Introduction to the A* Algorithm (2014)

https://www.redblobgames.com/pathfinding/a-star/introduction.html

by auraham • 1 year ago

288 103 1 year ago

redblobgames.com

JohnKemeny5• 1 year ago

It's that time of year again. I like A* as much as the next one, but it seems a bit excessive a times.

Title should have a (2014) in it: Introduction to the A* Algorithm (2014).

1 points, 8 months ago, 1 comments: Introduction to the a* Algorithm (https://news.ycombinator.com/item?id=41897736)

202 points, 3 years ago, 30 comments: Introduction to the A* Algorithm (2014) (https://news.ycombinator.com/item?id=30287733)

4 points, 5 years ago, 1 comments: Introduction to the a* Algorithm (https://news.ycombinator.com/item?id=24146045)

201 points, 7 years ago, 14 comments: Introduction to A* (2014) (https://news.ycombinator.com/item?id=18642462)

5 points, 7 years ago, 0 comments: Introduction to A* (https://news.ycombinator.com/item?id=16190604)

bandoti2• 1 year ago

Please consider some folks might be new to A*, and perhaps even HN, so maybe this is the first time they’ve seen it! :)

Also, I have ten books on perspective drawing, and my understanding isn’t complete without all ten of them

Or, if I’m teaching a subject on A*, perhaps ONE of those articles conveys the materials best for my students.

Thank you for providing links to the others though! I’m sure it will be helpful for someone.

add-sub-mul-div• 1 year ago

Yeah people live by this leaky abstraction that an article having been posted before means everyone was online that day and saw it and now it has expired. And for some reason they chase these hall monitor points for pointing it out. Let's see what a discussion would be like from today's point of view.

dspillett3• 1 year ago

I agree, though to be a pedant:

perhaps ONE of those articles

It is the same article each time, though the comments coming off the different postings of it might have unique nuggets of useful information to dig for.

Thank you for providing links to the others though! I’m sure it will be helpful for someone.

It isn't as prominent as on other sites, so it isn't difficult to miss sat right at the bottom of the main page, but HN does have a working search function. I find searching for older posts this way can be quite useful for the above reason, when something comes up that has existed for a few years.

schneems• 1 year ago

I wish there was a “evergreen” feature for social sites where it tracked resubmissions and would auto suggest them to people who haven’t seen them and periodically surface them to those who have and ask “is this still relevant” That way really good content keeps being recommended to those who need it and you get fewer complaints from old timers who don’t have to see it N times.

giancarlostoro• 1 year ago

Also, I have ten books on perspective drawing, and my understanding isn’t complete without all ten of them

Which books might those be? ;)

pmichaud• 1 year ago

I think it's just the first, most obvious thing to teach people just starting in pathfinding. It works in real life, it's easy to visualize and compute. Therefore all the tutorials are about it :)

msk-lywenn• 1 year ago

Please think of all the lucky few. (xkcd 1053)

devrandoom• 1 year ago

There are a lot of people on HN that aren't you.

dunham• 1 year ago

I was surprised to see this article because it's not that time of year. Typically A* shows up around December, because people discover it via Advent of Code. (And that's the only place I've used it.)

raincole• 1 year ago

How could the OP submit this link anyway?

When I submit a link that has been posted on HN before, it just redirects me to the old post.

random3• 1 year ago

Maybe there's a secret guide on how to build up karma :))

teo_zero• 1 year ago

I think the real reason this article keeps coming back is how good it is at teaching through examples and visualization, rather than the A* algorithm itself.

repiret• 1 year ago

I think A* deserves the popularity. It’s a simple variation on depth-first and breadth-first graph traversal that everyone learns in CS101, massively useful in some situations, yet for some reason isn’t a standard part of CS education. It’s a marvelous thing the first time you learn about it.

It’s even more marvelous if it helps you recognize that the difference between BFS and DFS is how you pick the next node to explore out of your bag of unexplored nodes. That symmetry is easily lost if DFS is only taught as a recursive algorithm.

Let it keep coming up every couple years to marvel a new generation of programmers.

hoseja2• 1 year ago

I don't like A*

It's a performance hack, not how entities trying to get somewhere behave.

herman_toothrot• 1 year ago

What is your preference?

FrustratedMonky• 1 year ago

just take all right turns?

hoseja2• 1 year ago

See the target/know which direction it is? Go that direction unless you see an obstacle, in that case go around the obstacle, eventually even backtracking if it turns out the obstacle was worse than you could see. Don't see/know the target? Brownian motion until you do or get tired. Have pathfinded to the target previously? The shortest path you saw while walking there.

Al these require deep and complicated simulation of the entity though instead of solving a graph problem from omniscient perspective. Many topdown games really break my immersion when I see things just blatantly a-staring places.

Basically, things usually have limited information and it's weird to see them behave as if they don't. Plus on grids there's the cringe "diagonal and then straight" movement pattern.

ecshafer4• 1 year ago

It works really wells for many situations. If I am making a top down strategy game (Think Civilization) then A* is exactly what I need, a fast performance hack that gives me the shortest path without anything weird going on. For different kind of environments, then yes it doesn't work. A* isn't very useful in a racing game.

smallstepforman• 1 year ago

It took me 3 hours to implement A* with hex tiles, got it working on first attempt (land tiles only), specifically for Civ type game. It gets complex when you want to group units so that they travel together. Adding water crossings with cargo ships and war ships is a different challenge.

diggan2• 1 year ago

It's a performance hack, not how entities trying to get somewhere behave.

Welcome to game development, where fun and performance tends to be more important than realism :)

JohnKemeny5• 1 year ago

It's neither a hack nor trying to "behave" like anything.

It is complete and optimal, with provable properties.

Whenever there exists an admissible heuristic, you should use A* over Dijkstra's algorithm.

HelloNurse2• 1 year ago

Even inadmissible heuristics can have their place, and it is easy to reason about how suboptimal they can be: you might want to trade off optimal results for performance (i.e. ignoring some part of the search space that should be searched) or to make an agent in a game a little stupid or stylized (e.g. prone to zig-zagging).

kccqzy3• 1 year ago

OP's point, I believe, is that inadmissible heuristics sometimes give behaviors that seem more natural, even if not optimal.

wat10000• 1 year ago

A* isn't how you get there, it's how you decide how to get there. The output is the shortest path, and that's what you actually follow.

It's pretty similar to how people find the shortest path in an unfamiliar environment. If you're looking at a map for the shortest route to a city to the north of your current position, you probably won't spend a lot of time looking at roads going south, and you'll concentrate on road segments that get you crow-flies closer to your destination.

ecshafer4• 1 year ago

Red Blob Games is a great blog if you are interested in game development. The explanations are solid, they have at least pseudo code or an implementation for most of their posts, and they have great animations on a lot of their bigger posts to help build intuition.

Barrin922• 1 year ago

I remember one of the Advent of Code challenges had a hex grid puzzle on it, and Red Blob Games hex grid guide was so good the site got hugged to death because of it for a while. Used that later when I built a civ clone too, it's a fantastic resource.

https://www.redblobgames.com/grids/hexagons/

lelandfe• 1 year ago

I still remember a decade on the little pop of joy of realizing that not only do all the graphics on that blog post change when moving from flat to pointy, the code animates too.

The interactive graphics evoke all the fun of being a child in a science museum.

vismit2000• 1 year ago

Problem: https://adventofcode.com/2020/day/24 My solution: https://github.com/vismit2000/compCode-II/blob/cb3041d2f5110...

S0y• 1 year ago

I came here to say the same thing. Red Blob Games is such a gold mine of resources for anyone looking to get into gamedev.

cmrdporcupine• 1 year ago

Red Blob Games is also "amitp", one of the earliest Google employees. #7 or something like that.

upghost2• 1 year ago

Interesting that this used to be called "AI". I'm still trying to figure out what to call the umbrella field of Artificial Intelligence now that "AI" has come to mean the genAI subset of DL which is a subset of ML which is a subset of what used to be called "AI".

diggan2• 1 year ago

The definition of "AI" has for a long time now been basically "We know it works somehow, but only few people really understand it", which is a moving target. At one point in the future, the LLMs we use today won't even be called AI anymore.

Al-Khwarizmi• 1 year ago

https://en.m.wikipedia.org/wiki/AI_effect

PS: the wiki article needs updating with more confirmation from the LLM era, if anyone's up for it... :)

ecshafer4• 1 year ago

I took a course in grad school on "Game AI" that put different path finding approaches and methods of making decisions into a useful bucket away from ML and AI.

dspillett3• 1 year ago

Interesting that this used to be called "AI".

What has been called AI in gaming in the past is rich and varied, and goes all the way down to a computer control opponent “seeing” a player and opening fire, moving towards, or moving away. Any code controlling NPC was referred to as “the AI of the game” even if all the code was doing was applying a few simple rote rules rather than following an exactly pre-specified sequence.

“AI” in gaming means (or has previously meant) a lot less than “AI” has meant in other fields, but with the increasing use of “AI” in all contexts this will soon no longer be the case.

kccqzy3• 1 year ago

There is also certainly more sophisticated AI in gaming. Remember the AI used by Deep Blue in chess? Or the AI used by DeepMind in Go against Lee Sedol? Classic games like chess or Go receive more attention from the AI community than contemporary video games.

bonoboTP• 1 year ago

Not just computer game AI. Literally university courses called "Artificial Intelligence" would teach A*, formal logic, planning, knowledge representation, etc. See for example the famous Russell-Norvig textbook. Since deep learning became dominant around 2012-2014, that conception of AI is now (somewhat deprecatingly) called GOFAI, or "good old-fashioned AI".

a-dub2• 1 year ago

a lot of early ai was simply applied classical data structures and algorithms.

although perceptrons go back decades and decades.

throwawayoldie3• 1 year ago

"Artificial intelligence" has always been a marketing term, not a technical one. John McCarthy coined the phrase when he was applying for a DARPA grant, and he picked it because he thought it would sound cool to the grant reviewers.

pantulis• 1 year ago

Interesting that this used to be called "AI".

I remember learning about A* in the AI lab at the University. Now these things sound trivial and we take them for granted. The joys of becoming old.

tsumnia• 1 year ago

The way I explain it to my students is through a venn diagram of "Traditional AI", "Machine Learning", and "Data Science" (though I suppose Gen AI is starting to form another circle). A* falls into the "Traditional AI" space, which is a mixed of state searching, logic representation, and statistics/probability (now called data science). What general public considers "AI" is where all the circles meet, and it means everything from Robots to if-else statements.

apnorton• 1 year ago

The article doesn't explicitly state it in this manner in one concise place, but the way I would always think about A* from a "practical/easy-to-remember" perspective back when I was doing competitive programming is that they're all the same algorithm, but with different priorities on the priority queue:

Breadth-first Search: Priority is order of discovery of edges (that is, no priority queue/just a regular queue)

Dijkstra: Priority is distance so far + next edge distance

A*: Priority is distance so far + next edge distance + estimate of distance to target node.

This also helps me remember whether the estimate must over- or under-estimate: Since Dijkstra is making the estimate "0", clearly the "admissible heuristic" criteria must be an under-estimation.

breckognize• 1 year ago

And depth first search is just a stack!

dietr1ch2• 1 year ago

Yes, but it doesn't come right away. When preferring deeper nodes for a moment you have a loop-safe Depth-first traversal, and then when simplifying things in case the Graph is a Tree you get your regular stack-based Depth-first traversal, in which if you settle for the first goal you get back a tail-call optimised DFS.

tonyhart7• 1 year ago

well they all just loop???

bcrosby95• 1 year ago

Simplifications help people memorize things but if you get too reductive it becomes useless.

salamanderman• 1 year ago

Breadth-first is a queue. Depth-first is a stack. A* is a priority queue.

cornstalks2• 1 year ago

More specifically Dijkstra's is a priority queue. A* is a priority queue with an added estimate to the cost function to prioritize searching nodes closer to the destination.

JohnKemeny5• 1 year ago

OP's point is that

· BFS is priority queue with key h(n) + g(n), where h(n) = 0, g(n) = #edges

· Dijkstra's is priority queue with key h(n) + g(n), where h(n) = 0, g(n) = sum over edges

· A* is priority queue with key h(n) + g(n), where h(n) = heuristic(n), g(n) = sum over edges

It's cute.

awesome_dude• 1 year ago

Which algorithm should I apply:

I have no information other than the fact that my agent has a decision to make (left or right).

- DFS or BFS

I have some information about the cost of the decision.

- UCS or Djikstra's algorithm

I have some idea of the cost of the decision, and a rough idea which direction the goal is in.

- A star

As well as knowing the cost, and a rough idea of the direction, I also know that I have a uniform cost grid.

- Jump point search

nostrademons3• 1 year ago

Another way I like to think about it is that every graph traversal can be represented as a white set of unknown nodes, a grey set of known but unvisited nodes, and a black set of visited nodes. The data structure used to represent the grey set defines the algorithm:

DFS = queue

BFS = stack

Dijstra's = priority queue keyed by edge weight

A* = priority queue with heuristic function

Beam search = bounded priority queue with heuristic function

Topological sort = priority queue keyed by number of unvisited inbound edges

Copying garbage collector = pointer address

Mark & sweep garbage collector = dirty bit on the object pointed to

Generational garbage collector = multi-level grey set represented by the write barriers between each generation.

a-dub2• 1 year ago

A* = priority queue with heuristic function _with specific properties_ ... in particular it must be an "admissible" hueristic that never overestimates the true value.

anitil• 1 year ago

This is very insightful and a handy mental model I'll be using thanks. A small nit is that I think you have DFS and BFS swapped?

djmips• 1 year ago

Any other tips for competitive coding ie a book or source of wisdom in a similar vein?

amy_petrik• 1 year ago

leetcode will strengthen the youngblood programmer for competitive matters such as competitions or interviews

dataflow• 1 year ago

This also helps me remember whether the estimate must over- or under-estimate: Since Dijkstra is making the estimate "0", clearly the "admissible heuristic" criteria must be an under-estimation.

You're thinking too hard. :-) Just think of a map the same way a 10-year-old would.

Straight-line (Euclidean) distance is the most obvious heuristic on a map for estimating distance, and it's admissible.

Straight lines minimize distances i.e. they never overestimate. Which is enough to remind you that you want an underestimate.

the way I would always think about A* from a "practical/easy-to-remember" perspective back when I was doing competitive programming is that they're all the same algorithm, but with different priorities on the priority queue

A much less obvious but much more fascinating (IMO) way to look at A* is that A* is actually Dijkstra but with a modified graph, where you adjust the heuristic delta between each edge's vertices to the edge's weight.

To remember the sign of the adjustment with this method, just imagine the next vertex getting super close to the destination, and then work out whether the weight needs to increase or decrease significantly in that case. (It needs to decrease, given you're getting closer to the destination.)

noqc• 1 year ago

I think probably the easiest way to remember under/over is just to remember that euclidean distance is a very common admissible heuristic.

k2xl• 1 year ago

As a game developer for a grid based puzzle game (https://thinky.gg - one of the games Pathology is a game where you have to go from Point A to Point B in shortest amount of steps).

I have found A* fascinating not because of the optimization but also from the various heuristics that can be built on top of it make it more generalized for other types of pathfinding.

Some devs have built solvers that use techniques like bidirectional search, precomputed pattern databases, and dead locking detection.

ncr100• 1 year ago

Just a note about bidirectional, or "double-ended" as I learned it - this can be very useful (i read 30% speed-up) for City / National Road searches.

One also has multiple layers of roadways of varying arterial significance, allowing higher speed (lower weight) travel, with real-world roads.

It was used at a mapping job to great boon by our backend.

0manrho• 1 year ago

I have a perhaps overly simplistic question, but how is this meant to be pronounced? A-star? Ah-sterisk?

Ao7bei3s• 1 year ago

Ay-star.

cryptomic22• 1 year ago

Path finding visualization, highlighting A*: https://youtube.com/shorts/L8t0tt1Vrsw

deadbabe• 1 year ago

A* is simple enough, but how do you handle pathfinding when the environment isn’t known to the entity?

ViscountPenguin2• 1 year ago

Iirc, the best approaches for this nowadays are machine learning based. Otherwise, you probably want to do an exploration step first, and bake in the environment.

gradstudent2• 1 year ago

ML methods are poor pathfinders. I'm unaware of any pathfinding problem where SOTA is not dominated by state-space search.

SnowflakeOnIce• 1 year ago

https://en.m.wikipedia.org/wiki/Simultaneous_localization_an...

gradstudent2• 1 year ago

SLAM is for mapping. For planning (and execution) in unknown environments, probably something like D* Lite.

leftnode• 1 year ago

I have a deep love of A* because it was the first complex algorithm I fully understood. In my first data structures and algorithms in college (early 2000's), we had to pick an algorithm to study, code, and write a paper on and I picked A*.

I spent hours painstakingly drawing similar grids that the author of this article made and manually doing the calculations [0]. I still have these notes somewhere, even though they're over 20 years old at this point, because I was so proud of the work I put into it.

At any rate, thanks for this article and the trip down memory lane.

[0] https://imgur.com/a/zRYaodL (apologies for the Imgur link)

mysteria• 1 year ago

Funny enough I got a hit of nostalgia seeing this as I learned A* for a school project many years ago using this exact same tutorial.

Kye2• 1 year ago

Not to be confused with Sagittarius A*.

https://en.wikipedia.org/wiki/Sagittarius_A*

lblume• 1 year ago

The star appears to have been cut off from the link, leaving an article about the corresponding radio source, Sagittarius A. This (official Wikipedia) short link leads to the desired article: https://w.wiki/5A7e

Kye2• 1 year ago

HN escaped the * when I posted. I tried to escape the escape, but it kept adding escapes to my escapes. After a while I conceded defeat and decided folks might enjoy taking the scenic route through disambiguation.

I never got in the habit of using share links, so it's interesting to learn Wikipedia has a URL shortener.

kazinator• 1 year ago

Someone send this to the idiots at Garmin, because their navigators will tell you to drive in a straight line across mountains or water water to an unreachable destination. It's like AI that never says "I don't know".

kriro• 1 year ago

Best introduction to A* in my opinion is travelling in Romania :)

AI a Modern Approach.

JohnKemeny5• 1 year ago

I always use A* when going to Bucharest.

throwawayoldie3• 1 year ago

"This problem assumes that the reader is not familiar with Romanian geography. We apologize to those who are unable to take advantage of this pedagogical device."

(Or words to that effect, it's been 20+ years since I read it.)

lukeyoo• 1 year ago

Relavants of mine

- https://lukeyoo.fyi/recap/2025/5/a-star

- https://github.com/micttyoid/quantized-pathfinding

90s_dev• 1 year ago

I have this bookmarked under "path finding" along with

https://juhrjuhr.itch.io/deep-space-exploitation/devlog/9454...

and https://www.redblobgames.com/grids/hexagons/

Introduction to the A* Algorithm (2014)

Related Stories