l A set is a collection of objects of the same l {6,9,11,-5} and {11,9,6,-5} are equivalent. l There is no first element, and no successor of 9.

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1 Sets & Hash Tabes Week 13 Weiss: chapter 20 CS 5301 Spring 2018 What are sets? A set is a coection of objects of the same type that has the foowing two properties: - there are no dupicates in the coection - the order of the objects in the coection is irreevant {6,9,11,-5 and {11,9,6,-5 are equivaent Ji Seaman 1 There is no first eement, and no successor of 9 2 Set Operations Set Operations Set construction - the empty set (0 eements in the set) is() - True, if the set is empty; fase, otherwise Insert(eement) - If eement is aready in the set, do nothing; otherwise add it to the set Deete(eement) - If eement is not a member of the set, do nothing; Member(eement): booean - True, if eement is a member of the set; fase, otherwise Union(Set1,Set2): Set - returns a Set containing a eements of the two Sets, no dupications Intersection(Set1,Set2): Set - returns a Set containing a eements common to both sets otherwise remove it from the set 3 4

2 Set Operations Difference(Set1,Set2): Set - returns a Set containing a eements of the first set except for the eements that are in common with the second set Subset(Set1,Set2): booean - True, if Set1 is a subset of Set2 (if a eements of the Set1 are aso eements of Set2) Equas(Set1,Set2): booean - True, if both sets contain exacty the same eements 5 Impementation Array of eements impementation - each eement of the set wi occupy a position in the array - the member (find) operation wi be inefficient, must use inear search cass IntSet { int count; //number of eements in the set, set to 0 in constr int intset[100]; //stores the eements in positions 0count - insert must not add dupicates: void insert(int x) { if (!member(x) && count<100) { intset[count] = x; count++; 6 Impementation Array of eements impementation: member boo member(int x) { for (int i=0; i<count; i++) { if (intset[i]==x) return true; return fase; Array of eements impementation: union IntSet operator+(intset rhs) { IntSet newset; for (int i=0; i<count; i++) newsetinsert(intset[i]); for (int i=0; i<rhscount; i++) newsetinsert(rhsintset[i]); return newset; - Exercise: impement a of the set operations for 7 the IntSet What are hash tabes? A Hash Tabe is used to impement a set (or a search tabe), providing basic operations in constant time (no oops/recursion): - insert - deete (optiona) - find (aso caed member ) - make (need not be constant time) It uses a function that maps an object in the set (a key) to its ocation in the tabe The function is caed a hash function 8

3 Using a hash function Pacing eements in the array vaues HandyParts company makes no more than 100 different parts But the parts a have four digit numbers This hash function can be used to store and retrieve parts in an array vaues Next pace part number 6702 in the array Hash(partNum) = partnum % % 100 = 2 10 Hash(partNum) = partnum % 100 Use the hash function to pace the eement with part number in the array 41 But vaues[2] is aready occupied COLLISION OCCURS 43 How to resove the coision? Coision resoved vaues One way is by inear probing This uses the foowing function (HashVaue + 1) % 100 repeatedy unti an empty ocation is found for part number 6702 vaues (Hash(6702) + 1) % 100 = 3 But vaues[3] is aready occupied (Hash(6702) + 2) % 100 = 4 Part 6702 can be paced at the ocation with index

4 Coision resoved Hashing concepts vaues 6702 Part 6702 is paced at the ocation with index 4 Where woud the part with number 4598 be paced using inear probing? Hash Tabe: (usuay an array) where objects are stored according to their key - key: attribute of an object used for searching/ sorting - number of vaid keys usuay greater than number of sots in the tabe - number of keys in use usuay much smaer than tabe size Hash function: maps a key to a Tabe index Coision: when two separate keys hash to the same ocation Impementation Simpe array impementation - keys are ints, a greater than or equa to 0: cass HashTabe { private: int *array; // array of int eements // use -1 to indicate empty sot int size; // size of array int hash (int key) ; // maps key to position in array pubic: HashTabe (int size); //initiaize a eements to -1 ~HashTabe(); ; boo find(int); void insert (int); void dispay(); //return true if int in tabe //add int to tabe //show eements in tabe 15 Impementation Simpe array impementation: HashTabe::HashTabe (int s) { size = s; array = new int[size]; //dynamic aocation for (int i=0; i<size; i++) { //set a vaues to -1 array[i] = -1; int HashTabe::hash(int key) { return key % size; //maps keys to array position void HashTabe::insert ( int eement) { int index = hash(eement); //inear probing, if not at index whie (array[index]!=-1 && array[index]!= eement) { index = (index+1)%size; array[index] = eement; //puts eement at first open sot 16

5 Coision Resoution: Linear Probing Insert: When there is a coision, search sequentiay for the next open sot (-1) - Put the vaue in the tabe at that position Find: if the key is not at the hashed ocation, keep searching sequentiay for it - if it reaches an open sot (-1), the key is not found Remove: if the key is not at the hashed ocation, keep searching sequentiay for it - if the key is found, set the status to -1 Probem: Removing an eement in the midde of a chain The Find method needs to know to 17 keep searching to the end of the chain Coision Resoution: Separate chaining Use an array of inked ists for the hash tabe Each inked ist contains a objects that hashed to that ocation - no coisions Hash function is sti: h(k) = k % Impementation Separate Chaining Array of inked ists impementation - The data structure: cass ChainedTabe { private: static const int SIZE = 10; struct Node { int key; node *nextnode; ; Node* tabe[size]; //array of pointers to Nodes int hash (int key) ; // maps key to position in array pubic: ChainedTabe(); //inits a pointers in array to NULL boo find(int); //return true if int in tabe void insert (int); //add int to tabe ; 19 To insert a an object: - if the object is not aready in the ist at that ocation, insert the object into the ist To find an object: - search the inked ist there for the key of the object To deete an object: - search the inked ist there for the key of the object - if found, remove it 20