10.1 Recursion
A method calling itself to repeat a certain number of times.
Popcorn Hack: What is recursion?
Vocabulary
- Base Case - this sets the termination condition for when the recursion will stop
- Formal Parameters - arguments within the method (local variables)
Method Calling Itself
We will first be looking at how a method can call itself.
/*
DO NOT RUN THIS
if you do run this accidentally press `ctrl + c` twice
*/
public class Recursion1 {
public static void recursionCount(int n) { // original method definition
System.out.println(n);
recursionCount(n - 1); // calling the method again
}
public static void main(String[] args) {
recursionCount(3); // original call of the method
}
}
Recursion1.main(null);
In this example, you will receive an overflow error because the method is calling itself infinitely. To avoid this you would create a __ __.
A base case usually uses an if-then statement to turn the loop off after a certain condition is met.
public class Recursion2 {
public static void recursionCount(int n) {
if (n <= 0) { // added base case, telling the program to stop when n is less than or equal to 0
System.out.println("done");
} else {
System.out.println(n);
recursionCount(n - 1); // method recursion
}
}
public static void main(String[] args) {
recursionCount(3); // calling original function
}
}
Recursion2.main(null);
3
2
1
done
Popcorn Hack: Create your own quick recursive program.
public class Calculator {
public static int factorial(int n) {
if (n == 0) {
return 1;
} else {
return n * factorial(n - 1);
}
}
public static void main(String[] args) {
int result = factorial(5);
System.out.println("factor of 5 is " + result);
}
}
Calculator.main(null)
factor of 5 is 120
Calling With Different Parameters
You can also call the method with multiple different different arguments (values or variables), also known as local variables.
In this example we call an integer and string.
public class Recursion3 {
public static void recursionPrint(int n, String a) { // formal parameters are within this method
if (n <= 0) { // base case
System.out.println("done");
return;
}
System.out.println("n = " + n + " a = " + a); // printing each var with each call of the method
String addChar = a + "!"; // mutating string with each recursion
int addN = n - 1; // mutating integer with each recursion
recursionPrint(addN, addChar); // recursive function
}
public static void main(String[] args) {
recursionPrint(5, "test"); // initial call
}
}
Recursion3.main(null);
n = 5 a = test
n = 4 a = test!
n = 3 a = test!!
n = 2 a = test!!!
n = 1 a = test!!!!
done
Here we just concatenate each string and subtract one from every previous number.
Capturing Progression Through Recursion
You can also capture the progression through recursion. This is useful for doing specific tasks in specific steps through recursion.
public class Recursion4 {
public static void recursionCount(int n, int progress) {
if (n <= 0) {
System.out.println("done");
} else {
if (n == progress) {
System.out.println("Reached halfway through the array."); // special action done only when the program has reached a certain point
}
System.out.println(n);
recursionCount(n - 1, progress); // recursive call
}
}
public static void main(String[] args) {
int n = 6;
int progress = n / 2; // progress is set to half of the array length
recursionCount(n, progress); // initial call
}
}
Recursion4.main(null);
6
5
4
Reached halfway through the array.
3
2
1
done
Popcorn Hack: What other situations could we use capturing progression for?
Algorithms:Implementing recursive versions of sorting algorithms, like quicksort or merge sort.
Game Development:Handling game state and logic in certain game development scenarios, such as maze generation or AI decision-making.
Converting Recursion to Iteration
Any recursive process can be made iterative, however this is not needed for the AP test. It is, however, still useful.
Recursive Version
public class Recursion5 {
public static void recursionCount(int n) {
if (n <= 0){
System.out.println("done");
} else {
System.out.println(n);
recursionCount(n - 1); // recursive function
}
}
public static void main(String[] args) {
recursionCount(3); // initial call
}
}
Recursion5.main(null);
3
2
1
done
Iterative Version
public class Iteration {
public static void iterationCount(int n) {
for (int i = n; i >= 1; i--) { // using a for loop to iterate, notice we go only so i is greater than 1 and not 0
System.out.println(i);
}
System.out.println("done");
}
public static void main(String[] args) {
iterationCount(3); // same initial call
}
}
Iteration.main(null);
3
2
1
done
Popcorn Hacks: We we go only so i is greater than or equal to 1 and not 0. Why?
Because indexing starts at 0 and not 1, we don’t need iteration with i equal to 0. only n to 1
Traversing With Recursion
You can traverse many things with recursion. Here I will show traversal through a string, array, and an ArrayList object.
String Traversal
You can traverse through a string, finding a specific character in the string.
public class FindChar {
public static boolean findChar(String s, char target, int index) {
if (index >= s.length()) { // checks for if the index we are looking for is greater than the word length
return false;
}
if (s.charAt(index) == target) { // checks for if there is a character the index we are searching
return true;
}
return findChar(s, target, index + 1); // recalls method, recursion
}
public static void main(String[] args) {
String sampleString = "words"; // example string
char targetChar = 's'; // character we are looking for
System.out.println("Sample String: " + sampleString);
boolean isCharFound = findChar(sampleString, targetChar, 0); // checks if the character we are looking for is found, based on boolean
if (isCharFound) {
System.out.println("Character '" + targetChar + "' is found."); // if it is found, show this
} else {
System.out.println("Character '" + targetChar + "' is not found."); // if not show this
}
}
}
FindChar.main(null);
Sample String: words
Character 's' is found.
You can also find and print a certain range of characters from the string.
public class PrintRange {
public static void printRange(String s, int startIndex, int endIndex) {
if (startIndex < 0 || endIndex > s.length() || startIndex >= endIndex) { // checks for invalid declaration or end (negative index, index larger than string length, index start has reached the end)
return;
}
System.out.print(s.charAt(startIndex)); // print the characters in the range
printRange(s, startIndex + 1, endIndex); // recursive calling
}
public static void main(String[] args) {
String sampleString = "example"; // string
int startIndex = 0;
int endIndex = 5;
System.out.println("Word: " + sampleString);
System.out.println("Start: " + startIndex + ", End: " + endIndex);
System.out.print("Characters in the specified range: ");
printRange(sampleString, startIndex, endIndex); // initial call
}
}
PrintRange.main(null);
Word: example
Start: 0, End: 5
Characters in the specified range: examp
Popcorn Hack: Make it so that we can find two different ranges and print them.
public class DiffRange {
public static void twoRange(String s, int startIndex, int endIndex) {
if (startIndex < 0 || endIndex > s.length() || startIndex >= endIndex) { // checks for invalid declaration or end (negative index, index larger than string length, index start has reached the end)
return;
}
System.out.print(s.charAt(startIndex)); // print the characters in the range
twoRange(s, startIndex + 1, endIndex); // recursive calling
}
public static void main(String[] args) {
String sampleString = "example"; // string
int startIndex = 0;
int endIndex = 5;
System.out.println("Word: " + sampleString);
System.out.print(startIndex + " to " + endIndex + ": ");
twoRange(sampleString, startIndex, endIndex); // initial call
int startIndex1 = 2;
int endIndex1 = 6;
System.out.print("\n" + startIndex1 + " to " + endIndex1 + ": ");
twoRange(sampleString, startIndex1, endIndex1);
}
}
DiffRange.main(null);
Word: example
0 to 5: examp
2 to 6: ampl
Array Traversal
You can also traverse arrays using recursion.
public class ArrayTraversal {
public static int findIndex(int[] arr, int targetIndex, int currentIndex) {
if (currentIndex < 0 || currentIndex >= arr.length) { // checks for invalid (index less than 0, index greater than array length)
return -1; // returns invalid
}
if (currentIndex == targetIndex) { // if the index is found
return currentIndex;
}
return findIndex(arr, targetIndex, currentIndex + 1); // recursion
}
public static void main(String[] args) {
int[] sampleArray = {1, 4, 6, 8, 4, 9, 12};
int targetIndex = 2; // target
int foundIndex = findIndex(sampleArray, targetIndex, 0);
if (foundIndex != -1) { // checks what has been outputted
int value = sampleArray[foundIndex]; // found index value
System.out.println("Index " + targetIndex + " is found in the array at position " + foundIndex + " with a value of " + value); // found
} else {
System.out.println("Index " + targetIndex + " is not found in the array."); // not found
}
}
}
ArrayTraversal.main(null);
Index 2 is found in the array at position 2 with a value of 6
ArrayList Object Traversal
We can also traverse the ArrayList object.
import java.util.ArrayList;
public class ArrayListTraversal {
public static int findIndex(ArrayList<Integer> list, int targetIndex, int currentIndex) {
if (currentIndex < 0 || currentIndex >= list.size()) { // checks for invalid
return -1;
}
if (currentIndex == targetIndex) { // finds index
return currentIndex;
}
return findIndex(list, targetIndex, currentIndex + 1); // recusion
}
public static void main(String[] args) {
ArrayList<Integer> sampleList = new ArrayList<>(); // created ArrayList object
sampleList.add(1);
sampleList.add(4);
sampleList.add(6);
sampleList.add(8);
sampleList.add(4);
sampleList.add(9);
sampleList.add(12);
int targetIndex = 2; // target
int foundIndex = findIndex(sampleList, targetIndex, 0);
if (foundIndex != -1) { // checks for if found or not
int value = sampleList.get(foundIndex); // gets found index from ArrayList object
System.out.println("Index " + targetIndex + " is found in the ArrayList at position " + foundIndex + " with a value of " + value);
} else {
System.out.println("Index " + targetIndex + " is not found in the ArrayList.");
}
}
}
ArrayListTraversal.main(null);
Index 2 is found in the ArrayList at position 2 with a value of 6
Conclusion
Must Knows:
- How to create recursive code (recalling a method within itself, with parameters that change)
- Record progression through recursion (completing certain tasks through the progression)
- Traversal of strings, arrays, and the
ArrayListobject
Overall, recursion is useful as an alternative to iteration. Recursion is usually chosen for problems that can naturally be divided into parts, like sorting which will be talked about in the next part.
10.2 Binary Search
Binary Search vs Linear Search
target number - 24
intArray - 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40
Popcorn hack:
How many times iterated through for Linear Search?
Answer:
Example of Binary Search with Recursion
public class BinarySearch {
public static void main(String[] args) {
int[] intArray = {0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40}; // Example array
int target = 24; // Example target value
int result = recBinarySearch(intArray, 0, intArray.length - 1, target);
if (result == -1) {
System.out.println("Target not found in the array.");
} else {
System.out.println("Target found at index: " + result);
}
}
public static int recBinarySearch(int[] intArray, int lowPosition, int highPosition, int target) { //method
if (lowPosition > highPosition) {
return -1; // Element not found in the array
} else {
int midPosition = (lowPosition + highPosition) / 2;
if (intArray[midPosition] < target) {
return recBinarySearch(intArray, midPosition + 1, highPosition, target);
} else if (intArray[midPosition] > target) {
return recBinarySearch(intArray, lowPosition, midPosition - 1, target);
} else {
return midPosition; // Element found at midPosition
}
}
}
}
BinarySearch.main(null);
Target found at index: 12
Call 1
Index = 0 - 20, midPosition = 10, midPosition value = 20
Since 24 > 20,
then…
lowPosition = midPosition + 1 = 11 highPosition = highPosition = 20
Call 2
Index = 11-20, midPosition index = 15, midPosition value = 30
Since 24 < 30,
then…
lowPosition = lowPosition = 11 high position = midPosition - 1 = 14
Call 3
Index = 11-14, midPosition index = 12, midPosition value = 24
Since 24 = 24,
then…
return midPosition;
In total, our recursive calls to the method 3 times.
Recursive Logic behind Merge Sort
What is Merge Sort? Merge sort is a recursive sorting algorithm that can be used to sort elements in an array or ArrayList
- Follows a divide and conquer approach
Notes:
List: [38,27,43,3,9,82,10]
sudocode version:
mergeSort(List) {
mergeSort(left)
mergeSort(right)
merge(left & right)
}
- Must finish call above it in order to finish the call
Call 1:
- Splitting List into half
- Left side: [38, 27, 43, 3]
- Must finish call 1 in order to do the right side and do the merge
- Recursively calls mergesort and splits the list in half again
Call 2:
- New Left side List: [38, 27]
- Method calls are stacking on top of each other
Notes:
- Element 5 can’t be split into the left or the right, nor can it be merged with itself
- Consider the left call complete!
Notes:
- Same thing applies with the right, element 25 can’t be split to the left or the right, nor can it merge with itself.
- Now we will merge them back in order: [5, 25]
Important concepts:
- When making new recursive call, we are NOT making a new list, array, or a new set of elements.
- Basically updating all the way back to the original list
Notes:
- When merging back together, it will merge back from least to greatest.
Popcorn Hack: What will the final list be?
Answer:least to greatest, -9, -2, 0, 2, 5, 8, 14, 25
The mergeSort Method
//sudocode
mergeSort(myArray, low, high) {
if (low < high) {
middle = (low + high)/2; //find middle
mergeSort(myArray, low, middle); //make a recursive call from low to middle (look at left hand side)
mergeSort(myArray, middle + 1, high); //once low is no longer less than high, make a new recursive call (look at right hand side)
merge (myArray, low, middle, high); //merge back together
}
}
int [] myArray = {3, 4, 6, 8, 1, 2, 5, 7};
Steps:
- Split the Array in half
- Left side: {3, 4, 6, 8}; Right side {1, 2, 5, 7};
- Compare the first indexes in each individual array (which would be index 0 and index 4)
- Since 1<3, our new index 0 value would be 1 when we merge the array back together
- Since 5>3, our new index 2 value would be 3 when we merge the array back together
Popcorn Hack: What will the final array return?
Answer: 1, 2, 3, 4, 5, 6, 7, 8
Wait, but there’s an issue…
- After comparing index 3 and index 7, we then need to compare index 3 and index, but there is no index 8…
- Will recieve an index out of bounds exception.
No worries! Since we are done with the sort, we can just move the rest of the elements to the end of the array because we are already done sorting.
Index 3 will now become index 7.
Hacks
- Create a 2D array either with just a regular array variable that you can recursively traverse through.
- Each value in this 2D array must be a string that you can individually traverse through.
- You must output a result of all string values that have a user inputted letter.
import java.util.Scanner;
public class String2DArrayTraversal {
public static void main(String[] args) {
Scanner scanner = new Scanner(System.in);
// Initialize a 2D array of strings
String[][] array = {
{"luna", "emma", "aliya"},
{"tay", "ekam", "raymond"},
{"ethan", "mort", "grace"}
};
System.out.print("enter a letter to search for: ");
char letter = scanner.nextLine().charAt(0);
System.out.println("strings containing the letter '" + letter + "':");
traverseString2DArray(array, letter, 0, 0);
}
public static void traverseString2DArray(String[][] array, char letter, int row, int col) {
if (row < array.length) {
if (col < array[row].length) {
String currentString = array[row][col];
if (currentString.contains(String.valueOf(letter))) {
System.out.println(currentString);
}
traverseString2DArray(array, letter, row, col + 1);
} else {
traverseString2DArray(array, letter, row + 1, 0);
}
}
}
}
String2DArrayTraversal.main(null)
enter a letter to search for: strings containing the letter 'a':
luna
emma
aliya
tay
ekam
raymond
ethan
grace
Create your own merge chart (like the first image in this 10.2 lesson) with your own values from a list, array, or arraylist (doesn’t matter). Explain how recursion works in the merge chat.
// code in here