📓10.2: Recursive Searching & Sorting

Table of Contents

• Recursive Searching & Sorting Algorithms
  • Recursive Binary Search
  • Recursive Merge Sort
• ⭐️ Summary

---

📖 This page is a condensed version of CSAwesome Topic 10.2

---

Recursive Searching & Sorting Algorithms

In this lesson, we will take a look at a recursive binary search algorithm and a recursive merge-sort algorithm.

Recursive Binary Search

In Unit 7, we learned about two search algorithms, Linear Search and Binary Search. Linear search searches for an element in an array or ArrayList by checking each element in order.

Binary search is more efficient (faster) because it starts at the middle of a sorted array or ArrayList and eliminates half of the array or ArrayList each pass through the algorithm.

Binary search only works on sorted data! It can be written with iteration (using a loop) like below, or recursively.

public static int binarySearch(int[] elements, int target) {
  int left = 0;
  int right = elements.length - 1;
  while (left <= right) {
    int middle = (left + right) / 2;
    if (target < elements[middle]) {
      right = middle - 1;
    }
    else if (target > elements[middle]) {
      left = middle + 1;
    }
    else {
      return middle;
    }
  }
  return -1;
}

Watch the iterative binary search code running in the Java Visualizer.

Note that you can write solutions to many problems using either recursion or iteration. Iteration is usually preferred and more efficient, but recursive solutions can be elegant and require less code.

Let’s write a recursive version of Binary Search with the method signature boolean binarySearch(int[] array, int startIndex, int endIndex, int target).

💬 DISCUSS:

• What recursive method call would search the array from index 0 to the middle index?
• What’s the base case for a recursive version of Binary Search (when we want the recursion to stop)? Remember that in binary search, we always check the middle element first when looking for a target element from a startIndex to an endIndex.

Here is the Java code for a recursive binary search:

public static int recursiveBinarySearch(int[] array, int start, int end, int target) {
  int middle = (start + end) / 2;

  // base case: check middle element
  if (target == array[middle]) {
    return middle;
  }
  // base case: check if we've run out of elements
  if (end < start) {
    return -1; // not found
  }
  // recursive call: search start to middle
  if (target < array[middle]) {
    return recursiveBinarySearch(array, start, middle - 1, target);
  }
  // recursive call: search middle to end
  if (target > array[middle]) {
    return recursiveBinarySearch(array, middle + 1, end, target);
  }
  return -1;
}

Try the recursive binary search code in this Java visualizer link.

Recursive Merge Sort

In Unit 7, we looked at two sorting algorithms, Selection Sort and Insertion Sort. In this lesson, we will look at a third sorting algorithm, Merge Sort, which uses recursion.

Merge Sort is actually more efficient (faster) than Selection Sort and Insertion Sort because it divides the problem in half each time like binary search. This is called a divide and conquer algorithm.

Here is a short video that describes how merge sort works:

Here’s a video that shows merge sort with cards:

A merge sort recursively splits the values to be sorted in half until there is only one value to be sorted, and then it merges the two sorted lists into one sorted list.

The code shown below uses a second array the same size as the original array for merging the values in order. Then it copies all of the sorted values back into the original array.

🔍 To identify a merge sort algorithm look for the following:

• 3 methods: mergeSort, mergeSortHelper, and merge
• mergeSortHelper is recursive

public class MergeSort {

  public static void mergeSort(int[] elements) {
    int n = elements.length;
    int[] temp = new int[n];
    mergeSortHelper(elements, 0, n - 1, temp);
  }

  private static void mergeSortHelper(int[] elements, int from, int to, int[] temp) {
    if (from < to)  {
      int middle = (from + to) / 2;
      mergeSortHelper(elements, from, middle, temp);
      mergeSortHelper(elements, middle + 1, to, temp);
      merge(elements, from, middle, to, temp);
    }
  }

  private static void merge(int[] elements, int from, int mid, int to, int[] temp) {
    int i = from;
    int j = mid + 1;
    int k = from;

    while (i <= mid && j <= to) {
      if (elements[i] < elements[j]) {
        temp[k] = elements[i];
        i++;
      }
      else {
        temp[k] = elements[j];
        j++;
      }
      k++;
    }

    while (i <= mid) {
      temp[k] = elements[i];
      i++;
      k++;
    }

    while (j <= to) {
      temp[k] = elements[j];
      j++;
      k++;
    }

    for (k = from; k <= to; k++) {
      elements[k] = temp[k];
    }
  }
}

You can see this executing using the Java visualizer for merge sort.

✍️ You can trace through a merge sort algorithm given an array by using parentheses or curly braces to show how the array is divided into subarrays and then merged.

For example, here is how you could write down the trace of mergeSort(arr1) where arr1 = {86, 3, 43, 5}:

1. Split 1: { {86, 3} , {43, 5} }
2. Split 2: { { {86},{3}} , { {43},{5}} }
3. Merge 1: { {3, 86} , {5,43} }
4. Merge 2: { 3, 5, 43, 86 }

---

⭐️ Summary

• The binary search algorithm can be written either iteratively or recursively.
  • Data must be in sorted order to use the binary search algorithm.
  • The binary search algorithm starts at the middle of a sorted array or ArrayList and eliminates half of the array or ArrayList in until the desired value is found or all elements have been eliminated.
  • Binary search can be more efficient than sequential/linear search.
• Merge sort is a recursive sorting algorithm that can be used to sort elements in an array or ArrayList.

---

Acknowledgement

Content on this page is adapted from Runestone Academy - Barb Ericson, Beryl Hoffman, Peter Seibel.