Sep 4:
4 more exercises were handed out:
(a) Prove lower bound for EP (extreme point problem)
(first part is same as for ED).
(b) Give an efficient algorithm to decide if a given polygon is
starshaped.
(c) What is the best description (least number of polynomial inequalities)
of a regular polygon as the intersection of sets defined by polynomial
inequalities.
(d) Given a set of line segments, given an algorithm that
outputs a set of linesegments that "cover" all intersections (i.e,
every intersection has atleast one participating linesegment in that output set,
and every linesegment in the output set participates in atleast 1 intersection).
Aug 30:
4 more exercises were handed out:
(a) [As a warm up to Aug 28(c)]
Prove the equivalence of 3 of the common definitions of
a convex polyhedron in arbitrary dimensions, or atleast in 2 dimensions.
(b)
Prove a polygon in 2D is convex, if and only if
for every internal point p, the order of vertices of its
convex hull is exactly the sorted order of their polar angle about
p.
(c) Design and Analyze (correctness/complexity) of an efficient algorithm
that finds the supporting lines of 2 separated convex hulls
on the plane
(d) Design and Analyze (correctness/complexity) of an efficient algorithm
that inserts (if necessary) a given point in a given convex hull;
and deletes a given point from a given convex hull.
This goes with maintaining the hull and the entire current point set
in an appropriate datastructure.
Aug 28:
4 exercises were handed out last thursday.
Prove that:
(a) an 8X8 board that is missing 2 diagonally across squares
needs 32 2X1 dominoes to cover it.
(b) splitting a 3X3X3 cube into 27 identical cubes
needs 6 plane cuts
(c) given an input set S of points on the plane,
the subset E of extreme points (i.e. the minimal subset that form
the vertices of some convex polygon P that contains S)
is exactly the minimal subset of points
such that each point in S can be written as a convex combination
of points in E, i.e, each p in S = sum_{q\in E} lambda_q q, where
the lambda_q >=0 and
sum_{q\in E} lambda_q = 1.
(d) given an input set S of points on the plane,
the problem EP of finding
the subset E of extreme points
is atleast as hard as finding the convex hull of S, i.e.,
the order in which the points in E appear in the polygon P in (c)
above.