1 / 7

CS 330: Algorithms 2-3-4 and Red- Black Trees

CS 330: Algorithms 2-3-4 and Red- Black Trees. Gene Itkis. 2-3-4 trees. x y z. x y. x. > z. > z. < x. > y < z. < x. > y < z. > x < y. < x. > x < y. > x. > y. < x. > x. > x < y. > y. > x < y. < x. < x. y. y. x. x. x. z. x. y. 2-3-4 nodes:

eleanor-parker
Télécharger la présentation

CS 330: Algorithms 2-3-4 and Red- Black Trees

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. CS 330: Algorithms2-3-4andRed-BlackTrees Gene Itkis

  2. 2-3-4 trees x y z x y x >z >z <x >y<z <x >y<z >x<y <x >x<y >x >y <x >x >x<y >y >x<y <x <x y y x x x z x y • 2-3-4 nodes: • Red-black implementation: or Gene Itkis

  3. 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 • Black node = “root” of a 2-3-4 node • Black depth = depth in 2-3-4 tree Gene Itkis

  4. 2-3-4 & Red-Black trees • 2-3-4 trees • Perfectly balanced • height ≤ lg n • Red-Black trees • 2-3-4 node = • =Red-Blacksubtree of height ≤ 2 • Red-Black tree height ≤ 2 lg n • Search (both 2-3-4 & Red-Black): O(lg n) Gene Itkis

  5. 2-3-4 tree example 2 5 2 5 7 5 5 2 6 7 8 2 7 8 5 7 8 9 6 2 6 9 8 5 2 7 5 2 7 5 5 7 2 6 8 Gene Itkis

  6. Zoom-in: one node A A A A A C B B E D B C D B B D E D C C 4 3 5 +6 • Simple inserts: • Case 0 • No fixing needed • …even inside tree • More complex: • Case 1: • Split (3,4,5) • 4 will try to join higher • 6 can now join as case 0 • …same inside tree 4 5 4 6 5 5 5 6 4 4 6 +6 4 4 3 5 6 3 5 4 6 5 3 6 Gene Itkis

  7. Zoom-in: one node 4 4 5 6 5 4 +6 5 4 5 4 6 5 5 • Case 0: • Case 3: • Rotate 4-5… • To get case 0 above • Case 2: • Rotate 6-5 • To get case 3 above 4 6 6 4 +6 4 4 5 5 6 5 6 4 6 6 +5 4 4 6 5 6 5 6 4 5 Gene Itkis

More Related