![Stacks
• INSERT operation on a stack PUSH
• DELETE operation POP.
• top[S] that indexes the most recently inserted
element.
• The stack consists of elements S[1..top[S]], where
S[1] is the element at the bottom of the stack and
S[top[S]] is the element at the top.
• When top [S] = 0, the stack contains no elements
and is empty
• If an empty stack is popped, stack underflows,
which is normally an error.
• If top[S] exceeds n, the stack overflows.](https://image.slidesharecdn.com/chapter4-230317140923-9a6cea92/85/Chapter-4-pptx-5-320.jpg)
![What is a Hash Table ?
Each record has a special
field, called its key.
In this example, the key is a
long integer field called
Number.
[ 0 ] [ 1 ] [ 2 ] [ 3 ] [ 4 ] [ 5 ]
. . .
[ 700]
[ 4 ]
Number 506643548
Each record has a special
field, called its key.
In this example, the key is a
long integer field called
Number.
Each record has a special
field, called its key.
In this example, the key is a
long integer field called
Number.](https://image.slidesharecdn.com/chapter4-230317140923-9a6cea92/85/Chapter-4-pptx-13-320.jpg)
![Collisions
This is called a collision,
because there is already
another valid record at [2].
[ 0 ] [ 1 ] [ 2 ] [ 3 ] [ 4 ] [ 5 ] [ 700]
Number 506643548
Number 233667136
Number 281942902
Number 155778322
. . .
Number 580625685
Number 701466868
When a collision
occurs,
move forward until you
find an empty spot.](https://image.slidesharecdn.com/chapter4-230317140923-9a6cea92/85/Chapter-4-pptx-14-320.jpg)
![AUGMENTING DATA STRUCTURES
It introduced the notion of Dynamic order
statistics
size[x] = size[left[x]] + size[right[x]] + 1 .](https://image.slidesharecdn.com/chapter4-230317140923-9a6cea92/85/Chapter-4-pptx-26-320.jpg)
Chapter 4.pptx
- 2. Chapter about.. • Elementary data structures • Hash tables • Binary search trees • Red black trees • Augmenting data structures
- 3. Elementary data structures • Stacks and queues • Linked lists • Implementing pointers and objects • Binary trees • Representing rooted trees • The representation of dynamic sets by simple data structures that use pointers. • Complex data structures can be fashioned using pointers.
- 4. Stacks and queues • In a stack, the element deleted from the set is the one most recently inserted: the stack implements a last-in, first-out, or LIFO, policy. • In a queue , the element deleted is always the one that has been in the set for the longest time: the queue implements a first-in, first- out, or FIFO, policy.
- 5. Stacks • INSERT operation on a stack PUSH • DELETE operation POP. • top[S] that indexes the most recently inserted element. • The stack consists of elements S[1..top[S]], where S[1] is the element at the bottom of the stack and S[top[S]] is the element at the top. • When top [S] = 0, the stack contains no elements and is empty • If an empty stack is popped, stack underflows, which is normally an error. • If top[S] exceeds n, the stack overflows.
- 6. An array implementation of a stack S. Stack elements appear only in the lightly shaded positions. (a) Stack S has 4 elements. The top element is 9. (b) Stack S after the calls PUSH(S, 17) and PUSH(S, 3). (c) Stack S after the call POP(S) has returned the element 3, which is the one most recently pushed. Although element 3 still appears in the array, it is no longer in the stack; the top is element 17.
- 7. Queue INSERT operation on a queue ENQUEUE, DELETE operation DEQUEUE; like the stack operation POP, DEQUEUE takes no element argument. The FIFO property of a queue causes it to operate like a line of people in the registrar's office. The queue has a head and a tail
- 9. Linked lists • A linked list is a data structure in which the objects are arranged in a linear order. • Unlike an array, though, in which the linear order is determined by the array indices, the order in a linked list is determined by a pointer in each object. • Linked lists provide a simple, flexible representation for dynamic sets
- 11. Hash Table • A hash table is an effective data structure for implementing dictionaries. Although searching for an element in a hash table can take as long as searching for an element in a linked list. • Direct-address tables
- 12. • Hash tables are a common approach to the storing/searching problem.
- 13. What is a Hash Table ? Each record has a special field, called its key. In this example, the key is a long integer field called Number. [ 0 ] [ 1 ] [ 2 ] [ 3 ] [ 4 ] [ 5 ] . . . [ 700] [ 4 ] Number 506643548 Each record has a special field, called its key. In this example, the key is a long integer field called Number. Each record has a special field, called its key. In this example, the key is a long integer field called Number.
- 14. Collisions This is called a collision, because there is already another valid record at [2]. [ 0 ] [ 1 ] [ 2 ] [ 3 ] [ 4 ] [ 5 ] [ 700] Number 506643548 Number 233667136 Number 281942902 Number 155778322 . . . Number 580625685 Number 701466868 When a collision occurs, move forward until you find an empty spot.
- 15. Hash functions uniform hashing A good hash function satisfies (approximately) the assumption of simple uniform hashing: each key is equally likely to hash to any of the m slots. Hashing by division Hashing by multiplication Universal hashing.
- 16. Hashing by division division method for creating hash functions, we map a key k into one of m slots by taking the remainder of k divided by m. That is, the hash function is h(k) = k mod m
- 17. Hashing by multiplication The multiplication method for creating hash functions operates in two steps. First, we multiply the key k by a constant A in the range 0 < A < 1 and extract the fractional part of kA. Then, we multiply this value by m and take the floor of the result. In short, the hash function is h(k) = m (k A mod 1)
- 18. Universal hashing choose the hash function randomly in a way that is independent of the keys that are actually going to be stored. This approach, called universal hashing h(x) = h(y) is precisely be a finite collection of hash functions that map a given universe U of keys into the range {0,1, . . . , m - 1}.
- 19. BINARY SEARCH TREES Search trees are data structures that support many dynamic-set operations, including SEARCH, MINIMUM, MAXIMUM, PREDECESSOR, SUCCESSOR, INSERT, and DELETE.
- 20. Thus, a search tree can be used both as a dictionary and as a priority queue. Basic operations on a binary search tree take time proportional to the height of the tree. For a complete binary tree with n nodes, such operations run in (lg n) worst-case time. Example:
- 21. Querying a binary search tree the SEARCH operation, binary search trees can support such queries as MINIMUM, MAXIMUM, SUCCESSOR, and PREDECESSOR.
- 22. RED-BLACK TREES the set operations are fast if the height of the search tree is small; but if its height is large, their performance may be no better than with a linked list. Red-black trees are one of many search-tree schemes that are "balance" in order to guarantee that basic dynamic-set operations take O(lg n) time in the worst case.
- 23. Properties of red-black trees Each node of the tree now contains the fields color, key, left, right, and p. If a child or the parent of a node does not exist, the corresponding pointer field of the node contains the value NIL. red-black properties: 1) Every node has a color either red or black. 2) Root of tree is always black. 3) There are no two adjacent red nodes (A red node cannot have a red parent or red child). 4) Every path from root to a NULL node has same number of black nodes
- 24. Example
- 25. Rotations
- 26. AUGMENTING DATA STRUCTURES It introduced the notion of Dynamic order statistics size[x] = size[left[x]] + size[right[x]] + 1 .