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POTest point for its position relative to the convex hull in log(n)
 

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    <p>I have a collection of 2D points <code>S</code> and I need to test if an input point <code>q</code> is inside or outside the convex hull of <code>S</code>.</p> <p>Since it's about a binary decision, I was thinking I could theoretically achieve <code>O(log(N))</code> by using a decision tree.</p> <p>However I have no idea how to organize the data and how the algorithm would look like to really get an answer in <code>O(log(N))</code>.</p> <p>While researching with this idea in mind, I've found this:</p> <blockquote> <p>How can we find these two cases more quickly? Binary search. Just search for x in the first coordinates of points in the two chains. If it is in the chain, you have found a crossing through a vertex (and you don't have to be as careful to tell what kind of crossing, either). If x is not the coordinate of a vertex in the chain, the two nearest values to it tell you which segment the ray from (x,y) might cross. So we can test whether a point is in a convex polygon in time <strong>O(log n)</strong>.</p> <p>It turns out that there are <strong>data structures</strong> that can test whether a point is in an arbitrary polygon (or which of several polygons it's in) in the same <strong>O(log n)</strong> time bound. But they're more complicated, so I don't have time to describe them here; I'll talk about them at some point in ICS 164.</p> </blockquote> <p>(<a href="http://www.ics.uci.edu/~eppstein/161/960307.html">http://www.ics.uci.edu/~eppstein/161/960307.html</a>)</p> <p>So do you have any ideas:</p> <ol> <li>How the data structure should look like to get it down in <code>O(log(N))</code>?</li> <li>How the algorithm should look like?</li> </ol>
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    1. COBuilding the tree would be at least O(n log(n)), no?
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      1. USMark Ping
    2. CO@MarkPing Sure, but that can be memoized. What I need is `log(N)` for the test itself.
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    3. CO[Possible duplicate?](http://stackoverflow.com/questions/16750618/whats-an-efficient-way-to-find-if-a-point-lies-in-the-convex-hull-of-a-point-cl/16807944#16807944). In any case, I provided an answer there that should suit your needs.
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