Today is raining. Usual.

Yesterday Sariel Har-Paled has started is series of talks about data structures for

geometric approximation. He has presented various topics like grids, quadtrees

and a technique for representing (approximately) the distance between points in

the space (well separated pairs decomposition).

Talking about grids, it is interesting how this data structure can be used to find the

closest pair in a set of points on the plane. The data structure is able to

determine in linear time, given , whether its closest pair has distance

equal, smaller or larger than a real number .

A grid of cells of size is built on top of . The data structure

maintains, for each cell, a linked list of points falling in that cell. This data structure

allows for constant time insertions (using hashing) and each time a new point is

added, it can check in constant time if the smaller distance between points changes.

Since insertions are needed, the result follows.

Now, using this data structure in a randomized algorithm, the closest pair of a set

can be found in expected linear time bounding the expected number of times the

closest pair changes (note that the model of computation relies on hashing in

constant time, true randomization and a constant time computation for the floor

function).

Today there have also been the first lectures of John Iacono concerning the point

location problem, that is, given a space and a partition of it in polygons, find the

polygon containing a point given as the input.

A technique for facing this problem is the Kirkpatrick’s method. Let’s look at the

method in the case of where we have a triangulation of the space. The

technique relies on removing an independent set of vertices. This operation could

break the triangulation, so a new one has to be computed (and this can be done

in constant time in this particular case). Then it is just sufficient to proceed in the

same way on the resulting instance, until we end up with a single, big, triangle.

Then we look at the point we want to localize and start the process of rebuilding

the triangulation keeping track of the triangle the points lies in at each step (this

can be done easily since the data structure maintains pointers from triangles at

various levels). It turns out that the number of steps needed in order to carry out

the search of the point is , where is the number of vertices, while

building the structure (as pointed out before there are pointers in the game that have

to be set up) takes . Moreover the algorithm is completely deterministic.

Tomorrow is the last day of the summer school, heading back to København.

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