kartik8800's blog

Dynamic Programming on Trees
By kartik8800, history, 4 years ago, In English

Hello Codeforces!!

In this blog, I want to present to you a beginner-friendly video lecture series on dynamic programming on trees/an editorial for the CSES tree algorithms section.

CSES is a brilliant problemset for people wanting to get started at competitive programming and get good at it.

My aim till now has been to make the explanations intuitive, crisp and clear. I feel even a beginner will be able to benefit from these video lectures.

So let's get started.
Link to the problems

Subordinates

Explanation : https://youtu.be/fGznXJ-LTbI
AC code : https://github.com/kartik8800/CSES/blob/master/Subordinates

Tree Matching

Somehow I feel some other nice approaches are also possible, please do share.
Explanation : https://youtu.be/RuNAYVTn9qM
AC code : https://github.com/kartik8800/CSES/blob/master/Tree%20Matching

Tree Diameter

Explanation : https://youtu.be/qNObsKl0GGY
AC code : https://github.com/kartik8800/CSES/blob/master/Tree%20Diameter

Tree Distances I

This problem uses the rerooting technique, we evaluate the answer for every node assuming it to be the root of the tree. If we do this naively this will be O(N^2) time but if we do this using something known as the reroorting technique and propogate partial answers of parent node with respect to the child nodes we can optimize our solution to O(N) overall time complexity.
I feel Tree Distances II is easier compared to Tree Distances I and would recommend to try it first.

Explanation : https://youtu.be/N7e4CTfimkU
AC code : https://github.com/kartik8800/CSES/blob/master/Tree%20Distances%201

Tree Distances II

This problem also uses the rerooting technique.
Explanation : https://youtu.be/nGhE4Ekmzbc
AC code : https://github.com/kartik8800/CSES/blob/master/Tree%20Distances%202

Distance in Tree

This problem also uses the rerooting technique.
Problem link: https://codeforces.net/contest/161/problem/D
Explanation : https://youtu.be/SOhZqL6HPjQ

Company Queries I

This problem uses a technique(the technique itself uses DP) known as Binary lifting. I will be adding a detailed lecture on binary lifting with code.

Explanation : https://youtu.be/FAfSArGC8KY

Company Queries II

This problems requires us to know a technique to calculate LCA of two nodes in a tree in O(logN) time. Evaluation of LCA in O(logN) can be done using binary lifting. I will add a more detailed video soon.

LCA using binary Search: https://youtu.be/qPxS_rY0OJw
LCA slightly different from standard algorithm: https://youtu.be/s9zZOVsF_eo

Distance Queries

Problem Statement : what is the distance between nodes a and b?
Short explanation : Let LCA(a,b) be node x, what is distance between a and x?
Preprocess the levels of all the nodes in the tree.

lvl[i] : level of node i in the tree.
Ans to query distance(a,b) = (lvl[a] — lvl[x]) + (lvl[b] — lvl[x]) where x is the LCA(a,x).

Again finding LCA of two nodes can be done in O(logN) time and levels of all nodes can be found in O(N) time preprocessing.
Explanation: https://youtu.be/i9ctLWyVSsA

Appleman and tree

problem statement: https://codeforces.net/problemset/problem/461/B
solution video: https://youtu.be/gj0zp--fBL8

Binary Tree Cameras Leetcode

problem statement: https://leetcode.com/problems/binary-tree-cameras/
solution video: https://www.youtube.com/watch?v=VBxiavZYfoA

I am also planning to add video lecture series on more topics which might be helpful to beginners as well intermediates. I will be open to feedback and suggestions.

I hope people find this helpful :)

UPD: added detailed explanation for binary lifting and video solution to Company Queries I.
UPD: added detailed explanation for LCA techniques.
UPD: added solution to distance queries(CSES) and Distance in Tree(CF, VKCup,Problem D).
UPD: added solution to appleman and tree from codeforces.
UPD: added solution to binary tree cameras from leetcode.

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Comments Comments (32)
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4 years ago, # |
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Very usefull!! Thanks!

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4 years ago, # |
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Nice I was looking for some tutorials and I found this!!

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4 years ago, # |
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Auto comment: topic has been updated by kartik8800 (previous revision, new revision, compare).

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4 years ago, # |
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Your Range Queries CSES Editorials, now these. You are surely doing a great help to all of us. Thanks a lot :)

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4 years ago, # |
  Vote: I like it +9 Vote: I do not like it

Tree Distances I can be solved without rerooting. The idea is to calculate the two longest paths that go only downward from each node, and then to start another dp to consider also the paths that go upward, being careful not to choose the same node twice on the same path. Submission

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4 years ago, # |
← Rev. 2   Vote: I like it +7 Vote: I do not like it

Checkout thumbnail for my new video on Binary lifting.

PS: will upload by tonight.

UPD : check it out here: https://youtu.be/FAfSArGC8KY

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4 years ago, # |
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for Tree Distances II, a good article , https://leetcode.com/articles/sum-of-distances-in-tree/

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    4 years ago, # ^ |
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    looks somewhat like the reroorting technique only, not sure though didn't read it thoroughly.
    My video solution is similar to this I guess.

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4 years ago, # |
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All hail _LeMur_ the God of DP on trees.

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4 years ago, # |
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Auto comment: topic has been updated by kartik8800 (previous revision, new revision, compare).

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4 years ago, # |
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Is tree distance 1 can be solved like this — find the maximum of (distance between node and diameter end point1,distance between node and diameter end point2).

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    4 years ago, # ^ |
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    Yes it looks alright to me.
    nice Algorithm. I think the proof will be similar to tree diameter BFS approach proof.

    Did you try and Implement your idea?

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      4 years ago, # ^ |
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      Yes, it's working but I prefer rerooting technique as rerooting is used in many questions.

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        4 years ago, # ^ |
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        Hi5, same here.
        I try to avoid Algorithms which I cannot prove mathematically.
        still i did like your approach:)

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4 years ago, # |
← Rev. 2   Vote: I like it +1 Vote: I do not like it

Interested people may also checkout detailed video explanations for the DP section of CSES.

Introduction to Dynamic Programming

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4 years ago, # |
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Carry on brother!!

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4 years ago, # |
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I think "tree distances I" can be solved in an easier way by using the diameter of the tree. I guess the maximum distance for each node is max of distance between the node and the two ends of diameter. This passed for me.

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4 years ago, # |
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Auto comment: topic has been updated by kartik8800 (previous revision, new revision, compare).

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4 years ago, # |
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Auto comment: topic has been updated by kartik8800 (previous revision, new revision, compare).

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4 years ago, # |
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Auto comment: topic has been updated by kartik8800 (previous revision, new revision, compare).

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4 years ago, # |
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Auto comment: topic has been updated by kartik8800 (previous revision, new revision, compare).

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