Mathematics at Google

http://research.google.com/pubs/pub38331.html

by PierreMage • 14 years ago

188 50 14 years ago

research.google.com

danso• 14 years ago

My day-to-day programming consists so much of process and simple boolean logic that I hardly ever use math more challenging than 1 + 1 and 1 != 0. It's great to review how math can greatly influence the potential of your code.

wickedchicken• 14 years ago

> simple boolean logic

A SAT solver[1] can automatically figure out a range of acceptable solutions for your conditions. 'I want this goal, is there a way to make that happen?' Security people love this. SMT solvers are even crazier[2] and can automatically determine solutions to linear equations or inequalities (this comes in handy when analyzing loops). Z3 has an online version at [3].

Also, if you have a lot of nasty cascaded if-statements, a K-map[4] and espresso[5] can reduce those down to a minimal set of branches. Note that fewer branches means fewer branch prediction penalties, which means faster code!

[1] http://en.wikipedia.org/wiki/Boolean_satisfiability_problem

[2] http://media.tumblr.com/tumblr_m8ywn1iesH1r6uu3b.gif

[3] http://rise4fun.com/z3/tutorial/guide

[4] http://en.wikipedia.org/wiki/Karnaugh_map

[5] http://embedded.eecs.berkeley.edu/pubs/downloads/espresso/in... and ftp://ftp.cs.man.ac.uk/pub/amulet/balsa/other-software/espresso-ab-1.0.tar.gz

btilly2• 14 years ago

My experience is that people who are aware of opportunities to use math find them. And people who are not aware conclude that they weren't really there.

For example are you a web developer? Have you started using A/B testing yet? If yes, that's math. If no, then you are in a position to make your employer a substantial amount of money.

For another example my very first project in my first programming job required me to build a set of canned reports. There was one user request which turned into, "We want to do this, but don't know if it is possible." I wound up using inclusion-exclusion techniques that I'd learned in grad school to solve that problem! (The problem was how to ship a local database without detailed information they shouldn't have about the rest of the company, and yet still be able to present reports comparing themselves to the rest of the company. And those reports had to drill down. So if you were responsible for 5 sites you'd be able to generate those reports for everyone that you could see, or for each site individually.)

j2kun8• 14 years ago

Look at the slide entitled Gmail (5), and compare the picture with the first graph on my blog post http://jeremykun.wordpress.com/2011/08/11/the-perceptron-and...

It just goes to show, Google steals content without attribution just like everyone else.

pav3l3• 14 years ago

Wait, they just used your graph and didn't even contact you about it?

j2kun8• 14 years ago

Shocking, right? I demand compensation! :)

gms3• 14 years ago

It's one guy who happens to work for Google, rather than the whole company.

j2kun8• 14 years ago

I imagine they have a release process to get something on research.google.com, though. So the company is endorsing it.

Muzza• 14 years ago

Using the same logic, a company can never be said to "do" anything. Everything is (ultimately) done by a real person.

tlammens2• 14 years ago

I am trying really hard to ignore the Futurama picture. Really hard!

j2kun8• 14 years ago

At least I gave the source :) And I don't claim I'm any better.

pav3l3• 14 years ago

Do they still heavily rely on PageRank? With the amount of traffic data Google has, I would expect more statistical approaches based on what users click (rather than graph algorithms based on how the web is linked) to be the backbone for ranking their results.

j2kun8• 14 years ago

PageRank is definitely still used, although it's doubtfully the sole determinant of ranking results. See http://jeremykun.wordpress.com/2011/06/21/googles-page-rank-...

technel• 14 years ago

I agree with your statement, but I'm confused by the story in the link... I haven't come across a message board that doesn't use rel="nofollow" in a few years (that NYT article is from 2010). How would these negative reviews bolster his PageRank?

iskander• 14 years ago

It's a closely guarded secret but at the very least we know that every page is annotated with many different signals, which are combined with magical secret sauce.

franze• 14 years ago

i'm always disgusted when a googler publishes anything about pagerank. yes, google is still using links as a single in their search result, but the original pagerank paper is from 1998, we can sure as hell be sure that they iterrated/rewrote it a few thousand times since then.

every time a (any) googler now publishes a paper (or presentation) about (the old) page rank a horde of too well paid SEOs is pointing to it for the importance of pagerank, and why linking out is bad, and whatever bullshit (i.e.: later in this thread the based on nothing hypothesis about "nofollow") they come up with. everytime it happens, the SEO bullshit dance starts anew. (in my opinion pagerank is thought-cancer, please see my old TC article for more about this http://techcrunch.com/2010/07/07/startups-linking-to-your-co... )

cjdrake• 14 years ago

This is a fantastic publication! I do some part-time mathematics tutoring, and kids are always wondering where math is used in "real life". Since kids are all familiar with Google, this should resonate with them.

btilly2• 14 years ago

Seeing PageRank discussed reminds me of a piece of fun trivia. The idea for PageRank came out of the success of the Science Citation Index, which ranks papers according to how often they have been cited. The idea of trying to study the structure of citations in academia came out of people who were inspired by a 1948 essay, As We May Think.

But that essay's main topic was an imagined technology called memex, to be implemented with an automated indexing system and microfilm. This technology is the first description of hypertext, which inspired multiple technologies. The second successful consumer application that I'm aware of that used hypertext was the web. (The first was HyperCard from Apple.)

Thus Google started as the application of one set of techniques inspired by As We May Think to a technology that was also inspired by As We May Think.

See http://www.theatlantic.com/magazine/archive/1945/07/as-we-ma... for the essay itself. Do keep in mind that it was written one year after the transistor was invented, but the author already had 2 decades of experience with computing.

killerbat002• 14 years ago

Vannevar Bush's ideas about information organization and consumption in the future were eerily accurate. Reading about the history of Memex and the roots of the Information Architecture field in general is something I highly recommend for anyone interested in Information Science, etc.

JumpCrisscross• 14 years ago

Recommended book(s)?

pav3l3• 14 years ago

Someone just started a discussion: http://news.ycombinator.com/item?id=4577865

pdog• 14 years ago

Previous discussion (two years ago, with comments):

http://news.ycombinator.com/item?id=1565764

tantalor• 14 years ago

My mathematician friend pointed out that all that "research at google" requires "experience with large data sets and quantitative analysis". They want statisticians, not mathematicians.

micro_cam• 14 years ago

I'm not a googler but I do know a bit of Math and Statistics and many of the approaches they take can be cleverly reduced to optimization problems or matrix math both of which are more the domain of applied mathematicians then statisticians.

dxbydt2• 14 years ago

The article has a section on the math used in Google Maps, which points to

http://algo2.iti.kit.edu/schultes/hwy/esaHwyHierarchies.pdf

which says - there are 24 million places in the USA, connected by 29 million roads. You need 4 hours 15 minutes to pre-process this information. From then on, it only takes 7 milliseconds to find the shortest path from one place to another by running the Multilevel Query Algorithm, which is a souped up version of Dijkstra and runs 2000 times faster than Dijkstra's Shortest Path algorithm.

Is that right ? 24 million choose 2 is 288 trillion, so do an all paths search, then have a lookup table with 288 trillion entries, store that in HDFS, slap an LRU caching layer atop that, and you wouldn't have to run any graph query algorithm at all, so should be able to do much better than 7 ms ... just thinking out loud.

psykotic4• 14 years ago

Good luck precomputing all those pairwise shortest paths. Storing the table might not be too bad. But standard algorithms like Floyd-Warshall are O(n^3) in the number of vertices. There are faster algorithms based on fast matrix multiplication but the precomputation time would still be prohibitive. Keeping it up to date would be even worse. The construction of a new highway could require updating the shortest paths for an enormous number of pairs.

The economic argument would go like this: The revenue generated from your maps business is proportional to the number of queries actually processed, not the total number of conceivable queries. The queries processed is such a tiny subset of the possible queries that you want your computational expenses to track the former, not the latter.

With a hierarchical shortest paths algorithm, you can still precompute all pairwise shortest paths at the coarser level. For road navigation you might precompute the shortest path between every pair of interstate highway exits in the USA like this paper is doing. In an open-world game like Skyrim you might precompute the shortest paths between all towns and other major hubs and points of interest. That might not yield truly optimal paths. For game use it's close enough and has the benefit of corresponding to how humans naturally navigate.

Sidebar: The old Crash Bandicoot games used precomputed shortest paths in a neat way. Their navigation was based on a triangular mesh, so every triangle had up to three edgewise neighbors. Thus for a navmesh with n triangles, they needed lg(3) n(n-1)/2 <= n(n-1) bits to store the table. For convenience they probably stored this in a redundant form requiring 2n^2 bits = n^2/4 bytes. But with only 64KB of additional memory, this still let them support 512 navmesh triangles per level with lightning-fast path finding.

dxbydt2• 14 years ago

>For road navigation you might precompute the shortest path between every pair of interstate highway exits in the USA like this paper is doing.

Oh, so they are populating a lookup table, not for the entire nodeset but just the tuples denoting interstate highways. All I was advocating was a much larger ( and more expensive) lookup table :) btw, you've edited your response like 6 times now...everytime I try to reply there's new info in there!

sb• 14 years ago

Hm, thanks for the nice remark on the game stuff, I kind of like that.

While agree with your primary criticism, I think the principle of locality applies well to this problem. While there are 24 million places (supposedly, never checked that from the paper), I'm positive that most maps queries are served to the areas with higher population density. Therefore, it might make sense to use the look-up table approach for densly populated areas to reduce the search time. (OTOH, the computer scientist in me reminds me that the 7ms might pale in comparison to latency times for mobile phones receiving the map information...)

azth• 14 years ago

...and use up tons of hardware when you can get by without doing that?

Mathematics at Google

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