Chapter 12Advanced Data Structures

Red-Black Trees

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add the attribute of _________ (red or black) to links/nodesred-black trees used in

C++ Standard Template Library (STL)Java to implement maps (or _________________, as inPython)

Red-Black Trees

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a red-black tree is a ______ with the following properties:every node is either red or blackthe root is __________if a node is red, its children must be _________every path from the root to a null link contains the samenumber of black nodes

perfect black ___________the height of an -node red-black BST is at most

, sosearch, insertion, and deletion are operations

Red-Black Trees

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building a red-black treein order to maintain perfect black balance, any new nodeadded to the tree must be ______if the parent of the new node is _________, all is wellif the parent of the new node is ______, this violates thecondition that red nodes have only black children

fix with ___________ similar to those for AVL and splaytreescan be used to maintain the red-black structure at anypoint in the tree, not just at insertion of a new node

Red-Black Trees

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top-down insertion: color flipsto preserve perfect black balance, a newly inserted nodemust be redin top-down insertion, we change the tree as we movedown the tree to the point of ______________

the changes we make insure that when we insert thenew node, the parent is _________if we encounter a node X with two red children, wemake X red and its children black

if X is the root, we change the color back to blacka color flip can cause a red-black violation (a red child

______

Red-Black Trees

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red-black tree rotationscase 1: the ___________black (or missing)

this is a _________ rotation and a color swap for P and G

Red-Black Trees

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red-black tree rotations

(or missing)

this is a _________ rotation and a color swap for X and G

Red-Black Trees

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red-black tree rotations______

and its sibling are both redwe changed all such pairs to _________ on the waydown

Red-Black Trees

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example: GOTCHA

Red-Black Trees

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example: GOTCHA (cont.)

Red-Black Trees

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example: GOTCHA (cont.)

Red-Black Trees

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example: GOTCHA (cont.)

the standard ______ (rightmost) for GOTCHA is slightlyshorter

Red-Black Trees

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example: ISOGRAM

Red-Black Trees

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example: ISOGRAM (cont.)

Red-Black Trees

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example: ISOGRAM (cont.)

Red-Black Trees

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example: ISOGRAM (cont.)

Red-Black Trees

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example: ISOGRAM (cont.) 2-3-4 treesuseful because we can insert new items while maintaining___________ balancea 2-3-4 tree consists of

2-nodes: one key, ______ children3-nodes: two keys, __________ children4-nodes: three keys, ________ children

2-3-4 Trees

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2-3-4 Trees

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insertion into 2-3-4 treesinsert the new key into the ___________ existing nodereached in the search

2-3-4 Trees

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what about a 4-node?top-down insertion

as we move down the tree, whenever we encounter a4-node, we move the __________ element up into theparent node and break up the remainder into _______2-nodes

2-3-4 Trees

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what about a 4-node?top-down insertion (cont.)

insertion, if done here, now reduces to the case of a__________ or ___________

2-3-4 Trees

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top-down insertionas we move down the tree, we split up 4-nodes as weencounter them through the following process

move the middle key up to the parentsplit the remaining keys into 2-nodes

this action ______________ that the parent of any 4-nodewe encounter is a 2-node or 3-node

therefore, the tree will always have room to accept the___________ element of the 4-node

2-3-4 Trees

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example: insert 42, 9, 39, 11, 27, 13, 33, 16, 28the first three insertions are straightforward

when inserting 11, we encounter a 4-node, which we ______

39 is first promoted as a new root nodeperfect balance is maintained in 2-3-4 trees by growing atthe ________

2-3-4 Trees

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example: insert 42, 9, 39, 11, 27, 13, 33, 16, 28 (cont.)insert 27

insert 13: first split 4-node

2-3-4 Trees

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example: insert 42, 9, 39, 11, 27, 13, 33, 16, 28 (cont.)insert 33

insert 16: split 4-node

2-3-4 Trees

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example: insert 42, 9, 39, 11, 27, 13, 33, 16, 28 (cont.)insert 28: split 4-node at root

once again, ___________ at the root maintains perfectbalance

2-3-4 Trees

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complexity of 2-3-4 tree operationsthe height of an -node 2-3-4 tree is between

and

searching and inserting are both operationsrather than splitting 4-nodes on the way down, we couldalso perform _______________ insertion, starting at theinsertion node and moving upwardsdeletion involves ___________ nodes (and is also )

2-3-4 Trees as Red-Black Trees

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red-black trees are a way of realizing 2-3-4 trees as binarysearch trees

allows us to re-use an implementation of a BST, andsimplifies _______________add the attribute of ________ (red or black) to links/nodes

2-3-4 Trees as Red-Black Trees

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encoding 2-3-4 trees as red-black trees

2-3-4 Trees as Red-Black Trees

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encoding 2-3-4 trees as red-black trees

2-3-4 Trees as Red-Black Trees

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a red-black tree is a BST with the following properties:every node is either red or blackthe root is blackif a node is red, its children must be blackevery path from the root to a null link contains the samenumber of black nodes

in the encoding of 2-3-4 trees from red-black trees, the _____links in the red-black tree correspond to the links in the 2-3-4tree, while the ____ links denote a split of a 2-node or 3-nodecondition 4 corresponds to the perfect balance of 2-3-4 trees

the height of an -node red-black BST is at most ,so search, insertion, and deletion are operations

2-3-4 Trees as Red-Black Trees

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building a red-black treein order to maintain perfect black balance, any new nodeadded to the tree must be redif the parent of the new node is black, all is wellif the parent of the new node is red, this violates thecondition that red nodes have only black children

fix with rotations similar to those for AVL and splay treescan be used to maintain the red-black structure at anypoint in the tree, not just at insertion of a new node

2-3-4 Trees as Red-Black Trees

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top-down insertion: color flipswe will follow a top-down insertion scheme as we did with2-3-4 treesas we move down the tree to insert a node, if weencounter a node X with two red children, we make X redand its children black

if X is the root, we change the color back to blacka color flip can cause a red-black violation (a red child with

2-3-4 Trees as Red-Black Trees

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color flips correspond to splitting 4-nodes

2-3-4 Trees as Red-Black Trees

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red-black tree rotations

(or missing)

this is a single rotation and a color swap for P and G

2-3-4 Trees as Red-Black Trees

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color flips correspond to splitting 4-nodes

2-3-4 Trees as Red-Black Trees

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red-black tree rotations

(or missing)

this is a double rotation and a color swap for X and G

2-3-4 Trees as Red-Black Trees

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color flips correspond to splitting 4-nodes

2-3-4 Trees as Red-Black Trees

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red-black tree rotations

and its sibling are part of a ____________we split all the 4-nodes we encountered on the waydown

2-3-4 Trees as Red-Black Trees

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example: GOTCHA

2-3-4 Trees as Red-Black Trees

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example: GOTCHA (cont.)

2-3-4 Trees as Red-Black Trees

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example: GOTCHA (cont.)

2-3-4 Trees as Red-Black Trees

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example: GOTCHA (cont.)

the standard BST (rightmost) for GOTCHA is slightlyshorter

2-3-4 Trees as Red-Black Trees

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example: ISOGRAM

2-3-4 Trees as Red-Black Trees

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example: ISOGRAM

2-3-4 Trees as Red-Black Trees

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example: ISOGRAM

2-3-4 Trees as Red-Black Trees

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example: ISOGRAM

2-3-4 Trees as Red-Black Trees

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example: ISOGRAM