UMass Lowell Computer Science 91.503 Analysis of Algorithms Prof. Karen Daniels Spring, 2005

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UMass Lowell Computer Science 91.503 Analysis
of Algorithms Prof. Karen Daniels Spring, 2005

• Lecture 1 (Part 2)
• How to Make an Algorithm Sandwich
• adapted from the fall, 2000 talk
• Finding the Largest Area Axis-Parallel Rectangle
  in a Polygon in O(n log2 n) Time
• given at UMass Lowell MATHEMATICAL SCIENCES
  COLLOQUIUM

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Context Taxonomy of ProblemsSupporting Apparel
Manufacturing
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Whats the Problem?

• Given a 2D polygon that
• does not intersect itself
• may have holes
• has n vertices
• Find the Largest-Area Axis-Parallel Rectangle
• How hard is it?
• How fast can we find it?

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Approach

• Explore the problem to gain intuition
• Describe it What are the assumptions? (model of
  computation, etc...)
• Has it already been solved?
• Have similar problems been solved? (more on this
  later)
• What does best-case input look like?
• What does worst-case input look like?
• Establish worst-case upper bound on the problem
  using an algorithm
• Design a (simple) algorithm and find an upper
  bound on its worst-case asymptotic running time
  this tells us problem can be solved in a certain
  amount of time. Algorithms taking more than this
  amount of time may exist, but wont help us.
• Establish worst-case lower bound on the problem
• Tighten each bound to form a worst-case
  sandwich

increasing worst-case asymptotic running time as
a function of n
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Upper Bound

• First Attempt

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Designing an Algorithmto Provide Upper Bound

• First attempt uses a straightforward approach
• brute-force
• naïve
• Will most likely produce a loose upper bound
• Characterize rectangle based on how it can grow

Contacts reduce degrees of freedom
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O(n5) LR Algorithm

• Find_LR(Polygon P)
• area0 Find_LR_0_RC(P)
• area1 Find_LR_1_RC(P)
• area2 Find_LR_2_RC(P)
• area3 Find_LR_3_RC(P)
• area4 Find_LR_4_RC(P)
• return maximum(area0, area1, area2, area3,
  area4)
• Find_LR_0_RC(P)
• for i 1 to n (for each edge of P)
• for j 1 to n
• for k 1 to n
• for l 1 to n
• area area of LR for 0-RC
  determining set for (i,j,k,l)
• if LR is empty, then update maximum area
• return maximum area

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First Upper Bound What can we really conclude?

• Algorithms worst-case running time is in O(n5).
• Problem can be solved in O(n5) time, even for
  worst-case inputs.
• Note there might exist algorithms for this
  problem that take more than n5 time, but they
  arent useful to us!
• If a worst-case input exists that causes
  algorithm to actually use time proportional to
  n5, then algorithms worst-case running time is
  also in W(n5).
• In this case, we can say algorithms worst-case
  running time is in Q(n5).

An inefficient algorithm for the problem might
exist that takes this much time, but would not
help us.
n
2n
1
n5
n2
increasing worst-case asymptotic running time as
a function of n
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Lower Bound

• First Attempt

An inefficient algorithm for the problem might
exist that takes this much time, but would not
help us.
n
1
2n
n5
n2
increasing worst-case asymptotic running time as
a function of n
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First Attempt

• First attempt will most likely produce a loose
  lower bound that can be improved later.
• W(1) and W(n) are not hard to achieve
• Rationale for W(n) is that we must examine every
  vertex at least once in order to solve the
  problem. This holds for every algorithm that
  solves the problem, so it gives us a lower bound
  on the problem in an even stronger sense than our
  upper bound.
• Remember that our upper bound does not guarantee
  that no algorithm exists for this problem that
  takes more than O(n5) time, but if one exists it
  will not help us!

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Know the Difference!
Strong Bound This worst-case lower bound on the
problem holds for every algorithm that solves the
problem and abides by our problems assumptions.
Weak Bound This worst-case upper bound on the
problem comes from just considering one
algorithm. Other, less efficient algorithms that
solve this problem might exist, but we dont care
about them!
Both the upper and lower bounds are probably
loose (i.e. probably can be tightened later on).
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Upper Bound

• Tightening It

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Approach

• Think harder
• Design an algorithm that takes only O(n2) time
  for worst-case inputs

tighter upper bound
2n
1
n5
Now we no longer care about this one!
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Approach (continued)

• Think even harder
• Run into brick wall!
• Characterize the brick wall this type of case
• Go around brick wall by looking at similar
  problems.
• This requires ability to compare functions!

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Some Related Problems
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Some Related Problems (continued)
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Approach (continued)

• Reduce the O(n2) bound to O(n log2 n)
• Adapt a technique that worked for a similar
  problem in order to develop a general framework
  for the problematic case

n log2 n
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Lower Bound

• Tightening It

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Approach

• Go around lower bound brick wall by
• examining strong lower bounds for some similar
  problems
• transforming a similar problem to our problem
• this process is similar to what we do when we
  prove problems NP-complete

worst-case bounds on problem
n5
2n
1
n2
n log2 n
n
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Lower Bounds for Related Problems
LargestInnerRectangle W (n)
polygon
LargestInnerCircle Q (n log n)
point set
Largest circle containing no points of the set
whose center is in convex hull of the points
worst-case bounds on our problem
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Lower Bound of W(n log n) by Transforming a
(seemingly unrelated) Problem
MAX-GAP instance given n real numbers x1, x2,
... xn find the maximum difference between 2
consecutive numbers in the sorted list.
O(n) time transformation
Rectangle area is a solution to the MAX-GAP
instance
specialized polygon
Rectangle algorithm must take as least as much
time as MAX-GAP. MAX-GAP is known to be in W(n
log n).
Rectangle algorithm must take W(n log n) time for
specialized polygons. Transforming yet another
different problem yields bound for unspecialized
polygons.
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Summary

• First attempt
• Establish weak (and loose) worst-case upper bound
  on the problem using an algorithm
• Establish strong (and loose) worst-case lower
  bound on the problem
• Tighten bounds to make an algorithm sandwich
• Establish weak (but tighter) worst-case O(n log2
  n) upper bound on the problem using an algorithm
• Establish strong (and tighter) worst-case lower
  bound on the problem by transforming a problem
  with known lower bound

bounds on problem
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Recent Improvement!

• Finding the Largest Axis-Aligned Rectangle in a
  Polygon in O(n log n) Time
• Ralph Boland, Jorge Urrutia
• Canadian Conference on Computational Geometry,
  August 13-15, 2001

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For More Information

• Computational Geometry
• Graduate CS course in Computational Geometry
  offered at UMass Lowell in Spring 01, Spring 04
• Introductory texts
• Computational Geometry in C (ORourke)
• Computational Geometry An Introduction
  (Preparata Shamos)
• Bibliography ftp//ftp.cs.usask.ca/pub/geometry/
• Softwarehttp//www.geom.umn.edu/software/cglist/
• My research
• http//www.cs.uml.edu/kdaniels
• Journal paper Finding the largest area
  axis-parallel rectangle in a polygon
• (Computational Geometry Theory and Applications)
• Prof. Victor Milenkovic Frequent co-author and
  former PhD advisor
• http//www.cs.miami.edu/vjm