Raft: Understandable Distributed Consensus (2014)(thesecretlivesofdata.com)
185 points by Hrun0 20 hours ago | 16 comments
_russross 17 hours ago
I am in the minority who thinks Raft is overrated.

I tried teaching Raft one year instead of Paxos but ended up switching back. While it was much easier to understand how to implement Raft, I think my students gained deeper insight when focusing on single-decision Paxos. There is a lightbulb moment when they first understand that consensus is a property of the system that happens first (and they can point at the moment it happens) and then the nodes discover that it has been achieved later. Exploring various failure modes and coming to understand how Paxos is robust against them seems to work better in this setting as well.

I think this paper by Heidi Howard and Richard Mortier is a great way to move on to Multipaxos:

https://arxiv.org/abs/2004.05074

They present Multipaxos in a similar style to how Raft is laid out and show that Multipaxos as it is commonly implemented and Raft are almost the same protocol.

Raft was a great contribution to the engineering community to make implementing consensus more approachable, but in the end I don't think the protocol itself is actually more understandable. It was presented better for implementers, but the implementation focus obscures some of the deep insights that plain Paxos exposes.

sriram_malhar 2 hours ago
I have had the opposite trajectory. Used to teach Paxos, but was so relieved to switch to Raft when the paper came out.

I discuss distributed data structures in the context of maps and sequences.

For maps, I discuss key-value stores (NUMA, Redis). I have them implement cache coherence (MESI protocol, TARDIS 2.0), then linearizable, fault-tolerant, wait-free shared memory registers (the Attiya/Bar-Noy/Dolev algorithm[1]).

For sequences, I cover shared logs and state machine replication, including database log shipping, Kafka, queues and Raft.

I like Raft because it cuts down the design space by making certain very intuitive and pragmatic choices, like using timeouts (which are almost beneath Lamport to discuss :), or idioms like "follow the leader", "if the leader is unreachable, stand for election", "elect the latest & most informed leader", (how I wish that was true in real life!), "always append" etc. There are simple mechanisms to preserve invariants.

The problem with Paxos is that there is such a large range of papers that there is no one paper that makes the leap in easy digestible chunks from Basic to MultiPaxos. When I got students to implement MultiPaxos, I never could get sufficient confidence that it was done right (esp. the "disorderly" filling in of log slots).

Paxos is like Monads; when you get it, you feel compelled to write a "Paxos explained" paper :)

[1]https://groups.csail.mit.edu/tds/papers/Attiya/PODC90.pdf

hintymad 11 hours ago
Is there any paper/handouts/video that explains Paxos in depth, especially its implementations and intuitions? Paxos Made Simple gave intuitive explanations, but I feel it still misses a lot of intricate details if I were to build Praxos for production use.
withinboredom 16 hours ago
Know that I join you in Raft being overrated. I’m working on a multipaxos implementation right now. There are some really neat capabilities/properties that paxos has that Raft can never achieve (see wpaxos, for example, that lets keys migrate to nodes near the client).
alexchamberlain 15 hours ago
I've read both the Paxos and Raft papers a few times, and hacked on some implementations, but never quite got one over the line to working...

Raft strikes me as a particular set of decisions made within a Paxos framework, such as having 1 entity for Proposers, Acceptor and Followers. It's frustrating that there isn't a clearly written defacto paper on Paxos - the story style confused the monkeys out of me.

jimbokun 14 hours ago
> but never quite got one over the line to working...

I've never implemented something like this. But my first thought is "how do you implement the testing system?"

I feel like once you had a robust testing system that can verify things work correctly in all the different network partition and other scenarios, and allowing rapid iteration of setting up those scenarios, the implementation would be comparatively easy.

heromal 3 hours ago
Check out maelstrom
alexchamberlain 14 hours ago
Yeah, you kind of can’t test any of it until you test all of it…
weinzierl 16 hours ago
This is an interesting insight into the educational side, but now I am curious about the implementation side. Raft is easier to implement but that's just one factor. Looking at real world usages there seems to be a draw. I could easily count as many Paxos implementations as Raft. Is this just historical or are there good reasons for a new project to still implement Oaxos?
bfdes 15 hours ago
Paxos and Multi-Paxos have been around much longer than Raft. The paper that introduced Raft was published in 2014.
senderista 14 hours ago
Agree, Raft is less modular and therefore harder to understand than MultiPaxos:

https://maheshba.bitbucket.io/blog/2021/12/14/Modularity.htm...

alexnewman 12 hours ago
You are telling me your students can safely implement single decree paxos.... I worked on a few paxos production implementations before the RAFT paper, single and multi decree. The idea that paxos, as it is to be implemented, is easy for students to understand... Well let me re-read the paper, but i assure you, raft was a big deal
benbjohnson 18 hours ago
Author here. I'm happy to answer any questions although this project was from 10+ years ago so I could be a little rusty.

Over the years I've been trying to find better ways to do this kind of visualization but for other CS topics. Moving to video is the most realistic option but using something like After Effects takes A LOT of time and energy for long-form visualizations. It also doesn't produce a readable output file format that could be shared, diff'd, & tweaked.

I spent some time on a project recently to build out an SVG-based video generation tool that can use a sidecar file for defining animations. It's still a work in progress but hopefully I can get it to a place where making this style of visualizations isn't so time intensive.

huntaub 16 hours ago
I just want you to know how much this visualization was appreciated. In my time working at AWS, I recommended this website to every one of our new hires to learn how distributed consensus works. Know that this has taught probably 50+ people. Thank you for what you’ve built.
benbjohnson 12 hours ago
Thanks so much for letting me know! It's always hard to tell when I put something out there if it just gets lost in the ether. I'm glad to hear it helped so many folks.
mgenglder 17 hours ago
This is wonderful. Can I ask how you created it? Stack used and sour e code? I'd love to create something like this to help visualize things I'm working with currently.
benbjohnson 12 hours ago
It's all done with d3 and JavaScript. The visualizations aren't deterministic so I ended up writing a shitty Raft implementation in JS. Overall it was a terrible approach because it was so time consuming but I made it work. You can find all the source code in this repo: https://github.com/benbjohnson/thesecretlivesofdata
kfrzcode 17 hours ago
What are your thoughts on Dr. Leemon Baird's Hedera Hashgraph?

https://www.swirlds.com/downloads/SWIRLDS-TR-2016-01.pdf

benbjohnson 12 hours ago
I haven't read that paper but it seems like it's fixing a different problem of Byzantine fault tolerance. Most consensus systems that are internal for an organization don't have the Byzantine issue so it simplifies the problem.
MarkMarine 19 hours ago
This is one of my favorite pieces of software engineering because it took something difficult and tried to design something easy to understand as a main criteria for success. The PHD Thesis has a lot more info about this if anyone is curious, it is approachable and easy to read:

https://web.stanford.edu/~ouster/cgi-bin/papers/OngaroPhD.pd...

I think this was core to Raft’s success, and I strive to create systems like this with understandability as a first goal.

throwawaymaths 17 hours ago
Weirdly it's also kinda worse is better: raft is non-deterministic and has an unboundedly long election cycle time. IIRC:

- it assumes no hysteresis in network latencies and if there is a hysteresis it's possible that elections can be deterministically infinite.

- this fact and the use of raft in production has caused real, large scale network outages.

Paxos is of course a beast and hard to understand. There is an alternative, VSR (which was developed ~time of paxos) which is easy to understand and does not have the issues caused by election nondeterminism in raft.

Of course everyone uses raft so raft dominates.

eatonphil 17 hours ago
> - this fact and the use of raft in production has caused real, large scale network outages.

While this has surely happened, I am not so confident about what the reasons were for this. If you've got links on details I'd love to read.

> which is easy to understand

I've implemented core bits of Raft twice now and have looked at VSR a couple of times and VSR wasn't easier for me to understand. I'm sure I could implement VSR and would like to some day, but just comparing the papers alone I personally felt like Raft was better presented (i.e. easier to understand).

Also keep in mind that nobody ships consensus implementations exactly in line with the original paper. There are dozens or hundreds of papers on variations and extensions of Raft/Paxos and every actual implementation is going to implement some selection of these extensions/variations. You have to look at each implementation carefully to know how it diverges from the original paper.

throwawaymaths 16 hours ago
> A known limitation of the base Raft protocol is that partial/asymmetric network partitions can cause a loss of liveness [27, 32]. For instance, if a leader can no longer make progress because it cannot receive messages from the other nodes, it continues to send AE heartbeats to followers, preventing them from timing out and from electing a new leader who can make progress.

(Howard, Abram et al)

Me: note this can also occur if there isn't a complete outage, if the latency back to the shit leader is different from the latency out of the shit leader.

> nobody ships consensus implementations exactly in line with the original paper. There are dozens or hundreds of papers on variations

As the paper above explains once you add extensions you might have broken the correctness proofs in raft. More to the original point, you're now in a state where it's no longer "simple"... I would go so far as to say if you have to consider the extensions, which are distributed over several papers and sometimes not even papers at all, you're in "deceptively simple" land.

As a pedagogical tool, raft is valuable because it can be a launching ground for conversations like these... But maybe we shouldn't use it in prod when there are better, straightforward options? I get the feeling that being hard sold as simple nerdsniped devs into writing it and someone r/very smart put it into prod and with social proof more people did and now here we are

lifeinthevoid 16 hours ago
> A known limitation of the base Raft protocol is that partial/asymmetric network partitions can cause a loss of liveness [27, 32]. For instance, if a leader can no longer make progress because it cannot receive messages from the other nodes, it continues to send AE heartbeats to followers, preventing them from timing out and from electing a new leader who can make progress.

Real-world raft implementations make the leader step down if it hasn’t heard from a quorum for a while. Not part of vanilla raft though.

eatonphil 16 hours ago
The thesis does describe doing this fwiw while the paper does not.
MarkMarine 8 hours ago
> this fact and the use of raft in production has caused real, large scale network outages.

Paxos as well, I remember full cloud GCP outage that had something to do with Paxos, and I can’t find the data on it but I thought there was a nasty bug in zookeeper paxos implementation.

That isn’t to say any of these are perfect or bug free, it’s made by humans and we’re going to make mistakes, but my experience implementing both was I had a working raft implementation and paxos baked my brain until I gave up.

I think everyone uses raft _because_ it was possible to implement for a working dev, so there are a number of implementations, and it’s easier to understand the phases the application is in.

I’ll check out VSR I appreciate the rec.

lanstin 6 hours ago
My zookeeper outages are always due to simple things like all the workers having the same basic image and then having increased write rate filling up the disk of all workers at the same time.
convolvatron 16 hours ago
don't forget about calm. its a shame that that isn't the default we reach for, and only struggle when we really need stronger latency.

https://arxiv.org/abs/1901.01930

prydt 15 hours ago
I think the CALM theorem and this whole line of research is so interesting and it is still carried on by the CRDT people. But I would love to see more of this.

I feel like it doesn't get as much attention as it deserves.

kfrzcode 13 hours ago
I've made a longer comment in another thread; but have you investigated the hashgraph algorithm? Gossip-about-gossip and virtual voting combine to result in leaderless consensus, fair ordering and aBFT. It's very performant with over 10k TPS on-chain. I'm learning about DLT from the perspective of hashgraph which is why I don't understand why it doesn't get love - it seems to have all of the good and none of the bad.
prydt 15 hours ago
I've run a reading group for distributed systems for the last 2 years now and I do think that Raft is a better introduction to Consensus than any Paxos paper I have seen (I mean the Paxos Made Simple paper literally has bugs in it). But when I learned consensus in school, we used Paxos and Multi-Paxos and I do believe that there was a lot to be gained by learning both approaches.

Heidi Howard has several amazing papers about how the differences between Raft and Multi-Paxos are very surface level and that Raft's key contribution is its presentation as well as being a more "complete" presentation since there are so many fragmented different presentations of Multi-Paxos.

As a bonus, one of my favorite papers I have read recently is Compartmentalized Paxos: https://vldb.org/pvldb/vol14/p2203-whittaker.pdf which is just a brilliant piece on how to scale Multi-Paxos

senderista 14 hours ago
There are several Multi-Paxos papers (some of them dating before Raft) that are intended as guidance for implementers:

https://paper-notes.zhjwpku.com/assets/pdfs/paxos_for_system...

https://www.cs.cornell.edu/home/rvr/Paxos/paxos.pdf

https://www.scs.stanford.edu/~dm/home/papers/paxos.pdf

prydt 12 hours ago
Ah thank you. That is a good list although I personally dislike the "Paxos Made Moderately Complex" paper... I think it adds too many different roles for very little benefit. When implementing multi-Paxos for class, I used that paper and felt it was more trouble than it needed to be.

I'll check out the other two papers though! Also just looking around and I found this paper https://arxiv.org/pdf/1103.2408 [PDF] which looks useful as well.

shepherdjerred 8 hours ago
What’s your reading group?

I took a DS class and (poorly) implemented Paxos a few years ago. I’m curious about how others continue learning about DS.

mbivert 16 hours ago
In case this is of interest, MIT's 6.5840[0], distributed systems, has a series of labs, implementing Raft in Go. Haven't made it through the whole thing yet, but it's quite entertaining so far.

The teachers provide you with some code templates, a bunch of tests, and a progressive way to implement it all.

[0]: https://pdos.csail.mit.edu/6.824/index.html

lifeinthevoid 2 minutes ago
really liked the course, it was also instrumental in landing me my previous job :-)
dang 15 hours ago
Related:

Raft Consensus Animated (2014) - https://news.ycombinator.com/item?id=32484584 - Aug 2022 (67 comments)

Raft Visualization - https://news.ycombinator.com/item?id=25326645 - Dec 2020 (35 comments)

Raft: Understandable Distributed Consensus - https://news.ycombinator.com/item?id=8271957 - Sept 2014 (79 comments)

quelltext 5 hours ago
In the log replication example, after healing the partition the uncommitted log changes in the minority group are rolled back and the leader's log is used.

However it's not clear how that log is transmitted. Until this point only heartbeats via append entry were discussed, so it's not clear if the followers pull that information from the leader somehow via a different mechanism, or whether it's the leader's responsibility to detect followers that are left behind and replay everything. That would seem rather error prone and a lot of coordination effort. So how's it actually done?

eatonphil 17 hours ago
Ben's visualization here is great.

The other biggest help to me aside from the paper and the thesis was Ongaro's TLA+ spec: https://github.com/ongardie/raft.tla/blob/master/raft.tla. It's the only super concise "implementation" I found that is free of production-grade tricks, optimizations, and abstractions.

And for building an intuition, TigerBeetle's sim.tigerbeetle.com is great. What happens to consensus when there's high latency to disk or network? Or as processes crash more frequently? It demonstrates.

pa7ch 13 hours ago
Interesting that TigerBeetle uses Viewstamp Replication over Paxos/Raft. TB says viewstamp replication lends itself to a more performant implementation and doesn't rely on disk storage as much. I'm surprised I'm not seeing this brought up more in Paxos/Raft discussions.
skilning 18 hours ago
I was asked to click "continue" after each of the first two sentences, and the fade-in of the text took longer than reading the text.

This may be a great article, but I'll never know because it's frustrating to try and read.

jhanschoo 12 hours ago
Just earlier this month I was going through https://github.com/jepsen-io/maelstrom and following the demo implementing (a cut-corners version of) Raft, and I found it quite elucidating. The sample (ruby) code contains some bugs, and I had to use some understanding to fix them. (The bugs were of the kind where a dummy implementation from an earlier step isn't correctly changed)
ko_pivot 19 hours ago
The writing and the visualizations are great. The ‘continue’ button is way too frequent.
shiredude95 14 hours ago
"Paxos Made Moderately Complex" by Robert van Renesse and Deniz Altinbuken: http://www.cs.cornell.edu/courses/cs7412/2011sp/paxos.pdf is a great starting point for implementing multi-paxos. The authors also provide a working python implementation.
rapsey 18 hours ago
While understandable, implementing it is however far from easy.
jimbokun 14 hours ago
How do you write a testing framework/system for evaluating distributed algorithms like this?

Doing that in a way that's easy to automate, set up, evaluate, and tear down various scenarios seems like the hardest part to me.

ukd1 17 hours ago
Having implemented it twice (for fun/learning, though) I think you're right - yet it's also the easiest, imho. The paper, and resources around implementing it (posts, other implementations) are great. Also, the authors still reply on github when issues are opened, which is great.
MarkMarine 18 hours ago
Right, but Paxos is double hard in comparison. I’ve read both papers multiple times, tried to implement and failed, and I still don’t think I understand Paxos.
mrkeen 17 hours ago
I'm on the other side.

I think Leslie Lamport asserted that Paxos is minimal, and that "all other consensus algorithms are just Paxos with more steps". I'm inclined to believe him.

I've implemented Paxos but I can't get through "Raft for dummies" style blog posts.

Regarding Raft [1]:

  > The consensus problem is divided into three sub-problems: Leader election, Replication and Safety.
What is leader election? It's a distributed system coming to consensus on a fact (i.e. who the leader is.) Then once you have the leader, you do additional steps. The entirety of Paxos is a distributed system coming to consensus on a fact.

When I read these posts, i see things like "timeout", "heartbeat", and I think: timeout according to whom? I read "once the leader has been elected", um, hangon, according to whom? Has node 1 finally agreed on the leader, just while node 3 has given up and started another election? I don't doubt that Raft is correct, but the writing about it seems simple by glossing over details.

Paxos, on the other hand, seems timeless. (And the writing about it doesn't trigger my "distributed system fallacies" reaction)

[1] https://www.brianstorti.com/raft/

spmurrayzzz 17 hours ago
I tend to agree that many explanations of raft dont get into the useful details and handwave some of the hard problems. But the original paper does do a good job of this and is pretty accessible to read IMO.

> I read "once the leader has been elected", um, hangon, according to whom? Has node 1 finally agreed on the leader, just while node 3 has given up and started another election?

The simple response I think to "according to whom" is "the majority of voting nodes". When the leader assumes its role, it sends heartbeats which are then accepted by the other nodes in the cluster. Even if (in your example) node 3 starts a new election, it will only succeed if it can get a majority of votes. If node 2 has already acknowledged a leader, it won't vote for node 3 in the same term.

There's some implicit concessions inherent there around eventual consistency, but I don't think thats novel to Raft compared to other distributed consensus protocols.

ivankelly 17 hours ago
100% agree. I haven't read the raft paper in years, but I remember thinking there's just too much stuff in there. That stuff in important, but if you want people to understand what's happening they internalize the fundamental idea of being able to block other writers by bumping a number. Which is all covered in the single decree paxos section in part time parliment.
kfrzcode 17 hours ago
Paxos is nice, sure, but Hedera does DLT with aBFT and much more efficient, as well as being faster and ensuring fairness. It's leaderless, and achieves incredible TPS (10k+ in practice, 100k+ in theory).

I am curious on your thoughts here.

Vervious 17 hours ago
Raft has a problem where, in the protocol description, sometimes I have no idea why some line is there, but if you take it out the protocol comes to a grinding halt... it's really a mumble jumble. It has good intuition, but the details are really messy, it's edge cases all the way down.

Whereas I think each line of pseudocode in Paxos is much more motivated.

In other words, if a philosopher had to design a crash-fault protocol from scratch, without having seen any before, I think 80% of the time it would look exactly like Paxos.

candiddevmike 18 hours ago
IMO, Paxos has a lot less edge cases than Raft, mostly because of the complexity of the implementation covers them/forces you to think about how to handle them.
hinkley 17 hours ago
How many people actually understand Paxos?

The assertion at the time was that only a few people understood it well enough to make a correct implementation, the others were full of bugs.

The problem we have with Lamport is that he’s very good at talking to computers but not so good at talking to humans. I think the world would be a better place today if someone had forced him to learn to speak human.

He did a presentation at MS just after he won the Turing Award. He prefaces it with how important writing is to thinking, and how you don’t really know what you think until you write it down. Those are the words and thinking of an introvert. Writing is still the shallow end of understanding. The deep end is teaching. If you understand something and you teach it to others, then you have proven that you understand it, and caused that understanding not to be lost to posterity. If you only write about it, it might work as instructional material, or it may require some very clever people who can teach themselves using your words. But they may also get it wrong, and not have you for feedback.

The latter is where seem to be with Leslie’s works. The consensus is that few people actually understand what he’s talking about well enough to implement it correctly.

I had a coworker once who was shocked to learn I read the ACM SIGPLAN proceedings. “You can read those??” I knew what he meant and yeah, a lot of those were very unapproachable and I understood two thirds of them and only half of each of the rest. Before I committed to using Raft I gave Lamport’s paper a try. It was a slog and he doesn’t sell the why of each part. He’s just giving you a very, very long recipe without the mental models necessary to reproduce it robustly.

jeffbee 17 hours ago
Students of raft say they understand it but when they have to implement it they make as many mistakes as students of paxos. The simplicity is misleading.
matthewfcarlson 11 hours ago
This really teaches Raft well. Is there a good example of this but for Paxos?
cedws 16 hours ago
Can't proof-of-work be used as a leader election algorithm? If the proof is hard enough to generate then one node should be able to generate one and broadcast it before the other nodes can, then that node becomes the leader.
withinboredom 16 hours ago
There’s a paper about that, using paxos as the base. Can’t find it right now, it is called “chained paxos” or “block paxos” or something like that.
ryanthemadone 16 hours ago
Proof of work is a leader election algorithm!
kfrzcode 16 hours ago
DLT technology discussions are entirely incomplete without consideration of Hedera Hashgraph [0], an aBFT, leaderless, fair and fast DLT using a gossip-about-gossip consensus mechanism. It's absolutely a more robust and scalable technology than Paxos or any other DLT for that matter. I'd love to know what the HN crowd thinks about Hedera as the trust layer of the internet but.... nobody around here seems to have any. It's like ignoring Linux while comparing Mac and Windows based computing.

[0]: https://www.swirlds.com/downloads/SWIRLDS-TR-2016-01.pdf

ryanthemadone 16 hours ago
Paxos isn't a DLT, it's a consensus algorithm — granted, DLTs tend to require a consensus algorithm, but they're not the same things.

As for Hedera Hashgraph being the trust layer of the internet, we tend to build the internet through the IETF and standards setting. Unfortunately HH is an endeavour from a private company so isn't especially likely to be taken on in that context.

I'd also wonder what you mean by the trust layer of the internet, what are the use cases that you'd like to see solved with such a trust layer?

kfrzcode 13 hours ago
Great point, I wasn't being precise. My point stands, however! I must also correct your understanding re: "private company."

The DLT in question - Hedera - is built on the unique consensus algorithm, the "hashgraph". The hashgraph algorithm combines a gossip-about-gossip protocol with virtual voting.

In fact, services development is now in the hands of the largest open source foundation in the world. These implementations are entirely open source [0], and very recently the codebase has been donated in whole to the Linux Foundation.

Furthermore, Hedera is a Pioneer member of the newly founded Linux Foundation Decentralized Trust organization [1] - along with Chainlink, Deloitte, Hitachi, many other major organizations, some of whom are also on the Hedera governing council [2]. This foundation will be a big player in the future of decentralized web, and Hedera is the only L1 that I know of which is both primed for this future and actually scalable.

I understand the IETF and standards approach; and am not aware of a current draft or intention for such a draft. The idea of Hedera being the 'trust layer' of the internet is more about use cases like decentralized recovery, process validation, carbon offsets, extremely granular supply chain auditing, and any other application you might imagine that would benefit from having extremely fast (10k+ TPS), 100% guaranteed aBFT consensus on-chain. I'd love to hear what you might think up or where this could be particularly useful. Strong governance with 39 council members - including Google, IBM, Boeing, Tata, AP+, Hitachi and more... with decentralized network operation and stable fees (essential for enterprise application).

> Additional use cases include, but are not limited to, financial markets, matching engines (like those used in Uber or AirBnb), or supply chain negotiations (e.g., several competing factories bidding on parts from several competing parts suppliers).

So, admittedly the use cases maybe aren't evident or interesting to you and I right now at the TCP/IP layer, but I can certainly say there are a plethora of trust-based problems that could be solved with consensus only needing a few thousand ms. Think digital identity, healthcare, financial markets, IoT, supply chain, real-world asset tokenization... For any real-world, scaled application, a DLT must have very high throughput at high performance. It's the absolute highest performance and security possible in a leaderless consensus-based DLT as far as I know.

Literally carbon neutral or negative because of buybacks but even without Hedera buying carbon credits, it's the single most "green" i.e. power-efficient DLT on the market. Does HN still care about Bitcoin using too much power? They would like Hedera, to that extent. Predictable, very low fixed fees. Long list of the biggest tech players leading the open development process. What's not to love?

I urge you to help me invalidate these claims as it's pretty important I understand the tech here... But I'm very bullish on the token price as the technology is proven, robust, and overall extremely undervalued by retail - in my opinion. NFA.

> The Hashgraph consensus algorithm is an algorithm for asynchronous Byzantine fault tolerance intended for distributed shared ledgers. Its main distinguishing characteristic is it achieves consensus without exchanging any extra messages; each participant’s votes can be determined from public information, so votes need not be transmitted.

For more rigorous explanation, see [3] and the associated Coq proof of the algorithm [4].

[0]: https://github.com/hashgraph/hedera-services

[1]: https://www.lfdecentralizedtrust.org/

[2]: https://hedera.com/ecosystem/governing-council

[3]: https://hedera.com/papers

[4]: https://www.cs.cmu.edu/~crary/papers/2021/hashgraph.pdf