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 total. Prove that for any...
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...
(2) Define the set A C 2 by s) n-0 (a) Prove that for any N 2 0 the set is compact. (b) Prove that for any є > 0 there exists some N > 0 so that for any x E A we have (c) Prove that A is totally bounded. (d) Prove that A is compact (2) Define the set A C 2 by s) n-0 (a) Prove that for any N 2 0 the set is compact....
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.
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)...
Let S be a finite set with cardinality n>0. a. Prove, by constructing a bijection, that the number of subsets of S of size k is equal to the number of subsets of size n- k. Be sure to prove that vour mapping is both injective and surjective. b. Prove, by constructing a bijection, that the number of odd-cardinality subsets of S is equal to the number of even-cardinality subsets of S. Be sure to prove that your mapping is...
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...
Can you solve No.6 6. Let (a.)and b) be bounded sequences in R .a. Prove that lima. +İimb, siim (a, + b.) s ima, + İim br b. Prove that lim (-a)lima . Given an example to show that equality need not hold in (a) If o, and b, are positive for all n, prove that lim (a)s(im a)mb). provided the product on the right is 7. not of the form 0 oo. b. Need equality hold in (a)? 6....
Input: a directed grid graph G, a set of target points S, and an integer k Output: true if there is a path through G that visits all points in S using at most k left turns A grid graph is a graph where the vertices are at integer coordinates from 0,0 to n,n. (So 0,0, 0,1, 0,2, ...0,n, 1,0, etc.) Also, all edges are between vertices at distance 1. (So 00->01, 00->10, but not 00 to any other vertex....