2 3 4 Trees

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2 3 4 Trees

• Multi-Way Search Tree

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Binary Search Tree

• BSTs are efficient for storage retrieval but
  have (potentially) poor worst case performance
• Insert, delete, find all are O(h) hheight of
  tree
• hn in worst case
• All three are O(n)

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Complete Binary Search Tree?

• If we could keep a binary search tree complete h
  would be equal to log n
• All operations would be O(log n) like in our heap
• How can we keep BST complete?

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Complete Binary Search Tree?

• Rebuild tree after every insert/delete
• Takes extra time O(n)
• Have insert/delete operations ensure completeness
• Heap used this option
• BST we talked about did neither

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Complete BST Performance

• Dont necessarily want complete BST
• Want search behavior of BST with performance of
  complete tree
• Multiple implementations of this
• Splay trees, AVL trees (most commonly)
  red-black trees

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Before Red-Black

• Before we talk about Red-Black trees we will try
  to get an understanding of how they work using
  new kind of tree
• 2-3-4 Trees (also called 2-4 trees)

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2-4 Trees

• Generalization of BST
• Called multi-way search tree
• Ordered tree with special nodes
• Each node holds one or more entries
• Entry is (key, value) pair

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Nodes

• Previously nodes only had one element
• 2-4 Tree nodes have anywhere from 1-3 elements
• Need secondary data structure to hold elements
• Can be as simple as unsorted list
• 2-4 nodes have children, but number of children
  based on number of elements

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Example Nodes

• 2-node
• 1 element
• 2 children (2 subtrees)
• Like binary search tree

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k
k 8
10
3-Node

• 2 elements
• 3 children

8 19
k
k 19
8
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4-Node

• 3 elements
• 4 children

8 19 44
k
k 44
19
8
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Cannot Have

• Examples of nodes that cannot be
• 1 element 3 children
• 2 elements 1 child
• 3 elements 2 children

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Properties

• Size
• Every internal node has at least 2, at most 4
  children
• Depth
• All external nodes (dummy nodes) have same depth
• As with BST, will use dummy nodes as external
  nodes
• Enforces runtime bound of O(log n)

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Ordering Properties

• 2-nodes
• ?c,p LeftDescendant(c,p) ? (? k ? keys(c) ? k key(p))
• ?c,p RightDescendant(c,p) ? (? k ? keys(c) ? k
  key(p))
• 3-nodes
• ?c,p LeftDescendant(c,p) ? (? k ? keys(c) ? k leftKey(p))
• ? c,p MiddleDescendant(c,p) ? (? k ? keys(c) ? (k
  leftKey(p)) ? (k
• ?c,p RightDescendant(c,p) ? (? k ? keys(c) ? k
  rightKey(p))
• 4-nodes
• ? c,p LeftDescendant(m,n) ? (? k ? keys(m) ? k leftKey(n))
• ? c,p MiddleLeftDescendant(c,p) ? (? k ? keys(c)
  ? (k leftKey(p)) ? (k
• ? c,p MiddleRightDescendant(c,p) ? (? k ? keys(c)
  ? (k middleKey(p)) ? (k
• ? c,p RightDescendant(c,p) ? (? k ? keys(c) ? k
  rightKey(p))

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Example 2-4 Tree
More entries (15) than nodes (9)
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2-4 Tree Height

• Worst case
• All nodes 2 nodes
• Complete binary tree
• Height O(log n)

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Proposition 10.8

• Proposition The height of a 2-4 tree sorting n
  entries is O(log n)
• Justification Assume proposition correct

What is BEST height of 2-4 tree?
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Find

• Works very much like in BST
• Runtime O(h)
• but we know h is log n so it is O(log n)

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Insert

• To insert a key, value pair we cant just add in
  new node at end of tree
• Would violate depth property
• Since nodes can have multiple elements most of
  time we can just find correct node and add it

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Insert Example
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Insert Time

• How long did that example take?
• Find insert location O(h)
• Insert O(1)
• Since h log n it took O(log n)
• But, what if the node is full?

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Insert

• If node is full, but need to add there
• Called overflow
• Move up one of elements to parent node
• Promote it
• And split node
• Split into two nodes

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Insert
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Insert

• Important Observation
• Promote and split strategy rows tree from bottom
  up
• Ensures depth property (log n height) is
  maintained

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Insert (with promote split) Time

• How long?
• Finding insert location O(h)
• Add entry find overflow O(1)
• Promote entry O(1)
• Split node O(1)
• Since h log n it is O(log n)

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Insert Example 3
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What If Parent is Full?

• If you try to promote entry to parent and parent
  is full promote split again
• If we get to the root and there is no node to
  promote to
• make a new node
• promote to that node
• make it the root

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How Long Does THAT Take?

• Finding insert location O(h)
• Add entry find overflow O(1)
• Promote entry O(1)
• Split node O(1)
• O(h)O(h) O(h), but hlog n, so O(log n)

From bottom to top (h levels) so O(h)
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Worksheet