# What are competitive programming problems that require a tree data structure to solve them?

I want to practice my problem-solving skills that involve the usage of a tree data structure (e.g. Binary Search Tree, Heap, Fenwick Tree, etc.)

The problem should make me search for the right tree data structure and apply it in solving the problem, but not manipulate the tree itself. Hence, problems like the lowest common ancestor, inorder traversal, etc. do not count.

One nice problem that I know is as follows:

N pixels in a row are originally coloured as white:
WWWWWWWWWWWWWWW // N=15 in this case

We now keep recolouring all pixels between position i and position j
with a different colour c for T times:

T = 1: WWWBBBBBWWWWWWW // i = 4, j = 8, c = B
T = 2: WWWBBGGGGWWWWWW // i = 6, j = 9, c = G
T = 3: YYYBBGGGGWWWWWW // i = 1, j = 3, c = Y
T = 4: YBBBBBGGGWWWWWW // i = 2, j = 5, c = B
...

At the end, you are asked to print the colour of each pixel.


Since N and T can be large, one requires a tree data structure to efficiently update the colour pattern of the pixels.

Can anyone suggest good programming problems of this nature?

• Is there a reason for the restriction on updating the tree? How much practice does it take to learn tree traversal? – Ben I. Aug 26 '19 at 18:07
• Well updating the tree is already implemented, I'm looking more for ways to practice the usage of a tree. Same goes for tree traversal, in modern languages you traverse a tree with the same piece of code as traversing a list. – mercury0114 Aug 26 '19 at 19:10
• It's like the difference between learning to sort vs using a sorting algorithm to solve a problem. – mercury0114 Aug 26 '19 at 19:12
• I don't understand why you're talking about tree data structures but mention std::set and std::map. These might work with tree data structures in the background, but they don't actually model trees, they model sets and (key,value) pairs. So: Are you interested in using tree structures or learning how sets and (key,value) associative containers work? – Pascal Sep 18 '19 at 14:15
• @Pascal I modified the question. I am interested in using tree data structures. I want to practice my skill at spotting what data structure fits to solve the problem effectively. – mercury0114 Sep 19 '19 at 15:32

Given a text that uses an alphabet of $$n$$ unique characters, how can we uniquely encode the alphabet so that the text uses the smallest amount of bits?

More formally for Huffman encodings (as formulated by Tardos & Kleinberg in their book Algorithm Design):

Given an alphabet and a set of frequencies for letters, we would like to produce a prefix code that is as efficient as possible - namely, a prefix code that minimises the average number of bits per letters $$ABL(\gamma) = \sum\limits_{x\in S} f_x|\gamma(x)|$$. We will call such a prefix code optimal.

This has an $$O(n \log n)$$ solution.

One nice problem that I found is:

Given n segments in 2D Euclidean space, find two segments that intersect.


Seek for $$O(n \cdot log(n))$$ solution.

https://cp-algorithms.com/geometry/intersecting_segments.html

Another one is the following job scheduling problem:

N jobs are arriving with priorities:

priority_1  arrival_time_1  execution_time_1
priority_2  arrival_time_2  execution_time_2
...
priority_N  arrival_time_N  execution_time_N

The arriving jobs are queued. When the CPU can process the next job, he will pick
the one that has already arrived and has the highest priority in the queue. The
job will then be processed for a time equal to its execution time.

Determine the maximum time some job will have to wait in the queue until being
processed.


An efficient implementation of the queue will give $$O(N \cdot log(N))$$ solution.

You have a list $$[l_1, l_2, ..., l_N]$$ of $$N$$ distinct integers.

Then you receive $$K$$ queries where the query $$q_i = (a_i, b_i)$$ asks you to find the minimal element in the sublist $$[l_{a_i}, l_{a_i+1}, ..., l_{b_i}]$$

Preprocess the list to serve $$K$$ queries in less than $$O(K \cdot N)$$ time.