Generic Set Algorithms

1
Generic Set Algorithms

• Andy Wang
• Data Structures, Algorithms, and Generic
  Programming

2
Generic Set Algorithms

• Very useful software tools
• A part of the STL specification
• Implement set theory
• Union
• Intersection
• Difference
• Containment
• Merge
• Runtime complexity in O(size)

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Set AlgorithmsAssumptions and Outcomes

• Assumptions
• Input ranges determined by input iterators
• Output start determined by output iterator
• Input ranges are sorted
• Outcomes
• Output range is sorted
• Output range is the set operation applied to the
  input ranges

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Set Algorithm Complexity

• Unsorted input ranges
• O(size2) to iterate through range for each
  element
• g, c, m, y, x, h, a union h, w, a, b
• Sorted input rages
• O(size) to iterate through each range once
• a, c, g, h, m, x, y union a, b, h, w

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Sorted Range Control Structure

• Compare current elements from each input range
• Perform action based on comparison
• Increment past elements used for action
• Continue until an input range is exhausted
• Deal with tail of remaining range

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Set Union

• template ltclass I1, class I2, class I3gt
• void g_set_union(I1 B1, I1 E1, I2 B2, I2 E2, I3
  D)
• for ( B1 ! E1 B2 ! E2 D)
• if (B1 lt B2)
• D B1
• B1
• else if (B2 lt B1)
• D B2
• B2
• else // disallow duplicates
• D B1
• B1, B2
• while (B1 ! E1) D B1
• while (B2 ! E2) D B2

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Set Union Illustrated

• Set 1 a, c, g, m
• Set 2 a, b, d
• Union Set

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Set Union Illustrated

• Set 1 a, c, g, m
• Set 2 a, b, d
• Union Set a

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Set Union Illustrated

• Set 1 a, c, g, m
• Set 2 a, b, d
• Union Set a

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Set Union Illustrated

• Set 1 a, c, g, m
• Set 2 a, b, d
• Union Set a, b

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Set Union Illustrated

• Set 1 a, c, g, m
• Set 2 a, b, d
• Union Set a, b

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Set Union Illustrated

• Set 1 a, c, g, m
• Set 2 a, b, d
• Union Set a, b, c

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Set Union Illustrated

• Set 1 a, c, g, m
• Set 2 a, b, d
• Union Set a, b, c

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Set Union Illustrated

• Set 1 a, c, g, m
• Set 2 a, b, d
• Union Set a, b, c, d

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Set Union Illustrated

• Set 1 a, c, g, m
• Set 2 a, b, d
• Union Set a, b, c, d

B2, E2
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Set Union Illustrated

• Set 1 a, c, g, m
• Set 2 a, b, d
• Union Set a, b, c, d, g

B2, E2
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Set Union Illustrated

• Set 1 a, c, g, m
• Set 2 a, b, d
• Union Set a, b, c, d, g

B2, E2
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Set Union Illustrated

• Set 1 a, c, g, m
• Set 2 a, b, d
• Union Set a, b, c, d, g, m

B2, E2
19
Set Union Illustrated

• Set 1 a, c, g, m
• Set 2 a, b, d
• Union Set a, b, c, d, g, m

B1, E1
B2, E2
20
Set Merge

• template ltclass I1, class I2, class I3gt
• void g_set_merge(I1 B1, I1 E1, I2 B2, I2 E2, I3
  D)
• while (B1 ! E1 B2 ! E2)
• if (B2 lt B1)
• D B2
• else // allow duplicates
• D B1
• while (B1 ! E1) D B1
• while (B2 ! E2) D B2

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Set Merge Illustrated

• Set 1 a, c, g, m
• Set 2 a, b, d
• Merged Set

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Set Merge Illustrated

• Set 1 a, c, g, m
• Set 2 a, b, d
• Merged Set a

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Set Merge Illustrated

• Set 1 a, c, g, m
• Set 2 a, b, d
• Merged Set a

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Set Merge Illustrated

• Set 1 a, c, g, m
• Set 2 a, b, d
• Merged Set a, a

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Set Merge Illustrated

• Set 1 a, c, g, m
• Set 2 a, b, d
• Merged Set a, a

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Set Merge Illustrated

• Set 1 a, c, g, m
• Set 2 a, b, d
• Merged Set a, a, b

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Set Merge Illustrated

• Set 1 a, c, g, m
• Set 2 a, b, d
• Merged Set a, a, b

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Set Merge Illustrated

• Set 1 a, c, g, m
• Set 2 a, b, d
• Merged Set a, a, b, c

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Set Merge Illustrated

• Set 1 a, c, g, m
• Set 2 a, b, d
• Merged Set a, a, b, c

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Set Merge Illustrated

• Set 1 a, c, g, m
• Set 2 a, b, d
• Merged Set a, a, b, c, d

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Set Merge Illustrated

• Set 1 a, c, g, m
• Set 2 a, b, d
• Merged Set a, a, b, c, d

B2, E2
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Set Merge Illustrated

• Set 1 a, c, g, m
• Set 2 a, b, d
• Merged Set a, a, b, c, d, g

B2, E2
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Set Merge Illustrated

• Set 1 a, c, g, m
• Set 2 a, b, d
• Merged Set a, a, b, c, d, g

B2, E2
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Set Merge Illustrated

• Set 1 a, c, g, m
• Set 2 a, b, d
• Merged Set a, a, b, c, d, g, m

B2, E2
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Set Merge Illustrated

• Set 1 a, c, g, m
• Set 2 a, b, d
• Merged Set a, a, b, c, d, g, m

B1, E1
B2, E2
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Set Intersection

• template ltclass I1, class I2, class I3gt
• void g_set_intersection(I1 B1, I1 E1, I2 B2, I2
  E2, I3 D)
• while (B1 ! E1 B2 ! E2)
• if (B2 lt B1) // B2 not in set 1
• B2
• else if (B1 lt B2) // B1 not in set 2
• B1
• else
• D B1
• B2

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Set Intersection Illustrated

• Set 1 a, c, g, m
• Set 2 a, b, d
• Intersection Set

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Set Intersection Illustrated

• Set 1 a, c, g, m
• Set 2 a, b, d
• Intersection Set a

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Set Intersection Illustrated

• Set 1 a, c, g, m
• Set 2 a, b, d
• Intersection Set a

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Set Intersection Illustrated

• Set 1 a, c, g, m
• Set 2 a, b, d
• Intersection Set a

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Set Intersection Illustrated

• Set 1 a, c, g, m
• Set 2 a, b, d
• Intersection Set a

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Set Intersection Illustrated

• Set 1 a, c, g, m
• Set 2 a, b, d
• Intersection Set a

B2, E2
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Set Intersection Illustrated

• Set 1 a, c, g, m
• Set 2 a, b, d
• Intersection Set a

B2, E2
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Set Intersection Illustrated

• Set 1 a, c, g, m
• Set 2 a, b, d
• Intersection Set a

B1, E1
B2, E2
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Set Difference

• template ltclass I1, class I2, class I3gt
• void g_set_difference(I1 B1, I1 E1, I2 B2, I2 E2,
  I3 D)
• while (B1 ! E1 B2 ! E2)
• if (B2 lt B1) // B2 not in set 1
• B2
• else if (B1 lt B2) // B1 not in set 2
• D B1
• else
• B1
• B2
• while (B1 ! E1) D B1

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Set Difference Illustrated

• Set 1 a, c, g, m
• Set 2 a, b, d
• Difference Set

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Set Difference Illustrated

• Set 1 a, c, g, m
• Set 2 a, b, d
• Difference Set

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Set Difference Illustrated

• Set 1 a, c, g, m
• Set 2 a, b, d
• Difference Set

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Set Difference Illustrated

• Set 1 a, c, g, m
• Set 2 a, b, d
• Difference Set c

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Set Difference Illustrated

• Set 1 a, c, g, m
• Set 2 a, b, d
• Difference Set c

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Set Difference Illustrated

• Set 1 a, c, g, m
• Set 2 a, b, d
• Difference Set c

B2, E2
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Set Difference Illustrated

• Set 1 a, c, g, m
• Set 2 a, b, d
• Difference Set c, g

B2, E2
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Set Difference Illustrated

• Set 1 a, c, g, m
• Set 2 a, b, d
• Difference Set c, g

B2, E2
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Set Difference Illustrated

• Set 1 a, c, g, m
• Set 2 a, b, d
• Difference Set c, g, m

B2, E2
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Set Difference Illustrated

• Set 1 a, c, g, m
• Set 2 a, b, d
• Difference Set c, g, m

B1, E1
B2, E2
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Set Containment

• template ltclass I1, class I2, class I3gt
• void g_subset_of(I1 B1, I1 E1, I2 B2, I2 E2)
• while (B1 ! E1 B2 ! E2)
• if (B1 lt B2) // B1 not in set 2
• return 0
• else if (B2 lt B1) // B2 not in set 1
• B2
• else
• B1
• B2
• if (B1 E1) return 1
• return 0

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Set Containment Illustrated

• Set 1 a, c, g, m
• Set 2 a, b, d

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Set Containment Illustrated

• Set 1 a, c, g, m
• Set 2 a, b, d

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Set Containment Illustrated

• Set 1 a, c, g, m
• Set 2 a, b, d

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Set Containment Illustrated

• Set 1 a, c, g, m
• Set 2 a, b, d
• Return 0