How to determine how many elements from a range are within another given range?

Question

I need a little help trying to figure out something:

Given a sequence of unordered numbers (less than 15.000) - A - I must answer Q queries (Q <= 100000) of the form i, j, x, y that translate as the following:

How many numbers in range [i,j] from A are bigger (or equal) than x but smaller than y with all numbers in the sequence smaller than 5000.

I am under the impression this requires something like O(logN) because of the big length of the sequence and this got me thinking about BIT (binary indexed trees - because of the queries) but a 2D BIT is too big and requires way to much time to run even on the update side. So the only solution I see here should be 1D BIT or Segment Trees but I can't figure how to work out a solution based on these data structures. I tried retaining the positions in the ordered set of numbers but I can't figure out how to make a BIT that responds to queries of the given form.

Also the algorithm should fit in like 500ms for the given limits. Edit 1: 500ms for all of the operations on preprocessing and answering the queries

EDIT 2: Where i, j are positions of first and last element in the sequence A to look for elements bigger than x and smaller than y

EDIT 3: Example: Let there be 1, 3, 2, 4, 6, 3 and query 1, 4, 3, 5 so between positions 1 and 4 (inclusive) there are 2 elements (3 and 4) bigger (or equal) than 3 and smaller than 5

Thank you in advance! P.S: Sorry for the poor English!

Is that 500 ms per query, or 500 ms for all queries combined? Does preprocessing time (i.e. construction of the segment tree, for example) count against that 500 ms? — Jim Mischel, Feb 29 '16 at 16:33
It's for the whole algorithm (so all queries and preprocessing) — Tudor Berbinschi, Feb 29 '16 at 16:37
So your sequence of max 15.000 numbers contain numbers within a range of 0 to 5000, right? — Vilmantas Baranauskas, Feb 29 '16 at 16:42
Not necesarilly bigger than 0, but yeah I shoud have no problem translating negative integers above 0. L.E. take it like this: x is from A and -5000 <= x <= 5000 — Tudor Berbinschi, Feb 29 '16 at 16:45

score 2 · Accepted Answer · answered Feb 29 '16 at 19:31

Implement 2D-range counting by making a BIT-organized array of sorted subarrays. For example, on the input

[1, 3, 2, 4, 6, 3]

the oracle would be

[[1]
,[1, 3]
,[2]
,[1, 2, 3, 4]
,[6]
,[3, 6]
].

The space usage is O(N log N) (hopefully fine). Construction takes O(N log N) time if you're careful, or O(N log^2 N) time if not (no reason to be for your application methinks).

To answer a query with maximums on sequence index and value (four of these can be used to answer the input queries), do the BIT read procedure for the maximum index, using binary search in the array to count the number of elements not exceeding the maximum value. The query time is O(log^2 N).

How to determine how many elements from a range are within another given range?

1 Answers1