13 A: External Algorithms II; Disjoint Sets; Java API Support

Transcription

1 13 A: External Algorithms II; ; Java API Support Martin Henz April 15, 2009 Generated on Friday A: th External April, 2009, Algorithms 12:37 II; ; Java API Support 1

2 1 External Sorting A: External Algorithms II; ; Java API Support 2

3 Model for External Sorting The Simple Algorithm Multiway Merge 1 External Sorting Model for External Sorting The Simple Algorithm Multiway Merge A: External Algorithms II; ; Java API Support 3

4 Model for External Sorting The Simple Algorithm Multiway Merge Tapes as Storage Similar to disks Access time many orders of magnitude slower than main memory Additional characteristics Large amounts of data can be read sequentially quite efficiently Access of previous locations is extremely slow, as it requires re-winding the tape! 13 A: External Algorithms II; ; Java API Support 4

5 Model for External Sorting The Simple Algorithm Multiway Merge External Sorting Main idea Use tapes sequentially, and read one block from each input tape tape Merge blocks Sort the blocks Use merge procedure from mergesort to merge 13 A: External Algorithms II; ; Java API Support 5

6 Model for External Sorting The Simple Algorithm Multiway Merge The Simple Algorithm: Overview Four tapes Two input tapes; two output tapes Read and write runs Read runs from input tape, sort them and write alternatively to output tapes Continue, writing larger runs Read two runs from each output tape, and merge them on the fly, writing alternatively to input tapes Continue until one tape has all sorted data 13 A: External Algorithms II; ; Java API Support 6

7 Model for External Sorting The Simple Algorithm Multiway Merge Multiway Merge Why only four tapes? If we have more than four tapes, we can take advantage of them by using multiway merge How finding the smallest element during merge? Priority queue! Each iteration of inner loop deletemin to find smallest element insert new element from tape from which element was deleted 13 A: External Algorithms II; ; Java API Support 7

8 Model for External Sorting The Simple Algorithm Multiway Merge Polyphase Merge and Replacement Selection Polyphase merge: main idea Make use of fewer tapes, by re-using tapes for reading and writing Leading to tape organization using kth order Fibonacci numbers Replacement selection: main idea Make use of input tape as output tape, reusing the tapes on the fly 13 A: External Algorithms II; ; Java API Support 8

9 Equivalence Relations The Dynamic Equivalence Problem Basic Data Structure Variants 1 External Sorting 2 Equivalence Relations The Dynamic Equivalence Problem Basic Data Structure Variants Applications A: External Algorithms II; ; Java API Support 9

10 Equivalence Relations The Dynamic Equivalence Problem Basic Data Structure Variants Equivalence Relations Definition An equivalence relation is a relation R that satisfies three properties: 1 (Reflexive) ara, for all a S. 2 (Symmetric) arb if and only if bra. 3 (Transitive) arb and brc implies arc. Examples Electrical connectivity (metal wires between points) Cities belonging to same country 13 A: External Algorithms II; ; Java API Support 10

11 The Dynamic Equivalence Problem Equivalence Relations The Dynamic Equivalence Problem Basic Data Structure Variants Initial setup Collection of N disjoint sets, each with one element Operations find(a): return the set of which x is element union(a, b): merge the sets to which a and b belong, so that find(a) = find(b) 13 A: External Algorithms II; ; Java API Support 11

12 Equivalence Relations The Dynamic Equivalence Problem Basic Data Structure Variants Strategies Fast Find, Slow Union Use array repres to store equivalence class for each element find(a): return repres[a] union(a, b): if repres[x] = repres[b] then set repres[x] to repres[a] Fast Union, Reasonable Find Union/find data structure 13 A: External Algorithms II; ; Java API Support 12

13 Equivalence Relations The Dynamic Equivalence Problem Basic Data Structure Variants Basic Data Structure Idea Maintain forest corresponding to equivalence relation Union Merge trees Find Return root of tree Observe Only upward direction needed! 13 A: External Algorithms II; ; Java API Support 13

14 Example External Sorting Equivalence Relations The Dynamic Equivalence Problem Basic Data Structure Variants Initial setup: After union(4, 5) 13 A: External Algorithms II; ; Java API Support 14

15 Example External Sorting Equivalence Relations The Dynamic Equivalence Problem Basic Data Structure Variants After union(4, 5) After union(6, 7) 13 A: External Algorithms II; ; Java API Support 15

16 Example External Sorting Equivalence Relations The Dynamic Equivalence Problem Basic Data Structure Variants After union(6, 7) After union(4, 6) 13 A: External Algorithms II; ; Java API Support 16

17 Equivalence Relations The Dynamic Equivalence Problem Basic Data Structure Variants Representation Idea Remember parent node only; mark root with 1 13 A: External Algorithms II; ; Java API Support 17

18 Equivalence Relations The Dynamic Equivalence Problem Basic Data Structure Variants Variants Problem How to choose root for union? Bad choice can lead to long paths Union-by-size Always make the smaller tree a subtree of the larger tree Analysis When depth increases, the tree is smaller than the other side. Thus, after union, it is at least twice as large. Height less than or equal to log N 13 A: External Algorithms II; ; Java API Support 18

19 Variants External Sorting Equivalence Relations The Dynamic Equivalence Problem Basic Data Structure Variants Union-by-height Always make the shorter tree a subtree of the higher tree Height As with union-by-size: O(log N) 13 A: External Algorithms II; ; Java API Support 19

20 Path Compression Equivalence Relations The Dynamic Equivalence Problem Basic Data Structure Variants During find make every node point to root after find(14) 13 A: External Algorithms II; ; Java API Support 20

21 A Very Slowly Growing Function Equivalence Relations The Dynamic Equivalence Problem Basic Data Structure Variants Definition log N is the number of times log needs to be applied to N until N 1. Examples log 2 = 1 log 4 = 2 log 16 = 3 log = A: External Algorithms II; ; Java API Support 21

22 Equivalence Relations The Dynamic Equivalence Problem Basic Data Structure Variants Runtime Consider variant Union-by-height combined with path compression Theorem The running time of M unions and finds is O(M log N). 13 A: External Algorithms II; ; Java API Support 22

23 Collections, Lists, Iterators Trees Hashing PriorityQueue Sorting 1 External Sorting 2 3 Collections, Lists, Iterators Trees Hashing PriorityQueue Sorting 4 13 A: External Algorithms II; ; Java API Support 23

24 The Top-level Collection Interface Collections, Lists, Iterators Trees Hashing PriorityQueue Sorting public interface C o l l e c t i o n <Any> extends I t e r a b l e <Any> { i n t size ( ) ; boolean isempty ( ) ; void c l e a r ( ) ; boolean contains ( Any x ) ; boolean add ( Any x ) ; / / s i c boolean remove ( Any x ) ; / / s i c java. u t i l. I t e r a t o r <Any> i t e r a t o r ( ) ; } 13 A: External Algorithms II; ; Java API Support 24

25 Collections, Lists, Iterators Trees Hashing PriorityQueue Sorting The List Interface in Collection API public interface L i s t <Any> extends C o l l e c t i o n <Any> { Any get ( i n t i d x ) ; Any set ( i n t idx, Any newval ) ; void add ( i n t idx, Any x ) ; void remove ( i n t i d x ) ; } L i s t I t e r a t o r <Any> l i s t I t e r a t o r ( i n t pos ) ; 13 A: External Algorithms II; ; Java API Support 25

26 ArrayList and LinkedList Collections, Lists, Iterators Trees Hashing PriorityQueue Sorting public class A r r a y L i s t <Any> implements L i s t <Any> {... } public class LinkedList<Any> implements L i s t <Any> {... } 13 A: External Algorithms II; ; Java API Support 26

27 Iterators External Sorting Collections, Lists, Iterators Trees Hashing PriorityQueue Sorting public interface I t e r a t o r <Any> { boolean hasnext ( ) ; Any next ( ) ; void remove ( ) ; } 13 A: External Algorithms II; ; Java API Support 27

28 ListIterators External Sorting Collections, Lists, Iterators Trees Hashing PriorityQueue Sorting public interface L i s t I t e r a t o r <Any> extends I t e r a t o r <Any> { boolean hasprevious ( ) ; Any previous ( ) ; void add ( Any x ) ; void set ( Any newval ) ; } 13 A: External Algorithms II; ; Java API Support 28

29 TreeSet External Sorting Collections, Lists, Iterators Trees Hashing PriorityQueue Sorting Implements Collection Guarantees O(log N) time for add, remove and contains 13 A: External Algorithms II; ; Java API Support 29

30 AbstractMap<K,V> Collections, Lists, Iterators Trees Hashing PriorityQueue Sorting Basic operations V get(k key): Returns the value to which the specified key is mapped. V put(k key, V value): Associates the specified value with the specified key in this map. Other operations containskey(key), containsvalue(val), remove(key) 13 A: External Algorithms II; ; Java API Support 30

31 TreeMap External Sorting Collections, Lists, Iterators Trees Hashing PriorityQueue Sorting Extends AbstractMap Guarantees O(log N) time for put, get, containskey, containsvalue, remove 13 A: External Algorithms II; ; Java API Support 31

32 HashMap External Sorting Collections, Lists, Iterators Trees Hashing PriorityQueue Sorting Extends AbstractMap Uses separate chaining with rehashing Rehashing is governed by initial capacity and load factor, set in constructor 13 A: External Algorithms II; ; Java API Support 32

33 HashSet External Sorting Collections, Lists, Iterators Trees Hashing PriorityQueue Sorting Implements Collection using HashMap 13 A: External Algorithms II; ; Java API Support 33

34 PriorityQueue External Sorting Collections, Lists, Iterators Trees Hashing PriorityQueue Sorting Implements Collection Efficient implementation of heap data structure Operation names: deletemin is called poll insert is called add 13 A: External Algorithms II; ; Java API Support 34

35 Sorting External Sorting Collections, Lists, Iterators Trees Hashing PriorityQueue Sorting Generic sorting supported by class Collections Uses mergesort in order to minimize number of comparisons Sorting of built-in numerical types supported by class Arrays Uses efficient implementation of quicksort, to take advantage of tight inner loop. 13 A: External Algorithms II; ; Java API Support 35

36 1 External Sorting A: External Algorithms II; ; Java API Support 36

37 Last Puzzler: Package Deal package c l i c k ; public class CodeTalk { public void d o I t ( ) { printmessage ( ) ; } void printmessage ( ) { System. out. p r i n t l n ( C l i c k ) ; } } 13 A: External Algorithms II; ; Java API Support 37

38 Last Puzzler: Package Deal package hack ; import c l i c k. CodeTalk ; public class TypeIt { private s t a t i c class C l i c k I t extends CodeTalk { void printmessage ( ) { System. out. p r i n t l n ( Hack ) ; } } public s t a t i c void main ( S t r i n g [ ] args ) { C l i c k I t c l i c k i t = new C l i c k I t ( ) ; c l i c k i t. d o I t ( ) ; } } What does clickit. doit () print? Click or Hack? 13 A: External Algorithms II; ; Java API Support 38

39 Java s Access Modifiers public : available wherever the class is available private : only available within the class protected : available in subclasses and within same package none : available within the same package 13 A: External Algorithms II; ; Java API Support 39

40 Access Modifiers Govern Inheritance Overriding only available methods A method can be overridden only when it is available according to the modifier rules. Package visibility Method printmessage can only be overridden within package click. Result Click is printed. 13 A: External Algorithms II; ; Java API Support 40

41 This Week and Beyond Thursday tutorial: Assignment 9 Friday lecture: CS1102S summary, outlook; questions? Next week: Reading week, consultation by appointment 27/4 and 28/4: no consultation 29/4, 5pm: Final 13 A: External Algorithms II; ; Java API Support 41