Fenwick Tree (Binary Indexed Tree) and Segment Tree are both data structures used for efficient range query and update operations on an array. Here’s a tabular comparison of these two data structures.

Similarities between the Fenwick tree and Segment tree

Here are some of the areas where Fenwick Tree is similar to Segment Tree:

• Array type: Both Fenwick Tree and Segment Tree are based on 1D arrays.
• Construction time: Both structures require O(N * log(N)) time to construct.
• Query time (range sum): Both structures offer O(log(N)) time complexity for range sum queries.
• Update time (point): Both structures support point updates in O(log(N)) time.

Let us see these in the form of a table.

Difference between Fenwick tree and Segment tree

Now let us look into the areas where Fenwick tree is different from Segment Tree:

• Purpose: Fenwick Tree is primarily used for efficient prefix sum queries and point updates, while Segment Tree is more versatile and can be used for a wide range of range query and update operations.
• Space complexity: Fenwick Tree has a space complexity of O(N), whereas Segment Tree typically requires O(4*N) space.
• Update time (range): Segment Tree efficiently supports range updates with O(log(N) + K) time complexity, where K is the size of the range.
• Query range type: Fenwick Tree is primarily used for prefix sum queries, whereas Segment Tree can handle arbitrary range queries.
• Range updates: Segment Tree supports efficient range updates, while Fenwick Tree does not.
• Lazy propagation: Segment Tree supports lazy propagation for efficient range updates, which Fenwick Tree does not require.
• Additional operations: Segment Trees can be customized for various operations beyond simple range queries and updates, whereas Fenwick Trees are typically used for prefix sum calculations.
• Memory-efficient: Fenwick Trees are more memory-efficient than Segment Trees.

Let us see these in the form of table:

Conclusion:

In summary, Fenwick Trees are more memory-efficient and are suitable for prefix sum queries with point updates, while Segment Trees are more versatile and support a wider range of operations, including efficient range updates and custom operations. The choice between them depends on the specific requirements of your application.