GATE-CS-2005 Exam: Chapter 56

This section of the GATE-CS-2005 exam focuses on data structures and algorithms. Specifically, Chapter 56 covers topics related to computational geometry, which involves the study of algorithms for solving geometric problems.

In this chapter, you will learn about several fundamental concepts in computational geometry, including:

• Convex hull - the smallest convex polygon that encloses a set of points
• Line segment intersection - determining if two line segments intersect and where
• Closest pair of points - finding the two points in a set that are closest together
• Triangulation - dividing a polygon into triangles
• Voronoi diagrams - a way of dividing a space based on a set of points

You will also learn about several algorithms for solving these problems, including:

• Gift wrapping algorithm - an algorithm for finding the convex hull of a set of points
• Bentley-Ottman algorithm - an algorithm for finding the intersections of line segments
• Divide-and-conquer algorithm - an algorithm for finding the closest pair of points
• Delaunay triangulation - an algorithm for triangulating a set of points
• Fortune's algorithm - an algorithm for constructing Voronoi diagrams

These concepts and algorithms are important for a wide range of applications, including computer graphics, geographic information systems, and robotics. In addition to learning about the theory behind these algorithms, you will also gain practical experience in implementing them in code.

Code Examples

Here are some code examples for implementing these algorithms:

Convex hull using gift wrapping algorithm:

def gift_wrapping(points):
  hull = []
  n = len(points)
  if n < 3:
    return None
  l = 0
  for i in range(1, n):
    if points[i][0] < points[l][0]:
      l = i
  p = l
  while True:
    hull.append(points[p])
    q = (p + 1) % n
    for i in range(n):
      if orientation(points[p], points[i], points[q]) == 2:
        q = i
    p = q
    if p == l:
      break
  return hull

Line segment intersection using Bentley-Ottman algorithm:

def line_intersection(segments):
  events = []
  for segment in segments:
    events.append((segment[0][0], segment[0][1], "left", segment))
    events.append((segment[1][0], segment[1][1], "right", segment))
  events.sort()
  tree = []
  intersections = []
  for event in events:
    x, y, typ, segment = event
    if typ == "left":
      node = insert(segment, tree)
      if get_left(node) is not None and intersect(segment, get_left(node)):
        intersections.append((segment, get_left(node)))
      if get_right(node) is not None and intersect(segment, get_right(node)):
        intersections.append((segment, get_right(node)))
    else:
      if get_left(segment) is not None and get_right(segment) is not None and intersect(get_left(segment), get_right(segment)):
        intersections.append((get_left(segment), get_right(segment)))
      delete(segment, tree)
  return intersections

Closest pair of points using divide-and-conquer algorithm:

def closest_pair(points):
  n = len(points)
  if n <= 1:
    return float("inf")
  points.sort(key=lambda x: x[0])
  mid = n // 2
  dl = closest_pair(points[:mid])
  dr = closest_pair(points[mid:])
  d = min(dl, dr)
  strip = []
  for point in points:
    if abs(point[0] - points[mid][0]) < d:
      strip.append(point)
  strip.sort(key=lambda x: x[1])
  for i in range(len(strip)):
    for j in range(i+1, len(strip)):
      if abs(strip[j][1] - strip[i][1]) >= d:
        break
      d = min(d, dist(strip[i], strip[j]))
  return d

Delaunay triangulation using divide-and-conquer algorithm:

def delaunay_triangulation(points):
  n = len(points)
  if n <= 3:
    return [tuple(p) for p in points]
  points.sort()
  mid = n // 2
  left_triangulation = delaunay_triangulation(points[:mid])
  right_triangulation = delaunay_triangulation(points[mid:])
  return merge_triangulations(left_triangulation, right_triangulation)

Voronoi diagram using Fortune's algorithm:

def voronoi_diagram(points):
  fortunes_algorithm(points)
  edges = []
  for arc in arcs:
    if arc[2] is not None:
      edge = (arc[2], arc[0])
      edges.append(edge)
  return edges

以上是本章的简介和代码片段。要深入理解这些概念和算法，您需要仔细研究它们以及它们的实现方式。然后，您可以通过自己编写代码来应用这些知识，从而更好地掌握它们。