Comparison between Fenwick Tree vs Segment Tree – with Similarities and Differences

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.

Feature	Fenwick Tree (BIT)	Segment Tree
Array type	1D array	1D array
Construction time	O(N*log(N))	O(N*log(N))
Query time (range sum)	O(log(N))	O(log(N))
Update time (point)	O(log(N))	O(log(N))

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:

Feature	Fenwick Tree (BIT)	Segment Tree
Purpose	Primarily used for efficient range queries (sums) and point updates.	Primarily used for efficient range queries and updates.
Space complexity	O(N)	O(4*N)
Update time (range)	Not efficient	O(log(N) + K)
Query range type	Prefix sum queries	Arbitrary range
Range updates	Not efficient	Supported
Lazy propagation	Not applicable	Supported
Additional operations	Not common	Customizable
Memory-efficient	Yes	No

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.