段树、区间树、二叉索引树和范围树有什么区别？

Question

What are differences between segment trees, interval trees, binary indexed trees and range trees in terms of:

• Key idea/definition
• Applications
• Performance/order in higher dimensions/space consumption

Please do not just give definitions.

Answer 1

All these data structures are used for solving different problems:

• Segment tree stores intervals, and optimized for " which of these intervals contains a given point " queries.
• Interval tree stores intervals as well, but optimized for " which of these intervals overlap with a given interval " queries. It can also be used for point queries - similar to segment tree.
• Range tree stores points, and optimized for " which points fall within a given interval " queries.
• Binary indexed tree stores items-count per index, and optimized for " how many items are there between index m and n " queries.

Performance / Space consumption for one dimension:

• Segment tree - O(n logn) preprocessing time, O(k+logn) query time, O(n logn) space
• Interval tree - O(n logn) preprocessing time, O(k+logn) query time, O(n) space
• Range tree - O(n logn) preprocessing time, O(k+logn) query time, O(n) space
• Binary Indexed tree - O(n logn) preprocessing time, O(logn) query time, O(n) space

(k is the number of reported results).

All data structures can be dynamic, in the sense that the usage scenario includes both data changes and queries:

• Segment tree - interval can be added/deleted in O(logn) time (see here )
• Interval tree - interval can be added/deleted in O(logn) time
• Range tree - new points can be added/deleted in O(logn) time (see here )
• Binary Indexed tree - the items-count per index can be increased in O(logn) time

Higher dimensions (d>1):

• Segment tree - O(n(logn)^d) preprocessing time, O(k+(logn)^d) query time, O(n(logn)^(d-1)) space
• Interval tree - O(n logn) preprocessing time, O(k+(logn)^d) query time, O(n logn) space
• Range tree - O(n(logn)^d) preprocessing time, O(k+(logn)^d) query time, O(n(logn)^(d-1))) space
• Binary Indexed tree - O(n(logn)^d) preprocessing time, O((logn)^d) query time, O(n(logn)^d) space

Answer 2

Not that I can add anything to Lior's answer , but it seems like it could do with a good table.

One Dimension

k is the number of reported results

Operation	Segment	Interval	Range	Indexed
Preprocessing	n logn	n logn	n logn	n logn
Query	k+logn	k+logn	k+logn	logn
Space	n logn	n	n	n
Insert/Delete	logn	logn	logn	logn

Higher Dimensions

d > 1

Operation	Segment	Interval	Range	Indexed
Preprocessing	n(logn)^d	n logn	n(logn)^d	n(logn)^d
Query	k+(logn)^d	k+(logn)^d	k+(logn)^d	(logn)^d
Space	n(logn)^(d-1)	n logn	n(logn)^(d-1))	n(logn)^d

Answer 3

The bounds for preprocessing and space for segment trees and binary indexed trees can be improved:

• Segment trees can be stored in 2n space and subsequently built in 2n = O(n) using dynamic programming, if you forgo adding intervals at any arbitrary point: https://cp-algorithms.com/data_structures/segment_tree.html#toc-tgt-6
• BIT can be built in n , see this answer: Is it possible to build a Fenwick tree in O(n)?
• BIT also supports append to end as an operation in O(log(n)) time