On Varying Perspectives of Problem Decomposition

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On Varying Perspectivesof Problem Decomposition

• David Ginat
• Tel-Aviv University
• February 2002

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Majority in a Large Set

• N integers
• N may be very large
• The range of integers is large
• Is there an integer that appears more then N/2
  times?
• 7 8 3 3 7 3 3 22 3
• - Cannot keep a counter for each possible integer
  value
• - Sorting will be very inefficient

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Range Decomposition

• Quick-Sort array partitioning
• dynamic partitioning, relative to the
  pivot value
• Here, we may fix the partition a-priori, by
  possible values
• Range-partitioning,
• into equal-size sub-ranges
• 0..100, 101.. 200, 201.. 300,

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Range Decomposition

• If majority element then majority
  sub-range
• 17 18 3 3 17 3 3 22 3
• Coarse-grain counting Count of
  representatives of each sub-range in the input.
• 2. Is there a majority sub-range?
• 3. Fine-grain counting If there is, zoom
  into it.

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Range Decomposition

• Two-stage counting
• Array of counters for each stage
• Optimal array size?
• R the range of integers
• K sub-ranges
• R/K values in each sub-range
• Optimal value for minimizing both K and R/K?
  ?R

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Range Decomposition

• Coarse-Grain Fine-Grain
• - B-Trees
• Skip-Lists Pugh 1990
• Distributed mutual exclusion scheme Maekawa
  1985 - message complexity of O(?N)

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Element Decomposition

• Bucket-Sort element partitioning into atomic
  (digit) components
• Here, we may partition in the same way atomic
  components may be bits
• 011 101 110 101 101
• 3 5 6 5 5

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Element Decomposition

• If majority element then for each
  bit Bi of the majority element, its value will
  dominate the Bi-th bit values across the input
• 011 101 110 101 101
• 3 5 6 5 5

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Element Decomposition

• Zi the number of times 0 appears in bit i
• 3 5 6 5 5
• 011 101 110 101 101
• Z01, Z13, Z21
• 3 5 6 5 5 4
• 011 101 110 101 101 100
• Z02, Z14, Z21

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Element Decomposition

• Compute the Zi values across the input, in one
  input scan.
• Is there a majority candidate?
• If there is a candidate, then count its number of
  appearances in the input.

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Element Decomposition

• Partitioning and processing by atomic components
• Orthogonal processing across the input
• The N-th power of a number
• De-Bruijn sequences Euler circuits
• Hypercube paths
• Bucket-Sort, Radix-Sort

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Inductive Decomposition

• Insertion-Sort reduce problem instances
  inductively
• Here, we may examine what we can gain by
  eliminating input elements inductively
• 3 5 6 5 5 7 5
• 4 3 5 6 5 5 5 7

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Inductive Decomposition

• If two input elements X, Y are different and X is
  majority then X will remain majority after
  removing both
• K/N lt K-1/N-2 e.g., 6/10 lt 5/8
• Boyer Moore 1980
• Variables Candidate, Counter
• 3 3 5 5 5 7 5 7 5

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Inductive Decomposition

• Eliminate pairs of different elements, in one
  input scan.
• Is there a majority candidate?
• If there is a candidate, then count its number of
  appearances in the input.

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Inductive Decomposition

• Inductive processing
• Mutual cancellation
• Many algorithms are designed inductively e.g.,
  Manber 1988 - Greedy algorithms
• Elections with two candidates
• Extension Misra Gries 1982

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Variety of Perspectives

• Case study
• Diverse decompositions
• Underlying mathematical patterns
• Space-Efficiency improvements O(R) ?
  O(?R) ? O(log R) ? O(1)
• Relation to sorting schemes
• Binary representation, induction, various
  counting schemes, two-stage processing