## Search For A Range In A Sorted Array Difficulty: Easy

#### Understanding the Problem

Given a sorted array A[] with possibly duplicate elements, write a program to find indexes of first and last occurrences of an element k in the given array.

Problem Note

• The algorithm’s runtime complexity must be in the order of O(log n).
• If the target is not found in the array, return [-1, -1].

Example 1

``````Input: A[] = [1, 3, 5, 5, 5, 5 ,28, 37, 42], target = 5
Output: [2, 5]
Explanation: First Occurrence = 2 and Last Occurrence = 5``````

Example 2

``````Input: A[] = [1, 3, 5, 5, 5, 5 ,7, 28, 37], target = 7
Output: [6, 6]``````

Example 3

``````Input: A[] = [5,7,7,8,8,10], target = 6
Output: [-1,-1]``````

#### Solutions

We will be discussing two different approaches to solve this problem:-

1. Linear Search: Iterate over the array and maintain `lower_range` and `upper_range` variables while comparing the value at each index with `target` .
2. Binary Search: Perform binary search twice for the `target`.

You may try to solve this problem here.

#### 1. Linear Search

We have to search for the start and end indexes for the `target` value. As the array is sorted then it means that all the target values will be seen together. If we will iterate over the input array, then we could check if the element at any index is equal to the target or not. On the first occurrence of the `target` element update the `lower_range` and `upper_range` with that index. Upon further iteration, on each encounter of the `target` value, update `upper_range` .

Solution Steps

• Initialize `lower_bound` and `upper_bound` with `-1.`
• On the first occurrence of `target` element update these variables with the index.
• For the next occurrence of the `target` element, update the `upper_bound `with the index.

Pseudo Code

``````int[] searchRange(int[] arr, int target) {
lower_bound = -1
upper_bound = -1
for(i = 0 to i < arr.length){
if(arr[i] == target){
if(lower_bound == -1){
lower_bound = i
}
upper_bound = i
}
}
return [lower_bound, upper_bound]
}``````

Complexity Analysis

Time Complexity: O(n)

Space Complexity: O(1)

Critical Ideas To Think

• Why we are updating `upper_bound` at each occurrence of `target`?

#### 2. Binary Search

To find the upper and lower range we could call Binary Search two times for the target value. To get the lower range, we would want to slide to the “Left” of the array overtime (Since it's sorted). A slight modification in the Binary search would give us the left-most value:

``````while(lower_bound < upper_bound)
mid = (upper_bound + lower_bound)/2
if (target <= arr[mid])
upper_bound = mid
else:
lower_bound = mid+1``````

`(upper_bound + lower_bound)/2` would give us floor value of the integer. So, for every iteration, if `arr[mid]` is `<=` the target number, the starting range of the number is either at `arr[mid]` or is at its left of it.

So, we set the upper bound at mid (not mid-1, as the current element could itself be the upper limit). If the number is greater then the mid, the lower bound should be `mid+1` .

Next, we need to find the upper bound, so we need to slide to the right of the array. `mid = (upper_bound+lower_bound)/2 + 1`

Also, we won’t change the value stored at `lower_bound`. Since the `upper_bound` will be at `lower_bound` index or at the right of it.

``````while (lower_bound < upper_bound)
mid = (lower_bound+upper_bound)/2 + 1
if target < arr[mid]:
upper_bound = mid-1
else:
lower_bound = mid``````

So, if the target is lower, `upper_bound = mid-1` (it's at the lower part of the array).

If the target element is `≥ arr[mid]`, we would set the `lower_bound `as `mid` (not `mid+1`, as this index could itself hold the upper range of the number).

Solution Steps

• Find the leftmost index of the target in the array using binary search
• Find the rightmost index of the target in the array using another binary search
• Return the lower and upper index.

Pseudo Code

``````// Driver function
int[] searchRange(int[] arr, int target) {
int upperLowerBounds
start_index = 0
end_index = arr.length - 1
left_most_index = leftmost(start_index, end_index, target)
right_most_index = rightmost(start_index, end_index, target)
upperLowerBounds = left_most_index
upperLowerBounds = right_most_index
return upperLowerBounds
}
// Binary search function for leftmost target
int leftmost(int min, int max, int target) {
if (min == max)
return min
int mid = (min + max) / 2
if (array[mid] < target)
return leftmost(mid + 1, max, target)
else
return leftmost(min, mid, target)
}
// Binary search function for rightmost target
int rightmost(int min, int max, int target)
{
if (min == max)
return min
int mid = (min + max + 1) / 2
if (array[mid] > target)
return rightmost(min, mid - 1, target)
else
return rightmost(mid, max, target)
}``````

Complexity Analysis

Time Complexity: O(log n)

Space Complexity: O(log n).

We can reduce the space complexity by using the iterative function for binary search instead of recursive to O(1) space.

``````// Range search using itarative binary search
public int[] searchRange(int[] arr, int target)
{
int[] upperLowerBounds
int startIndex = 0;
int endIndex = arr.Length - 1
while (startIndex < endIndex) {
int midIndex = (startIndex + endIndex) / 2
if (target <= arr[midIndex]) {
endIndex = midIndex
}
else {
startIndex = midIndex + 1
}
}
// saving startIndex in the result array that will be  returned...
upperLowerBounds = startIndex
endIndex = arr.Length - 1
while (startIndex < endIndex) {
int midIndex = (startIndex + endIndex) / 2 + 1
if (target < arr[midIndex]) {
endIndex = midIndex  -1
}
else {
startIndex = midIndex;
}
}
upperLowerBounds = endIndex
return upperLowerBounds
}``````

Critical Ideas to Think

• How do the two binary search functions differ?
• How are we reducing space complexity?
• For what does the upperLowerBounds array is used for?

#### Comparison of different solutions #### Suggested Problems to Solve

• Find the element that appears once in a sorted array
• Find the minimum element in a sorted and rotated array
• Find the only repeating element in a sorted array of size n
• Find the Kth smallest element in the sorted generated array
• Find the missing element in a sorted array of consecutive numbers

If you have any more approaches or you find an error/bug in the above solutions, please comment down below.

Happy Coding! Enjoy Algorithms!