What are the differences between segment trees, interval trees, binary indexed trees and range trees?

Question

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

• Key idea/definition
• Applications

Answer 1

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

|              | 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

|              | 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 |

These tables are created in Github Formatted Markdown - see this Gist if you want the tables formatted nicely.

Answer 2

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