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![Example: RED-BLACK-TREE
• For convenience, we add NIL nodes and refer to them as
the leaves of the tree.
– Color[NIL] = BLACK
26
17 41
30 47
38 50
NIL NIL
NIL
NIL NIL NIL NIL
NIL
4
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![Claim 2 (cont’d)
Proof: By induction on h[x]
Basis: h[x] = 0 ⇒
x is a leaf (NIL[T]) ⇒
bh(x) = 0 ⇒
# of internal nodes: 20 - 1 = 0
Inductive Hypothesis: assume it is true for
h[x]=h-1
Inductive step:
• Prove it for h[x]=h
NIL
x
10
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![Claim 2 (cont’d)
Prove it for h[x]=h
internal nodes at x=
internal nodes at l +
internal nodes at r + 1
Using inductive hypothesis:
internal nodes at x ≥ (2bh(l) – 1) + (2bh(r) – 1) + 1
x
l r
h
h-1
11
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![23
RB-INSERT-FIXUP
Case 1: z’s “uncle” (y) is red
(z could be either left or right child)
Idea:
• p[p[z]] (z’s grandparent) must be black
• color p[z] ← black
• color y ← black
• color p[p[z]] ← red
• z = p[p[z]]
– Push the “red” violation up the tree
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![RB-INSERT-FIXUP
Case 2:
• z’s “uncle” (y) is black
• z is a left child
Case 2
Idea:
• color p[z] ← black
• color p[p[z]] ← red
• RIGHT-ROTATE(T, p[p[z]])
• No longer have 2 reds in a row
• p[z] is now black
24
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![25
RB-INSERT-FIXUP
Case 3:
• z’s “uncle” (y) is black
• z is a right child
Idea:
• z ← p[z]
• LEFT-ROTATE(T, z)
⇒ now z is a left child, and both z and p[z] are red ⇒ case 2
Case 3 Case 2
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![Example
11
Insert 4
2 14
1 15
7
8
5
4
y
11
2 14
1 15
7
8
5
4
z
Case 1
y
z and p[z] are both red
z’s uncle y is red
z
z and p[z] are both red
z’s uncle y is black
z is a right child
Case 3
11
2
14
1
15
7
8
5
4
z
y
Case 2
z and p[z] are red
z’s uncle y is black
z is a left child
11
2
14
1
15
7
8
5
4
z
26
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![RB-INSERT-FIXUP(T, z)
1. while color[p[z]] = RED
2. do if p[z] = left[p[p[z]]]
3. then y ← right[p[p[z]]]
4. if color[y] = RED
5. then Case1
6. else if z = right[p[z]]
7. then Case2
8. Case3
9. else (same as then clause with “right” and “left”
exchanged)
10. color[root[T]] ← BLACK
The while loop repeats only when
case1 is executed: O(logN) times
Set the value of x’s “uncle”
We just inserted the root, or
The red violation reached the root
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docsity-red-black-trees-data-structures-lecture-slides_2.pdf
- 1. Data Structures andAlgorithms Amrita Vishwa Vidyapeetham 36 pag. Document shared on www.docsity.com Downloaded by: AayushAdhikari ([email protected])
- 2. Red-Black Trees Docsity.com Document sharedon www.docsity.com Downloaded by: AayushAdhikari ([email protected])
- 3. 2 Red-Black Trees • “Balanced”binary trees guarantee an running time on the basic dynamic-set operations • Red-black tree – Binary tree with an additional attribute for its nodes: color which can be red or black – The nodes inherit all the other attributes from the binary-search trees: key, left, right, p Docsity.com Document shared on www.docsity.com Downloaded by: AayushAdhikari ([email protected]) O(lgn)
- 4. Red-Black Trees • Constrainsthe way nodes can be colored on any path from the root to a leaf. – Ensures that no path is more than twice as long as another ⇒ the tree is balanced Balanced tree O(logN) 3 Docsity.com Document shared on www.docsity.com Downloaded by: AayushAdhikari ([email protected])
- 5. Example: RED-BLACK-TREE • Forconvenience, we add NIL nodes and refer to them as the leaves of the tree. – Color[NIL] = BLACK 26 17 41 30 47 38 50 NIL NIL NIL NIL NIL NIL NIL NIL 4 Docsity.com Document shared on www.docsity.com Downloaded by: AayushAdhikari ([email protected])
- 6. Red-Black-Trees Properties (**Binary searchtree property is satisfied**) 1. Every node is either red or black 2. The root is black 3. Every leaf (NIL) is black 4. If a node is red, then both its children are black • No two consecutive red nodes on a simple path from the root to a leaf 5. For each node, all paths from that node to a leaf contain the same number of black nodes 5 Docsity.com Document shared on www.docsity.com Downloaded by: AayushAdhikari ([email protected])
- 7. Definitions • Height ofa node: the number of edges in the longest path to a leaf. • Black-height bh(x) of a node x: the number of black nodes (including NIL) on the path from x to a leaf, not counting x. 26 17 41 30 47 38 50 NIL NIL NIL NIL NIL NIL NIL NIL h = 4 bh = 2 h = 3 bh = 2 h = 2 bh = 1 h = 1 bh = 1 h = 1 bh = 1 h = 2 bh = 1 h = 1 bh = 1 6 Docsity.com Document shared on www.docsity.com Downloaded by: AayushAdhikari ([email protected])
- 8. Height of Red-Black-Trees Ared-black tree with n internal nodes has height at most 2log(N+1) • Need to prove two claims first … 7 Docsity.com Document shared on www.docsity.com Downloaded by: AayushAdhikari ([email protected])
- 9. Claim 1 • Anynode x with height h(x) has bh(x) ≥ h(x)/2 • Proof – By property 4, at most h/2 red nodes on the path from the node to a leaf – Hence, at least h/2 are black 26 17 41 30 47 38 50 NIL NIL NIL NIL NIL NIL NIL NIL h = 4 bh = 2 h = 3 bh = 2 h = 2 bh = 1 h = 1 bh = 1 h = 1 bh = 1 h = 2 bh = 1 h = 1 bh = 1 8 Docsity.com Document shared on www.docsity.com Downloaded by: AayushAdhikari ([email protected])
- 10. Claim 2 • Thesubtree rooted at any node x contains at least 2bh(x) - 1 internal nodes 26 17 41 30 47 38 50 NIL NIL NIL NIL NIL NIL NIL NIL h = 4 bh = 2 h = 3 bh = 2 h = 2 bh = 1 h = 1 bh = 1 h = 1 bh = 1 h = 2 bh = 1 h = 1 bh = 1 9 Docsity.com Document shared on www.docsity.com Downloaded by: AayushAdhikari ([email protected])
- 11. Claim 2 (cont’d) Proof:By induction on h[x] Basis: h[x] = 0 ⇒ x is a leaf (NIL[T]) ⇒ bh(x) = 0 ⇒ # of internal nodes: 20 - 1 = 0 Inductive Hypothesis: assume it is true for h[x]=h-1 Inductive step: • Prove it for h[x]=h NIL x 10 Docsity.com Document shared on www.docsity.com Downloaded by: AayushAdhikari ([email protected])
- 12. Claim 2 (cont’d) Proveit for h[x]=h internal nodes at x= internal nodes at l + internal nodes at r + 1 Using inductive hypothesis: internal nodes at x ≥ (2bh(l) – 1) + (2bh(r) – 1) + 1 x l r h h-1 11 Docsity.com Document shared on www.docsity.com Downloaded by: AayushAdhikari ([email protected])
- 13. Claim 2 (cont’d) •Let bh(x) = b, • then any child y of x has: – bh (y) = – bh (y) = b (if the child is red), or b - 1 (if the child is black) 26 17 41 30 47 38 50 x y1 y2 bh = 2 bh = 2 bh = 1 NIL NIL NIL NIL NIL NIL bh(y)≥bh(x)-1 12 Docsity.com Document shared on www.docsity.com Downloaded by: AayushAdhikari ([email protected])
- 14. Claim 2 (cont’d) •So, back to our proof: internal nodes at x ≥ (2bh(l) – 1) + (2bh(r) – 1) + 1 ≥ (2bh(x) - 1 – 1) + (2bh(x) - 1 – 1) + 1 = 2 · (2bh(x) - 1 - 1) + 1 = 2bh(x) - 1 internal nodes x l r h h-1 13 Docsity.com Document shared on www.docsity.com Downloaded by: AayushAdhikari ([email protected])
- 15. Height of Red-Black-Trees(cont’d) A red-black tree with N internal nodes has height at most 2log(N+1). Proof: N • Solve for h: N + 1 ≥ 2h/2 log(N + 1) ≥ h/2 ⇒ h ≤ 2 log(N + 1) root l r height(root) = h bh(root) = b number of internal nodes ≥ 2b - 1 ≥ 2h/2 - 1 Claim 2 Claim 1 14 Docsity.com Document shared on www.docsity.com Downloaded by: AayushAdhikari ([email protected])
- 16. Operations on Red-Black-Trees •Non-modifying binary-search-tree operations (e.g., RetrieveItem, GetNextItem, ResetTree), run in O(h)=O(logN) time • What about InsertItem and DeleteItem? – They will still run on O(logN) – Need to guarantee that the modified tree will still be a valid red-black tree 15 Docsity.com Document shared on www.docsity.com Downloaded by: AayushAdhikari ([email protected])
- 17. 16 InsertItem What color tomake the new node? • Red? Let’s insert 35! – Property 4 is violated: if a node is red, then both its children are black • Black? Let’s insert 14! – Property 5 is violated: all paths from a node to its leaves contain the same number of black nodes 26 17 41 30 47 38 50 Docsity.com Document shared on www.docsity.com Downloaded by: AayushAdhikari ([email protected])
- 18. Rotations • Operations forre-structuring the tree after insert and delete operations – Together with some node re-coloring, they help restore the red-black-tree property – Change some of the pointer structure – Preserve the binary-search tree property • Two types of rotations: – Left & right rotations 17 Docsity.com Document shared on www.docsity.com Downloaded by: AayushAdhikari ([email protected])
- 19. Left Rotations • Assumptionsfor a left rotation on a node x: – The right child y of x is not NIL • Idea: – Pivots around the link from x to y – Makes y the new root of the subtree – x becomes y’s left child – y’s left child becomes x’s right child 18 Docsity.com Document shared on www.docsity.com Downloaded by: AayushAdhikari ([email protected])
- 20. 19 Example: LEFT-ROTATE Docsity.com Document sharedon www.docsity.com Downloaded by: AayushAdhikari ([email protected])
- 21. Right Rotations • Assumptionsfor a right rotation on a node x: – The left child x of y is not NIL • Idea: – Pivots around the link from y to x – Makes x the new root of the subtree – y becomes x’s right child – x’s right child becomes y’s left child 20 Docsity.com Document shared on www.docsity.com Downloaded by: AayushAdhikari ([email protected])
- 22. InsertItem • Goal: – Inserta new node z into a red-black tree • Idea: – Insert node z into the tree as for an ordinary binary search tree – Color the node red – Restore the red-black tree properties • i.e., use RB-INSERT-FIXUP 21 Docsity.com Document shared on www.docsity.com Downloaded by: AayushAdhikari ([email protected])
- 23. RB Properties Affectedby Insert 1. Every node is either red or black 2. The root is black 3. Every leaf (NIL) is black 4. If a node is red, then both its children are black 5. For each node, all paths from the node to descendant leaves contain the same number of black nodes OK! If z is the root ⇒ not OK OK! 26 17 41 47 38 50 If p(z) is red ⇒ not OK z and p(z) are both red OK! Docsity.com Document shared on www.docsity.com Downloaded by: AayushAdhikari ([email protected])
- 24. 23 RB-INSERT-FIXUP Case 1: z’s“uncle” (y) is red (z could be either left or right child) Idea: • p[p[z]] (z’s grandparent) must be black • color p[z] ← black • color y ← black • color p[p[z]] ← red • z = p[p[z]] – Push the “red” violation up the tree Docsity.com Document shared on www.docsity.com Downloaded by: AayushAdhikari ([email protected])
- 25. RB-INSERT-FIXUP Case 2: • z’s“uncle” (y) is black • z is a left child Case 2 Idea: • color p[z] ← black • color p[p[z]] ← red • RIGHT-ROTATE(T, p[p[z]]) • No longer have 2 reds in a row • p[z] is now black 24 Docsity.com Document shared on www.docsity.com Downloaded by: AayushAdhikari ([email protected])
- 26. 25 RB-INSERT-FIXUP Case 3: • z’s“uncle” (y) is black • z is a right child Idea: • z ← p[z] • LEFT-ROTATE(T, z) ⇒ now z is a left child, and both z and p[z] are red ⇒ case 2 Case 3 Case 2 Docsity.com Document shared on www.docsity.com Downloaded by: AayushAdhikari ([email protected])
- 27. Example 11 Insert 4 2 14 115 7 8 5 4 y 11 2 14 1 15 7 8 5 4 z Case 1 y z and p[z] are both red z’s uncle y is red z z and p[z] are both red z’s uncle y is black z is a right child Case 3 11 2 14 1 15 7 8 5 4 z y Case 2 z and p[z] are red z’s uncle y is black z is a left child 11 2 14 1 15 7 8 5 4 z 26 Docsity.com Document shared on www.docsity.com Downloaded by: AayushAdhikari ([email protected])
- 28. RB-INSERT-FIXUP(T, z) 1. whilecolor[p[z]] = RED 2. do if p[z] = left[p[p[z]]] 3. then y ← right[p[p[z]]] 4. if color[y] = RED 5. then Case1 6. else if z = right[p[z]] 7. then Case2 8. Case3 9. else (same as then clause with “right” and “left” exchanged) 10. color[root[T]] ← BLACK The while loop repeats only when case1 is executed: O(logN) times Set the value of x’s “uncle” We just inserted the root, or The red violation reached the root Docsity.com Document shared on www.docsity.com Downloaded by: AayushAdhikari ([email protected])
- 29. Analysis of InsertItem •Inserting the new element into the tree O(logN) • RB-INSERT-FIXUP – The while loop repeats only if CASE 1 is executed – The number of times the while loop can be executed is O(logN) • Total running time of InsertItem: O(logN) 28 Docsity.com Document shared on www.docsity.com Downloaded by: AayushAdhikari ([email protected])
- 30. DeleteItem • Delete asusually, then re-color/rotate • A bit more complicated though … • Demo – http://gauss.ececs.uc.edu/RedBlack/redblack.html 29 Docsity.com Document shared on www.docsity.com Downloaded by: AayushAdhikari ([email protected])
- 31. DeleteItem What color wasthe node that was removed? Red? 1. Every node is either red or black 2. The root is black 3. Every leaf (NIL) is black 4. If a node is red, then both its children are black 5. For each node, all paths from the node to descendant leaves contain the same number of black nodes OK! OK! OK! OK! Does not create two red nodes in a row Not OK! Could change the black heights of some nodes 26 17 41 30 47 38 50 30 Docsity.com Document shared on www.docsity.com Downloaded by: AayushAdhikari ([email protected])
- 32. DeleteItem What color wasthe node that was removed? Black? 1. Every node is either red or black 2. The root is black 3. Every leaf (NIL) is black 4. If a node is red, then both its children are black 5. For each node, all paths from the node to descendant leaves contain the same number of black nodes OK! OK! Not OK! Could create two red nodes in a row Not OK! Could change the black heights of some nodes 26 17 41 30 47 38 50 Not OK! If removing the root and the child that replaces it is red 31 Docsity.com Document shared on www.docsity.com Downloaded by: AayushAdhikari ([email protected])
- 33. Red-Black Trees -Summary • Operations on red-black trees: – SEARCH O(h) – PREDECESSOR O(h) – SUCCESOR O(h) – MINIMUM O(h) – MAXIMUM O(h) – INSERT O(h) – DELETE O(h) • Red-black trees guarantee that the height of the tree will be O(lgn) CS 477/677 - Lecture 12 Docsity.com Document shared on www.docsity.com Downloaded by: AayushAdhikari ([email protected])
- 34. Problems 33 Docsity.com Document shared onwww.docsity.com Downloaded by: AayushAdhikari ([email protected])
- 35. Problems • What red-blacktree property is violated in the tree below? How would you restore the red-black tree property in this case? – Property violated: if a node is red, both its children are black – Fixup: color 7 black, 11 red, then right-rotate around 11 11 2 14 1 15 7 8 5 4 z 34 Docsity.com Document shared on www.docsity.com Downloaded by: AayushAdhikari ([email protected])
- 36. Problems • Let a,b, c be arbitrary nodes in subtrees α, β, γ in the tree below. • How do the depths of a, b, c change when a left rotation is performed on node x? – a: increases by 1 – b: stays the same – c: decreases by 1 35 Docsity.com Document shared on www.docsity.com Downloaded by: AayushAdhikari ([email protected])
- 37. Problems • When weinsert a node into a red-black tree, we initially set the color of the new node to red. • Why didn’t we choose to set the color to black? • Would inserting a new node to a red-black tree and then immediately deleting it, change the tree? 36 Docsity.com Document shared on www.docsity.com Downloaded by: AayushAdhikari ([email protected])