7 ms·

Mathematics at Google

188 points by PierreMage 14y ago

danso 14y 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 14y 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 http://en.wikipedia.org/wiki/Boolean_satisfiability_problem [2] http://media.tumblr.com/tumblr_m8ywn1iesH1r6uu3b.gif http://media.tumblr.com/tumblr_m8ywn1iesH1r6uu3b.gif [3] http://rise4fun.com/z3/tutorial/guide http://rise4fun.com/z3/tutorial/guide [4] http://en.wikipedia.org/wiki/Karnaugh_map http://en.wikipedia.org/wiki/Karnaugh_map [5] http://embedded.eecs.berkeley.edu/pubs/downloads/espresso/index.htm 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
- btilly 14y 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.)
j2kun 14y 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-all-the-things-it-cant-perceive/ http://jeremykun.wordpress.com/2011/08/11/the-perceptron-and... It just goes to show, Google steals content without attribution just like everyone else.
- pav3l 14y ago
  Wait, they just used your graph and didn't even contact you about it?
  - j2kun 14y ago
    Shocking, right? I demand compensation! :)
- gms 14y ago
  It's one guy who happens to work for Google, rather than the whole company.
  - j2kun 14y ago
    I imagine they have a release process to get something on research.google.com, though. So the company is endorsing it.
    - gms 14y ago
      What do you expect them to do? Somehow do an image-similarity search throughout the web for every image used in papers by their employees?
      - nine_k 14y ago
        BTW completely feasible with Google's resources. Can be fully automated, too.
      - jackpirate 14y ago
pav3l 14y 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.
- j2kun 14y 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-why-it-doesnt-work-anymore/ http://jeremykun.wordpress.com/2011/06/21/googles-page-rank-...
  - technel 14y 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?
    - wahnfrieden 14y ago
      Google seems to still weight those links, just differently or with less weight.
- iskander 14y 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 14y 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-competition-will-help-you-no-really/ http://techcrunch.com/2010/07/07/startups-linking-to-your-co... )
cjdrake 14y 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.
btilly 14y 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-may-think/303881/?single_page=true 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.
- killerbat00 14y 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 14y ago
    Recommended book(s)?
    - andrewcooke 14y ago
      bush is mentioned in mirowski's "machine dreams", and it's not very complementary. that book is one of my favourites, but it's a post-modern, opinionated, soft-science (he's a history / philosophy / economics guy) take on post-war economics (and a whole pile of surrounding subjects). i suspect most people will hate the book as much as i love it, but you might be interested (perhaps you can find a library copy to check out....)
tantalor 14y 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 14y 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.
dxbydt 14y 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 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.
- psykotic 14y 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.

188 points by PierreMage 14y ago

danso 14y 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 14y 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 http://en.wikipedia.org/wiki/Boolean_satisfiability_problem [2] http://media.tumblr.com/tumblr_m8ywn1iesH1r6uu3b.gif http://media.tumblr.com/tumblr_m8ywn1iesH1r6uu3b.gif [3] http://rise4fun.com/z3/tutorial/guide http://rise4fun.com/z3/tutorial/guide [4] http://en.wikipedia.org/wiki/Karnaugh_map http://en.wikipedia.org/wiki/Karnaugh_map [5] http://embedded.eecs.berkeley.edu/pubs/downloads/espresso/index.htm 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
btilly 14y 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.)

j2kun 14y 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-all-the-things-it-cant-perceive/ http://jeremykun.wordpress.com/2011/08/11/the-perceptron-and... It just goes to show, Google steals content without attribution just like everyone else.

pav3l 14y ago
Wait, they just used your graph and didn't even contact you about it?
- j2kun 14y ago
  Shocking, right? I demand compensation! :)
gms 14y ago
It's one guy who happens to work for Google, rather than the whole company.
- j2kun 14y ago
  I imagine they have a release process to get something on research.google.com, though. So the company is endorsing it.
  - gms 14y ago
    What do you expect them to do? Somehow do an image-similarity search throughout the web for every image used in papers by their employees?
    - nine_k 14y ago
      BTW completely feasible with Google's resources. Can be fully automated, too.
    - jackpirate 14y ago

pav3l 14y 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.

j2kun 14y 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-why-it-doesnt-work-anymore/ http://jeremykun.wordpress.com/2011/06/21/googles-page-rank-...
- technel 14y 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?
  - wahnfrieden 14y ago
    Google seems to still weight those links, just differently or with less weight.
iskander 14y 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 14y 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-competition-will-help-you-no-really/ http://techcrunch.com/2010/07/07/startups-linking-to-your-co... )

cjdrake 14y 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.

btilly 14y 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-may-think/303881/?single_page=true 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.

killerbat00 14y 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 14y ago
  Recommended book(s)?
  - andrewcooke 14y ago
    bush is mentioned in mirowski's "machine dreams", and it's not very complementary. that book is one of my favourites, but it's a post-modern, opinionated, soft-science (he's a history / philosophy / economics guy) take on post-war economics (and a whole pile of surrounding subjects). i suspect most people will hate the book as much as i love it, but you might be interested (perhaps you can find a library copy to check out....)

tantalor 14y 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 14y 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.

dxbydt 14y 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 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.

psykotic 14y 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.