ISYS 300 Trees: Especially BTrees

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ISYS 300 Trees Especially B-Trees
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How we got here?

• Created an inverted index from text
• Used the index for a text searching (vector
  model)
• There can be tens of thousands on items in an
  index How can we access them quickly?
• Note These are important techniques beyond text
  searching there are many places they can be
  used.

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Graph Theory

• A graph is a set of nodes and edges
• Some common graph problems include
• Layout of highways, telephone lines
• The structure of the Web
• Social networks

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Finding Items in an Inverted Index

• Sorted List
• Binary search
• Linked List
• Hard to search without an index
• Easy to insert and delete items
• Binary Trees
• Always two choices (20 questions)
• Multi-way Trees (B-Trees and B Trees)
• Several choices at choice point
• High fan-out. Not as deep as binary trees
• Other techniques to remember but probably not a
  good choice here
• Hashing
• Content-based retrieval (neural networks)

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Trees and Hierarchies Some Terminology

• A Tree is a data structure in which one node
  links to other nodes in exactly one way.
• We often think of a tree as a hierarchy
• For a hierarchy, we often talk about parents
  (up the hierarchy) and children (lower in the
  hierarchy).
• We often pick one central node as the root of a
  tree and the nodes with no children as the
  leaves
• The depth of a tree is the (average) distance
  from the root to the children
• Lots of times we say tree when we mean
  hierarchies
• Trees can be used for searching with keys.
  That is, an element providing directions for
  moving around in the tree.

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Balanced, Binary Trees

• Binary tree (there are always 2 choices)
• Each node contains a key
• Left sub-trees stored all keys smaller than the
  parent key
• Right sub-trees stored all keys larger than the
  parent key
• Balanced trees
• Every parent has an approximately equal number of
  left-and right sub-trees

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The Dewey Decimal System

• Has a fan-out of 10 (decimal)
• No attempt to be balanced.

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Advantages of Storing Data in a Balanced Binary
Tree

• Why is a balanced tree helpful?
• If its perfectly balanced, the number of steps
  to find an answer (the depth) is
• NumStepssqrt(NumItems)

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Operations on Search Trees

• Search
• Create a New Tree
• Insert New Items
• Delete Items

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Constructing Balanced Trees (after insertion or
deletion)

• Check if the original tree is balanced
• Check if the left child is balanced
• If it is not balanced, go to step two
• Check if the right child is balanced
• If it not balanced, go to step 2 (next page)

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Constructing Balanced Trees (Contd)

• 2. Rotate the unbalanced tree
• If the left branch is deeper
• Move the left child of the root to become the new
  root, move the right branch of new root to become
  the left branch of the old root
• Make the old root become the right child of new
  root
• If the right branch is deeper
• 3. Go back to Step 1 to check if the new tree is
  balanced or not

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B Trees

• Description
• Each node can have more than one key (multi-way
  tree)
• If a node has m keys, it will have m1 children
  branches.
• All keys in i-1 branch are smaller than key I
  This makes it useful for searching, it is the
  search tree property.
• All leaves are at the same depth.
• Advantages of B-Tree over Binary Tree
• Useful when reading long records off a disk

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B tree
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25
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8
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39
51
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3
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B Tree

• B Trees
• B-tree that stores all data in the leaves.
• That way, the B-Tree can be kept in a separate
  file from the data

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B tree
(F, M)
(Ap, Bs, E)
(Gr, H, L)
(P, Ru, T)
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Searching a Tree that is Not Well Structured

• The moves in some games can be organized as a
  tree (a game space)
• Tic-tac-toe
• Chess
• Have to make estimates of the value of going each
  choice
• Can do learning (remember what good choices were)