Find minimum element of a subsequence

Question

Given a sequence S of n integer elements, I need a function min(i,j) that finds the minimum element of the sequence between index i and index j (both inclusive) such that:

Initialization takes O(n);
Memory space O(n);
min(i,j) takes O(log(n)).

Please suggest an algorithm for this.

score 0 · Answer 1 · edited May 23 '17 at 10:33

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This TopCoder tutorial: An < O(n), O(1) > approach discusses your problem in a more detail way. In the notation, means the approach takes f(n) complexity to setup, and g(n) complexity to query.

Also, this post chews the algorithm again: Range Minimum Query <O(n), O(1)> approach (from tree to restricted RMQ).

Hope them clarifies your question :)

edited May 23 '17 at 10:33

Community

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answered Sep 07 '14 at 12:23

nevets

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Thanks. It didn't occur to me that binary trees could solve the problem so easily. – Palak Jain Sep 07 '14 at 12:33

score 0 · Answer 2 · answered Sep 07 '14 at 12:36

Segment tree is just what you need(it can be build in O(n) time and one query takes O(log n) time). Here is an article about it: http://wcipeg.com/wiki/Segment_tree.
Even though there is an algorithm that uses O(n) time for initialization and O(1) time per query, segment tree can be a good choice because it is much simpler.

score 0 · Accepted Answer · answered Sep 07 '14 at 12:50

Segmenttree is that what you need because it fulfils all your requirements.

Initialisation takes O(n) with Segment Tree
Memory is also O(n)
Queries can be done in O(log n)

Beside this, the tree is dynamic and can support updating in O(log n). This means one can modify the element of some element i in O(log n) and still retrieve the minimum.

Find minimum element of a subsequence

3 Answers3