The Traveling Salesman Problem: When Good Enough Beats Perfect

[@Reducible]

Thanks to CuriosityStream for sponsoring this video! If you want to watch their content and support the channel while doing so, go to curiositystream.com/reducible and use code "reducible" to get access for $14.99/year.

[@tuongnguyen9391]

Can you do Max cut problem :)))

[@hexeddecimals]

Congrats on the sponsorship dude!

[@deadhorseak]

Hello. I've academically researched the Travelling Salesman Problem for years now (mostly centered around finding better algorithms than existing branch-and-bound approaches). I can say you did quite a good job explaining it. Even better than I could do in some of my own presentations. If you're looking for a mind-bogglingly difficult problem, you could do one on the Bandwidth Minimization Problem. If I remember correctly one of the algorithms for it had complexity ≈ O(20^N).

[@dodimantakimpo5743]

What software do you use to create animations?

[@RakeshKumar-te1tr]

Tell me a good digital signal processing text book

[@KazeSenshi2929]

I know the optimal algorithm for a graph with two nodes.

[@shivanshmishra752]

Lol

[@cyanmargh]

Believe it or not, I have an algorithm even for three vertices!

[@gemigames154]

Prove it!

[@Shxrpay]

@@gemigames154 too long to write in a comment

[@no_name4796]

@@cyanmargh i am such a genious that i may be able to solve 4 vertices graph!

[@Nononoya009]

TSP is a NP hard problem but so is making this video. Every seconds of the video I see an exponential effort invested. Thank you so much!

[@cit0110]

@@abdulhfhd ohh nooo why did you correct him? you didn't owe him anything

[@cit0110]

@@abdulhfhd fs bestie, remember what you like to say. you don't owe anyone anything

[@cit0110]

@@abdulhfhd yeah but you still don't owe anyone anything. remember that. thats your motto 😂

[@cit0110]

@@abdulhfhd 💦 don't give it up hunny

[@cit0110]

@@abdulhfhd ;p

[@wafikiri_]

In the '80's, I used a different strategy to solve this problem, just for my satisfaction. I based my solution on the way Nature would create an inner expanding bubble until touching all the nodes with minimal tensional energy. I used Basic in an 80286 PC, with about 100 random nodes. A short program with less than 100 lines of code was enough to get satisfactory results. Edit: minor grammatical correction (English is not my first language, sorry for that).

[@kemsekov6331]

This is cool idea!

[@viniciusfriasaleite8016]

Really clever!

[@wafikiri_]

@@viniciusfriasaleite8016 Obrigado.

[@wafikiri_]

@@kemsekov6331 Thank you.

[@angeldude101]

I mean, a bubble tries to minimize tension around the outside, which is basically just a 3D version of the same problem.

[@breckandolley8904]

I'm quite impressed that you managed to avoid mentioning entropy while discussing the annealing approach and the pheromone approach. I once tried to explain pheromone optimization to a friend of mine who wasn't familiar with using entropy to backtrack a search, and I simply wasn't able to explain the process. Well done!

[@breckandolley8904]

Post Script: My specialty is particle physics, and his is machine learning, so maybe I'm just too in love with entropy to shut up about it. (I have been told that it is not normal to use the Bose Statistics to explain how I organize eggs in a carton.)

[@Reducible]

Interesting, the entropy connection feels natural with Simulated Annealing, and it makes some sense with Ant Colony Optimization. I couldn't quite find an appropriate place to add it into the script without it feeling a little forced though. Thanks for bringing up this connection.

[@herseem]

@@breckandolley8904 Even without having ever heard of Bose-Einstein statistics before, but knowing already it's an obscure branch of advanced maths, this made me laugh out loud. Nevertheless, I did check out what it is afterwards, so thanks for bringing it's existence to my attention.

[@TAP7a]

@@herseem Bose-Einstein is like second year Physics undergraduate material, hardly arcane forbidden knowledge

[@herseem]

@@TAP7a I believe you, and that is still pretty obscure for most people who struggle to understand basic statistics. And still far more advanced than is required for filling an egg carton

[@EricKolotyluk]

I have an MSc in Computing Science, but I have never seen a treatment of TSP and friends as beautiful as this. Thank you.

[@brain5853]

So I had this job at a water-selling firm (pumping artesian water source and distributing it either in bottles or through dispensing machines). I took care of machine maintenance and also the logistics of refilling the machines with water. Basically making routes for the drivers that did it. Then some time later got into coding trying to change my career, and in search of ideas to build a portfolio, had this "brilliant" idea of making a routing algorithm. "Should be easy, I can do the routes with my eyes closed by now, just need to put the logic in my brain into code". Or so I thought. After some research, you guessed it, it turned out that I was trying to tackle the TSP. After lots of hours trying to get it to output routes that actually make sense scaled it down to a small web-app that just sorted stuff by priorities, while still doing routing myself, still saved some time and made a nice entry into the portfolio.

[@Mutual_Information]

Been excited for this video for awhile. I’ve heard of TSP superficially many times.. but never got a deep dive like this before. Very motivated, exceptionally beautiful and naturally narrated. Content like this is the gem of KZbin. Just to add: As impossible as this problem is.. it’s wild to know we have a record solution with a tour length of 7.5+ billion.

[@schwingedeshaehers]

there are harder problems

[@tenet-rotas]

Never would I have thought that there is a connection between crystallurgy and the travelling salesman problem. What a beautiful video. Thank you for making this accessible for everyone!

[@tkzubaran]

PSPACE problems, common in planning, are arguably "harder", as TSP is "only" NP-hard.

[@theprofessionalfence-sitter]

And then you have infinitely many degrees of uncomputable problems on top of that.

[@mafrasi2]

According to the time hierarchy theorem, there even are infinitely many *computable* classes harder than NP. I think the video title is a bit misleading (or wrong really)

[@HaoSunUW]

It also has a 3/2 approx in general and PTAS in the planar and euclidean cases. Further real world instances are often solved to within 1-2%

[@Vaaaaadim]

Assuming NP != PSPACE

[@schwingedeshaehers]

@@Vaaaaadim i think that that is proven, but i am not sure. but the Problem, if a program runs infinity, is unsolveable.

[@rubensleite5838]

I will definitely apply this to my thesis problem: Point cloud registration and 3D reconstruction. Thank you very much.

[@ThePlayfulJoker]

Wow, that's a lot better than the 10 slides I got in university for this problem. Even took me less time to go through your video because it made so much more sense. Online universities with teachers like you would be so much better for everyone involved. Mad respect for you to make videos at this level of quality!

[@dfkjbdfondfngg]

The sheer amount of work that goes into creating these simulations.... Thank you!

[@CursedByManga]

Never knew there were so many similarities between solving TSPs and reinforcement learning! Simulated annealing is so similar to the explore-exploit problem/bandits/epsilon-greedy and the ant colony optimization also reminds me so much of genetic algorithms and reinforcement learning😻 Cool video!!

[@dr_davinci]

I was thinking the same, Ant colony optimization reminded me of genetic algorithm.

[@Arthurt120]

It's interesting to see how a "simplifyed" (two way edges, linear paths) can become arbitrarily hard to solve. Imagine bringing this problem to a urban scenario, where edges are streets that are not necessarily 2 way and are non linear and trafic is involved.

[@Agedude]

In a lot of cases that might be more difficult, but in many cases it might actually be easier to solve. Adding additional rules to the system could make most potential solutions very obviously bad or eliminate them entirely until there are only a few possibilities to check.

[@chri-k]

The ant solution might be the best in that case, and it would work without any extra modification. ( for one-way roads just initialize the appropriate entry in the weight matrix as zero instead of one ) Plus it can adapt to variable conditions well. ( Adding as an additional input an array that weighs the randomness on each node may help with sudden changes. )

[@thomascoolidge2161]

I did a report in Linear Algebra discussing using Adjacency Matrices to find paths from any point to any other point by raising matrices to their power step by step. I managed to make it work for minimum hop count and shortest path between two points. Unfortunately it doesnt solve the TSP because it does not go back to it starting point.. only that it can hit every point. We had another idea of using something similar to what we called gravitic weighting to image creating the shortest path by picking any point and then "lifting it" and seeing the other points fall based upon distance.. we ran into problems with pruning of ineffecient routes.. if we could identify routes that were not going to be used we think it would have worked.. unfortunately our advisor didnt seem to care to help us at all... to him TSP was just an impossible problem and we were waisting our time.

[@jimboli9400]

In love with the new intro. Amazing content as always! Doing computer science proud!

[@kaishang6406]

your videos are always a treat and often expose me to more knowledge that I didn't know exists before. thank you.

[@madhes23]

How to like a video multiple times? The explanation and animation was phenomenal. Great job!

[@rohitd09]

Great insight on TSP. The research is vast and the information is often overwhelming.. but the way everything seems connected at the end is just amazing. Great intuitive ways to solve the TSP, would love to see more content on NP hard and NP complete problems. I started watching your content almost 2 years back when I first saw your ‘Towers of Hanoi’ video. Informative and concise.. love the animations

[@Ruchir205]

A classical problem from the field of Operations Research and one the most studied

[@sathirasilva4958]

Woah this is a video that I was waiting for since a long time. I've attempted to solve many variations of TSPs but I couldn't find any good and intuitive video regarding TSPs and heuristic algorithms on KZbin. Thank you so much!

[@ricardocardoso6523]

This video was very well done! I come from a data science background, and so I was more familiar with the traveling salesman problem in the context of evolucionary optimization and multi objective optimization. The heuristic aproaches presented in this video are super interesting, and the presentation is amazing, it was really easy to understand.

[@Boringpenguin]

Very well done! TSP is one of the problems that interests me a lot but I just never got the time to learn it properly. This is the perfect introduction on TSP for all of us to enjoy :)

[@JordanMetroidManiac]

A perfect TSP 101

[@scatkit]

You're the only cs channel using manim at that level. Respect bro, keep going

[@stacksmasherninja7266]

I'm a grad student studying this and honestly this is probably the best online explanation on this topic. 10/10 would watch again for reference

[@rupen42]

Annealing also reminds me of stochastic descent. Find the direction that decreases what you're trying to minimize, then add some fuzz to help to escape local minima.

[@tellurian7999]

One of my favorite approximations is using a blob-like shape that stretches itself to just barely touch all the nodes while taking up as little space as possible, and then going around the edges of the blob.

[@rubensleite5838]

But this is a geometric aproach to a topological problem. The vertices positions are not important in a graph

[@O5680]

A general solution to the Travelling Salesman problem for ANY graph: 1.) Throw away the graph. 2.) Create a single node. 3.) Since there is only one node, the solution for this graph is optimal. This algorithm runs in O(1) time. I can't overstate how important this development is to the world.

[@papagunit]

What an incredible intro to this problem. Wow, so clearly laid out. Thank you for all your work on this!!

[@srinivasyeleti]

Quality of the Content is Mind Bending .

[@ralphwiggum3134]

I had trouble programming a tic-tac-toe game where I didn't want to rely on simple strategies for the computer's moves. I wanted to create a logic function that could be expanded to 4x4, 5x5, or more grids. I gave up for the time, but plan to try again some day. I think this video is very informative and can give me some clues to creating the logic that I would like to create.

[@cubiccentral4480]

This is one of the greatest computer science videos I have ever seen. Wow.

[@AmaroqStarwind]

1. Massively parallel computers 2. Digital/Analog hybrid computers 3. Fully-Associative memory

[@officialaalamjotsingh9465]

Dropshipping is the answer to this problem.Use supply chain management to maximize profits form the products and scale business using social media. A different approach to solving the TSP

[@Amplefii]

I was reading grokkings algorithms book last night and ended up looking for a video on TSP and didnt find anything good, then you uploaded this. Gotta love when that happens.

[@linuxdragon57]

My CS Capstone is centered around this problem. Thank you so much. I am considering a modification of simulated annealing and reinforcement learning to optimize the cooling schedule and the probability that it accepts a solution at any given temperature.

[@fugoogle_was_already_taken]

Thank you for this video, it has been a fine "survey" on TSP in aestheticaly pleasing and compact way.

[@elijahberegovsky8957]

I loooove discrete optimization, very happy you made a video on it. The problem with ACO as opposed to SA is that every single step of ACO is, in my (small) experience, slower. SA usually makes very local changes to the current solution, so calculating the makeup and cost of the next solution is very efficient, sometimes, like with 2-opt, even independent of size of the problem. ACO, on the other hand, has to compute its solutions from scratch, which makes it incredibly slow.

[@user-hl2yj8kp2s]

I love this channel. This is better presented than a college education. Super high quality video. Thanks!

[@BoazNahumPlus]

Making this video must been a hell of a work. Thank you!

[@LiborTinka]

I remember simulated annealing from learning nonlinear numerical optimization - it's a popular way to discover local minima that wouldn't be otherwise found with systematic approach.

[@gabotron94]

Your narration has improved a lot! These keep getting better.

[@ayoCC]

this video was specially intuitive and engaging. Extremely entertaining presentation!

[@gaspump]

I always felt a bit daft at formal math but I'm amazed at how this video visualized my own intuition

[@_spartan11796]

Always great to see you post!

[@Joker22593]

Complexity Theorist here. Did you know you can solve this in polynomial time using parallel computing on exponential processors? You just try every route at the same time. Since we all have video cards with thousands of cores, it's mostly practical to use this method for small data sets.

[@harold2718]

And yet in real life we (in Operations Research) solve such problems and harder ones such as vehicle routing daily, using solvers that are based on linear programming. Finding optimal solutions is very practical actually, it's mostly *theoretically* impossible, but the worst cases do not arise organically (you can construct them though).

[@JobvanderZwan]

Just so you know, the choice of green and red at 19:40 is completely indistinguishable to people with red-green colorblindness, with is roughly five to ten percent of all men.

[@10produz90]

The visualizations are just awesome and add so much to the already awesome explanation.

[@raykos4257]

Phenomenal video. This deserves more views.

[@nsiepmann]

Thank you - this was SO helpful. I watched it once this morning, and came back to rewatch and make notes in the afternoon, and I finally feel like I have a good understanding of the topic! Subscribed.

[@laurenswissels8480]

The production quality of your videos is insane. Keep it up!

[@rodgerwilliam198]

Hello Lauren

[@dustinm2717]

Within the first 30 seconds of this video believe i have encountered this exact problem organically on my own, I've not watched the video to confirm its exactly the same problem but i figured I'd write this now if anybody wants a look into the thought process of somebody who encountered at least a similar problem at some point without knowing anything about this problem the solutions that have been devised for it or even what to call it (i had no idea it had a name, the closest thing i could think of to call it was non spatial pathfinding but at the time i didn't find much when i tried searching for it) so in effect i had to try figure it out on my own in a vacuum without knowing anything of it The encounter is basically this, i had a list of arbitrary colors of arbitrary and wanted to sort them in an order that creates a smooth gradient, which put into abstract can be viewed as being simply just a set of points that contain a set of its distances from every other point (I precalculated all the distances, so i was just working with a 2d table with each cell being it's distance between its 2 indexes) with the goal being to find an ordering of these points with the least total distance So again i didn't even know what to call it at the time but i figured out very quickly that it wasn't as simple a problem to solve as you might think at first look, and i ended up trying 3 solutions and deriving a 4th that i haven't bothered to test, in the order i thought of them - single pass selection - conventional sorting - brute force - single pass selection + smoothing pass I quickly figured out that conventional sorting algorithms wouldn't help so that one didn't last long and there isn't really much to talk about on it The third one, trying to brute force the absolute shortest order i quickly found is quite possibly the least efficient thing you could do with a computer that's not deliberately designed to be inefficient with any more than like 8 points, with larger ones having the potential to take up more computing power to complete than the entire planet has available even with my attempt to optimize the brute forcing (it did work at finding the objectively shortest route, but i was expecting to put up to 100 points into it and being the time to try every single order increases exponentially i determined it'd very quickly enter taking years, when i needed it to be measured in seconds at most) eventually i just settled on what was my first instinct solution which was to take a starting point and then simply pick the shortest not yet traveled point from it and traverse there and repeat until there were none left unvisited, being it was good enough and only took one pass so it was quite quick, the only real problem being it'll often shove a few odd ones on the end one improvement I thought of a couple days after decided on that (though I've not actually implemented or tested this) was to do basically do a second pass where it goes through the list again a few times after selecting the initial order to see if moving any to be between other points would make the total shorter to smooth out the straggler that usually ends up on the end, basically a reinsertion or smoothing pass And that's where i ended up after trying to figure it out on my own, figured it could be interesting for maybe at least someone to read the perspective of solving it as someone who doesn't really know a whole lot about algorithms or math or logic or data science

[@dustinm2717]

I would say i wish i had this video when i was working on that, but the solution i found was good enough and honestly i think the perspective i have of trying to solve a pretty significant algorithmic problem in a vacuum unaware of any of the methods already derived or even what the problem is called kinda makes it more interesting than if i just knew when i was doing that i think

[@ferdiaz8703]

Incredible explanation! I learned about simmulated annealling in a stochastic simulation methods course but never got the key idea behind it, but thanks to you now I get it!

[@Elitea20]

I am doing a dissertation about visualising these graph algorithms , if anyone wants me to remind when I finish and want me to send the deployed version let me know.

[@sanchitsinghiitg]

Awesome video! Nice summarisation of very complex ideas.

[@KurtWoloch]

I once coded a similar algorithm. My version started with a single point (the origin point) and then added the other points one by one, in no particular order, but for each point an existing section of the path was broken up and the point inserted into it, and it checked for each existing segment how much length would be added if the new point was inserted into that segment, and the segment with the shortest length increase would be chosen to insert the new point into. Don't know how good of a solution this was though... By the way, I used this for a simulation of a scooter service man employed by an e-scooter sharing company who had to service (collect, deploy or battery swap) several given scooters on the map, and it would give a list of directions, a series of coordinates where to go to and instructions of what to do there, and the coordinates were sorted by this algorithm.

[@identityelement7729]

I am thinking that I saw a video where to calculate a lower bound was done by Linear Programming. Then the next task was to find a route which has this minimal cost. And from that video I remember that 100 nodes are paractically computable to get an optimal route.

[@JordanMetroidManiac]

Amazingly easy to understand, thanks to you! One idea: What if we initialize R in the AS algorithm to correspond to the MST? I think that would yield faster convergence to a good solution and/or better solutions on average.

[@fredericmazoit1441]

There is an algorithm which is based on a simple idea that I like. The reason why the brute force algorithm does not work is not really the number of nodes but the number of edges. Of course, in the initial setting, the 2 are linked but it need not be the case. One reason why finding a minimum spanning tree is easy is because we can drastically reduce the number of edges. In Prim's algorithm, at each step, most if not all but one edges can be discarded. The idea is thus the following. - Take k tours obtains by good heuristics. - Construct the union of these k tours. - Brute force the solution in this reduced graph.

[@9034833838]

Alpha Pheonix's Voting Block algorithm was a beautiful demonstration of this

[@Igigog]

What an absolute banger of a video. Liked every second of it.

[@hacker2ish]

Great, very comprehensive overview with all the important details, learned a lot

[@Funzelwicht]

Sooooo well explained and animated, so good!

[@gosunov]

Great video, simulated annealing is really interesting and easy to implement idea

[@_AvaGlass]

This video is so good that I want more people to see it, so I'm commenting to drive engagement.

[@JohSmith]

This video is awesome. Great explanation and great montage (it's a lot of work to show all this nodes and highlight them)

[@BillyBob-wh4sq]

This video is super well-made and really interesting, even as a person who already has some knowledge of the concepts mentioned. You did a great job taking a topic such as the TSP and simplifying it in a way that can be shown as an intro-level lesson.

[@daemitrious]

what an incredibly well done video mate

[@simpletongeek]

You can brute force it easy, but takes a long time. Or, you can do it fast, but not optimal. It's only difficult if you want a quick optimal solution with large N. Notice the value of N is always increasing!

[@rodgerwilliam198]

Hello

[@研治石橋]

6:15 Just pet peeve of mine. It doesn't beg the question. It raises the question. Begging the question means something completely different (circular reasoning).

[@roberthildebrand1780]

Absolutely amazing video! Fantastic resources are posted in the information as well.

[@VincentKun]

Coding Adventure did the ants colony so i was familiar but the simulated annealing is just so pretty as a natura approach to this problem, i want to try to implement these two as soon i've some spare time.

[@kaifscarbrow]

I'm probably having a bias from the examples shown but I kept thinking that the optimal path is when you imagine a circular string that surrounds all the points starts shrinking onto the points. Ofc, this will only connect the outermost vertices and the inner ones will be left out. [Edit: So how do we get to the inner nodes from here? Let the strings grow until it hits one of the inner nodes. DO NOT RECURSE from here. Move to the next segment. Do this until all the inner nodes are visited] So I had another idea. Imagine making a polygon out of the vertices. I'm guessing that the polygon whose sum of external angles is the least will be the optimum solution. Well I could just be guessing. Very nice video. Thanks for introducing the topic.

[@HiAdrian]

Great presentation on TSP, this was a pleasure to follow!

[@yakir11114]

here's an idea: use only nearest neighbor , but make all the possible paths each starting with a different node. then select the best one.

[@Habib-ov3nv]

Such an great video! You're doing an amazing job explaining those problems! Can't wait for your other videos!

[@Veptis]

Finding a global minimum among many local minimum strongly recommends me about the various optimizers in machine learning. Usually gradient descent or some interactive version (momentum, aggregation). I wondered if there might be a better solution that's computationally viable. And perhaps this annealing simulation could do it. But the dimensionality is crazy

[@rustix3]

12:30 From wikipedia "Christofides algorithm ... is named after Nicos Christofides and Anatoliy I. Serdyukov, who discovered it independently in 1976." I read about Christofides, and was somehow surprised that there wasn't much "glory" for him for such groundbreaking algorithm. I was expecting that he will then became a professor at some of the best known universities in Computer Science etc. He became a professor at Imperial College London only in 2009, at the age of 67 and then died 10 years later on 2019. Regarding Anatoliy I. Serdyukov there was no article in Wikipedia, and when I tried google it it was some other random guy.

[@user-iu1xg6jv6e]

I had issue understanding your pronunciation of "accept", so I did a huge mistake and went searching for how to pronounce it! I'm more confused now! Anyway, great video.

[@mastershooter64]

it's pronounced "accept" you're welcome

[@betterbelle29]

Annealing sounds a lot like an idea used in Reinforcement Learning where during training you actually allow the agent to select a suboptimal choice (according to what the agent knows at the time) in order to explore other states. Funny enough, this exact idea is being used to train networks that can solve TSPs and other NP-Hard problems.

[@anshumandash1304]

I think you do amazing work, you have a talent at showing your passion for this beauty, it's truly infectious. Amazing work as always!

[@RecursionIs]

Great intro to the TSP! You should check out the IPPA algorithm applied to this problem, it can create some great animations 😁

[@PowerhouseCell]

Beautiful explanation of the Travelling Salesman Problem! 💛

[@tinymints3134]

You did a great job on this video. I learned a lot and it solidified other understandings I had. Awesome!

[@gerryakbar]

Never realized traveling salesman is a computer scientist problem, not a salesman manager problem

[@afjer]

Always less than or equal to. If A, B, and C are colinear (on the same line), then length(A->C) = length (A->B->C).

[@SabaSa6a]

Simulated annealing is actucally a "clever brute-force search": it is guaranteed that all states can be visited, but we have a much higher posiblity to visit the optimal state by using simulated annealing than normal brute-force search.

[@JimLeonard]

Happy to see simulated annealing here. I improved the quality of an audio encoder I wrote that used vector quantization by introducing simulated annealing to refine the initial codebook.

[@staa1337]

also a nice solution: take the mid point out of all points, imagine a homogeneous stretching ideal baloon on that point. you pump the ballon up and somewhen it touches all the points. in a 2d plane you get the points to travel one by one on the line of the circle. this works great if you are into puzzles and wanna do it by hand.

[@Feroste]

'Ant colony optimization' I literally came here to improve my Screeps.

[@awwastor]

5:30 At one point we might get a good solution for the optimal road, if P =NP, then we will get it for large enough N anyway but even if not, there are some very fast solutions to many complex, NP-Complete problems. Like the N queens problem or the exact cover problem (or the generalised cover problem)

[@qm3ster]

Crazy how the ants actually did achieve the 2-opt swap at 28:19

[@Chizypuff]

I had an idea for this problem at the start of the video, Draw a shape that represents the average location of the remaining nodes before each, then choose between the closest nodes favoring nodes that are furthest from the average location border. I think it would help link those outer nodes so they don't have multiple long jumps to them

[@rubensleite5838]

You are thinking geometricaly, but this is a topological problem

[@tru1988]

Great exploration of the problem space and amazing explanations of common approaches to solving it.

[@louighi91]

Such a high quality content! Thank you :) remembers me to the time where I learned that during studying

[@fa-pm5dr]

btw if you run simulated annealing with temperature decreasing sufficiently slow, the algorithm is guaranteed, with probability 1 to reach the optimum (although it make take ages) since the search space is a nonhomogeneous positive recurrent markov chain

[@herseem]

Absolutely top-quality video. Thanks for creating it. Has stimulated me to find an algorithmic solution that is better than what is currently available. Will be a useful problem for learning Python

[@akshayuppala9295]

If I could make presentations like this, I could die happy.