Let S be a set of disjoint simple polygons in the plane with n edges in total. Prove that for any start and goal position the number of segments on the shortest path is bounded by O(n). Give an example where it is Θ(n).
Let S be a set of disjoint simple polygons in the plane with n edges in...
Let S be a set of disjoint simple polygons in the plane with n edges in total. Prove that for any start and goal position the number of segments on the shortest path is bounded by O(n). Give an example where it is Θ(n).
The degree of a point in a triangulation is the number of edges incident to it. Give an example of a set of n points in the plane such that, no matter how the set is triangulated, there is always a point whose degree is n-1 9.2 The degree of a point in a triangulation is the number of edges incident to it. Give an example of a set of n points in the plane such that, no matter how...
Let s = {k=1CkXAz be a simple function, where {A1, A2, ... , An} are disjoint. Prove that for every p>0, |CK|PXAR
The intersection detection problem for a set S of n line segments is to determine whether there exists a pair of segments in S that intersect. Give a plane sweep algorithm that solves the intersection detection problem in O(nlogn) time. Prove it only requires O(nlogn) time.
2. If S:= {1/n - 1/min, me N}, find inf S and sup S. 4. Let S be a nonempty bounded set in R. (a) Let a > 0, and let aS := {as : S ES). Prove that inf(as) = a infs, sup(as) = a sup S. (b) Let b <0 and let b = {bs : S € S}. Prove that inf(bs) = b supS, sup(bs) = b inf S. 6. Let X be a nonempty set and...
Let Σ = {0, 1). (a) Give a recursive definition of Σ., the set of strings from the alphabet Σ. (b) Prove that for every n E N there are 2" strings of length n in '. (c) Give a recursive definition of I(s), the length of a string s E Σ For a bitstring s, let O(s) and I(s) be number of zeroes and ones, respectively, that occur in s. So for example if s = 01001, then 0(s)...
1. Let U be the universal set with disjoint subsets A and B, where n(U-46, n(A-15, and n(B-14. Find nAn B 2. A merchant surveyed 300 people to determine the way they leaned about an upcoming sale. The survey showed that 180 learned about the sale from the radio, 170 from television, 130 from the newspaper, 120 from radio and television, 70 from radio and newspapers, 80 from television and newspapers, and 60 from all three sources. How many people...
Let S = {n ∈ N | 1 ≤ n < 6} and R = {(m, n) ∈ S × S | m ≡ n mod 3} a. List all numbers of S. b. List all ordered pairs in R. c. Does R satisfy any of the following properties: (R), (AR), (S), (AS), and/or (T)? d. Draw the digraph D presenting the relation R where S are the vertices, and R determines the directed edges. e. Give each edge in...
Example: Let x, y ∈ Rn, where n ∈ N. The line segment joining x to y is the subset {(1 − t)x + ty : 0 ≤ t ≤ 1 } of R n . A subset A of Rn, where n ∈ N, is called convex if it contains the line segment joining any two of its points. It is easy to check that any convex set is path-connected. (a) Let f : X → Y be an...
Let L be a set of n lines in the plane. Give an O(nlogn) time algorithm to compute an axis-parallel rectangle that contains all the intersection points of those n lines in the plane.