1. Computer Science Dictionaries: 2-3-4 Red-Black CS 330: Algorithms Dictionaries: 2-3-4 and Red-Black Trees Gene Itkis

2. Computer Science CS-330: Algorithms, Fall 2004Gene Itkis2 Dictionary: Implementations Simple/naïve Lists Lists Better implementations Hash Tables Hash Tables Probabilistic methods, Expected values Ordered trees  “Good enough” approximations; Augmenting (  order stats )  Other Skip-lists

3. Computer Science CS-330: Algorithms, Fall 2004Gene Itkis3 Hashing: Collision Resolution Methods Comparison  Chaining  Requires extra space  As with linked list  Stretchable  Can degenerate to linked-list search  Open Addressing  No extra space used   Has size limit  Also can degenerate to unordered list search  Perfect Hashing:  Must know the set ahead of time (no/little dynamics)  O(n) space overhead

4. Computer Science CS-330: Algorithms, Fall 2004Gene Itkis4 Dictionary Interface  isEmpty()  size()  find(key)  insert()  insert(elem)  remove()  remove(key) Dynamic

5. Computer Science CS-330: Algorithms, Fall 2004Gene Itkis5 Ordered Trees  Order  x< y, z : min at root – heap (  )  y < x < z  y < x < z : search   Depth  Shallow  Balanced  There might be exceptions:  e.g., Leftist heaps  “Strong” balance  Approximate   E.g., depth of leaves within factor of 2 (R-B trees) heap x z y

6. Computer Science CS-330: Algorithms, Fall 2004Gene Itkis6 Ordered Trees  Searching  Easy  Insert/Delete  Naïve: destroys balance  Fix balance  How?  AVL trees Nodes keep children heights Rotate when needed: when children heights are >1 apart  Red-Black2-3-4  Red-Black / 2-3-4 trees “Almost balanced”

7. Computer Science CS-330: Algorithms, Fall 2004Gene Itkis7 2-3-4 trees  Nodes:  Redblack  Red-black implementation: x y z >y<z>y<z >z>z >x<y>x<y <x<x >y>y >x<y>x<y <x<x x y >x>x<x<x x >x>x<x<x x >y>y >x<y>x<y <x<x x y y x or >y<z>y<z >z>z >x<y>x<y <x<x y z x

8. Computer Science CS-330: Algorithms, Fall 2004Gene Itkis8 Red-Black trees  Properties:  Every node is either red or black  Root: black  Leaf (nil): black  Children of red node are black  Any root-leaf path has same # of black nodes  Black depth of node v  = “# of black nodes on the path from root to v  same for all leaves  Black2-3-4  Black node = “root” of a 2-3-4 node  Black2-3-4  Black depth = depth in 2-3-4 tree

9. Computer Science CS-330: Algorithms, Fall 2004Gene Itkis9 2-3-4 2-3-4 tree example52 7 8 6 9 5 225225 7 2 5 7 7 2 5 878878 2 5 6 6 7 8 2 5 2 5 8 6 7 5757 2 8 6 9

10. Computer Science CS-330: Algorithms, Fall 2004Gene Itkis10 2-3-4 tree  Alwaysperfectly balanced  Always remains perfectly balanced  Your assignment: Red-Black  Sketch your own Red-Black tree insertion pseudo-code Red-Black  Read Red-Black tree insertion pseudo-code in the book  Compare  2-3-4 trees are a special case of B-trees  Used in databases and disk memory management