Different Tree Data Structures for Different Problems

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Different Tree Data Structures for Different
Problems
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Data Structures

• Data Structures have to be adapted for each
  problem
• A specific subset of operations
• (Example Search is needed, Insert is not)
• A specific set of requirements or usage patterns
• (Example Search occurs frequently, Insert occurs
  rarely)
• Special Data Structures
• Hybrid data structures
• Augmented data structures

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Problem 1

• Problem Dictionary, where every key has the same
  probability to be searched for and the structure
  must be dynamic (inserts and deletes)
• Solution we have seen that we can use Balanced
  Binary Search Trees

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Problem 2

• Problem Dictionary, where every key has a
  different, known probability to be searched for
  the structure is not dynamic.
• Example A dictionary must contain words bag,
  cat, dog, school. It is known that school gets
  searched 40 of the time, bag 30 of the time,
  dog 20 of the time and cat 10 of the time.
• The keys which are more frequent searched for
  must be near the root. Balancing for reducing
  height does not help here !
• Solution build the Optimal Binary Search Tree

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Optimal Binary Search Tree

• Balanced BST

• Optimal BST

0.1
0.4
cat
school
0.3
0.3
0.2
bag
dog
bag
0.2
0.4
dog
school
0.1
cat
0.1 0 0.31 0.2 1 0.42 1.3
0.4 0 0.31 0.2 2 0.13 1.0
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Problem 3

• Problem A dictionary structure, used to store a
  dynamic set of words that may contain common
  prefixes.
• Solution
• We can exploit the common prefixes of the words,
  and associate the words with paths in a tree
  instead of nodes of a tree
• Solution Trie trees (Prefix trees, Retrieval
  trees)
• Multipath trees
• If the alphabet has R symbols, a node may have R
  children

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Trie Tree
B
S
E
E
H
Y
A
L
E
E
N
L
Word set BE, BY, SEA, SEE, SEEN, SELL, SHE
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Problem 4

• Problem we need a data structure having features
  similar to the search trees (search, insert,
  delete) but on the secondary storage (disk).
• What is different with disks?
• Disks are slow, thus read/write happens in bulk
  (several items at once, forming a page)
• Solution Adapt the BST data structure to a
  search tree that may contain in a node several
  keys, up to fill up a page gt B-Trees

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B-Tree
8 13 17
11
15
22 25 27
4 5 6
Minimum degree of a B-tree tgt2, such that every
node, except the root, must have n keys and n1
children, where tltn1lt2t 2-3-4 trees are
actually B-trees of min degree 2 The value of t
must be in concordance with the size of the disk
page
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B-Trees formal definition

• A B-tree T is a rooted tree (whose root is
  T.root) having the properties
• Every node x has the following attributes
• x.n, the number of keys currently stored in node
  x,
• the x.n keys themselves, x.key1, x.key2, ,
  x.keyx.n, stored in nondecreasing order, so that
  x.key1 lt x.key2 lt lt x.keyx.n
• x.leaf , a boolean value that is TRUE if x is a
  leaf and FALSE if x is an internal node.
• Each internal node x also contains x.n1 pointers
  x.c1, x.c2, x.cx.n1 to its children. Leaf
  nodes have no children, and so their ci
  attributes are undefined.
• The keys x.keyi separate the ranges of keys
  stored in each subtree if ki is any key stored
  in the subtree with root x.ci, then k1 lt x.key1
  lt k2 lt x.key2 lt lt X.keyx.n lt kx.n1
• All leaves have the same depth, which is the
  trees height h.
• Nodes have lower and upper bounds on the number
  of keys they can contain. We express these bounds
  in terms of a fixed integer t gt2 called the
  minimum degree of the B-tree
• Every node other than the root must have at least
  t -1 keys. Every internal node other than the
  root thus has at least t children. If the tree is
  nonempty, the root must have at least one key.
• Every node may contain at most 2t -1 keys.
  Therefore, an internal node may have at most 2t
  children.

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Problem 5

• Problem we need a data structure combining the
  features of both Binary Search Trees and Heaps,
  holding nodes that have both a search key and a
  priority
• Binary Search Trees and Heaps both come in form
  of Binary Trees
• Solution Treap a Binary Tree where the nodes
  satisfy both the BST property and the Heap
  property !
• Why is this structure important ?
• Building a Treap by using random generated values
  as priorities is a method for building a balanced
  binary tree

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Treap