Priority Search Trees

1
Priority Search Trees

• Keys are distinct ordered pairs (xi, yi).
• Basic operations.
• get(x,y) return element whose key is (x,y).
• delete(x,y) delete and return element whose key
  is (x,y).
• insert(x,y,e) insert element e, whose key is
  (x,y).
• Rectangle operations.

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minXinRectangle(xL,xR,yT)

• Return element with min x-coordinate in the
  rectangle defined by the lines, x xL, x xR, y
  0, y yT, xL lt xR, 0 lt yT.

• I.e., return element with min x such that
• xL lt x lt xR and 0 lt y lt yT.

3
maxXinRectangle(xL,xR,yT)

• Return element with max x-coordinate in the
  rectangle defined by the lines, x xL, x xR, y
  0, y yT, xL lt xR, 0 lt yT.

• I.e., return element with max x such that
• xL lt x lt xR and 0 lt y lt yT.

4
minYinXrange(xL,xR)

• Return element with min y-coordinate in the
  rectangle defined by the lines, x xL, x xR, y
  0, y infinity, xL lt xR.

• I.e., return element with min y such that
• xL lt x lt xR.

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enumerateRectangle(xL,xR,yT)

• Return all elements in the rectangle defined by
  the lines, x xL, x xR, y 0, y yT, xL lt xR,
  0 lt yT.

• I.e., return all elements such that
• xL lt x lt xR and 0 lt y lt yT.

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Complexity

• O(log n) for each operation except for
  enumerateRectangle, where n is the number of
  elements in the tree.
• Complexity of enumerateRectangle is O(log n
  s), where s is the number of elements in the
  rectangle.

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Applications Visibility

• Dynamic set of semi-infinite vertical line
  segments.
• Vertical lines with end points (xi,infinity) and
  (xi,yi).

Opaque/translucent lines.
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Translucent Lines

• Eye is at (x,y).
• Priority search tree of line end points.
• enumerateRectangle(x, infinity, y).

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Opaque Lines

• Eye is at (x,y).
• Priority search tree of line end points.
• minXinRectangle(x, infinity, y).

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Bin Packing

• First fit.
• Best fit.
• Combination.
• Some items packed using first fit.
• Others packed using best fit.
• Memory allocation.

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Combined First And Best Fit

• Bins are numbered 0, 1, , n 1.
• Capacity of each is c.
• Initialize priority search tree with the pairs
  (c, j), 0 lt j lt n.

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First Fit

• minYinXrange(neededSize, infinity)

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Best Fit

• minXinRectangle(neededSize, infinity, infinity)

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Online Intersection Of Line Intervals

• Intervals are of the form i,j, i lt j.
• i,j may, for example represent the fact that a
  machine is busy from time i to time j.
• Answer queries of the form which intervals
  intersect/overlap with a given interval u,v,
  u lt v.
• List all machines that are busy at any time
  between u and v.

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Example

• Machine a is busy from time 1 to time 3.
• Interval is 1,3.
• Machines a, b, c, e, and f are busy at some time
  in the interval 2,4.

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Example

• Interval i,j corresponds to the pair (x,y),
  where x j and y i.

• enumerateRectangle(u, infinity, v).
• enumerateRectangle(2, infinity, 4).

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Compare With Segment Tree

• Min and max interval end points must be known in
  advance to define the root range of the segment
  tree.
• Search interval size is 1.
• Must break larger search intervals into unit
  intervals and remove duplicate responses.

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Interval Containment

• List all intervals i,j that contain the
  interval u,v.
• i,j contains u,v iff i lt u lt v lt j.

u,v 5,6
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Interval Containment

• Interval i,j corresponds to the pair (x,y),
  where x j and y i.

• enumerateRectangle(v, infinity, u).
• enumerateRectangle(6, infinity, 5).

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Intersecting Rectangle Pairs

• (A,B), (A,C), (D, G), (E,F)
• Online interval intersection.

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Algorithm

• Examine horizontal edges in sorted y order.
• Bottom edge gt insert interval into a priority
  search tree.
• Top edge gt report intersecting segments and
  delete the top edges corresponding bottom edge.

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Complexity

• Examine edges in sorted order.
• Bottom edge gt insert interval into a priority
  search tree.
• Top edge gt report intersecting segments and
  delete the top edges corresponding bottom edge.

• O(n log n) to sort edges by y, where n is of
  rectangles.
• Insert n intervals O(n log n).
• Report intersecting segments O(n log n s).
• Delete n intervals O(n log n).

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IP Router Table

• Longest-prefix matching
• 10, 101, 1001
• Destination address d 10100
• Longest matching-prefix is 101
• Prefix is an interval
• d is 5 bits gt 101 10100, 10111 20,23
• 2 prefixes may nest but may not have a proper
  intersection

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IP Router Table

• p(d) prefixes that match d.
• Use online interval intersection mapping.
• p(d) enumerateRectangle(d,infinity,d)

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IP Router Table

• lmp(d) maxStart(p(d)), minFinish(p(d))
• minXinRectangle(d,infinity,d) finds lmp(d) except
  when gt1 prefixes have same finish point.

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IP Router Table

• Remap finish points so that all prefixes have
  different finish point.
• f 2wf s 2w 1 , w length of d
• f is smaller when s is bigger

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IP Router Table

• Complexity is O(log n) for insert, delete, and
  find longest matching-prefix.