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Red-Black Tree Presentation By Mobin Nesari.pdf
- 1. Red-Black Trees Intro to Presentedby: Mobin Nesari
- 2. Overview Introduction & Motivation 1. Propertiesof Red-Black Trees 2. Tree Rotations 3. Insertion 4. Deletion 5. Examples 6. Common Questions & Pitfalls 7. Summary 8. References 9.
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- 4. Why Balanced Trees? LOOKSLIKE A LINKED LIST HEIGHT = N OPERATIONS: O(N) HEIGHT ≈ LOG₂N OPERATIONS: O(LOG N) Balanced trees ensure consistently fast operations.
- 5. Red-Black Trees: SmartBSTs with Color BST + one extra bit: color (RED or BLACK) Enforces 5 color-based properties Ensures the tree is approximately balanced Height is at most 2·log(n+1) ⇒ O(log n)
- 6. Tree Type BalanceStrategy Worst-Case Height Pros & Cons AVL Tree Strict height check O(log n) ✅Faster search ❌more rotations Red-Black Tree Color rules (loose) ≤ 2·log(n+1) ✅Fewer rotations, good insert/delete B-Tree Multi-way node degrees O(log n) ✅Great for disk-based systems Why Use Red-Black Trees? Red-Black Trees strike a practical balance between strictness and performance.
- 7. Red-Black Trees: SmartBSTs with Color 🔺BSTs can become unbalanced ⇒ O(n) operations 🧠We want logarithmic height to stay fast 🌈Red-Black Trees enforce 5 properties using node color ⏱️Guaranteed: O(log n) for SEARCH, INSERT, and DELETE 🎯Widely used in real-world systems (e.g., Java TreeMap, C++ STL map)
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- 9. What Is aRed-Black Tree? A RED-BLACK TREE IS A BINARY SEARCH TREE WHERE EACH NODE HAS AN EXTRA ATTRIBUTE: COLOR ∈ {RED, BLACK}. Definition
- 10. The 5 Red-BlackProperties Every node is either red or black. 1. The root is black. 2. Every leaf (NIL) is black. 3. If a node is red, then both its children are black. 4. For each node, all paths to descendant leaves contain the same number of black nodes. 5.
- 11. Property 5 ensures black-depthconsistency across subtrees Together, these rules guarantee balance — but loosely How Do These Properties Help? Limit Height Restricting red children prevents long chains of red nodes Depth Consistency Balance is All You Need ! Red-black trees allow more flexibility than AVL, with fewer rotations.
- 12. Black-Height: A KeyConcept The black-height of a node x is the number of black nodes (not including x) on any path from x down to a leaf. Notation: bh(x) Black-height of tree = bh(root) Definition
- 13. Black-Height: A KeyConcept
- 14. Lemma 13.1 (HeightBound) Lemma 13.1: A red-black tree with n internal nodes has height ≤ 2·log(n + 1).
- 15. Lemma 13.1 (HeightBound) Lemma 13.1 (Proof): Show: Subtree rooted at x contains ≥ 2^bh(x) - 1 internal nodes (proved by induction). 1. Use Property 4: 2. At least half the nodes on any root-to-leaf path must be black ⇒ bh(root) ≥ h/2 Then: 3. n ≥ 2^(h/2) - 1 ⇒ log(n + 1) ≥ h/2 ⇒ h ≤ 2·log(n + 1)
- 16. Consequence: Efficient SetOperations Because red-black trees have height O(log n), the following operations are guaranteed to be: SEARCH, MINIMUM, MAXIMUM, PREDECESSOR, SUCCESSOR ➜ O(log n) worst-case time Note: We’ll later see that even after insertion/deletion, red-black properties can be restored efficiently.
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- 18. Fixing Tree Structurewith Rotations Red-black tree invariants may be violated during insertion or deletion. Rotations help restore balance without violating the binary search tree property. Two types: LEFT-ROTATE and RIGHT-ROTATE
- 19. LEFT-ROTATE(x) Time complexity: O(1) Updatesonly a few pointers
- 20. LEFT-ROTATE(x) ✅BST structure preserved ✅Heightand black-height may change locally
- 21. RIGHT-ROTATE(x) Time complexity: O(1) Updatesonly a few pointers
- 22. RIGHT-ROTATE(x) ✅Again, BST structureis preserved ✅These operations are local but key to restoring red-black properties
- 23. Summary: Rotations Simple pointerchanges (no key swaps) Used in RB-INSERT-FIXUP and RB-DELETE-FIXUP Maintain BST invariants Run in O(1) time Only affect a small part of the tree
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- 25. How Insertion Works First,insert the new node as in a normal BST. Color it red (to avoid violating black-height property immediately). Then call RB-INSERT-FIXUP to fix any red-black violations. Note: We color new nodes red to avoid increasing the black-height along any path.
- 26. RB-INSERT(T, z) Note: This partis just BST logic + coloring the new node red.
- 27. What Can GoWrong After Insertion? If z.p.color == BLACK: No problem. If z.p.color == RED: Red-red violation! (violates Property 4) This triggers RB-INSERT-FIXUP, which resolves these problems.
- 28. RB-INSERT-FIXUP: Fixing Violations KeyIdea: We look at the uncle y of the inserted node z: Three cases depending on z’s uncle: Case 1: Uncle y is red Case 2: Uncle y is black and z is a right child Case 3: Uncle y is black and z is a left child Plus symmetric versions if z.p is a right child.
- 29. RB-INSERT-FIXUP(T, z) Note: This partis just BST logic + coloring the new node red.
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- 31. Efficiency of Insertion Insertion:O(log n) to find the location Fixup: O(1) rotations per level, so total O(log n) ✅All red-black properties restored ✅Tree remains balanced
- 32. Summary: Insertion Insert nodelike BST, color red Fix red-red issues with RB-INSERT-FIXUP Use rotations and recoloring as needed Height remains O(log n)
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- 34. Deletion: The HardestOperation Just like in BSTs, we use the successor if the node has two children. But deleting a node might violate red-black properties. Especially problematic if we delete a black node → might change black-height.
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- 36. What Violations CanOccur? Deleting a black node can violate: Property 1: Root is black (if black root deleted) Property 5: All paths must have same # of black nodes
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- 38. Fixup: The 4Key Cases Each based on color of sibling w and its children: Case 1: Sibling w is red → rotate & convert to another case Case 2: Sibling w and both its children are black → recolor and move up Case 3: w is black, w.left red, w.right black → rotate to convert to Case 4 Case 4: w is black, w.right is red → rotate and recolor, fix complete
- 39. Fixup: The 4Key Cases
- 40. Efficiency of Deletion LikeBST: O(log n) to locate node Fixup: O(log n) (at most 3 rotations) ✅All red-black properties restored ✅Tree remains balanced
- 41. Summary: Deletion Start withBST-style deletion If deleted node is black → fix with RB-DELETE-FIXUP Four main cases depending on sibling and its children Tree height remains O(log n)
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- 43. Property # Description 1Every node is red or black 2 The root is black 3 Every leaf (NIL) is black 4 Red nodes have black children (no two reds in a row) 5 Every path from a node to its descendant NILs contains the same number of black nodes Red-Black Tree Properties Recap
- 44. Lemma: Height isO(log n) A red-black tree with n internal nodes has height at most
- 45. Lemma: Height isO(log n) Lemma(Proof): Show: Subtree rooted at x contains ≥ 2^bh(x) - 1 internal nodes (proved by induction). 1. Use Property 4: 2. At least half the nodes on any root-to-leaf path must be black ⇒ bh(root) ≥ h/2 Then: 3. n ≥ 2^(h/2) - 1 ⇒ log(n + 1) ≥ h/2 ⇒ h ≤ 2·log(n + 1)
- 46. Operation Property 1Property 2 Property 3 Property 4 Property 5 Insert-Fixup ✅ ✅ ✅ ✅(via rotations) ✅ Delete-Fixup ✅ ✅ ✅ ✅ ✅ Why Insertion and Deletion Maintain Red-Black Properties Insert and delete call fixup routines that: Recolor Rotate Propagate problems up if needed
- 47. Summary: Properties andProofs Lemma: Height = O(log n) All operations preserve 5 key properties Fixups use only local changes Proves red-black trees are reliable balanced BSTs
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- 49. Motivation for Red-BlackTrees in Practice 🌲Self-balancing: Guaranteed O(log n) for: Search Insert Delete ⚖️Less rigid than AVL trees → fewer rotations ✅Used when balance and performance are needed 📦Real-world uses: C++ STL map, set Java TreeMap, TreeSet Linux kernel scheduler
- 50. Feature Red-Black AVLTree Splay Tree Balance Factor Loose Strict Self-adjusting Rotations Needed Fewer More Depends Worst-case time O(log n) O(log n) O(log n) Use Case General use Fast search Cache locality Red-Black vs AVL vs Others
- 51. Red-Black Trees: KeyTakeaways A type of binary search tree with added color and balance properties Maintains 5 crucial invariants using: Rotations Recoloring Insertions & deletions require fixups, but always keep height = O(log n) Supported by rigorous lemmas and proofs Widely used in practice because of their efficiency and reliability
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