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.
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. The Fenwick tree can only be used for those range query and updates which can be converted into a prefix sum query like range sum. For such problems, it performs better than Segment Tree.
- 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.
- Memory-efficient: Fenwick Trees are more memory-efficient than Segment Trees.
