1. What is a Map (or Function)? Maps. CITS2200 Data Structures and Algorithms. Some definitions... Topic 16

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1 1. What is a Map (or Function)? CITS2200 Data Structures an Algorithms Topic 16 Maps Definitions what is a map (or function)? Specification List-base representation (singly linke) Sorte block representation Some efinitions... relation setofn-tuples eg. { 1,i,a, 2,ii,b, 3,iii,c, 4,iv,,... binary relation setofpairs(2-tuples) eg. { lassie, og, babushka, cat, benji, og, babushka, human,... omain setofvalueswhichcanbetakenonbythefirstitemofabinaryrelation eg. {lassie, babushka, benji, felix, tweety coomain setofvalueswhichcanbetakenonbytheseconitemofabinary relation eg. {og, cat, human, bir Reaing: Weiss, Section 6.8 c Tim French CITS2200 Maps Slie 1 c Tim French CITS2200 Maps Slie 2 Example tweety lassie babushka felix benji omain og is calle the image of lassie uner the relation og cat human bir coomain map (or function) binaryrelationinwhicheachelementintheomainismappe to at most one element in the coomain (many-to-one) eg. Shorthan notation: Affiliation = { Turing, Manchester Von Neumann, Princeton Knuth, Stanfor Minsky, MIT Dijkstra, Texas McCarthy, Stanfor eg. affiliation(knuth) = Stanfor partial map noteveryelementoftheomainhasanimageunerthemap(ie, the image is unefine for some elements) c Tim French CITS2200 Maps Slie 3 c Tim French CITS2200 Maps Slie 4

2 2. Asie: Why Stuy Maps? 3. Map Specification AJavamethoisafunctionormap whyimplementourownmapasan ADT? Create, moify, an elete maps uring use. eg. a map of affiliations may change over time Turing starte in Cambrige, but move to Manchester after the war. AJavaprogramcannotmoify itself(antherefore its methos) uring execution (some languages, eg Prolog, can!) Java methos just return a result we want more functionality (eg. ask is the map efine for a particular omain element? ) Constructor 1. Map(): createanewmapthatisunefineforallomainelements. Checkers 2. isempty(): return true if the map is empty (unefine for all omain elements), false otherwise. 3. isdefine(): returntrue if the image of is efine, false otherwise. Manipulators 4. assign(,c): assign c as the image of. 5. image(): return the image of if it is efine, otherwise throw an exception. 6. eassign(): iftheimageof is efine return the image an make it unefine, otherwise throw an exception. c Tim French CITS2200 Maps Slie 5 c Tim French CITS2200 Maps Slie 6 4. List-base Representation Implementation... Amapcanbeconsieretobealistofpairs.Proviingthislist is finite, itcanbe implemente using one of the techniques use to implement the list ADT. Better still, it can be built using the list ADT! (Proviing it can be one efficiently recall the example of overwrite, usinginsert an elete, inatexteitorbaseonthelistadt.) public class MapLinke { private ListLinke list; public MapLinke () { list = new ListLinke(); Question: Which List ADT shoul we use? Require arbitrarily many assignments. Do we nee previous? c Tim French CITS2200 Maps Slie 7 c Tim French CITS2200 Maps Slie 8

3 4.1 Pairs We sai a (finite) map coul be consiere a list of pairs nee to efine a Pair object... public class Pair { public Object item1; public Object item2; public Pair (Object i1, Object i2) { item1 = i1; item2 = i2; // the first item (or omain item) // the secon item (or coomain item) // etermine whether this pair is the same as the object passe // assumes appropriate equals methos for the components public boolean equals(object o) { if (o == null) return false; else if (!(o instanceof Pair)) return false; else return item1.equals( ((Pair)o).item1) && item2.equals( ((Pair)o).item2); // generate a string representation of the pair public String tostring() { return "< "+item1.tostring()+", "+item2.tostring()+" >"; c Tim French CITS2200 Maps Slie 9 c Tim French CITS2200 Maps Slie Example Implementation of image Notes: public Object image (Object ) throws ItemNotFoun { WinowLinke w = new WinowLinke(); list.beforefirst(w); list.next(w); while (!list.isafterlast(w) &&!((Pair)list.examine(w)).item1.equals() ) list.next(w); if (!list.isafterlast(w)) return ((Pair)list.examine(w)).item2; else throw new ItemNotFoun("no image for object passe"); 1.!list.isAfterLast(w) must precee list.examine(w) in the conition for the loop why?? 2. Note use of parentheses aroun casting so that the fiel reference (eg.item1) applies to the cast object (Pair rather than Object). 3. Assumes appropriate equals methos for each of the items in a pair. 4.3 Performance Map an isempty make trivial calls to constant-time list ADT commans. The other four operations all require a sequential search within the list in the size of the efine omain (O(n)) Performance using (singly linke) List ADT Operation Map 1 isempty 1 isdefine n assign n image n eassign n linear If the maximum number of pairs is preefine, an we can specify atotalorering on the omain, better efficiency is possible... c Tim French CITS2200 Maps Slie 11 c Tim French CITS2200 Maps Slie 12

4 5. Sorte-block Representation 5.1 Binary Search Some of the above operations take linear time because they nee to search for a omain element. The above program oes a linear search. Q: Are any more efficient searches available for arbitrary linke list? An algorithm for binary search... u m u u m l l =u l m l c Tim French CITS2200 Maps Slie 13 c Tim French CITS2200 Maps Slie 14 Assume block is efine as: private Pair[] block; Then binary search can be implemente as follows... Note: compareto is an instance metho of String returns 0 if its argument matches the String, a value < 0 if the String is lexicographically less than the argument, an a value > 0 otherwise. Exercise: Can bsearch be implemente using only the abstract operations of the list ADT? protecte int bsearch (Object, int l, int u) { if (l == u) { if (.tostring().compareto(block[l].item1.tostring()) ==0) return l; else return -1; else { int m = (l + u) / 2; if (.tostring().compareto(block[m].item1.tostring()) <=0) return bsearch(,l,m); else return bsearch(,m+1,u); c Tim French CITS2200 Maps Slie 15 c Tim French CITS2200 Maps Slie 16

5 5.2 Performance of Binary Search 6. Comparative Performance of Operations One way of looking at the problem, to get a feel for it, is to consier the biggest list of pairs we can fin a solution for with m calls to bsearch. Calls to bsearch Size of list m (2 m 2 +2 m )+1 =(2 m 1 1) + 1 =2 m 1 It can be shown (see Exercises) that T n is O(log n). isdefine an image simply require binary search, therefore they are O(log n) much better than singly linke list representation. However, since the block is sorte, both assign an eassign may nee to move blocks of items to maintain the orer. Thus they are In summary... max(o(log n),o(n)) = O(n). Operation Linke List Sorte Block Map 1 1 isempty 1 1 isdefine n log n assign n n image n log n eassign n n c Tim French CITS2200 Maps Slie 17 c Tim French CITS2200 Maps Slie 18 Sorte block may be best choice if: 1. map has fixe maximum size 2. omain is totally orere 3. map is fairly static mostly reaing (isdefine, image) ratherthanwriting (assign, eassign) Otherwise linke list representation is probably better. 7. Summary A map (or function) is a many-to-one binary relation. Implementation using linke list can be arbitrarily large reaing from an writing to the map takes linear time Sorte block implementation fixe maximum size requires orere omain reaing is logarithmic, writing is linear c Tim French CITS2200 Maps Slie 19 c Tim French CITS2200 Maps Slie 20