Fixed Point in Python

A fixed point in an array is an index where the element's value equals its position. Given a sorted array of unique integers, we need to find the smallest index i such that A[i] == i.

For example, in the array [-10, -5, 0, 3, 7], index 3 has value 3, making it a fixed point.

Algorithm

To solve this problem, we follow these steps ?

• Iterate through each index from 0 to length of array
• If i == A[i], return i as the fixed point
• If no fixed point is found, return -1

Linear Search Approach

The straightforward approach checks each element sequentially ?

class Solution:
    def fixedPoint(self, A):
        for i in range(len(A)):
            if i == A[i]:
                return i
        return -1

# Test the solution
solution = Solution()
result = solution.fixedPoint([-10, -5, 0, 3, 7])
print(f"Fixed point index: {result}")

Fixed point index: 3

Optimized Binary Search Approach

Since the array is sorted, we can use binary search for better efficiency ?

class Solution:
    def fixedPointBinary(self, A):
        left, right = 0, len(A) - 1
        result = -1
        
        while left <= right:
            mid = (left + right) // 2
            
            if A[mid] == mid:
                result = mid
                right = mid - 1  # Continue searching for smaller index
            elif A[mid] < mid:
                left = mid + 1
            else:
                right = mid - 1
                
        return result

# Test both approaches
solution = Solution()
test_array = [-10, -5, 0, 3, 7]

linear_result = solution.fixedPoint(test_array)
binary_result = solution.fixedPointBinary(test_array)

print(f"Linear search result: {linear_result}")
print(f"Binary search result: {binary_result}")

Linear search result: 3
Binary search result: 3

Multiple Test Cases

Let's test with different arrays to understand the behavior ?

def test_fixed_point():
    solution = Solution()
    
    test_cases = [
        [-10, -5, 0, 3, 7],    # Fixed point at index 3
        [0, 2, 5, 8, 17],      # Fixed point at index 0
        [-10, -5, 3, 4, 7, 9], # No fixed point
        [1, 2, 3, 4, 5]        # No fixed point
    ]
    
    for i, arr in enumerate(test_cases):
        result = solution.fixedPoint(arr)
        print(f"Array {i+1}: {arr}")
        print(f"Fixed point: {result}")
        print()

test_fixed_point()

Array 1: [-10, -5, 0, 3, 7]
Fixed point: 3

Array 2: [0, 2, 5, 8, 17]
Fixed point: 0

Array 3: [-10, -5, 3, 4, 7, 9]
Fixed point: -1

Array 4: [1, 2, 3, 4, 5]
Fixed point: -1

Comparison

Conclusion

A fixed point occurs when A[i] == i. Use linear search for simplicity or binary search for optimal performance on large sorted arrays. The binary search approach leverages the sorted property to achieve O(log n) time complexity.