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vgr's profile
Venkatesh Rao
Venkatesh Rao
Venkatesh Rao
@vgr

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Venkatesh Rao

@vgr

Conversational account. For work follow @ribbonfarm, @breaking_smart, @artofgig. Tweets are 90% vacuous views, apathetically held. Mediocritopian. IKEA builder.

Los Angeles, CA
venkateshrao.com
Joined August 2007

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    Venkatesh Rao‏ @vgr 28 Sep 2019
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    Is there an easy algorithm to a) count the number of equivalent shortest paths on a manhattan grid b) enumerate them systematically? I wrote a dumb random algo and I could run it for a long time to count unique paths but I imagine there's a closed-form solution?pic.twitter.com/Moj8hm9NcM

    12:08 PM - 28 Sep 2019
    • 3 Retweets
    • 9 Likes
    • Giridhur Sriraman Austin Jeff Will Bickford Drew Austin Arun Olaf Nick Parker stephen hoban
    7 replies 3 retweets 9 likes
      1. Venkatesh Rao‏ @vgr 28 Sep 2019
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        I'm actually interested in a second-order problem: count the fraction of paths that go through a particular segment. If you walked to work on a manhattan grid everyday, making random choices, how long before you walked a specific block? A specific sequence of blocks?

        5 replies 1 retweet 6 likes
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      2. micah ⎎‏ @glowcoil 28 Sep 2019
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        Replying to @vgr

        all equivalent shortest paths have N x-steps and M y-steps, which you can take in any order, so i think it's just (N+M)!/N!M!

        1 reply 0 retweets 3 likes
      3. Venkatesh Rao‏ @vgr 28 Sep 2019
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        Replying to @glowcoil

        Ah ok, that's the formula I derived, but wasn't entirely sure of my logic, thanks... any idea how to enumerate them systematically?

        1 reply 0 retweets 0 likes
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      2. Mr. Eli W. Jones‏ @mreliwjones 28 Sep 2019
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        Replying to @vgr

        Feels like N! ways.. so that would explode as N increases.

        1 reply 0 retweets 0 likes
      3. Mr. Eli W. Jones‏ @mreliwjones 28 Sep 2019
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        Replying to @mreliwjones @vgr

        Enumerating them can be done recursively, but that will explode in running time.

        0 replies 0 retweets 0 likes
      4. End of conversation
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      2. Drew Austin‏ @kneelingbus 28 Sep 2019
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        Replying to @vgr

        The concept of betweenness centrality is kinda relevant to that https://en.m.wikipedia.org/wiki/Betweenness_centrality …

        1 reply 0 retweets 2 likes
      3. Venkatesh Rao‏ @vgr 28 Sep 2019
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        Replying to @kneelingbus

        Venkatesh Rao Retweeted Venkatesh Rao

        heh looks like I'm empircally rediscovering thathttps://twitter.com/vgr/status/1178027742467481600 …

        Venkatesh Rao added,

        Venkatesh Rao @vgr
        manhattan grid possible world paths are really pretty if you add a bit of jitter to the path rendering interesting how if you make random choices (weighted by bounding box side lengths) most paths sorta go through the middle pic.twitter.com/nxefENFmrf
        0 replies 0 retweets 2 likes
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      2. Matt Elder‏ @fiddlemath 28 Sep 2019
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        Replying to @vgr

        Yep. All this follows neatly from the same logic by which Pascal's Triangle enumerates binomial coefficients. https://en.wikipedia.org/wiki/Binomial_coefficient …

        2 replies 0 retweets 3 likes
      3. Venkatesh Rao‏ @vgr 28 Sep 2019
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        Replying to @fiddlemath

        how would I use pascal's triangle for enumeration here?

        1 reply 0 retweets 0 likes
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      2. Jeff‏ @jeffdeecee 28 Sep 2019
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        Replying to @vgr

        https://betterexplained.com/articles/navigate-a-grid-using-combinations-and-permutations …

        1 reply 0 retweets 1 like
      3. Jeff‏ @jeffdeecee 28 Sep 2019
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        Replying to @jeffdeecee @vgr

        so the key idea is this: When considering the possible paths (tracing them out with your finger), you might whisper "Up, right, up, right...". [...] Using the text interpretation, the question becomes "How many ways can we re-arrange the letters rrrrrruuuu?"

        0 replies 0 retweets 2 likes
      4. End of conversation

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