Find First and Last Position of Element in Sorted Array

Difficulty: Easy, Asked in: Google, Amazon, Microsoft

Key takeaway: This is an excellent interview problem to learn problem-solving using binary search. We apply binary search twice to improve the time complexity over a brute force approach.

Let's understand the problem

Given an array of integers sorted in ascending order, write code to find the first and last position of a given target value.

• If the target is not found in the array, return [-1, -1].
• We must design an algorithm with O(log n) time complexity.

Example 1

Input: A[] = [-1, 1, 2, 2, 2, 5, 6, 12, 12], target = 2

Output: First occurrence = 2, Last occurrence = 4

Example 2

Input: A[] = [21, 32, 51, 70, 71], target = 70

Output: First occurrence = 3, Last occurrence = 3

Discussed solution approaches

• Brute force approach  using linear search
• Efficient approach  by applying binary search twice

Brute force approach using linear search

Solution idea

The straightforward approach is to traverse the array and track the index of the first and last occurrence of the target. This approach is inefficient because we did not take advantage of the fact that the given array is sorted.

Solution steps

1. Take an extra constant size array firstLast[2] to store the first and last occurrence of the target element. We will store the first occurrence at firstLast[0] and the last occurrence at firstLast[1]. At the start, we initialize both values with -1.
2. Now we traverse the array linearly from i = 0 to n - 1 and update firstLast[0] when we encounter the target value for the first time. To keep track of the last occurrence, we keep updating the value of firstLast[1] every time when we encounter the target value.
3. By the end of the loop, we return the firstLast[] array.

Solution code C++

vector<int> firstLastOccurence(int A[], int n, int target)
{
    vector<int> firstLast = {-1, -1};
    for (int i = 0; i < n; i = i + 1)
    {
        if (A[i] == target)
        {
            if (firstLast[0] == -1)
                firstLast[0] = i;
            firstLast[1] = i;
        }
    }
    return firstLast;
}

Solution code Python

def first_last_occurence(A, n, target):
    first_last = [-1, -1]
    for i in range(n):
        if A[i] == target:
            if first_last[0] == -1:
                first_last[0] = i
            first_last[1] = i
    return first_last

Solution analysis

We are running a single loop n time and doing O(1) operations at each iteration. Time complexity = n * O(1) = O(n). We are using constant extra space, so space complexity = O(1).

An efficient approach using binary search

Solution idea and steps

Now the critical questions are: Can we improve the time complexity further? Can we use the sorted order property to enhance the efficiency of searching?

As we already know, binary search works perfectly for searching for any element in the sorted array. It would take O(logn) time to search for the target value. But the given problem is different from the binary search problem, where we need to search for two occurrences of the target element. So, how do we modify the standard binary search algorithm to solve it? Think.

The idea is simple: We can use binary search twice to solve the problem. The first binary search is for finding the first occurrence of the target, and the second binary search is for finding the last occurrence of the target. Let’s design an algorithm for both steps separately.

Binary search for finding the first occurrence

Let's use iterative binary search! We first initialize low = 0 and high = n - 1 to track the left and right ends. Now we run a loop till low ≤ high:

1. Calculate mid, i.e. mid = low + (high - low)/2
2. Now compare the value at mid with the target. The middle element will be the first occurrence in two situations: 1) target == A[mid] and target > A[mid - 1], i.e. when the first occurrence is present somewhere in the middle. 2) mid == 0 and A[mid] == target, i.e., when the first occurrence is present at the first position. In both situations, we return mid-index as the first occurrence.
3. If (target > A[mid]), we search for the first occurrence in the right part because all values in the left part are less than the target. Update low = mid + 1.
4. If both conditions are not satisfied, we need to search for the first occurrence in the left part (Think). We update high = mid - 1.
5. If we did not find the target value by the end of the loop, we return -1.

Pseudocode to find the first occurrence

int findFirstOccurrence(int A[], int n, int target)
{
    int low = 0, high = n - 1
    while (low <= high)
    {
        int mid = low + (high - low)/2
        if ((mid == 0 || A[mid - 1] < target) && A[mid] == target)
            return mid
        else if (target > A[mid])    
            low = mid + 1
        else
            high = mid - 1
    }
    return -1
}

Binary search for finding the last occurrence

Similarly, to find the last occurrence, we also modify the binary search algorithm. We first initialize low = 0 and high = n - 1 to track the left and right ends during the binary search loop. Now we run a loop until low ≤ high:

1. We calculate mid i.e. mid = low + (high - low)/2
2. Next, we compare the value at mid with the target. The mid element will be the last occurrence in two situations: 1) target == A[mid] and target < A[mid + 1] i.e. when the last occurrence is present somewhere in the middle 2) mid == n - 1 and A[mid] == target i.e. when the last occurrence is at the last position.
3. If (target < A[mid]), we need to search for the last occurrence in the left part because all values in the right part are greater than the target. Update high= mid - 1.
4. If both conditions are not satisfied, we need to search for the last occurrence in the right part (Think). Update low = mid + 1.
5. By the end of the loop, if we did not find the target value, return -1.

Pseudocode to find the last occurrence

int findLastOccurrence(int A[], int n, int target)
{
    int low = 0, high = n - 1
    while (low <= high)
    {
        int mid = low + (high - low)/2
        if ((mid == n - 1 || A[mid + 1] > target) && A[mid] == target)
            return mid
        else if (target < A[mid])    
            high = mid - 1
        else
            low = mid + 1
    }
    return -1
}

Solution code C++

int findLastOccurrence(int A[], int n, int target)
{
    int low = 0, high = n - 1;
    while (low <= high)
    {
        int mid = low + (high - low) / 2;
        if ((mid == n - 1 || A[mid + 1] > target) && A[mid] == target)
            return mid;
        else if (target < A[mid])    
            high = mid - 1;
        else
            low = mid + 1;
    }
    return -1;
}

int findFirstOccurrence(int A[], int n, int target)
{
    int low = 0, high = n - 1;
    while (low <= high)
    {
        int mid = low + (high - low) / 2;
        if ((mid == 0 || A[mid - 1] < target) && A[mid] == target)
            return mid;
        else if (target > A[mid])    
            low = mid + 1;
        else
            high = mid - 1;
    }
    return -1;
}

vector<int> findFirstLastOccurrence(int A[], int n, int target)
{
    vector<int> firstLast = {-1, -1};
    firstLast[0] = findFirstOccurrence(A, n, target);
    firstLast[1] = findLastOccurrence(A, n, target);
    return firstLast;
}

Solution code Python

def find_last_occurrence(A, n, target):
    low = 0
    high = n - 1
    while low <= high:
        mid = low + (high - low) // 2
        if (mid == n - 1 or A[mid + 1] > target) and A[mid] == target:
            return mid
        elif target < A[mid]:
            high = mid - 1
        else:
            low = mid + 1
    return -1

def find_first_occurrence(A, n, target):
    low = 0
    high = n - 1
    while low <= high:
        mid = low + (high - low) // 2
        if (mid == 0 or A[mid - 1] < target) and A[mid] == target:
            return mid
        elif target > A[mid]:
            low = mid + 1
        else:
            high = mid - 1
    return -1

def find_first_last_occurrence(A, n, target):
    first_last = [-1, -1]
    first_last[0] = find_first_occurrence(A, n, target)
    first_last[1] = find_last_occurrence(A, n, target)
    return first_last

Solution analysis

Inside the function firstLastOccurrence(A[], n, target), we are doing some constant extra operations and applying binary search twice. Time complexity = O(1) + Time complexity to search first occurrence + Time complexity to search last occurrence = O(1) + O(logn) + O(logn) = O(log n).

We are using the implementation of iterative binary search, which takes O(1) extra space. Space complexity = O(1)

Critical ideas to think!

• Can we solve this problem using a single binary search?
• How can we implement the above approach using a recursive binary search? What will be the space complexity?

Comparisons of time and space complexities

• Using linear search: Time = O(n), Space = O(1)
• Using binary search: Time = O(logn), Space = O(1)

Coding problems to practice using binary search

• Find maximum in bitonic array
• Min in rotated and sorted array
• Search in a sorted matrix
• Median of sorted arrays
• Find square root
• Find peak element in array
• Find smallest letter greater than target
• Search in rotated sorted array

Thanks to Navtosh for his contribution in creating the first version of this content. If you have any queries/doubts/feedback, please write us at contact@enjoyalgorithms.com. Enjoy learning, Enjoy algorithms!