1 an introduction to data structures and abstract data types

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An Introduction to Data Structures and Abstract Data Types

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The Need for Data Structures

Data structures organize data

more efficient programs.

More powerful computers more complex applications.

More complex applications demand more calculations.

Complex computing tasks are unlike our everyday experience.

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Organizing Data

Any organization for a collection of records can be searched, processed in any order, or modified.

The choice of data structure and algorithm can make the difference between a program running in a few seconds or many days.

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Efficiency

A solution is said to be efficient if it solves the problem within its resource constraints.– Space– Time

• The cost of a solution is the amount of resources that the solution consumes.

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Selecting a Data Structure

Select a data structure as follows:1. Analyze the problem to determine the

resource constraints a solution must meet.

2. Determine the basic operations that must be supported. Quantify the resource constraints for each operation.

3. Select the data structure that best meets these requirements.

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Some Questions to Ask

• Are all data inserted into the data structure at the beginning, or are insertions interspersed with other operations?

• Can data be deleted?

• Are all data processed in some well-defined order, or is random access allowed?

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Data Structure Philosophy

Each data structure has costs and benefits.

Rarely is one data structure better than another in all situations.

A data structure requires:– space for each data item it stores,– time to perform each basic operation,– programming effort.

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Data Structure Philosophy (cont)

Each problem has constraints on available space and time.

Only after a careful analysis of problem characteristics can we know the best data structure for the task.

Bank example:– Start account: a few minutes– Transactions: a few seconds– Close account: overnight

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Abstract Data Types

Abstract Data Type (ADT): a definition for a data type solely in terms of a set of values and a set of operations on that data type.

Each ADT operation is defined by its inputs and outputs.

Encapsulation: Hide implementation details.

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Data Structure

• A data structure is the physical implementation of an ADT.– Each operation associated with the ADT is

implemented by one or more subroutines in the implementation.

• Data structure usually refers to an organization for data in main memory.

• File structure is an organization for data on peripheral storage, such as a disk drive.

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Metaphors

An ADT manages complexity through abstraction: metaphor.– Hierarchies of labels

Ex: transistors gates CPU.

In a program, implement an ADT, then think only about the ADT, not its implementation.

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Logical vs. Physical Form

Data items have both a logical and a physical form.

Logical form: definition of the data item within an ADT.– Ex: Integers in mathematical sense: +, -

Physical form: implementation of the data item within a data structure.– Ex: 16/32 bit integers, overflow.

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Problems

• Problem: a task to be performed.– Best thought of as inputs and matching

outputs.– Problem definition should include constraints

on the resources that may be consumed by any acceptable solution.

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Problems (cont)

• Problems mathematical functions– A function is a matching between inputs (the

domain) and outputs (the range).– An input to a function may be single number,

or a collection of information.– The values making up an input are called the

parameters of the function.– A particular input must always result in the

same output every time the function is computed.

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Algorithms and Programs

Algorithm: a method or a process followed to solve a problem.– A recipe.

An algorithm takes the input to a problem (function) and transforms it to the output.– A mapping of input to output.

A problem can have many algorithms.

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Algorithm Properties

An algorithm possesses the following properties:– It must be correct.– It must be composed of a series of concrete steps.– There can be no ambiguity as to which step will be

performed next.– It must be composed of a finite number of steps.– It must terminate.

A computer program is an instance, or concrete representation, for an algorithm in some programming language.

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Mathematical Background

Set concepts and notation.

Recursion

Induction Proofs

Logarithms

Summations

Recurrence Relations

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Estimation Techniques

1. Determine the major parameters that effect the problem.

2. Derive an equation that relates the parameters to the problem.

3. Select values for the parameters, and apply the equation to yield and estimated solution.

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Estimation Example

How many library bookcases does it take to store books totaling one million pages?

Estimate:– Pages/inch– Feet/shelf– Shelves/bookcase

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Algorithm Efficiency

There are often many approaches (algorithms) to solve a problem. How do we choose between them?

At the heart of computer program design are two (sometimes conflicting) goals.

1. To design an algorithm that is easy to understand, code, debug.

2. To design an algorithm that makes efficient use of the computer’s resources.

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Algorithm Efficiency (cont)

Goal (1) is the concern of Software Engineering.

Goal (2) is the concern of data structures and algorithm analysis.

When goal (2) is important, how do we measure an algorithm’s cost?

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How to Measure Efficiency?

Critical resources:

Factors affecting running time:

For most algorithms, running time depends on “size” of the input.

Running time is expressed as T(n) for some function T on input size n.

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Examples of Growth Rate

Example 1:

// Find largest valueint largest(int array[], int n) {int currlarge = 0; // Largest value seen for (int i=1; i<n; i++) // For each val if (array[currlarge] < array[i]) currlarge = i; // Remember pos return currlarge; // Return largest}

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Examples (cont)

Example 2: Assignment statement.

sum = 0;for (i=1; i<=n; i++) for (j=1; j<n; j++) sum++;}

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Growth Rate Graph

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Best, Worst, Average Cases

Not all inputs of a given size take the same time to run.

Sequential search for K in an array of n integers:

• Begin at first element in array and look at each element in turn until K is found

Best case:

Worst case:

Average case:

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Which Analysis to Use?

While average time appears to be the fairest measure, it may be difficult to determine.

When is the worst case time important?

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Faster Computer or Algorithm?

What happens when we buy a computer 10 times faster?

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Binary Search

How many elements are examined in worst case?

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Binary Search// Return position of element in sorted// array of size n with value K.

int binary(int array[], int n, int K) { int l = -1; int r = n; // l, r are beyond array bounds while (l+1 != r) { // Stop when l, r meet int i = (l+r)/2; // Check middle if (K < array[i]) r = i; // Left half if (K == array[i]) return i; // Found it if (K > array[i]) l = i; // Right half } return n; // Search value not in array}

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Other Control Statements

while loop: Analyze like a for loop.

if statement: Take greater complexity of then/else clauses.

switch statement: Take complexity of most expensive case.

Subroutine call: Complexity of the subroutine.

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Analyzing Problems

Upper bound: Upper bound of best known algorithm.

Lower bound: Lower bound for every possible algorithm.

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Space Bounds

Space bounds can also be analyzed with complexity analysis.

Time: AlgorithmSpace Data Structure

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Space/Time Tradeoff Principle

One can often reduce time if one is willing to sacrifice space, or vice versa.

• Encoding or packing informationBoolean flags

• Table lookupFactorials

Disk-based Space/Time Tradeoff Principle: The smaller you make the disk storage requirements, the faster your program will run.

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Lists

A list is a finite, ordered sequence of data items.

Important concept: List elements have a position.

Notation: <a0, a1, …, an-1>

What operations should we implement?

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List Implementation Concepts

Our list implementation will support the concept of a current position.

We will do this by defining the list in terms of left and right partitions.• Either or both partitions may be empty.

Partitions are separated by the fence.

<20, 23 | 12, 15>

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List ADT

template <class Elem> class List {public: virtual void clear() = 0; virtual bool insert(const Elem&) = 0; virtual bool append(const Elem&) = 0; virtual bool remove(Elem&) = 0; virtual void setStart() = 0; virtual void setEnd() = 0; virtual void prev() = 0; virtual void next() = 0;

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List ADT (cont)

virtual int leftLength() const = 0; virtual int rightLength() const = 0; virtual bool setPos(int pos) = 0; virtual bool getValue(Elem&) const = 0; virtual void print() const = 0;};

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List ADT Examples

List: <12 | 32, 15>

MyList.insert(99);

Result: <12 | 99, 32, 15>

Iterate through the whole list:

for (MyList.setStart(); MyList.getValue(it);

MyList.next()) DoSomething(it);

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List Find Function

// Return true if K is in list

bool find(List<int>& L, int K) { int it; for (L.setStart(); L.getValue(it);

L.next()) if (K == it) return true; // Found it return false; // Not found}

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Array-Based List Insert

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Array-Based List Class (1)

class AList : public List<Elem> {private: int maxSize; // Maximum size of list int listSize; // Actual elem count int fence; // Position of fence Elem* listArray; // Array holding listpublic: AList(int size=DefaultListSize) { maxSize = size; listSize = fence = 0; listArray = new Elem[maxSize]; }

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Array-Based List Class (2)

~AList() { delete [] listArray; }void clear() { delete [] listArray; listSize = fence = 0; listArray = new Elem[maxSize];}void setStart() { fence = 0; }void setEnd() { fence = listSize; }void prev() { if (fence != 0) fence--; }void next() { if (fence <= listSize) fence++; }int leftLength() const { return fence; }int rightLength() const { return listSize - fence; }

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Array-Based List Class (3)

bool setPos(int pos) { if ((pos >= 0) && (pos <= listSize)) fence = pos; return (pos >= 0) && (pos <= listSize);}

bool getValue(Elem& it) const { if (rightLength() == 0) return false; else { it = listArray[fence]; return true; }}

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Insert

// Insert at front of right partition

bool AList<Elem>::insert(const Elem& item) {

if (listSize == maxSize) return false; for(int i=listSize; i>fence; i--)

// Shift Elems up to make room listArray[i] = listArray[i-1]; listArray[fence] = item;

listSize++; // Increment list size return true;}

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Append// Append Elem to end of the list

bool AList<Elem>::append(const Elem& item) {

if (listSize == maxSize) return false; listArray[listSize++] = item; return true;}

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Remove

// Remove and return first Elem in right// partition

AList<Elem>::remove(Elem& it) { if (rightLength() == 0) return false; it = listArray[fence]; // Copy Elem for(int i=fence; i<listSize-1; i++) // Shift them down listArray[i] = listArray[i+1]; listSize--; // Decrement size return true;}

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Link Class

Dynamic allocation of new list elements.

// Singly-linked list node

class Link {public: Elem element; // Value for this node Link *next; // Pointer to next node Link(const Elem& elemval, Link* nextval =NULL) { element = elemval; next = nextval; } Link(Link* nextval =NULL) { next = nextval; }};

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Linked List Position (1)

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Linked List Position (2)

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Linked List Class (1)

/ Linked list implementation

class LList:public List<Elem> {

private: Link<Elem>* head; // Point to list header Link<Elem>* tail; // Pointer to last Elem Link<Elem>* fence;// Last element on left

int leftcnt; // Size of left int rightcnt; // Size of right void init() { // Intialization routine fence = tail = head = new Link<Elem>; leftcnt = rightcnt = 0; }

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Linked List Class (2)

void removeall() { // Return link nodes to free store

while(head != NULL) { fence = head; head = head->next; delete fence; } }public: LList(int size=DefaultListSize) { init(); } ~LList() { removeall(); } // Destructor void clear() { removeall(); init(); }

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Linked List Class (3)void setStart() { fence = head; rightcnt += leftcnt; leftcnt = 0; }void setEnd() { fence = tail; leftcnt += rightcnt; rightcnt = 0; }void next() { // Don't move fence if right empty if (fence != tail) { fence = fence->next; rightcnt--; leftcnt++; }}int leftLength() const { return leftcnt; }int rightLength() const { return rightcnt; }bool getValue(Elem& it) const { if(rightLength() == 0) return false; it = fence->next->element; return true; }

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Insertion

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Insert/Append// Insert at front of right partition

bool LList<Elem>::insert(const Elem& item) { fence->next = new Link<Elem>(item, fence->next); if (tail == fence) tail = fence->next; rightcnt++;

return true;}

// Append Elem to end of the list

bool LList<Elem>::append(const Elem& item) { tail = tail->next = new Link<Elem>(item, NULL); rightcnt++; return true;}

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Removal

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Remove// Remove and return first Elem in right// partition

bool LList<Elem>::remove(Elem& it) { if (fence->next == NULL) return false; it = fence->next->element; // Remember val

// Remember link node Link<Elem>* ltemp = fence->next; fence->next = ltemp->next; // Remove if (tail == ltemp) // Reset tail tail = fence; delete ltemp; // Reclaim space rightcnt--; return true;}

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Prev// Move fence one step left;// no change if left is empty

void LList<Elem>::prev() { Link<Elem>* temp = head; if (fence == head) return; // No prev Elem

while (temp->next!=fence) temp=temp->next; fence = temp; leftcnt--; rightcnt++;}

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Setpos// Set the size of left partition to pos

bool LList<Elem>::setPos(int pos) { if ((pos < 0) || (pos > rightcnt+leftcnt))

return false; fence = head; for(int i=0; i<pos; i++) fence = fence->next; return true;}

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Comparison of Implementations

Array-Based Lists:• Array must be allocated in advance.• No overhead if all array positions are full.

Linked Lists:• Space grows with number of elements.• Every element requires overhead.

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Space Comparison

“Break-even” point:

DE = n(P + E);

n = DE P + E

E: Space for data value.P: Space for pointer.D: Number of elements in array.

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DictionaryOften want to insert records, delete records,

search for records.

Required concepts:• Search key: Describe what we are looking for• Key comparison

– Equality: sequential search– Relative order: sorting

• Record comparison

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Comparator Class

How do we generalize comparison?• Use ==, <=, >=: Disastrous• Overload ==, <=, >=: Disastrous• Define a function with a standard name

– Implied obligation– Breaks down with multiple key fields/indices

for same object• Pass in a function

– Explicit obligation– Function parameter– Template parameter

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Comparator Example

class intintCompare {public: static bool lt(int x, int y) { return x < y; } static bool eq(int x, int y) { return x == y; } static bool gt(int x, int y) { return x > y; }};

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Comparator Example (2)class PayRoll {public: int ID; char* name;};

class IDCompare {public: static bool lt(Payroll& x, Payroll& y) { return x.ID < y.ID; }};

class NameCompare {public: static bool lt(Payroll& x, Payroll& y) { return strcmp(x.name, y.name) < 0; }};

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Dictionary ADT

// The Dictionary abstract class.

class Dictionary {public: virtual void clear() = 0; virtual bool insert(const Elem&) = 0; virtual bool remove(const Key&, Elem&) = 0; virtual bool removeAny(Elem&) = 0; virtual bool find(const Key&, Elem&) const = 0; virtual int size() = 0;};

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Stacks

LIFO: Last In, First Out.

Restricted form of list: Insert and remove only at front of list.

Notation:• Insert: PUSH• Remove: POP• The accessible element is called TOP.

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Stack ADT// Stack abtract class

class Stack {public: // Reinitialize the stack virtual void clear() = 0; // Push an element onto the top of the stack. virtual bool push(const Elem&) = 0; // Remove the element at the top of the stack. virtual bool pop(Elem&) = 0; // Get a copy of the top element in the stack virtual bool topValue(Elem&) const = 0; // Return the number of elements in the stack. virtual int length() const = 0;};

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Array-Based Stack

// Array-based stack implementationprivate: int size; // Maximum size of stack int top; // Index for top element Elem *listArray; // Array holding elements

Issues:• Which end is the top?• Where does “top” point to?• What is the cost of the operations?

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Linked Stack// Linked stack implementationprivate: Link<Elem>* top; // Pointer to first elem int size; // Count number of elems

What is the cost of the operations?

How do space requirements compare to the array-based stack implementation?

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Queues

FIFO: First in, First Out

Restricted form of list: Insert at one end, remove from the other.

Notation:• Insert: Enqueue• Delete: Dequeue• First element: Front• Last element: Rear

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Queue Implementation (1)

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Queue Implementation (2)

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Binary Trees

A binary tree is made up of a finite set of nodes that is either empty or consists of a node called the root together with two binary trees, called the left and right subtrees, which are disjoint from each other and from the root.

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Binary Tree Example

Notation: Node, children, edge, parent, ancestor, descendant, path, depth, height, level, leaf node, internal node, subtree.

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Full and Complete Binary Trees

Full binary tree: Each node is either a leaf or internal node with exactly two non-empty children.

Complete binary tree: If the height of the tree is d, then all leaves except possibly level d are completely full. The bottom level has all nodes to the left side.

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Binary Tree Node Class// Binary tree node class

class BinNodePtr : public BinNode<Elem> {private: Elem it; // The node's value BinNodePtr* lc; // Pointer to left child BinNodePtr* rc; // Pointer to right child

public: BinNodePtr() { lc = rc = NULL; } BinNodePtr(Elem e, BinNodePtr* l =NULL, BinNodePtr* r =NULL) { it = e; lc = l; rc = r; }

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Traversals

Any process for visiting the nodes in some order is called a traversal.

Any traversal that lists every node in the tree exactly once is called an enumeration of the tree’s nodes.

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Traversal Example// Return the number of nodes in the tree

int count(BinNode<Elem>* subroot) { if (subroot == NULL) return 0; // Nothing to count return 1 + count(subroot->left()) + count(subroot->right());}

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Binary Tree Implementation (1)

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Binary Tree Implementation (2)

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Array Implementation

Position 0 1 2 3 4 5 6 7 8 9 10 11

Parent -- 0 0 1 1 2 2 3 3 4 4 5

Left Child 1 3 5 7 9 11 -- -- -- -- -- --

Right Child 2 4 6 8 10 -- -- -- -- -- -- --

Left Sibling -- -- 1 -- 3 -- 5 -- 7 -- 9 --

Right Sibling -- 2 -- 4 -- 6 -- 8 -- 10 -- --

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Array Implementation

Parent (r) =

Leftchild(r) =

Rightchild(r) =

Leftsibling(r) =

Rightsibling(r) =

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Binary Search Trees

BST Property: All elements stored in the left subtree of a node with value K have values < K. All elements stored in the right subtree of a node with value K have values >= K.

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Cost of BST Operations

Find:

Insert:

Delete:

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Heaps

Heap: Complete binary tree with the heap property:

• Min-heap: All values less than child values.• Max-heap: All values greater than child values.

The values are partially ordered.

Heap representation: Normally the array-based complete binary tree representation.

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Building the Heap

(a) (4-2) (4-1) (2-1) (5-2) (5-4) (6-3) (6-5) (7-5) (7-6)(b) (5-2), (7-3), (7-1), (6-1)

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Priority Queues

A priority queue stores objects, and on request releases the object with greatest value.

Example: Scheduling jobs in a multi-tasking operating system.

The priority of a job may change, requiring some reordering of the jobs.

Implementation: Use a heap to store the priority queue.

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Sorting

Each record contains a field called the key.– Linear order: comparison.

Measures of cost:– Comparisons– Swaps

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Insertion Sort

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Insertion Sort

void inssort(Elem A[], int n) { for (int i=1; i<n; i++) for (int j=i; (j>0) && (Comp::lt(A[j], A[j-1])); j--)

swap(A, j, j-1);}

Best Case:Worst Case:Average Case:

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Bubble Sort

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Bubble Sort

void bubsort(Elem A[], int n) { for (int i=0; i<n-1; i++) for (int j=n-1; j>i; j--) if (Comp::lt(A[j], A[j-1])) swap(A, j, j-1);}

Best Case:Worst Case:Average Case:

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Selection Sort

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Selection Sort

void selsort(Elem A[], int n) { for (int i=0; i<n-1; i++) { int lowindex = i; // Remember its index for (int j=n-1; j>i; j--) // Find least if (Comp::lt(A[j], A[lowindex])) lowindex = j; // Put it in place swap(A, i, lowindex); }}

Best Case:Worst Case:Average Case:

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Pointer Swapping

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Summary of Exchange Sorting

All of the sorts so far rely on exchanges of adjacent records.

What is the average number of exchanges required?– There are n! permutations– Consider permutation X and its reverse, X’– Together, every pair requires n(n-1)/2

exchanges.

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Golden Rule of File Processing

Minimize the number of disk accesses!1. Arrange information so that you get what you want

with few disk accesses.2. Arrange information to minimize future disk accesses.

An organization for data on disk is often called a file structure.

Disk-based space/time tradeoff: Compress information to save processing time by reducing disk accesses.

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Disk Drives

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Sectors

A sector is the basic unit of I/O.

Interleaving factor: Physical distance between logically adjacent sectors on a track.

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Terms

Locality of Reference: When record is read from disk, next request is likely to come from near the same place in the file.

Cluster: Smallest unit of file allocation, usually several sectors.

Extent: A group of physically contiguous clusters.

Internal fragmentation: Wasted space within sector if record size does not match sector size; wasted space within cluster if file size is not a multiple of cluster size.

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Seek Time

Seek time: Time for I/O head to reach desired track. Largely determined by distance between I/O head and desired track.

Track-to-track time: Minimum time to move from one track to an adjacent track.

Average Seek time: Average time to reach a track for random access.

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Buffers

The information in a sector is stored in a buffer or cache.

If the next I/O access is to the same buffer, then no need to go to disk.

There are usually one or more input buffers and one or more output buffers.

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Buffer Pools

A series of buffers used by an application to cache disk data is called a buffer pool.

Virtual memory uses a buffer pool to imitate greater RAM memory by actually storing information on disk and “swapping” between disk and RAM.

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Organizing Buffer Pools

Which buffer should be replaced when new data must be read?

First-in, First-out: Use the first one on the queue.

Least Frequently Used (LFU): Count buffer accesses, reuse the least used.

Least Recently used (LRU): Keep buffers on a linked list. When buffer is accessed, bring it to front. Reuse the one at end.

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Bufferpool ADT

class BufferPool { // (1) Message Passingpublic: virtual void insert(void* space, int sz, int pos) = 0; virtual void getbytes(void* space, int sz, int pos) = 0;};

class BufferPool { // (2) Buffer Passingpublic: virtual void* getblock(int block) = 0; virtual void dirtyblock(int block) = 0; virtual int blocksize() = 0;};

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Design Issues

Disadvantage of message passing:– Messages are copied and passed back and forth.

Disadvantages of buffer passing:– The user is given access to system memory (the

buffer itself)– The user must explicitly tell the buffer pool when

buffer contents have been modified, so that modified data can be rewritten to disk when the buffer is flushed.

– The pointer might become stale when the bufferpool replaces the contents of a buffer.

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Programmer’s View of Files

Logical view of files:– An a array of bytes.– A file pointer marks the current position.

Three fundamental operations:– Read bytes from current position (move file

pointer)– Write bytes to current position (move file

pointer)– Set file pointer to specified byte position.

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C++ File Functions

#include <fstream.h>

void fstream::open(char* name, openmode mode);– Example: ios::in | ios::binary

void fstream::close();

fstream::read(char* ptr, int numbytes);

fstream::write(char* ptr, int numbtyes);

fstream::seekg(int pos);fstream::seekg(int pos, ios::curr);

fstream::seekp(int pos);fstream::seekp(int pos, ios::end);

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External Sorting

Problem: Sorting data sets too large to fit into main memory.

– Assume data are stored on disk drive.

To sort, portions of the data must be brought into main memory, processed, and returned to disk.

An external sort should minimize disk accesses.

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Model of External Computation

Secondary memory is divided into equal-sized blocks (512, 1024, etc…)

A basic I/O operation transfers the contents of one disk block to/from main memory.

Under certain circumstances, reading blocks of a file in sequential order is more efficient. (When?)

Primary goal is to minimize I/O operations.

Assume only one disk drive is available.

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Key Sorting

Often, records are large, keys are small.– Ex: Payroll entries keyed on ID number

Approach 1: Read in entire records, sort them, then write them out again.

Approach 2: Read only the key values, store with each key the location on disk of its associated record.

After keys are sorted the records can be read and rewritten in sorted order.

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Breaking a File into Runs

General approach:

– Read as much of the file into memory as possible.

– Perform an in-memory sort.– Output this group of records as a single run.

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Approaches to Search

1. Sequential and list methods (lists, tables, arrays).

2. Direct access by key value (hashing)

3. Tree indexing methods.

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Searching Ordered Arrays

• Sequential Search

• Binary Search

• Dictionary Search

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Self-Organizing Lists

Self-organizing lists modify the order of records within the list based on the actual pattern of record accesses.

Self-organizing lists use a heuristic for deciding how to reorder the list. These heuristics are similar to the rules for managing buffer pools.

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Heuristics

1. Order by actual historical frequency of access.

2. Move-to-Front: When a record is found, move it to the front of the list.

3. Transpose: When a record is found, swap it with the record ahead of it.

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Indexing

Goals:– Store large files– Support multiple search keys– Support efficient insert, delete, and range

queries

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Terms

Entry sequenced file: Order records by time of insertion.– Search with sequential search

Index file: Organized, stores pointers to actual records.– Could be organized with a tree or other data

structure.

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Terms

Primary Key: A unique identifier for records. May be inconvenient for search.

Secondary Key: An alternate search key, often not unique for each record. Often used for search key.

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Linear IndexingLinear index: Index file organized as a

simple sequence of key/record pointer pairs with key values are in sorted order.

Linear indexing is good for searching variable-length records.

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Linear Indexing

If the index is too large to fit in main memory, a second-level index might be used.

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Tree Indexing

Linear index is poor for insertion/deletion.

Tree index can efficiently support all desired operations:– Insert/delete– Multiple search keys (multiple indices)– Key range search

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Graph Applications

• Modeling connectivity in computer networks• Representing maps• Modeling flow capacities in networks• Finding paths from start to goal (AI)• Modeling transitions in algorithms• Ordering tasks• Modeling relationships (families, organizations)

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Graphs

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Paths and Cycles

Path: A sequence of vertices v1, v2, …, vn of length n-1 with an edge from vi to vi+1 for 1 <= i < n.

A path is simple if all vertices on the path are distinct.

A cycle is a path of length 3 or more that connects vi to itself.

A cycle is simple if the path is simple, except the first and last vertices are the same.

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Connected Components

An undirected graph is connected if there is at least one path from any vertex to any other.

The maximum connected subgraphs of an undirected graph are called connected components.

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Graph ADTclass Graph { // Graph abstract classpublic: virtual int n() =0; // # of vertices virtual int e() =0; // # of edges // Return index of first, next neighbor virtual int first(int) =0; virtual int next(int, int) =0; // Store new edge virtual void setEdge(int, int, int) =0; // Delete edge defined by two vertices virtual void delEdge(int, int) =0; // Weight of edge connecting two vertices virtual int weight(int, int) =0; virtual int getMark(int) =0; virtual void setMark(int, int) =0;};

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Graph Traversals

Some applications require visiting every vertex in the graph exactly once.

The application may require that vertices be visited in some special order based on graph topology.

Examples:– Artificial Intelligence Search– Shortest paths problems

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Graph Traversals

To insure visiting all vertices:

void graphTraverse(const Graph* G) { for (v=0; v<G->n(); v++) G->setMark(v, UNVISITED); // Initialize for (v=0; v<G->n(); v++) if (G->getMark(v) == UNVISITED) doTraverse(G, v);}

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The End