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Hashing Priority Queues Puzzlers 06 B: Hashing and Priority Queues CS1102S: Data Structures and Algorithms Martin Henz February 26, 2010 Generated on Friday 26 th February, 2010, 09:23 CS1102S: Data Structures and Algorithms 06 B: Hashing

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Hashing Priority Queues Puzzlers

06 B: Hashing and Priority Queues

CS1102S: Data Structures and Algorithms

Martin Henz

February 26, 2010

Generated on Friday 26th February, 2010, 09:23 CS1102S: Data Structures and Algorithms 06 B: Hashing and Priority Queues 1

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Hashing Priority Queues Puzzlers

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Hashing

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Priority Queues

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Puzzlers

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Hashing Priority Queues Puzzlers Collision Resolution Strategies Double Hashing A Detail: Removal from Hash Table Hash Tables in the Java API

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Hashing Collision Resolution Strategies Double Hashing A Detail: Removal from Hash Table Hash Tables in the Java API

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Priority Queues

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Puzzlers

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Hashing Priority Queues Puzzlers Collision Resolution Strategies Double Hashing A Detail: Removal from Hash Table Hash Tables in the Java API

Recap: Main Ideas

Implement set as array Store values in array; compute index using a hash function. Spread The hash function should “spread” the hash keys evenly over the available hash values Collision Hash table implementations differ in their strategies of collision resolution: Two hash keys mapping to the same hash value

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Hashing Priority Queues Puzzlers Collision Resolution Strategies Double Hashing A Detail: Removal from Hash Table Hash Tables in the Java API

Separate Chaining

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Hashing Priority Queues Puzzlers Collision Resolution Strategies Double Hashing A Detail: Removal from Hash Table Hash Tables in the Java API

Hash Tables without Linked Lists

Idea Store items directly into array; use alternative cells if a collision

ccurs

More formally Try cells h0(x), h1(x), h2(x), . . . until an empty cell is found. How to define hi? hi(x) = (hash(x) + f(i)) mod TableSize where f(0) = 0. Load factor, λ Ratio of number of elements in hash table to table size.

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Hashing Priority Queues Puzzlers Collision Resolution Strategies Double Hashing A Detail: Removal from Hash Table Hash Tables in the Java API

Linear Probing

Conflict resolution f(i) = i Clustering As the load factor λ increases, occupied areas in the array tend to occur in clusters, leading to frequent unsuccessful insertion tries.

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Hashing Priority Queues Puzzlers Collision Resolution Strategies Double Hashing A Detail: Removal from Hash Table Hash Tables in the Java API

Quadratic Probing

Conflict resolution f(i) = i2 Theorem If quadratic probing is used, and the table size is prime, then a new element can always be inserted if the table is at least half empty.

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Hashing Priority Queues Puzzlers Collision Resolution Strategies Double Hashing A Detail: Removal from Hash Table Hash Tables in the Java API

Double Hashing

Idea Use a second hash function to find the jump distance Formally f(i) = i · hash2(x) Attention The function hash2 must never return 0. Why?

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Hashing Priority Queues Puzzlers Collision Resolution Strategies Double Hashing A Detail: Removal from Hash Table Hash Tables in the Java API

Double Hashing: Example

hash1(x) = x mod 10 hash2(x) = 7 − (x mod 7) hi(x) = hashi(x) + i · hash2(x)

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Hashing Priority Queues Puzzlers Collision Resolution Strategies Double Hashing A Detail: Removal from Hash Table Hash Tables in the Java API

A Detail: Removal from Hash Table

Removal from separate chaining hash table Straightforward: remove item from respective linked list (if it is there) Removal from Probing Hash Table: First idea Set the respective table entry back to null Problem This operation interrupts probing chains; elements can be “lost”

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Hashing Priority Queues Puzzlers Collision Resolution Strategies Double Hashing A Detail: Removal from Hash Table Hash Tables in the Java API

Solution

private static class HashEntry<AnyType> { public AnyType element ; public boolean i s A c t i v e ; public HashEntry ( AnyType e ) { this (e , true ) ; } public HashEntry ( AnyType e , boolean i ) { element = e ; i s A c t i v e = i ; } } public void remove ( AnyType x ) { int currentPos = findPos ( x ) ; i f ( i s A c t i v e ( currentPos ) ) array [ currentPos ] . i s A c t i v e = false ; }

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Hashing Priority Queues Puzzlers Collision Resolution Strategies Double Hashing A Detail: Removal from Hash Table Hash Tables in the Java API

Remember Sets?

Idea A Set (interface) is a Collection (interface) that does not allow duplicate entries. HashSet A HashSet is a hash table implementation of Set. class HashSet<E> implements Set<E>

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Hashing Priority Queues Puzzlers Motivation Binary Heaps Basic Heap Operations Priority Queues in Standard Library

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Hashing

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Priority Queues Motivation Binary Heaps Basic Heap Operations Priority Queues in Standard Library

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Puzzlers

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Hashing Priority Queues Puzzlers Motivation Binary Heaps Basic Heap Operations Priority Queues in Standard Library

Motivation

Operations on queues add(e): enter new element into queue remove(): remove the element that has been entered first A slight variation Priority should not be implicit, using the time of entry, but explicit, using an ordering Operations on priority queues insert(e): enter new element into queue deleteMin(): remove the smallest element

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Hashing Priority Queues Puzzlers Motivation Binary Heaps Basic Heap Operations Priority Queues in Standard Library

Application Examples

Printer queue: use number of pages as “priority” Discrete event simulation: use simulation time as “priority” Network routing: give priority to packets with strictest quality-of-service requirements

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Hashing Priority Queues Puzzlers Motivation Binary Heaps Basic Heap Operations Priority Queues in Standard Library

Simple Implementations

Unordered list: insert(e): O(1), deleteMin(): O(N) Ordered list: insert(e): O(N), deleteMin(): O(1) Search tree: insert(e): O(log N), deleteMin(): O(log N)

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Hashing Priority Queues Puzzlers Motivation Binary Heaps Basic Heap Operations Priority Queues in Standard Library

Binary Heaps

Rough Idea Keep a binary tree whose root contains the smallest element insert(e) and deleteMin() need to restore this property Completeness Keep binary tree complete, which means completely filled, with the possible exception of the bottom level, which is filled from left to right. Heap-order For every node X, the key in the parent of X is smaller than or equal to the key in X, with the exception of the root

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Hashing Priority Queues Puzzlers Motivation Binary Heaps Basic Heap Operations Priority Queues in Standard Library

Order in Binary Heap

Tree on the left is a binary heap; tree on the right is not!

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Hashing Priority Queues Puzzlers Motivation Binary Heaps Basic Heap Operations Priority Queues in Standard Library

Representation as Array

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Hashing Priority Queues Puzzlers Motivation Binary Heaps Basic Heap Operations Priority Queues in Standard Library

insert

Idea Add “hole” at bottom and “percolate” the hole up to the right place for insertion

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Hashing Priority Queues Puzzlers Motivation Binary Heaps Basic Heap Operations Priority Queues in Standard Library

Example: insert(14)

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Hashing Priority Queues Puzzlers Motivation Binary Heaps Basic Heap Operations Priority Queues in Standard Library

Example: insert(14), continued

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Hashing Priority Queues Puzzlers Motivation Binary Heaps Basic Heap Operations Priority Queues in Standard Library

Analysis

Worst case O(log N) Average 2.607 comparisons

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Hashing Priority Queues Puzzlers Motivation Binary Heaps Basic Heap Operations Priority Queues in Standard Library

deleteMin

Idea Remove root, leaving “hole” at top. “Percolate” the hole down to a correct place for insertion of bottom element

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Hashing Priority Queues Puzzlers Motivation Binary Heaps Basic Heap Operations Priority Queues in Standard Library

Example: deleteMin()

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Hashing Priority Queues Puzzlers Motivation Binary Heaps Basic Heap Operations Priority Queues in Standard Library

Example: deleteMin(), continued

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Hashing Priority Queues Puzzlers Motivation Binary Heaps Basic Heap Operations Priority Queues in Standard Library

Example: deleteMin(), continued

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Hashing Priority Queues Puzzlers Motivation Binary Heaps Basic Heap Operations Priority Queues in Standard Library

Analysis

Worst case O(log N) Average log N

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Hashing Priority Queues Puzzlers Motivation Binary Heaps Basic Heap Operations Priority Queues in Standard Library

buildHeap

Initial setup Build a heap from a given (unordered) collection of elements Idea “Percolate” every inner node down the tree

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Hashing Priority Queues Puzzlers Motivation Binary Heaps Basic Heap Operations Priority Queues in Standard Library

Example: buildHeap, percolateDown(7)

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Hashing Priority Queues Puzzlers Motivation Binary Heaps Basic Heap Operations Priority Queues in Standard Library

Example: buildHeap, percolateDown(6), ..(5)

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Hashing Priority Queues Puzzlers Motivation Binary Heaps Basic Heap Operations Priority Queues in Standard Library

Example: buildHeap, percolateDown(4), ..(3)

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Hashing Priority Queues Puzzlers Motivation Binary Heaps Basic Heap Operations Priority Queues in Standard Library

Example: buildHeap, percolateDown(2), ..(1)

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Hashing Priority Queues Puzzlers Motivation Binary Heaps Basic Heap Operations Priority Queues in Standard Library

Analysis

Bound The runtime is bounded by the sum of all heights of all nodes Theorem For perfect binary tree of height h, containing 2h+1 − 1 nodes, sum of heights of nodes is 2h+1 − 1 − (h + 1). Worst case O(N)

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Hashing Priority Queues Puzzlers Motivation Binary Heaps Basic Heap Operations Priority Queues in Standard Library

Other Heap Operations

decreaseKey(p,∆) Lowers the value of item at position p by a positive amount ∆. Implementation: Percolate up increaseKey(p,∆) Increases the value of item at position p by a positive amount ∆. Implementation: Percolate down delete(p) Remove value at position p Implementation: decreaseKey(p,∞), then deleteMin()

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Hashing Priority Queues Puzzlers Motivation Binary Heaps Basic Heap Operations Priority Queues in Standard Library

Priority Queues in Standard Library

class PriorityQueue<E> { boolean add (E e ) { . . . } / / add element E p o l l ( ) { . . . } / / remove smallest }

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Hashing Priority Queues Puzzlers Last Puzzler: “It’s Elementary” New Puzzler: The Last Laugh

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Hashing

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Priority Queues

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Puzzlers Last Puzzler: “It’s Elementary” New Puzzler: The Last Laugh

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Hashing Priority Queues Puzzlers Last Puzzler: “It’s Elementary” New Puzzler: The Last Laugh

Last Puzzler: It’s Elementary

What does the following program print? public class Elementary { public static void main ( String [ ] args ) { System . out . p r i n t l n (12345 + 5432 l ) ; } }

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Hashing Priority Queues Puzzlers Last Puzzler: “It’s Elementary” New Puzzler: The Last Laugh

Scary...

I’m so scared when try running these codes on

Eclipse. When I run the file downloaded from

the module homepage, the result is 17777. But when I type it myself , the result is 66666. Maybe the number in teacher’s file is not the normal number, right ?

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Hashing Priority Queues Puzzlers Last Puzzler: “It’s Elementary” New Puzzler: The Last Laugh

The “Fine” Print

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Hashing Priority Queues Puzzlers Last Puzzler: “It’s Elementary” New Puzzler: The Last Laugh

Constant Numbers in Java

(see Java Language Specification) 12345: int constant in decimal notation 0xff: int constant in hexadecimal notation 077: int constant in octal notation 45.23: double constant 5432l: long constant in decimal notation 0xffL: long constant in hexadecimal notation 077L: long constant in octal notation

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Hashing Priority Queues Puzzlers Last Puzzler: “It’s Elementary” New Puzzler: The Last Laugh

Useful Habit

Use “L ” (and not “l”) to indicate long literals: public class Elementary { public static void main ( String [ ] args ) { System . out . p r i n t l n (12345 + 5432L ) ; } }

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Hashing Priority Queues Puzzlers Last Puzzler: “It’s Elementary” New Puzzler: The Last Laugh

New Puzzler: The Last Laugh

What does the following program print? public class LastLaugh { public static void main ( String [ ] args ) { System . out . p r i n t l n ( ”H” + ” a ” ) ; System . out . p r i n t l n ( ’H ’ + ’a ’ ) ; } }

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Hashing Priority Queues Puzzlers Last Puzzler: “It’s Elementary” New Puzzler: The Last Laugh

Next Week

Sorting, sorting, and more sorting!

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