Solutions

Prerequisites

Before attempting this problem, you should be comfortable with:

Binary Search - Finding elements or boundaries in sorted arrays in O(log N) time
Lower/Upper Bound - Using binary search variants to find first and last occurrences of a target
Sorted Array Properties - Leveraging the fact that duplicate elements are contiguous in sorted arrays

1. Frequency Count

Intuition

A majority element appears more than half the time. The simplest approach is to count how many times the target appears in the array by scanning through all elements. If the count exceeds n/2, the target is a majority element.

Algorithm

Initialize a counter to 0.
Iterate through the array:
- Increment the counter each time we encounter the target.
Return true if the count is greater than half the array length, false otherwise.

class Solution:
    def isMajorityElement(self, nums: List[int], target: int) -> bool:
        count = 0
        for num in nums:
            count = count + 1 if num == target else count
        
        return count > len(nums) // 2

class Solution {
    public boolean isMajorityElement(int[] nums, int target) {
        int count = 0;
        for (int num : nums) {
            count = num == target ? count + 1 : count;
        }
        
        return count > nums.length / 2;
    }
}

class Solution {
public:
    bool isMajorityElement(vector<int>& nums, int target) {
        int count = 0;
        for (int num : nums) {
            count = num == target ? count + 1 : count;
        }
        
        return count > nums.size() / 2;
    }
};

class Solution {
    /**
     * @param {number[]} nums
     * @param {number} target
     * @return {boolean}
     */
    isMajorityElement(nums, target) {
        let count = 0;
        for (let num of nums) {
            count = num === target ? count + 1 : count;
        }
        
        return count > Math.floor(nums.length / 2);
    }
}

public class Solution {
    public bool IsMajorityElement(int[] nums, int target) {
        int count = 0;
        foreach (int num in nums) {
            count = num == target ? count + 1 : count;
        }

        return count > nums.Length / 2;
    }
}

class Solution {
    fun isMajorityElement(nums: IntArray, target: Int): Boolean {
        var count = 0
        for (num in nums) {
            count = if (num == target) count + 1 else count
        }

        return count > nums.size / 2
    }
}

class Solution {
    func isMajorityElement(_ nums: [Int], _ target: Int) -> Bool {
        var count = 0
        for num in nums {
            count = num == target ? count + 1 : count
        }

        return count > nums.count / 2
    }
}

Time & Space Complexity

Time complexity: $O(N)$
Space complexity: $O(1)$ constant space

Where $N$ is the size of nums.

2. Binary Search (Two Pass)

Intuition

Since the array is sorted, all occurrences of the target are contiguous. We can use binary search to find the first and last occurrence of the target. The count is simply (lastIndex - firstIndex + 1), which we compare against n/2.

Algorithm

Use lower_bound to find the index of the first element >= target.
Use upper_bound to find the index of the first element > target.
The count of target is (upper_bound - lower_bound).
Return true if this count is greater than n/2.

class Solution:
    def lower_bound(self, nums: List[int], target: int) -> int:
        """
        Returns the index of the first element equal to or greater than the target.
        If there is no instance of the target in the list, it returns the length of the list.
        """
        start = 0
        end = len(nums) - 1
        index = len(nums)
        
        while start <= end:
            mid = (start + end) // 2
            if nums[mid] >= target:
                end = mid - 1
                index = mid
            else:
                start = mid + 1
        
        return index
    
    def upper_bound(self, nums: List[int], target: int) -> int:
        """
        Returns the index of the first element greater than the target.
        If there is no instance of the target in the list, it returns the length of the list.
        """
        start = 0
        end = len(nums) - 1
        index = len(nums)
        
        while start <= end:
            mid = (start + end) // 2
            if nums[mid] > target:
                end = mid - 1
                index = mid
            else:
                start = mid + 1
        
        return index
    
    def isMajorityElement(self, nums: List[int], target: int) -> bool:
        first_index = self.lower_bound(nums, target)
        next_to_last_index = self.upper_bound(nums, target)
        return next_to_last_index - first_index > len(nums) // 2

class Solution {
    // Returns the index of the first element equal to or greater than the target.
    // If there is no instance of the target in the list, it returns the length of the list.
    int lower_bound(int[] nums, int target) {
        int start = 0;
        int end = nums.length - 1;
        int index = nums.length;
            
        while (start <= end) {
            int mid = (start + end) / 2;
            
            if (nums[mid] >= target) {
                end = mid - 1;
                index = mid;
            } else {
                start = mid + 1;
            }
        }
        
        return index;
    }
    
    // Returns the index of the first element greater than the target.
    // If there is no instance of the target in the list, it returns the length of the list.
    int upper_bound(int[] nums, int target) {
        int start = 0;
        int end = nums.length - 1;
        int index = nums.length;
        
        while (start <= end) {
            int mid = (start + end) / 2;
            
            if (nums[mid] > target) {
                end = mid - 1;
                index = mid;
            } else {
                start = mid + 1;
            }
        }
        
        return index;
    }
    
    public boolean isMajorityElement(int[] nums, int target) {
        int firstIndex = lower_bound(nums, target);
        int nextToLastIndex = upper_bound(nums, target);
        
        return nextToLastIndex - firstIndex > nums.length / 2;
    }
}

class Solution {
public:
    // Returns the index of the first element equal to or greater than the target.
    // If there is no instance of the target in the list, it returns the length of the list.
    int lower_bound(vector<int>& nums, int target) {
        int start = 0;
        int end = nums.size() - 1;
        int index = nums.size();
            
        while (start <= end) {
            int mid = (start + end) / 2;
            
            if (nums[mid] >= target) {
                end = mid - 1;
                index = mid;
            } else {
                start = mid + 1;
            }
        }
        
        return index;
    }
    
    // Returns the index of the first element greater than the target.
    // If there is no instance of the target in the list, it returns the length of the list.
    int upper_bound(vector<int>& nums, int target) {
        int start = 0;
        int end = nums.size() - 1;
        int index = nums.size();
        
        while (start <= end) {
            int mid = (start + end) / 2;
            
            if (nums[mid] > target) {
                end = mid - 1;
                index = mid;
            } else {
                start = mid + 1;
            }
        }
        
        return index;
    }
    
    bool isMajorityElement(vector<int>& nums, int target) {
        int firstIndex = lower_bound(nums, target);
        int nextToLastIndex = upper_bound(nums, target);
        
        return nextToLastIndex - firstIndex > nums.size() / 2;
    }
};

class Solution {
    /**
     * Returns the index of the first element equal to or greater than the target.
     * If there is no instance of the target in the list, it returns the length of the list.
     * @param {number[]} nums
     * @param {number} target
     * @return {number}
     */
    lowerBound(nums, target) {
        let start = 0;
        let end = nums.length - 1;
        let index = nums.length;
        
        while (start <= end) {
            let mid = Math.floor((start + end) / 2);
            if (nums[mid] >= target) {
                end = mid - 1;
                index = mid;
            } else {
                start = mid + 1;
            }
        }
        
        return index;
    }
    
    /**
     * Returns the index of the first element greater than the target.
     * If there is no instance of the target in the list, it returns the length of the list.
     * @param {number[]} nums
     * @param {number} target
     * @return {number}
     */
    upperBound(nums, target) {
        let start = 0;
        let end = nums.length - 1;
        let index = nums.length;
        
        while (start <= end) {
            let mid = Math.floor((start + end) / 2);
            if (nums[mid] > target) {
                end = mid - 1;
                index = mid;
            } else {
                start = mid + 1;
            }
        }
        
        return index;
    }
    
    /**
     * @param {number[]} nums
     * @param {number} target
     * @return {boolean}
     */
    isMajorityElement(nums, target) {
        const firstIndex = this.lowerBound(nums, target);
        const nextToLastIndex = this.upperBound(nums, target);
        return nextToLastIndex - firstIndex > Math.floor(nums.length / 2);
    }
}

public class Solution {
    private int LowerBound(int[] nums, int target) {
        int start = 0;
        int end = nums.Length - 1;
        int index = nums.Length;

        while (start <= end) {
            int mid = (start + end) / 2;
            if (nums[mid] >= target) {
                end = mid - 1;
                index = mid;
            } else {
                start = mid + 1;
            }
        }

        return index;
    }

    private int UpperBound(int[] nums, int target) {
        int start = 0;
        int end = nums.Length - 1;
        int index = nums.Length;

        while (start <= end) {
            int mid = (start + end) / 2;
            if (nums[mid] > target) {
                end = mid - 1;
                index = mid;
            } else {
                start = mid + 1;
            }
        }

        return index;
    }

    public bool IsMajorityElement(int[] nums, int target) {
        int firstIndex = LowerBound(nums, target);
        int nextToLastIndex = UpperBound(nums, target);
        return nextToLastIndex - firstIndex > nums.Length / 2;
    }
}

func isMajorityElement(nums []int, target int) bool {
    lowerBound := func(nums []int, target int) int {
        start, end := 0, len(nums)-1
        index := len(nums)

        for start <= end {
            mid := (start + end) / 2
            if nums[mid] >= target {
                end = mid - 1
                index = mid
            } else {
                start = mid + 1
            }
        }

        return index
    }

    upperBound := func(nums []int, target int) int {
        start, end := 0, len(nums)-1
        index := len(nums)

        for start <= end {
            mid := (start + end) / 2
            if nums[mid] > target {
                end = mid - 1
                index = mid
            } else {
                start = mid + 1
            }
        }

        return index
    }

    firstIndex := lowerBound(nums, target)
    nextToLastIndex := upperBound(nums, target)
    return nextToLastIndex-firstIndex > len(nums)/2
}

class Solution {
    private fun lowerBound(nums: IntArray, target: Int): Int {
        var start = 0
        var end = nums.size - 1
        var index = nums.size

        while (start <= end) {
            val mid = (start + end) / 2
            if (nums[mid] >= target) {
                end = mid - 1
                index = mid
            } else {
                start = mid + 1
            }
        }

        return index
    }

    private fun upperBound(nums: IntArray, target: Int): Int {
        var start = 0
        var end = nums.size - 1
        var index = nums.size

        while (start <= end) {
            val mid = (start + end) / 2
            if (nums[mid] > target) {
                end = mid - 1
                index = mid
            } else {
                start = mid + 1
            }
        }

        return index
    }

    fun isMajorityElement(nums: IntArray, target: Int): Boolean {
        val firstIndex = lowerBound(nums, target)
        val nextToLastIndex = upperBound(nums, target)
        return nextToLastIndex - firstIndex > nums.size / 2
    }
}

class Solution {
    func isMajorityElement(_ nums: [Int], _ target: Int) -> Bool {
        func lowerBound(_ nums: [Int], _ target: Int) -> Int {
            var start = 0
            var end = nums.count - 1
            var index = nums.count

            while start <= end {
                let mid = (start + end) / 2
                if nums[mid] >= target {
                    end = mid - 1
                    index = mid
                } else {
                    start = mid + 1
                }
            }

            return index
        }

        func upperBound(_ nums: [Int], _ target: Int) -> Int {
            var start = 0
            var end = nums.count - 1
            var index = nums.count

            while start <= end {
                let mid = (start + end) / 2
                if nums[mid] > target {
                    end = mid - 1
                    index = mid
                } else {
                    start = mid + 1
                }
            }

            return index
        }

        let firstIndex = lowerBound(nums, target)
        let nextToLastIndex = upperBound(nums, target)
        return nextToLastIndex - firstIndex > nums.count / 2
    }
}

Time & Space Complexity

Time complexity: $O(\log N)$
Space complexity: $O(1)$ constant space

Where $N$ is the size of nums.

3. Binary Search (One Pass)

Intuition

If the target is a majority element, it must occupy more than half the array positions. This means if we find the first occurrence at index i, the element at index i + n/2 must also be the target. We only need one binary search to find the first occurrence, then check this specific position.

Algorithm

Use lower_bound to find the first occurrence of the target at index i.
Calculate the position i + n/2 (where n is the array length).
If this position is within bounds and the element at that position equals the target, return true.
Otherwise, return false.

class Solution:
    def lower_bound(self, nums: List[int], target: int) -> int:
        """
        Returns the index of the first element that is equal to or greater than the target.
        If there is no instance of the target in the list, it returns the length of the list.
        """
        start = 0
        end = len(nums) - 1
        index = len(nums)
        
        while start <= end:
            mid = (start + end) // 2
            if nums[mid] >= target:
                end = mid - 1
                index = mid
            else:
                start = mid + 1
        
        return index
    
    def isMajorityElement(self, nums: List[int], target: int) -> bool:
        first_index = self.lower_bound(nums, target)
        return first_index + len(nums) // 2 < len(nums) and nums[first_index + len(nums) // 2] == target

class Solution {
    // Returns the index of the first element that is equal to or greater than the target.
    // If there is no instance of the target in the list, it returns the length of the list.
    int lower_bound(int[] nums, int target) {
        int start = 0;
        int end = nums.length - 1;
        int index = nums.length;
            
        while (start <= end) {
            int mid = (start + end) / 2;
            
            if (nums[mid] >= target) {
                end = mid - 1;
                index = mid;
            } else {
                start = mid + 1;
            }
        }
        
        return index;
    }
    
    public boolean isMajorityElement(int[] nums, int target) {
        int firstIndex = lower_bound(nums, target);
        
        return firstIndex + nums.length / 2 < nums.length && nums[firstIndex + nums.length / 2] == target;
    }
}

class Solution {
public:
    // Returns the index of the first element equal to or greater than the target.
    // If there is no instance of the target in the list, it returns the length of the list.
    int lower_bound(vector<int>& nums, int target) {
        int start = 0;
        int end = nums.size() - 1;
        int index = nums.size();
            
        while (start <= end) {
            int mid = (start + end) / 2;
            
            if (nums[mid] >= target) {
                end = mid - 1;
                index = mid;
            } else {
                start = mid + 1;
            }
        }
        
        return index;
    }
    
    bool isMajorityElement(vector<int>& nums, int target) {
        int firstIndex = lower_bound(nums, target);
        
        return firstIndex + nums.size() / 2 < nums.size() && nums[firstIndex + nums.size() / 2] == target;
    }
};

class Solution {
    /**
     * Returns the index of the first element that is equal to or greater than the target.
     * If there is no instance of the target in the list, it returns the length of the list.
     * @param {number[]} nums
     * @param {number} target
     * @return {number}
     */
    lowerBound(nums, target) {
        let start = 0;
        let end = nums.length - 1;
        let index = nums.length;
        
        while (start <= end) {
            let mid = Math.floor((start + end) / 2);
            if (nums[mid] >= target) {
                end = mid - 1;
                index = mid;
            } else {
                start = mid + 1;
            }
        }
        
        return index;
    }
    
    /**
     * @param {number[]} nums
     * @param {number} target
     * @return {boolean}
     */
    isMajorityElement(nums, target) {
        const firstIndex = this.lowerBound(nums, target);
        return firstIndex + Math.floor(nums.length / 2) < nums.length && nums[firstIndex + Math.floor(nums.length / 2)] === target;
    }
}

public class Solution {
    private int LowerBound(int[] nums, int target) {
        int start = 0;
        int end = nums.Length - 1;
        int index = nums.Length;

        while (start <= end) {
            int mid = (start + end) / 2;
            if (nums[mid] >= target) {
                end = mid - 1;
                index = mid;
            } else {
                start = mid + 1;
            }
        }

        return index;
    }

    public bool IsMajorityElement(int[] nums, int target) {
        int firstIndex = LowerBound(nums, target);
        return firstIndex + nums.Length / 2 < nums.Length && nums[firstIndex + nums.Length / 2] == target;
    }
}

func isMajorityElement(nums []int, target int) bool {
    lowerBound := func(nums []int, target int) int {
        start, end := 0, len(nums)-1
        index := len(nums)

        for start <= end {
            mid := (start + end) / 2
            if nums[mid] >= target {
                end = mid - 1
                index = mid
            } else {
                start = mid + 1
            }
        }

        return index
    }

    firstIndex := lowerBound(nums, target)
    return firstIndex+len(nums)/2 < len(nums) && nums[firstIndex+len(nums)/2] == target
}

class Solution {
    private fun lowerBound(nums: IntArray, target: Int): Int {
        var start = 0
        var end = nums.size - 1
        var index = nums.size

        while (start <= end) {
            val mid = (start + end) / 2
            if (nums[mid] >= target) {
                end = mid - 1
                index = mid
            } else {
                start = mid + 1
            }
        }

        return index
    }

    fun isMajorityElement(nums: IntArray, target: Int): Boolean {
        val firstIndex = lowerBound(nums, target)
        return firstIndex + nums.size / 2 < nums.size && nums[firstIndex + nums.size / 2] == target
    }
}

class Solution {
    func isMajorityElement(_ nums: [Int], _ target: Int) -> Bool {
        func lowerBound(_ nums: [Int], _ target: Int) -> Int {
            var start = 0
            var end = nums.count - 1
            var index = nums.count

            while start <= end {
                let mid = (start + end) / 2
                if nums[mid] >= target {
                    end = mid - 1
                    index = mid
                } else {
                    start = mid + 1
                }
            }

            return index
        }

        let firstIndex = lowerBound(nums, target)
        return firstIndex + nums.count / 2 < nums.count && nums[firstIndex + nums.count / 2] == target
    }
}

Time & Space Complexity

Time complexity: $O(\log N)$
Space complexity: $O(1)$ constant space

Where $N$ is the size of nums.

Common Pitfalls

Using >= Instead of > for Majority Check

The majority element must appear more than n/2 times, not greater than or equal to. Using count >= n/2 will incorrectly return true for elements that appear exactly half the time.

# Wrong: count >= len(nums) // 2
# Correct:
return count > len(nums) // 2

Not Leveraging the Sorted Property

Since the array is sorted, all occurrences of the target are contiguous. A common mistake is to use a linear scan when binary search can find the first and last occurrence in O(log N) time. Additionally, forgetting that you only need to check if nums[first_index + n//2] == target after finding the first occurrence wastes an extra binary search.