Lecture Notes 08

Chapter 8: Data Abstractions

Computer Science: An Overview Tenth Edition

by J. Glenn Brookshear

Presentation files modified by Farn Wang

• 8.1 Data Structure Fundamentals • 8.2 Implementing Data Structures • 8.3 A Short Case Study • 8.4 Customized Data Types • 8.5 Classes and Objects • 8.6 Pointers in Machine Language

• Homogeneous array • Heterogeneous array • List – Stack – Queue • Tree

• List: A collection of data whose entries are arranged sequentially • Head: The beginning of the list • Tail: The end of the list

Stack A list in which entries are removed and inserted only at the head • LIFO: Last-in-first-out • Top: The head of list (stack) • Bottom or base: The tail of list (stack) • Pop: To remove the entry at the top • Push: To insert an entry at the top

Queue: A list in which entries are removed at the head and are inserted at the tail • FIFO: First-in-first-out

Tree: A collection of data whose entries have a hierarchical organization • Node: An entry in a tree • Root node: The node at the top • Terminal or leaf node: A node at the bottom

• Parent: The node immediately above a specified node • Child: A node immediately below a specified node • Ancestor: Parent, parent of parent, etc. • Descendent: Child, child of child, etc. • Siblings: Nodes sharing a common parent

• Binary tree: A tree in which every node has at most two children • Depth: The number of nodes in longest path from root to leaf

• Static Data Structures: Size and shape of data structure does not change • Dynamic Data Structures: Size and shape of data structure can change • Pointers: – Used to locate data – Used for constructing dynamic structures

Novel records arranged by title but linked according to authorship

• Homogeneous arrays – Row-major order versus column major order – Address polynomial • Heterogeneous arrays – Components can be stored one after the other in a contiguous block – Components can be stored in separate locations identified by pointers

• Contiguous list: List stored in a homogeneous array • Linked list: List in which each entries are linked by pointers – Head pointer: Pointer to first entry in list – NIL pointer: A “non-pointer” value used to indicate end of list

• Stacks usually stored as contiguous lists • Queues usually stored as Circular Queues – Stored in a contiguous block in which the first entry is considered to follow the last entry – Prevents a queue from crawling out of its allotted storage space

• Linked structure – Each node = data cells + two child pointers – Accessed via a pointer to root node • Contiguous array structure – A[1] = root node – A[2],A[3] = children of A[1] – A[4],A[5],A[6],A[7] = children of A[2] and A[3]

• Ideally, a data structure should be manipulated solely by pre-defined procedures. – Example: A stack typically needs at least push and pop procedures. – The data structure along with these procedures constitutes a complete abstract tool.

procedure print_list(list) { c = list; while (c is not NIL) do { print the name pointed to by c; let c by the next element to c in the list; } }

Problem: Construct an abstract tool consisting of a list of names in alphabetical order along with the operations search, print, and insert.

• A template for a heterogeneous structure • Example: define type EmployeeType to be {char Name[25]; int Age; real SkillRating; }

• A user-defined data type with procedures for access and manipulation • Example: define type StackType to be {int StackEntries[20]; int top = 0; procedure push(value){ StackEntries[top] ← value; top = top + 1; } procedure pop . . . }

• An abstract data type with extra features – Characteristics can be inherited – Contents can be encapsulated – Constructor methods to initialize new objects

• Immediate addressing: Instruction contains the data to be accessed • Direct addressing: Instruction contains the address of the data to be accessed • Indirect addressing: Instruction contains the location of the address of the data to be accessed

• Direct addressing

PC PC 9C00 AB02

BR AB02 BR AB02

35AD AB02 35AD

• Indirect addressing 35AD

PC PC …….. 9C00 35AD

BR *AB02 BR AB02

35AD AB02 35AD

Tree applications • heirarchical network topology of network • cluster computing • circuit signal propagations

Trees • depth  log(# of nodes in the tree) • efficient membership checking • Questions ? Can we maintain the log depths with dynamic node insertion and deletion ?

Trees Maintaining the log depths with dynamic node insertion and deletion ? • trees implemented with pointers: • solutions – 2-3 trees – red-black trees – splay trees

Graphs • undirected

• directed

Graph problems • Find a path (with a cost ?) between two nodes. • Find a cycle in a graph. • Find the biconnected components • Check if a graph is planar. • ……

DAG (directed acyclic graphs) • Directed graphs without cycles • Can be used for Boolean formula presentation.

x1 x2 x3 F • Disjunctive normal form 0 0 0 0 (DNF) 0 0 1 0 F = (x x x ) (x x x ) 0 1 0 0 1 2 3 1 2 3 (x x x ) 0 1 1 1 1 2 3 1 0 0 0 • Conjunctive normal form 1 0 1 1 (CNF) 1 1 0 0 F = (x1 x2 x3)  (x1 x2x3)  1 1 1 1 (x1  x2 x3)  (x1x2x3) 

(x1  x2  x3)

• Same size as truth tables • Lots of redundancy: Out of 8 subtrees rooted at b2 only 3 are distinct!!! • Merge isomorphic subtrees  BDD

x1 0 0 0 0 1 1 1 1

x2 0 0 1 1 0 0 1 1

x3 0 1 0 1 0 1 0 1 x1 0 1 F 0 0 0 1 0 1 0 1

x2 x2 0 1 0 1

x3 x3 x3 x3 0 1 0 1 0 1 0 1 0 0 0 1 0 1 0 1

60 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Structure sharing in BDT function values (a1b1a2b2)  (a1  b1  a2  b2) determined at  (a1  b1  a2  b2) b1  (a1  b1  a2  b2)

minimumrepresentation size for Boolean formulas. representation• Motivation: (for a given var ordering)– Too much space redundancy in traditional representations – BDD is more compact than truth tables, conjunctive normal form, disjunctive normal form, binary decision trees, etc. – BDD has a canonical form

minimal canonical form of Boolean functions • minimal  efficiency (space & computation) – for a specific variable ordering x 0 • Canonical form: 1 unique BDD for a function y 0 z 1 0 1 (x  y)  z 0 0 1 0 1

Using path traversal to evaluate function

x 1 (x  y)  z x x 0 x  0 0 1  y 0 1 0 y y 0  y 0  z z z 0 1 1 1 z 0 1 0 1

0 1 0 1

at leaf level x1 x1 0 1 0 1 x x 2 x2  2 x2 1 0 1 1 0 1 0 0 x3 x3 x3 1 0 x3 0 1 0 0 1 1

0 1 0 1 0 1

at level x3 x1 x1 0 1 0 1 x x 2 x2  2 1 0 x2 1 0 1 1 0 0 x3 x3 x3 0 1 0 0 1 1

0 1 0 1

at level x2 x1 x1 0 0 1 1 x x 2 1 0 x2  2 1 1 0 0 x3 x3 0 1 0 1

0 1 0 1

x1 x2 x3 y1 y2 0 0 0 0 0 x 1 y1 0 0 1 0 1 0 1 0 0 1 x2 0 1 1 0 1 y2 x3 1 0 0 0 0 1 0 1 0 1 1 1 0 1 1 1 1 1 1 1

x1 x2 x3 y1 y2 F 0 0 0 0 0 1 x1

0 0 1 0 1 1 x2 0 1 0 0 1 1 x 0 1 1 0 1 1 3 1 0 0 0 0 1 y1 1 0 1 0 1 1 y 1 1 0 1 1 1 2

1 1 1 1 1 1 0 1 other 0

Ins CS CSC NS NSC 0 A 00 B 10 C 0,1 A 00 A 00 1 0 B 10 B 10 0 1 B 10 A 00 0 B 0 C 01 B 10 A 1 C 01 A 00 1 0 0,1

Relation i = ia1a2b1b2 a1 + a1a2b1b2 b1 + ia1a2b1b2 a + ia1a2b1b2 2

+ ia1a2b1b2 b2 0 1 + ia1a2b1b2

• Apply – NOT, AND, OR, EXOR, etc. • Quantification (existential, universal) • Replace • Compose • Specialized operators – Generalized cofactor (constrain, restrict) – Compatible projection, etc.

• Nodes (represent functions; complexity) • Terminal nodes (constant functions) • Edges (relationship between functions) • Paths (true and false var. assignments) • Cuts (variable partitions and subsets)

• Canonicity • Compactness (with some exceptions) • Representing a variety of discrete objects – Boolean functions – Compositional sets – Encodings and labelings • Facilitating symbolic methods – Two-level minimization – State traversal of FSMs – …