Computational Geometry 2D Convex Hulls continued

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Computational Geometry2D Convex Hulls
(continued)
Joseph S. B. Mitchell Stony Brook University
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Output-Sensitive O(n log h)

• The ultimate convex hull (in 2D)
• Marriage before Conquest O(n log h)
• Lower bound ?(n log h)
• Kirkpatrick Seidel86
• Simpler Chan95
• 2 Ideas
• Break point set S into groups of appropriate size
  m (ideally, m h)
• Search for the right value of m ( h, which we
  do not know in advance) by repeated squaring

h output size
So, O(n log h) is BEST POSSIBLE, as function of n
and h !!
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Chans Algorithm

• Break S into n/m groups, each of size m
• Find CH of each group (using, e.g., Graham scan)
• O(m log m) per group, so total O((n/m) m log m)
  O(n log m)
• Gift-wrap the n/m hulls to get overall CH
• At each gift-wrap step, when pivoting around
  vertex v
• find the tangency point (binary search, O(log
  m)) to each group CH
• pick the smallest angle among the n/m tangencies
  
• O( h (n/m) log m)
• Hope mh
• Try m 4, 16, 256, 65536,

O(n log h)
O( h (n/h) log h) O(n log h)
v
O(n ( log (221 ) log (222 ) log (223 )
log (22 log (log (h)) ) O(n (21 22 23
2log (log (h)) )) O(n (2 log h)) O(n log
h)
v
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CH in Higher Dimensions

• 3D Divide and conquer
• T(n) ? 2T(n/2) O(n)
• O(n log n)
• Output-sensitive O(n log h) Chan
• Higher dimensions (d ? 4)
• O(n ?d/2 ? ), which is worst-case OPT, since
  point sets exist with h?(n ?d/2 ? )
• Output-sensitive O((nh) logd-2 h), for d4,5

applet
merge
h O(n)
Qhull website
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Algorithmic Paradigms

• Divide and Conquer
• Sweep (sorting helps!)
• Binary search parametric search
• Incremental construction
• Randomized incremental (randomization helps!)
• Greedy can be good
• Dynamic programming
• Approximation
• ?-nets, discrepency theory, coresets, t-spanners,
  dimension-reduction, m-guillotine subdivisions
• Combinatorial analysis, topological analysis

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More Demos

• Various 2D and 3D algorithms in an applet