Circular Array Loop

To solve this coding challenge, we'll follow a systematic approach to detect cycles in a circular array. The detailed explanation and pseudocode below will guide you through the solution.

Explanation:

The problem involves detecting a cycle in a circular array where each element represents the number of steps to move forward or backward. Our goal is to identify if there's a cycle that meets the criteria:
  1. Every element in the sequence is either all positive or all negative.
  2. The cycle length must be greater than 1.
  3. Key approach :
  4. Traverse each element of the array to treat it as the starting point of a potential cycle.
  5. Use two pointers (slow and fast) to detect a cycle (similar to Floyd's Tortoise and Hare algorithm).
  6. If a cycle is detected, check if all elements in the cycle move in the same direction.
  7. If no valid cycle is detected from a starting point, mark all elements along the path as visited to avoid reprocessing.
  8. Key constraint : Since the array is circular, moving past the last element means wrapping around to the first element and vice versa when moving backward.

    Detailed Steps in Pseudocode:

  9. Define a function
    next_index
    to compute the next index in the circular array.
  10. Loop through each element of the array to check for cycles:
    • If the element has been visited (marked by setting to 0), skip it.
    • Use
      slow
      and
      fast
      pointers to traverse the array.
    • Move
      slow
      one step at a time and
      fast
      two steps at a time.
    • Check for cycle detection (when
      slow
      meets
      fast
      ).
    • Ensure the cycle length is greater than 1.
    • If no valid cycle is found, mark all traversed elements to avoid rechecking.

Pseudocode:

                                            
# Utility function to get the next index in a circular manner
def next_index(nums, current_index):
    n = length of nums
    next_index = (current_index + nums[current_index]) % n

    # Handle wrap-around for negative indices
    if next_index < 0:
        next_index += n
    
    return next_index

# Main function to detect a circular array loop
def circular_array_loop(nums):
    n = length of nums

    # Iterate through each element in the array
    for i in range(n):
        # Skip if the element is marked as visited
        if nums[i] == 0:
            continue

        # Initialize slow and fast pointers
        slow = i
        fast = next_index(nums, i)

        # Loop to detect cycle
        while nums[slow] * nums[fast] > 0 and nums[fast] * nums[next_index(nums, fast)] > 0:
            # Check if the slow and fast pointers meet
            if slow == fast:
                # Check if the cycle length is greater than 1
                if slow == next_index(nums, slow):
                    break
                return True

            # Move slow pointer by one step
            slow = next_index(nums, slow)

            # Move fast pointer by two steps
            fast = next_index(nums, next_index(nums, fast))

        # Mark all nodes along the current path as visited
        slow = i
        current_value = nums[i]
        while nums[slow] * current_value > 0:
            next_slow = next_index(nums, slow)
            nums[slow] = 0  # Mark as visited
            slow = next_slow

    # No valid cycle found
    return False

                                        
This pseudocode outlines a method to efficiently detect cycles in a circular array by leveraging the two-pointer technique and ensuring minimal additional space complexity by leveraging in-place modifications. By carefully managing traversal and marking visited nodes, we ensure that each node is processed optimally.