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How to effectively and range queries in an array of integers?

Queries are of one type only, which is, given a range [a,b], find the sum of elements that are less than x (here x is a part of each query, say of the form a b x).

Initially, I tried to literally go from a to b and check if current element is less than x and adding up. But, this way is very inefficient as complexity is O(n).

Now I am trying with segment trees and sort the numbers while merging. But now my challenge is if I sort, then I am losing integers relative order. So when a query comes, I cannot use the sorted array to get values from a to b.

user3243499
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1 Answers1

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Here are two approaches to solving this problem with segment trees:

Approach 1

You can use a segment tree of sorted arrays.

As usual, the segment tree divides your array into a series of subranges of different sizes. For each subrange you store a sorted list of the entries plus a cumulative sum of the sorted list. You can then use binary search to find the sum of entries below your threshold value in any subrange.

When given a query, you first work out the O(log(n)) subrange that cover your [a,b] range. For each of these you use a O(log(n)) binary search. Overall this is O(qlog^2n) complexity to answer q queries (plus the preprocessing time).

Approach 2

You can use a dynamic segment tree.

A segment tree allows you to answer queries of the form "Compute sum of elements from a to b" in O(logn) time, and also to modify a single entry in O(logn).

Therefore if you start with an empty segment tree, you can reinsert the entries in increasing order. Suppose we have added all entries from 1 to 5, so our array may look like:

[0,0,0,3,0,0,0,2,0,0,0,0,0,0,1,0,0,0,4,4,0,0,5,1]

(The 0s represent entries that are bigger than 5 so haven't been added yet.) At this point you can answer any queries that have a threshold of 5.

Overall this will cost O(nlog(n)) to add all the entries into the segment tree, O(qlog(q)) to sort the queries, and O(qlog(n)) to use the segment tree to answer the queries.

Peter de Rivaz
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  • Could you elaborate on approach 1? What subranges do you mean? All of the O(n^2) possible subranges? – Nico Schertler Jul 14 '17 at 17:53
  • @NicoSchertler Suppose we had an array of length 2^5, a segment tree will represent 1 range of length 2^5, 2 of length 2^4, 4 of length 2^3, 8 of length 2^2, 16 of length 2, and 32 of length 1. Overall there are 1+2+4+8+16+32=63=2*n-1 subranges represented. – Peter de Rivaz Jul 14 '17 at 18:18