For all of these problems, give pseudocode for your solution, and remember to include a proof of1.For all of these problems, give pseudocode for your solution, and remember to include a proof of
correctness and runtime. Note that in general, faster algorithms will receive more credit, so a brute
force O(n2) algorithm will not get many points if there is a faster O(n) or O(n log n) alterative.
1. In IR2, we define a slab to be a pair of parallel lines. Given a set of points P in IR2, find the
narrowest slab containing P , where the width of the slab is the vertical distance between its
bounding lines.
2. Let H be a set of at least 3 half-planes. We call a half-plane h redundant if it doesn’t
contribute an edge to the intersection of all half planes in H. Prove that for any redundant
half-plane h∈H, there are two other half-planes h′, h′′∈H such that h′∩h′′contains h. Use
this fact to give an algorithm (as fast as possible) to compute all redundant half-planes.
3. A simple polygon P is star-shaped if there is a point p∈P which can “see” every other point
in the polygon: in other words, for every other x∈P , the line segment xp is contained inP . Given an algorithm with expecting running time O(n) which decides if a polygon is star
shapedFor all of these problems, give pseudocode for your solution, and remember to include a proof of
correctness and runtime. Note that in general, faster algorithms will receive more credit, so a brute
force O(n2) algorithm will not get many points if there is a faster O(n) or O(n log n) alterative.
1. In IR2, we define a slab to be a pair of parallel lines. Given a set of points P in IR2, find the
narrowest slab containing P , where the width of the slab is the vertical distance between its
bounding lines.
2. Let H be a set of at least 3 half-planes. We call a half-plane h redundant if it doesn’t
contribute an edge to the intersection of all half planes in H. Prove that for any redundant
half-plane h ∈ H, there are two other half-planes h′, h′′∈ H such that h′∩h′′contains h. Use
this fact to give an algorithm (as fast as possible) to compute all redundant half-planes.
3. A simple polygon P is star-shaped if there is a point p ∈ P which can “see” every other point
in the polygon: in other words, for every other x ∈ P , the line segment xp is contained in
P . Given an algorithm with expecting running time O(n) which decides if a polygon is star
shaped.For all of these problems, give pseudocode for your solution, and remember to include a proof of
correctness and runtime. Note that in general, faster algorithms will receive more credit, so a brute
force O(n2) algorithm will not get many points if there is a faster O(n) or O(n log n) alterative.
1. In IR2, we define a slab to be a pair of parallel lines. Given a set of points P in IR2, find the
narrowest slab containing P , where the width of the slab is the vertical distance between its
bounding lines.
2. Let H be a set of at least 3 half-planes. We call a half-plane h redundant if it doesn’t
contribute an edge to the intersection of all half planes in H. Prove that for any redundant
half-plane h ∈ H, there are two other half-planes h′, h′′∈ H such that h′∩h′′contains h. Use
this fact to give an algorithm (as fast as possible) to compute all redundant half-planes.
3. A simple polygon P is star-shaped if there is a point p ∈ P which can “see” every other point
in the polygon: in other words, for every other x ∈ P , the line segment xp is contained in
P . Given an algorithm with expecting running time O(n) which decides if a polygon is star
shaped.correctness and runtime. Note that in general, faster algorithms will receive more credit, so a brute
force O(n2) algorithm will not get many points if there is a faster O(n) or O(n log n) alterative.
1. In IR2, we define a slab to be a pair of parallel lines. Given a set of points P in IR2, find the
narrowest slab containing P , where the width of the slab is the vertical distance between its
bounding lines.
2. Let H be a set of at least 3 half-planes. We call a half-plane h redundant if it doesn’t
contribute an edge to the intersection of all half planes in H. Prove that for any redundant
half-plane h ∈ H, there are two other half-planes h′, h′′∈ H such that h′∩h′′contains h. Use
this fact to give an algorithm (as fast as possible) to compute all redundant half-planes.
3. A simple polygon P is star-shaped if there is a point p ∈ P which can “see” every other point
in the polygon: in other words, for every other x ∈ P , the line segment xp is contained in
P . Given an algorithm with expecting running time O(n) which decides if a polygon is star
shaped.