Longest Consecutive Sequence

The Longest Consecutive Sequence problem is a popular algorithm challenge that tests how efficiently one can find the longest sequence of consecutive integers in an unsorted array. While it looks like a basic problem, digging deeper reveals connections to graph algorithms, sequence detection, and even dynamic programming strategies.

This problem solving is not just another walkthrough; this article is for those who want to learn how graph traversal, array connectivity, and pathfinding ideas from graph theory can influence solving problems outside the typical graph domain.

Longest Consecutive Sequence

You're given an unsorted array of integers. Your goal is to return the length of the longest consecutive sequence. The elements in the array may not be in order, and duplicates may exist.

Example:

Input: [100, 4, 200, 1, 3, 2]
Output: 4
Explanation: The longest consecutive sequence is [1, 2, 3, 4].

Naive Approach and Its Pitfalls

The first idea many consider is sorting the array and then doing a single pass to count consecutive numbers. While this works, it costs O(n log n) time due to sorting. That violates the linear-time expectation when tackling interview-level questions.

Optimal Solution Using Hashing and Graph Connectivity Insight

Here's where the interesting part begins. We can think of the integers as nodes in a graph. Each number has a potential edge to num + 1 and num - 1, making this a case of vertex connectivity and array connectivity. Our task reduces to identifying the longest path of connected nodes — which directly correlates with the longest consecutive sequence.

Efficient Algorithm Using HashSet

We use a HashSet to efficiently check if a number is the start of a sequence and extend from there. This method mimics graph traversal, where each node is visited only once, similar to DFS but done iteratively.

Python Code Example

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def longestConsecutive(nums):
    num_set = set(nums)
    longest = 0

    for num in num_set:
        # Check if this number is the start of a sequence
        if num - 1 not in num_set:
            current_num = num
            current_streak = 1

            while current_num + 1 in num_set:
                current_num += 1
                current_streak += 1

            longest = max(longest, current_streak)

    return longest

Code Explanation:

• Convert the list to a set for O(1) lookups.
• Iterate only over potential sequence starts (num - 1 not in set).
• For each such start, increment to count sequence length.
• Track the maximum sequence length.

This approach ensures O(n) time complexity and minimal space overhead.

Graph Perspective and Theory Connection

This problem might not involve a graph explicitly, but conceptually, we model:

• Nodes: each integer.
• Edges: between two integers a and b if |a - b| == 1.

This model helps us understand the problem as exploring connected components, reinforcing graph traversal intuitions.

Role of Dynamic Programming

Though not traditionally used here, dynamic programming can come into play if the problem is extended. For instance, in detecting consecutive sequences with gaps allowed or constraints on sequence patterns, DP arrays can track sequence length per index, connecting it with graph traversal.

Pathfinding and Sequence Detection

Once we think of each sequence as a path, we naturally borrow techniques from pathfinding. Even though we don't run DFS or BFS here, the mindset of expanding paths using neighbors is conceptually similar.

Final Thoughts

This is more than an array problem. By recognizing patterns and thinking in terms of vertex connectivity, graph traversal, and sequence detection, we unlock ways to optimize beyond brute force. This shift in perspective is what separates good developers from great problem solvers.

If you're preparing for technical interviews or want to level up your algorithmic thinking, mastering problems like Longest Consecutive Sequence is a perfect start. Happy coding!