Need conditions on C and \cD to make these possible
Each input z=(x_1,\ldots,x_n), where
x_i are all real values, or
x_i are all points in the plane
Particular proof structure is a particular way
to model work that a correct algorithm must do
A comparison yields at most one bit of information
Under these conditions: comparison-based algorithms, proof-carrying output, product distributions...
Sorting | Delaunay Triangulation |
---|---|
Intervals (x_i, x_{i'}) containing no values of z | Delaunay disks |
Typical set V | Range space \epsilon-net V [MSW90, CV07], Ranges are disks, \epsilon = 1/n |
\log n training sample points per bucket | \log n training sample points per disk |
Expect O(1) values of z in each bucket | Expect O(1) points in each D. disk of V |
Entropy-optimal search trees T_i | Entropy-optimal planar point location T_i [AMMW07] |
Sorted within each bucket yields f_S(V \cup z) in O(n) |
Triangulated within each small region yields f_T(V \cup z) in O(n) |
Removal of V from sorted V \cup z (trivial) | Build f_T(z) from f_T(V \cup z) [ChDHMST02] |
In analysis: merge of f_S(V) and f_S(z) to get f_S(V\cup z) | In analysis: merge of f_T(V) and f_T(z) to get f_T(V\cup z) [Ch92] |
In analysis: recover buckets b_i from sorted f_S(V \cup z) (trivial) | In analysis: recover triangles b_i in f_T(z) from f_T(V \cup z) |
x_1,\ldots,x_{n/2} are fixed (singleton distributions)
x_1,\ldots,x_{n/2} are fixed (singleton distributions)
x_1,\ldots,x_{n/2} are fixed (singleton distributions)