1. Maintaining Dynamic Tree with Min Subject to Constraints
Data Structures 2003
CS TAU

2. Multi Criteria Representation
Consider a set of elements
Each element is equipped with two keys (values):
K1, K2
For simplicity: Assume each key appears at most once.
Want to support: Insert(k1, k2), delete(k1,k2), find_min(k1), find_min(k2)
Implementation: Double data structure, with pointers between the structures.
2/24/2019
Min s.t. Constraints, Hanoch Levy CS, TAU

3. Min s.t. Constraints, Hanoch Levy CS, TAU
Example
For each person: height + Grade
Two 2-3 trees, HEIGHT, GRADE
Insert in both
Findmin on either
Pointers between the two for member, deletions
GRADE 8 50 81 97
HEIGHT
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2/24/2019
Min s.t. Constraints, Hanoch Levy CS, TAU

4. Maintaining Minimum (Max) subject to constraint
Consider a set of elements
Each element is equipped with two keys (values):
K1, K2
Want to support:
Insert
Delete
Find MIN(K1) subject to K2 > k
(or MAX)
2/24/2019
Min s.t. Constraints, Hanoch Levy CS, TAU

5. Min s.t. Constraints, Hanoch Levy CS, TAU
Example
Store student records
Each student has GRADE, HEIGHT
Assume all GRADEs are unique.
Want:
Insert (g,h): Insert the pair
Delete (g,h): Delete the pair
x= FINDMAX (height): FindMAX(grade) s.t. H<=h (find the maximal grade of the short students).
2/24/2019
Min s.t. Constraints, Hanoch Levy CS, TAU

6. Dual 2-3 Tree with leaf linked lists and mutual pointers
Problematic
Easy to find the set of students whose height >= h
Finding the max grade on this is O(N)
GRADE 8 50 81 97
HEIGHT
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160
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173
2/24/2019
Min s.t. Constraints, Hanoch Levy CS, TAU

7. Efficient Data Structure
Want: minimum grade subject to height >=h
E.g: height >= 165
Store constraint on a search tree (2-3 tree)
The red recursion allows us to identify all the sub-trees that obey the constraint
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185
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HEIGHT
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165
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189
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Min s.t. Constraints, Hanoch Levy CS, TAU

8. Representing the Function minimized (grade)
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Each leaf marked with its GRADE (not sorted!)
Internal nodes keep the minimum grade of the sub-tree.
Algorithm:
Bring up black values which are roots of red trees.
When two values brought up – take min and bring up 60 80 75 50 90 93 60 80 75 95 50
2/24/2019
Min s.t. Constraints, Hanoch Levy CS, TAU

9. Alternative Description of algorithm (recursive)90 93 60 80 75 95 50
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Children of node can be of 3 “types”:
Black (no node in sub-tree is relevant)
Red (all nodes in sub-tree are relevant)
“gray” (some nodes in sub-tree are relevant)
Algorithm
Red: bring its value
Black: disregard
Gray: bring value computed recursively
2/24/2019
Min s.t. Constraints, Hanoch Levy CS, TAU

10. Min s.t. Constraints, Hanoch Levy CS, TAU
Complexity 90 93 60 80 75 95 50
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Each node has at most one gray
Red – O(1)
Black – O(1)
Gray – continue deeper  path of gray is of length O(h) = O(log N).
 Over all complexity for operation O(log N)
2/24/2019
Min s.t. Constraints, Hanoch Levy CS, TAU