Priority Search Trees
• Keys are pairs (x,y).
• Basic (search, insert, delete) and rectangle
operations.
• Two varieties.
 Based on a balanced binary search tree such as a
red-black tree.
• Red-black Priority Search Tree (RBPST)
 Based on a radix search tree.
• Radix Priority Search Tree (RPST)
Radix Priority Search Tree
• All x values are different, integral, and in the
range [0, k – 1].
• Each node of the priority search tree has exactly
one element.
• Min tree on y.
• Radix tree on x.
Radix Priority Search Tree
• The y value of the element in node w is <= the y
value of all elements in the subtree rooted at w
(y values define a min tree).
• Root interval is [0,k).
• Interval for node w is [a,b).
 Left child interval is [a, floor((a+b)/2)).
 Right child interval is [floor((a+b)/2, b)).
Insert
•
•
•
•
Start with empty RPST.
k = 16.
Root interval is [0,16).
Insert (5,8).
[0,16)
5,8
[0,16)
• Insert (6,9).
5,8
• (5,8) remains in root, because 8 < 9.
• (6,9) inserted in left subtree, because[0,8)
6 is in the left child interval.
6,9
Insert
[0,16)
5,8
• Insert (7,1).
• (7,1) goes into the root, because 1 < 8. [0,8)
• (5,8) inserted in left subtree, because 5 6,9
[0,16)
is in the left child interval.
7,1
• (5,8) displaces (6,9), because 8 < 9.
• (6,9) inserted in right subtree, because
6 is in the right child interval.
[0,8)
5,8
[4,8) 6,9
Insert
[0,16)
[0,16)
7,1
7,1
[0,8)
[0,8)
5,8
5,8
[4,8) 6,9
[4,8) 6,9
• Insert (11,5).
[8,16)
11,5
Properties
[0,16)
7,1
[0,8)
5,8
[0,4) 2,12
[8,16)
11,5
[4,8) 6,9
• Height is O(log k).
• Insert time is O(log k).
Search
[0,16)
7,1
[0,8)
5,8
[0,4) 2,12
[4,8) 6,9
• Search time is O(log k).
[8,16)
11,5
Delete
[0,16)
7,1
[0,8)
5,8
[0,4) 2,12
[8,16)
11,5
[4,8) 6,9
• Similar to delete min of min heap.
• Delete time is O(log k).
minXinRectangle(xL,xR,yT)
[0,16)
12,1
[0,8)
5,8
[8,16)
11,5
[12,16)
[0,4) 2,12
[4,8) 6,9
• minXinRectangle(6,12,7).
• Time is O(log k).
14,6
Complexity
yok
yok
yok
ynotOK
yok
ynotOK
yok
ynotOK
yok
ynotOK
Profile of visited nodes.
yok
ynotOK ynotOK
Complexity
yok
yok
yok
ynotOK
yok
ynotOK
yok
ynotOK
yok
ynotOK
yok
ynotOK ynotOK
Yellow node is node closest to root with 2
examined children.
Complexity
yok
yok
yok
ynotOK
yok
ynotOK
yok
ynotOK
yok
ynotOK
yok
ynotOK ynotOK
No yellow node  O(log k) examined nodes
Yellow node  unique
Complexity
yok
yok
yok
ynotOK
yok
ynotOK
yok
ynotOK
yok
ynotOK
yok
ynotOK ynotOK
No node in the yellow node’s subtree can have two
children marked yok.
Complexity
yok
yok
yok
ynotOK
yok
ynotOK
yok
ynotOK
yok
ynotOK
# nodes visited < 4log k.
yok
ynotOK ynotOK
maxXinRectangle(xL,xR,yT)
[0,16)
12,1
[0,8)
5,8
[8,16)
11,5
[12,16)
[0,4) 2,12
[4,8) 6,9
• maxXinRectangle(6,12,7).
• Time is O(log k).
14,6
minYinXrange(xL,xR)
[0,16)
12,1
[0,8)
5,8
[8,16)
11,5
[12,16)
[0,4) 2,12
[4,8) 6,9
• minYinXrange(6,10).
• Time is O(log k).
14,6
enumerateRectangle(xL,xR,yT)
[0,16)
12,1
[0,8)
5,8
[8,16)
11,5
[12,16)
[0,4) 2,12
[4,8) 6,9
14,6
• enumerateRectangle(6,12,8).
• Time is O(log k + s), where s is #points in rectangle.