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 Fa 2017 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 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 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 Booean array impementation - size of the array must be equa to number of a possibe eements (the universe) //This array wi represent a set of days of the week // (Sunday, Monday, Tuesday, ) boo daysofweek[7] = {fase; //sets a eements to fase - Here is the set {Monday, Wednesday, Friday: FALSE TRUE FALSE TRUE FALSE TRUE FALSE Sunday 0 Monday 1 Tuesday 2 Wednesday - if daysofweek[1] is true, then Monday is in the Set 3 Thursday 4 Friday 5 Saturday 6 8

3 Booean array impementation - need a mapping function to convert an eement of the universe to a position in the array int map(string x) { if (x=="sunday") return 0; if (x=="monday") return 1; if (x=="tuesday") return 2; if (x=="wednesday") return 3; if (x=="thursday") return 4; if (x=="friday") return 5; if (x=="saturday") return 6; - if daysofweek[map( Monday )] is true, then Monday is in the Set 9 Booean array impementation: member boo member(string x) { int pos = map(x); if (0<=pos && pos<7) return daysofweek[pos]; return fase; Booean array impementation: union // c wi be the union of a and b: void union(boo a[], boo b[], boo c[]) { for (int pos=0; pos<7; pos++) // if either a or b is true for pos, make c true for pos c[pos] = (a[pos] b[pos]); - Exercise: impement a of the set operations for 10 the set impemented as a booean array 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 Using a hash function The function is caed a hash function 11 in the array 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 Hash(partNum) = partnum % 100 Use the hash function to pace the eement with part number

4 Pacing eements in the array How to resove the coision? Next pace part number 6702 in the array One way is by inear probing This uses the foowing function Hash(partNum) = partnum % % 100 = 2 But [2] is aready occupied (HashVaue + 1) % 100 repeatedy unti an empty ocation is found for part number 6702 COLLISION OCCURS Coision resoved Coision resoved (Hash(6702) + 1) % 100 = 3 But [3] is aready occupied (Hash(6702) + 2) % 100 = 4 Part 6702 can be paced at the ocation with index Part 6702 is paced at the ocation with index 4 Where woud the part with number 4598 be paced using inear probing? 46 47

5 Hashing concepts 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 boo find(int); //return true if int in tabe void insert (int); //add int to tabe void dispay(); //show eements in tabe // do not impement remove same ocation ; Hash function: maps a key to a Tabe index Coision: when two separate keys hash to the 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(); Simpe array impementation: HashTabe::HashTabe (int s) { size = s; array = new int[size]; //dynamic aocation for (int i=0; i<size; i++) { //set a to -1 array[i] = -1; int HashTabe::hash(int key) { return key % size; //maps keys to array position 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 void HashTabe::insert ( int eement) { int index = hash(eement); //inear probing, if not at index - if the key is found, set the status to -1 whie (array[index]!=-1 && array[index]!= eement) { index = (index+1)%size; array[index] = eement; //puts eement at first open sot 19 a chain The Find method needs to know to 20 Probem: Removing an eement in the midde of keep searching to the end of the chain

6 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 % 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 ; 22 Separate Chaining To insert a an object: - compute hash(k) - if the object is not aready in the ist at that ocation, insert the object into the ist To find an object: - compute hash(k) - search the inked ist there for the key of the object To deete an object: - compute hash(k) - search the inked ist there for the key of the object - if found, remove it 23