and public static int recursiveBinarySearch(

int[] list, int key, int low, int high) { . . . }The first method calls the second method. Is it recursive call?

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Recursion versus iteration

Iteration• Uses for or while loop for

repetition but no recursion

• Why use recursion?

Recursion• Recursion achieves

repetition without a loop• Price paid: memory

overhead and run time (Every time a method is called, some space in memory must be reserved or allocated tostore the method’s local variables and formal parameters, if any)

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public class NestedCalls { public static void m1() { int m1_x = 1; int m1_y = 2; m2(); // m1 calls m2 } // m1 method

public static void m2() { int m2_ = 3; int z = 4; z = m3(); //m2 calls m3 }// m2 method

public static int m3() { int m3_x =5; int m3_y = 6; return 1; }//m3 method}//NestedCalls class

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Run NestedCalls.java with Bluej Debugger

Observe the stack memory and calling sequences

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Stack frame for m1:m1_x: 1 m1_y: 2

Stack frame for m1:m1_x: 1 m1_y: 2

Stack frame for m1:m1_x: 1 m1_y: 2

Stack frame for m2:m2_x: 3m2_z: 4

Stack frame for m2:m2_x: 3m2_z: 4

Stack frame for m3:m3_x: 5m3_y: 6

Just before calling m2

Just before calling m3

Just before m3 returns

View of stack at various points of execution of NestedCalls.java

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Stack of Binary Search just before returning from the last recursive call

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Why use recursion?

In some problems, iterative solutions are hard to find.But recursion is easy - See the Towers of Hanoi problem in textbook. Animation by Michael Iverson

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Other problems that are neatly solved by recursion:

Fractals:• The Koch snowflake (demo) • Sierpinski triangle (demo)