Scientific Programming. Data structures
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1 Scientific Programming Data structures Alberto Montresor Università di Trento 2017/11/21 This work is licensed under a Creative Commons Attribution-ShareAlike 4.0 International License.
2 Table of contents 1 Abstract data types (PSADS 1.8, 1.13) Sequences Sets Dictionaries 2 Elementary data structures implementations Linked List (PSADS ) Dynamic vectors 3 Sets, Dictionary implementations in Python Direct access tables Hash functions Hashtable implementation 4 Stack and queues Stack (PSADS ) Queue (PSADS , ) 5 Some code examples
3 Abstract data types (PSADS 1.8, 1.13) Definitions Introduction Data In a programming language, a piece of data is a value that can be assigned to a variable. Abstract data type (ADT) A mathematical model, given by a collection of values and a set of operators that can be performed on them. Primitive abstract data types Primitive abstract data types are provided directly by the language Type Operators Notes int +,-,*,/, //, % bits (2 64 1) long +,-,*,/, //, % Unlimited integer precision float +,-,*,/, //, % bits boolean and, or, not True, False Alberto Montresor (UniTN) SP - Data structure 2017/11/21 1 / 69
4 Abstract data types (PSADS 1.8, 1.13) Definitions Data types "Specification" and "implementation" of an abstract data type Specification: its manual, hides the implementation details from the user Implementation: the actual code that realizes the abstract data type Example Real numbers vs IEEE-754 Don t try this at home (or try it?): >>> Alberto Montresor (UniTN) SP - Data structure 2017/11/21 2 / 69
5 Abstract data types (PSADS 1.8, 1.13) Definitions Data structure Data structures Data structures are collections of data, characterized more by the organization of the data rather than the type of contained data. How to describe data structures a systematic approach to organize the collection of data a set of operators that enable the manipulation of the structure Characteristics of the data structures Linear / Non linear (sequence) Static / Dynamic (content variation, size variation) Alberto Montresor (UniTN) SP - Data structure 2017/11/21 3 / 69
6 Abstract data types (PSADS 1.8, 1.13) Definitions Data structures Type Java C++ Python Sequences List, Queue, Deque list, forward_list list LinkedList, vector tuple ArrayList, Stack, ArrayDeque stack queue, deque deque Sets Dictionaries Set TreeSet, HashSet, LinkedHashSet Map HashTree, HashMap, LinkedHashMap set unordered_set map unordered_map Trees set, frozenset dict Graphs Alberto Montresor (UniTN) SP - Data structure 2017/11/21 4 / 69
7 Abstract data types (PSADS 1.8, 1.13) Sequences Sequence Sequence A dynamic data structure representing an "ordered" group of elements The ordering is not defined by the content, but by the relative position inside the sequence (first element, second element, etc.) Values could appear more than once Example: [0.1, "alberto", 0.05, 0.1] is a sequence Alberto Montresor (UniTN) SP - Data structure 2017/11/21 5 / 69
8 Abstract data types (PSADS 1.8, 1.13) Sequences Sequence Operators It is possible to add / remove elements, by specifying their position s = s 1, s 2,..., s n the element s i is in position pos i It is possible to access directly some of the elements of the sequence the beginning and/or the end of the list having a reference to the position It is possible to sequentially access all the other elements Alberto Montresor (UniTN) SP - Data structure 2017/11/21 6 / 69
9 Abstract data types (PSADS 1.8, 1.13) Sequences Sequence Specification Sequence % Return True if the sequence is empty boolean isempty() % Returns the position of the first element Pos head() % Returns the position of the last element Pos tail() % Returns the position of the successor of p Pos next(pos p) % Returns the position of the predecessor of p Pos prev(pos p) Alberto Montresor (UniTN) SP - Data structure 2017/11/21 7 / 69
10 Abstract data types (PSADS 1.8, 1.13) Sequences Sequence Specification Sequence (continue) % Inserts element v of type object in position p. % Returns the position of the new element Pos insert(pos p, object v) % Removes the element contained in position p. % Returns the position of the successor of p, which % becomes successor of the predecessor of p Pos remove(pos p) % Reads the element contained in position p object read(pos p) % Writes the element v of type object in position p write(pos p, object v) Alberto Montresor (UniTN) SP - Data structure 2017/11/21 8 / 69
11 Abstract data types (PSADS 1.8, 1.13) Sets Sets Set A dynamic, non-linear data structure that stores an unordered collection of values without repetitions. We can consider a total order between elements as the order defined over their abstract data type, if present. Operators Basic operators: insert delete contains Sorting operators Maximum Minimum Set operators union intersection difference Iterators: for x in S: Alberto Montresor (UniTN) SP - Data structure 2017/11/21 9 / 69
12 Abstract data types (PSADS 1.8, 1.13) Sets Sets Specifications Set % Returns the size of the set int len() % Returns True if x belongs to the set; Python: x in S boolean contains(object x) % Inserts x in the set, if not already present add(object x) % Removes x from the set, if present discard(object x) % Returns a new set which is the union of A and B Set union(set A, Set B) % Returns a new set which is the intersection of A and B Set intersection(set A, Set B) % Returns a new set which is the difference of A and B Set difference(set A, Set B) Alberto Montresor (UniTN) SP - Data structure 2017/11/21 10 / 69
13 Abstract data types (PSADS 1.8, 1.13) Dictionaries Dictionaries Dictionary Abstract data structure that represents the mathematical concept of partial function R : D C, or key-value association Set D is the domain (elements called keys) Set C is the codomain (elements called values) Operators Lookup the value associated to a particular key, if present, None otherwise Insert a new key-value association, deleting potential association that are already present for the same key Remove an existing key-value association Alberto Montresor (UniTN) SP - Data structure 2017/11/21 11 / 69
14 Abstract data types (PSADS 1.8, 1.13) Dictionaries Dictionaries Specification Dictionary % Returns the value associated to key k, if present; returns none otherwise object lookup(object k) % Associates value v to key k insert(object k, object v) % Removes the association of key k remove(object k) Alberto Montresor (UniTN) SP - Data structure 2017/11/21 12 / 69
15 Abstract data types (PSADS 1.8, 1.13) Dictionaries Comments Concepts of sequences, sets, dictionaries are linked Key sets/ value sets Iterate over the set of keys Some realizations are "natural" Sequence list Queue queue as a list Alternative implementations exist Set as boolean vector Queue as circular vector The choice of the data structure has implications on the performances Dictionary as a hash table: lookup O(1), minimum search O(n) Dictionary as a tree: lookup O(log n), minimum search O(1) Alberto Montresor (UniTN) SP - Data structure 2017/11/21 13 / 69
16 Contents 1 Abstract data types (PSADS 1.8, 1.13) Sequences Sets Dictionaries 2 Elementary data structures implementations Linked List (PSADS ) Dynamic vectors 3 Sets, Dictionary implementations in Python Direct access tables Hash functions Hashtable implementation 4 Stack and queues Stack (PSADS ) Queue (PSADS , ) 5 Some code examples
17 Elementary data structures implementations Linked List (PSADS ) List List (Linked List) A sequence of memory objects, containing arbitrary data and 1-2 pointers to the next element and/or the previous one Note Contiguity in the list contiguity in memory All the operations require O(1), but in some cases you need a lot of single operations to complete an action Possible implementations Bidirectional / Monodirectional With sentinel / Without sentinel Circular / Non-circular Alberto Montresor (UniTN) SP - Data structure 2017/11/21 14 / 69
18 Elementary data structures implementations Linked List (PSADS ) List L v1 v2 v3 vn nil Monodirectional L nil v1 v2 v3 vn nil Bidirectional L v1 v2 v3 vn Bidirectional, circular L v1 v2 vn nil Monodirectional, with sentinel Alberto Montresor (UniTN) SP - Data structure 2017/11/21 15 / 69
19 Elementary data structures implementations Linked List (PSADS ) Monodirectional list, Python class Node: def init (self,initdata): self.data = initdata self.next = None def getdata(self): return self.data def getnext(self): return self.next def setdata(self,newdata): self.data = newdata def setnext(self,newnext): self.next = newnext Alberto Montresor (UniTN) SP - Data structure 2017/11/21 16 / 69
20 Elementary data structures implementations Linked List (PSADS ) Monodirectional list, Python class UnorderedList: def init (self): self.head = None def add(self,item): temp = Node(item) temp.setnext(self.head) self.head = temp Alberto Montresor (UniTN) SP - Data structure 2017/11/21 17 / 69
21 Elementary data structures implementations Linked List (PSADS ) Monodirectional list, Python class UnorderedList: def init (self): self.head = None def add(self,item): temp = Node(item) temp.setnext(self.head) self.head = temp Alberto Montresor (UniTN) SP - Data structure 2017/11/21 18 / 69
22 Elementary data structures implementations Linked List (PSADS ) Monodirectional list, Python class UnorderedList: # Continued def search(self,item): current = self.head found = False while current!= None and not found: if current.getdata() == item: found = True else: current = current.getnext() return found Alberto Montresor (UniTN) SP - Data structure 2017/11/21 19 / 69
23 Elementary data structures implementations Linked List (PSADS ) Monodirectional list, Python def remove(self,item): current = self.head previous = None found = False while not found: if current.getdata() == item: found = True else: previous = current current = current.getnext() if previous == None: self.head = current.getnext() else: previous.setnext(current.getnext()) Alberto Montresor (UniTN) SP - Data structure 2017/11/21 20 / 69
24 Elementary data structures implementations Dynamic vectors Dynamic vectors Lists in Python implemented through dynamic vectors A vector of a given size (initial capacity) is allocated When inserting an element before the end, all elements have to be moved - cost O(n) When inserting an element at the end (append), the cost is O(1) (just writing the element at first available slot) Problem: Solution: It is not known how many elements have to be stored The initial capacity could be insufficient A new (larger) vector is allocated, the content is copied in the new vector, the old vector is released Alberto Montresor (UniTN) SP - Data structure 2017/11/21 21 / 69
25 Elementary data structures implementations Dynamic vectors Dynamic vectors Question Which is the best approach? Approach 1 If the old vector has size n, allocate a new vector of size dn. For example, d = 2 Approach 2 If the old vector has size n, allocate a new vector of size n + d, where d is a constant. For example, d = 16 Alberto Montresor (UniTN) SP - Data structure 2017/11/21 22 / 69
26 Elementary data structures implementations Dynamic vectors Amortized analysis - doubling Actual cost{ of an append() operation: i k Z + 0 c i = : i = 2k otherwise Assumptions: Initial capacity: 1 Writing cost: Θ(1) n cost = = = = = 17 Alberto Montresor (UniTN) SP - Data structure 2017/11/21 23 / 69
27 Elementary data structures implementations Dynamic vectors Amortized analysis - doubling Actual cost of n operations append(): n T (n) = i=1 c i log n = n + j=0 2 j = n + 2 log n +1 1 n + 2 log n+1 1 = n + 2n 1 = O(n) Amortized cost of a single append(): T (n)/n = O(n) n = O(1) Alberto Montresor (UniTN) SP - Data structure 2017/11/21 24 / 69
28 Elementary data structures implementations Dynamic vectors Amortized analysis - increment Actual cost of an append() operation: { i (i mod d) = 1 c i = 1 altrimenti Assumptions: Increment: d Initial size: d Writing cost: Θ(1) In the example: d = 4 n cost d = d = d = d = 17 Alberto Montresor (UniTN) SP - Data structure 2017/11/21 25 / 69
29 Elementary data structures implementations Dynamic vectors Amortized analysis - increment Actual cost of n operations append(): n T (n) = i=1 c i n/d = n + d j j=1 n/d = n + d j j=1 ( n/d + 1) n/d = n + d 2 (n/d + 1)n n + = O(n 2 ) 2 Amortized cost of a single append(): T (n)/n = O(n2 ) = O(n) n Alberto Montresor (UniTN) SP - Data structure 2017/11/21 26 / 69
30 Elementary data structures implementations Dynamic vectors Reality check Language Data structure Expansion factor GNU C++ std::vector 2.0 Microsoft VC vector 1.5 Python list Oracle Java ArrayList 2.0 OpenSDK Java ArrayList 1.5 Alberto Montresor (UniTN) SP - Data structure 2017/11/21 27 / 69
31 Elementary data structures implementations Dynamic vectors Python List Operator Worst case Worst case amortized L.copy() Copy O(n) O(n) L.append(x) Append O(n) O(1) L.insert(i,x) Insert O(n) O(n) L.remove(x) Remove O(n) O(n) L[i] Index O(1) O(1) for x in L Iterator O(n) O(n) L[i:i+k] Slicing O(k) O(k) L.extend(s) Extend O(k) O(n + k) x in L Contains O(n) O(n) min(l), max(l) Min, Max O(n) O(n) len(l) Get length O(1) O(1) Alberto Montresor (UniTN) SP - Data structure 2017/11/21 28 / 69
32 Contents 1 Abstract data types (PSADS 1.8, 1.13) Sequences Sets Dictionaries 2 Elementary data structures implementations Linked List (PSADS ) Dynamic vectors 3 Sets, Dictionary implementations in Python Direct access tables Hash functions Hashtable implementation 4 Stack and queues Stack (PSADS ) Queue (PSADS , ) 5 Some code examples
33 Sets, Dictionary implementations in Python Dictionary: possible implementations Unordered array Ordered array Linked List RB Tree Ideal impl. insert() O(1), O(n) O(n) O(1), O(n) O(log n) O(1) lookup() O(n) O(log n) O(n) O(log n) O(1) remove() O(n) O(n) O(n) O(log n) O(1) Ideal implementation: hash tables Choose a hash function h that maps each key k U to an integer h(k) The key-value k, v is stored in a list at position h(k) This vector is called hash table Alberto Montresor (UniTN) SP - Data structure 2017/11/21 29 / 69
34 Sets, Dictionary implementations in Python Hash table Definitions All the possible keys are contained in the universe set U of size u The table is stored in list T [0... m 1] with size m An hash function is defined as: h : U {0, 1,..., m 1} Keys Alberto Montresor Cristian Consonni Alessio Guerrieri Edsger Dijkstra Hash function Hash table m-3 m-2 m-1 Alberto Montresor (UniTN) SP - Data structure 2017/11/21 30 / 69
35 Sets, Dictionary implementations in Python Collisions When two or more keys in the dictionary have the same hash values, we say that a collision has happened Ideally, we want to have hash functions with no collisions Keys Hash functions Hash table Alberto Montresor Cristian Consonni Alessio Guerrieri Edsger Dijkstra Collision m-3 m-2 m-1 Alberto Montresor (UniTN) SP - Data structure 2017/11/21 31 / 69
36 Sets, Dictionary implementations in Python Direct access tables Direct access tables In some cases: the set U is already a (small) subset of Z + The set of year days, numbered from 1 to 366 The set of Kanto s Pokemons, numbered from 1 to 151 Direct access tables We use the identity function h(k) = k as hash function We select m = u Problems If u is very large, this approach may be infeasible If u is not large but the number of keys that are actually recorded is much smaller than u = m, memory is wasted Alberto Montresor (UniTN) SP - Data structure 2017/11/21 32 / 69
37 Sets, Dictionary implementations in Python Hash functions Perfect hash functions Definition A hash function h is called perfect if h is injective, i.e. k 1, k 2 U : k 1 k 2 h(k 1 ) h(k 2 ) Examples Students ASD N. matricola in [ , ] h(k) = k , m = Studentes enrolled in 2014 N. matricola in [ , ] h(k) = k , m = Problems Universe space is often large, sparse, unknown To obtain a perfect hash function is difficult Alberto Montresor (UniTN) SP - Data structure 2017/11/21 33 / 69
38 Sets, Dictionary implementations in Python Hash functions Hash functions If collisions cannot be avoided Let s try to minimize their number We want hash functions that uniform distribute the keys into hash indexes [0... m 1] Simple uniformity Let P (k) be the probability that a key k is inserted in the table Let Q(i) be the probability that a key ends up in the i-th entry of the table Q(i) = P (k) k U:h(k)=i An hash function h has simple uniformity if: i [0,..., m 1] : Q(i) = 1/m Alberto Montresor (UniTN) SP - Data structure 2017/11/21 34 / 69
39 Sets, Dictionary implementations in Python Hash functions Hash functions To obtain a hash function with simple uniformity, distribution P should be known the probability Example if U is given by real number in [0, 1[ and each key has the same probability of being selected, then H(k) = km has simple uniformity In the real world The key distribution may unknown or partially known Heuristic techniques are used to obtain an approximation of simple uniformity Alberto Montresor (UniTN) SP - Data structure 2017/11/21 35 / 69
40 Sets, Dictionary implementations in Python Hash functions How to realize a hash function Assumption Each key can be translated in a numerical, non-negative values, by reading their internal representation as a number. Example: string transformation ord(c): ordinal binary value of character c in ASCII bin(k): binary representation of key k, by concatenating the binary values of its characters int(b): numerical value associated to the binary number b int(k) = int(bin(k)) Alberto Montresor (UniTN) SP - Data structure 2017/11/21 36 / 69
41 Sets, Dictionary implementations in Python Hash functions Division method Division method Let m be a prime number H(k) = int(k) mod m Example m = 383 H( Alberto ) = mod 383 = 221 H( Alessio ) = mod 383 = 77 H( Cristian ) = mod 383 = 130 Alberto Montresor (UniTN) SP - Data structure 2017/11/21 37 / 69
42 Sets, Dictionary implementations in Python Hash functions Multiplication method Multiplication method Le m be any size, for example 2 k Let C be a real constant, 0 < C < 1 Let i = int(k) H(k) = m(c i C i ) Example m = 2 16 C = H( Alberto ) = = H( Alessio ) = = H( Cristian ) = = Alberto Montresor (UniTN) SP - Data structure 2017/11/21 38 / 69
43 Sets, Dictionary implementations in Python Tecniche di risoluzione delle collisioni Separate chaining Liste di trabocco (chaining) Idea Gli elementi con lo stesso The valore keys hash with h vengono the same value hmemorizzati are storedin inuna a lista monodirectional Si memorizza un puntatore list / dynamic alla testa della vector lista nello slot A[h] della tabella hash The H(k)-th slot in the hash Operazioni: table contains the list/vector associated Insert: to k inserimento in testa Lookup, Delete: richiedono di scandire la lista alla ricerca della chiave Hashtable implementation 0 k 1 k 4 k 5 k 2 k 6 k 7 k 3 k 8 m 1 Alberto Montresor 20 Alberto Montresor (UniTN) SP - Data structure 2017/11/21 39 / 69
44 Sets, Dictionary implementations in Python Hashtable implementation Separate chaining: complexity analysis n m α = n/m I(α) S(α) Number of keys stored in the hash table Size of the hash table Load factor Average number of memory accesses to search a key that is not in the table (insuccess) Average number of memory accesses to search a key that is not in the table (success) Worst case analysis All the keys are inserted in a unique list insert(): Θ(1) lookup(), remove(): Θ(n) Alberto Montresor (UniTN) SP - Data structure 2017/11/21 40 / 69
45 Sets, Dictionary implementations in Python Hashtable implementation eorema: Separate chaining: complexity analysis In tavola hash con concatenamento, una ricerca senza successo richiede un tempo Average atteso Ө(1 case + α) analysis imostrazione: Let s assume the hash function has simple uniformity Una chiave non presente nella tabella può essere collocata in uno qualsiasi degli m slot Hash function computation: Θ(1), to be added to all searches How Una ricerca long the senza lists successo are? tocca tutte le chiavi nella lista corrispondente Tempo di hashing: 1 + lunghezza The expected attesa lista: length α of Θ(1+α) a list is equal to α = n/m 1 k 1 k 4 α Montresor 2 Alberto Montresor (UniTN) SP - Data structure 2017/11/21 41 / 69
46 Sets, Dictionary implementations in Python Hashtable implementation Separate chaining: complexity analysis Insuccess Success When searching for a missing key, all the keys in the list must be read When searching for a key included in the table, on average half of the keys in the list must be read. Expected cost: Θ(1) + α Expected cost: Θ(1) + α/2 Alberto Montresor (UniTN) SP - Data structure 2017/11/21 42 / 69
47 Sets, Dictionary implementations in Python Hashtable implementation Separate chaining: complexity analysis What is the meaning of the load factor? The cost factor of every operation is influenced by the cost factor If m = O(n), α = O(1) In such case, all operations are O(1) in expectation If α becomes too large, the size of the hash table can be doubled through dynamic vectors Alberto Montresor (UniTN) SP - Data structure 2017/11/21 43 / 69
48 Sets, Dictionary implementations in Python Hashtable implementation Python and hash function/tables Python sets and dict Are implemented through hash tables Sets are degenerate forms of dictionaries, where there are no values, only keys Unordered data structures Order between keys is not preserved by the hash function; this is why you get unordered results when you print them Alberto Montresor (UniTN) SP - Data structure 2017/11/21 44 / 69
49 Sets, Dictionary implementations in Python Hashtable implementation Python set Operation Average case Worst case x in S Contains O(1) O(n) S.add(x) Insert O(1) O(n) S.remove(x) Remove O(1) O(n) S T Union O(n + m) O(n m) S&T Intersection O(min(n, m)) O(n m) S-T Difference O(n) O(n m) for x in S Iterator O(n) O(n) len(s) Get length O(1) O(1) min(s), max(s) Min, Max O(n) O(n) n = len(s), m = len(t) Alberto Montresor (UniTN) SP - Data structure 2017/11/21 45 / 69
50 Sets, Dictionary implementations in Python Hashtable implementation Python dict Operation Average case Worst case x in D Contains O(1) O(n) D[] = Insert O(1) O(n) = D[] Lookup O(1) O(n) del D[] Remove O(1) O(n) for x in S Iterator O(n) O(n) len(s) Get length O(1) O(1) n = len(s), m = len(t) Alberto Montresor (UniTN) SP - Data structure 2017/11/21 46 / 69
51 Sets, Dictionary implementations in Python Hashtable implementation Implementing hash functions Rule: If two objects are equal, then their hashes should be equal If you implement eq (), then you should implement function hash () as well Rule: If two objects have the same hash, then they are likely to be equal You should avoid to return values that generate collisions in your hash function. Rule In order for an object to be hashable, it must be immutable The hash value of an object should not change over time Further details what-happens-when-you-mess-with-hashing-in-python/ Alberto Montresor (UniTN) SP - Data structure 2017/11/21 47 / 69
52 Sets, Dictionary implementations in Python Hashtable implementation Example class Point: def init (self, x, y): self.x = x self.y = y def eq (self, other): return self.x == other.x and self.y == other.y def hash (self): return hash( (self.x,self.y) ) Alberto Montresor (UniTN) SP - Data structure 2017/11/21 48 / 69
53 Contents 1 Abstract data types (PSADS 1.8, 1.13) Sequences Sets Dictionaries 2 Elementary data structures implementations Linked List (PSADS ) Dynamic vectors 3 Sets, Dictionary implementations in Python Direct access tables Hash functions Hashtable implementation 4 Stack and queues Stack (PSADS ) Queue (PSADS , ) 5 Some code examples
54 Stack and queues Stack Stack Stack A linear, dynamic data structure, in which the operation "remove" returns (and removes) a predefined element: the one that has remained in the data structure for the least time (LIFO - Last-in, First-out) Stack % Returns True if the stack is empty boolean isempty() % Returns the size of the stack boolean size() % Inserts v on top of the stack push(object v) % Removes the top element of the stack and returns it to the caller object pop() % Read the top element of the stack, without modifying it object peek() Alberto Montresor (UniTN) SP - Data structure 2017/11/21 49 / 69
55 Stack and queues Stack Stack example Stack Operation Stack Contents Return Value s.isempty() [] True s.push(4) [4] s.push( dog ) [4, dog ] s.peek() [4, dog ] dog s.push(true) [4, dog,true] s.size() [4, dog,true] 3 s.isempty() [4, dog,true] False s.push(8.4) [4, dog,true,8.4] s.pop() [4, dog,true] 8.4 s.pop() [4, dog ] True s.size() [4, dog ] 2 Alberto Montresor (UniTN) SP - Data structure 2017/11/21 50 / 69
56 Stack and queues Stack Stack Possible uses In languages like Python: In the compiler: To balance parentheses In the the interpreter: A new activation record is created for each function call top In graph analysis: To perform visits of the entire graph Possible implementations Through bidirectional lists reference to the top element Through vectors limited size, small overhead top A Alberto Montresor (UniTN) SP - Data structure 2017/11/21 51 / 69
57 Stack and queues Stack Stack in procedural languages def func3(y): z = y y = y*y return z+y def func2(x): y = x x = x*3 y = x * func3(y) return y+x def func1(w): x = w w = w*2 x = w + func2(x) return x+w Alberto Montresor (UniTN) SP - Data structure 2017/11/21 52 / 69
58 Stack and queues Stack Stack in Python class Stack: def init (self): self.items = [] def size(self): return len(self.items) def isempty(self): return self.items == [] def pop(self): return self.items.pop() def peek(self): return self.items[len(self.items)-1] def push(self, item): self.items.append(item) Alberto Montresor (UniTN) SP - Data structure 2017/11/21 53 / 69
59 Stack and queues Stack Example of application: balanced parenthesis Check whether the following sets of parentheses are balanced { { ( [ ] [ ] ) } ( ) } [ [ { { ( ( ) ) } } ] ] [ ] [ ] [ ] ( ) { } ( [ ) ] ( ( ( ) ] ) ) [ { ( ) ] These parentheses could be associated to sets, lists, tuples and/or arithmetic operations Alberto Montresor (UniTN) SP - Data structure 2017/11/21 54 / 69
60 Stack and queues Stack Example of application: balanced parenthesis def parchecker(parstring): s = Stack() index = 0 for symbol in parstring: if symbol in "([{": s.push(symbol) else: if s.isempty(): return False else: top = s.pop() if not matches(top,symbol): return False return s.isempty() Alberto Montresor (UniTN) SP - Data structure 2017/11/21 55 / 69
61 Stack and queues Stack Example of application: balanced parenthesis def matches(openpar,closepar): opens = "([{" closers = ")]}" return opens.index(openpar) == closers.index(closepar) print(parchecker( {{([][])}()} )) print(parchecker( [{()] )) Alberto Montresor (UniTN) SP - Data structure 2017/11/21 56 / 69
62 Stack and queues Queue Queue Queue A linear, dynamic data structure, in which the operation "remove" returns (and removes) a predefined element: the one that has remained in the data structure for the longest time (FIFO - First-in, First-out) Queue % Returns True if queue is empty boolean isempty() % Returns the size of the queue int size() % Inserts v at the end of the queue enqueue(object v) % Extracts q from the beginning of the queue object dequeue() % Reads the element at the top of the queue object top() Alberto Montresor (UniTN) SP - Data structure 2017/11/21 57 / 69
63 Stack and queues Queue Queue example Queue Operation Queue Contents Return Value q.isempty() [] True q.enqueue(4) [4] q.enqueue( dog ) [ dog,4] q.enqueue(true) [True, dog,4] q.size() [True, dog,4] 3 q.isempty() [True, dog,4] False q.enqueue(8.4) [8.4,True, dog,4] q.dequeue() [8.4,True, dog ] 4 q.dequeue() [8.4,True] dog q.size() [8.4,True] 2 Alberto Montresor (UniTN) SP - Data structure 2017/11/21 58 / 69
64 Stack and queues Queue Queue Possible uses To queue requests performed on a limited resource (e.g., printer) To visit graphs head tail Possible implementations Through lists add to the tail remove from the head Through circular array limited size, small overhead Alberto Montresor (UniTN) SP - Data structure 2017/11/21 59 / 69
65 Stack and queues Queue Queue in Python (wrong) class Queue: def init (self): self.items = [] def isempty(self): return self.items == [] def enqueue(self, item): self.items.insert(0,item) def dequeue(self): return self.items.pop() def size(self): return len(self.items) Alberto Montresor (UniTN) SP - Data structure 2017/11/21 60 / 69
66 Stack and queues Queue Queue based on circular array Implementation based on the modulus operation Pay attention to overflow problems (full queue) head+n head head+n enqueue(12) head head+n dequeue() 3 head+n head enqueue(15,17,33) head Alberto Montresor (UniTN) SP - Data structure 2017/11/21 61 / 69
67 Stack and queues Queue Queue based on circular array By MuhannadAjjan [CC BY-SA 4.0] ( via Wikimedia Commons Alberto Montresor (UniTN) SP - Data structure 2017/11/21 61 / 69
68 Stack and queues Queue Queue based on circular array By MuhannadAjjan [CC BY-SA 4.0] ( via Wikimedia Commons Alberto Montresor (UniTN) SP - Data structure 2017/11/21 61 / 69
69 Stack and queues Queue Queue based on circular array By MuhannadAjjan [CC BY-SA 4.0] ( via Wikimedia Commons Alberto Montresor (UniTN) SP - Data structure 2017/11/21 61 / 69
70 Stack and queues Queue Queue based on circular array By MuhannadAjjan [CC BY-SA 4.0] ( via Wikimedia Commons Alberto Montresor (UniTN) SP - Data structure 2017/11/21 61 / 69
71 Stack and queues Queue Queue based on circular array By MuhannadAjjan [CC BY-SA 4.0] ( via Wikimedia Commons Alberto Montresor (UniTN) SP - Data structure 2017/11/21 61 / 69
72 Stack and queues Queue Queue based on circular array By MuhannadAjjan [CC BY-SA 4.0] ( via Wikimedia Commons Alberto Montresor (UniTN) SP - Data structure 2017/11/21 61 / 69
73 Stack and queues Queue Queue based on circular array By MuhannadAjjan [CC BY-SA 4.0] ( via Wikimedia Commons Alberto Montresor (UniTN) SP - Data structure 2017/11/21 61 / 69
74 Stack and queues Queue Queue based on circular array By MuhannadAjjan [CC BY-SA 4.0] ( via Wikimedia Commons Alberto Montresor (UniTN) SP - Data structure 2017/11/21 61 / 69
75 Stack and queues Queue Queue based on circular array By MuhannadAjjan [CC BY-SA 4.0] ( via Wikimedia Commons Alberto Montresor (UniTN) SP - Data structure 2017/11/21 61 / 69
76 Stack and queues Queue Queue based on circular array By MuhannadAjjan [CC BY-SA 4.0] ( via Wikimedia Commons Alberto Montresor (UniTN) SP - Data structure 2017/11/21 61 / 69
77 Stack and queues Queue Queue based on circular array By MuhannadAjjan [CC BY-SA 4.0] ( via Wikimedia Commons Alberto Montresor (UniTN) SP - Data structure 2017/11/21 61 / 69
78 Stack and queues Queue Queue based on circular array By MuhannadAjjan [CC BY-SA 4.0] ( via Wikimedia Commons Alberto Montresor (UniTN) SP - Data structure 2017/11/21 61 / 69
79 Stack and queues Queue Queue based on circular array By MuhannadAjjan [CC BY-SA 4.0] ( via Wikimedia Commons Alberto Montresor (UniTN) SP - Data structure 2017/11/21 61 / 69
80 Stack and queues Queue Queue based on circular array By MuhannadAjjan [CC BY-SA 4.0] ( via Wikimedia Commons Alberto Montresor (UniTN) SP - Data structure 2017/11/21 61 / 69
81 Stack and queues Queue Queue based on circular array By MuhannadAjjan [CC BY-SA 4.0] ( via Wikimedia Commons Alberto Montresor (UniTN) SP - Data structure 2017/11/21 61 / 69
82 Stack and queues Queue Queue based on circular array By MuhannadAjjan [CC BY-SA 4.0] ( via Wikimedia Commons Alberto Montresor (UniTN) SP - Data structure 2017/11/21 61 / 69
83 Stack and queues Queue Queue based on circular array By MuhannadAjjan [CC BY-SA 4.0] ( via Wikimedia Commons Alberto Montresor (UniTN) SP - Data structure 2017/11/21 61 / 69
84 Stack and queues Queue Queue based on circular array By MuhannadAjjan [CC BY-SA 4.0] ( via Wikimedia Commons Alberto Montresor (UniTN) SP - Data structure 2017/11/21 61 / 69
85 Stack and queues Queue Queue based on circular array By MuhannadAjjan [CC BY-SA 4.0] ( via Wikimedia Commons Alberto Montresor (UniTN) SP - Data structure 2017/11/21 61 / 69
86 Stack and queues Queue Queue based on circular array By MuhannadAjjan [CC BY-SA 4.0] ( via Wikimedia Commons Alberto Montresor (UniTN) SP - Data structure 2017/11/21 61 / 69
87 Stack and queues Queue Queue based on circular array By MuhannadAjjan [CC BY-SA 4.0] ( via Wikimedia Commons Alberto Montresor (UniTN) SP - Data structure 2017/11/21 61 / 69
88 Stack and queues Queue Queue based on circular array By MuhannadAjjan [CC BY-SA 4.0] ( via Wikimedia Commons Alberto Montresor (UniTN) SP - Data structure 2017/11/21 61 / 69
89 Stack and queues Queue Queue based on circular array By MuhannadAjjan [CC BY-SA 4.0] ( via Wikimedia Commons Alberto Montresor (UniTN) SP - Data structure 2017/11/21 61 / 69
90 Stack and queues Queue Queue based on circular array By MuhannadAjjan [CC BY-SA 4.0] ( via Wikimedia Commons Alberto Montresor (UniTN) SP - Data structure 2017/11/21 61 / 69
91 Stack and queues Queue Queue based on circular array By MuhannadAjjan [CC BY-SA 4.0] ( via Wikimedia Commons Alberto Montresor (UniTN) SP - Data structure 2017/11/21 61 / 69
92 Stack and queues Queue Queue based on circular array By MuhannadAjjan [CC BY-SA 4.0] ( via Wikimedia Commons Alberto Montresor (UniTN) SP - Data structure 2017/11/21 61 / 69
93 Stack and queues Queue Queue based on circular array By MuhannadAjjan [CC BY-SA 4.0] ( via Wikimedia Commons Alberto Montresor (UniTN) SP - Data structure 2017/11/21 61 / 69
94 Stack and queues Queue Queue based on circular array By MuhannadAjjan [CC BY-SA 4.0] ( via Wikimedia Commons Alberto Montresor (UniTN) SP - Data structure 2017/11/21 61 / 69
95 Stack and queues Queue Queue based on circular array By MuhannadAjjan [CC BY-SA 4.0] ( via Wikimedia Commons Alberto Montresor (UniTN) SP - Data structure 2017/11/21 61 / 69
96 Stack and queues Queue Queue based on circular array By MuhannadAjjan [CC BY-SA 4.0] ( via Wikimedia Commons Alberto Montresor (UniTN) SP - Data structure 2017/11/21 61 / 69
97 Stack and queues Queue Queue based on circular array By MuhannadAjjan [CC BY-SA 4.0] ( via Wikimedia Commons Alberto Montresor (UniTN) SP - Data structure 2017/11/21 61 / 69
98 Stack and queues Queue Queue based on circular array By MuhannadAjjan [CC BY-SA 4.0] ( via Wikimedia Commons Alberto Montresor (UniTN) SP - Data structure 2017/11/21 61 / 69
99 Stack and queues Queue Queue based on circular array Pseudocode Queue A size head cap % Elements % Current size % Head of the queue % Maximum size Queue(self, dim) self.a new int[0... dim 1] self.cap dim self.head 0 self.size 0 top() if size > 0 then return A[head] dequeue() if size > 0 then temp A[head] head (head + 1)%cap size size 1 return temp enqueue(v) if size < cap then A[(head + size)%cap] v size size + 1 size() return size isempty() return size = 0 Alberto Montresor (UniTN) SP - Data structure 2017/11/21 62 / 69
100 Stack and queues Queue Python data structure - Dequeue >>> from collections import deque >>> Q = deque(["eric", "John", "Michael"]) >>> Q.append("Terry") # Terry arrives >>> Q.append("Graham") # Graham arrives >>> Q.popleft() # The first to arrive now leaves Eric >>> Q.popleft() # The second to arrive now leaves John >>> Q # Remaining queue in order of arrival deque([ Michael, Terry, Graham ]) Alberto Montresor (UniTN) SP - Data structure 2017/11/21 63 / 69
101 Contents 1 Abstract data types (PSADS 1.8, 1.13) Sequences Sets Dictionaries 2 Elementary data structures implementations Linked List (PSADS ) Dynamic vectors 3 Sets, Dictionary implementations in Python Direct access tables Hash functions Hashtable implementation 4 Stack and queues Stack (PSADS ) Queue (PSADS , ) 5 Some code examples
102 Some code examples Reverse string def reverse(s): n = len(s)-1 res = "" while n >= 0: res = res + s[n] n -= 1 return res Alberto Montresor (UniTN) SP - Data structure 2017/11/21 64 / 69
103 Some code examples Reverse string def reverse(s): n = len(s)-1 res = "" while n >= 0: res = res + s[n] n -= 1 return res Complexity: Θ(n 2 ) n string sums Each sum copies all the characters in a new string Alberto Montresor (UniTN) SP - Data structure 2017/11/21 64 / 69
104 Some code examples Reverse string def reverse(s): res = [] for c in s: res.insert(0, c) return "".join(res) Alberto Montresor (UniTN) SP - Data structure 2017/11/21 65 / 69
105 Some code examples Reverse string def reverse(s): res = [] for c in s: res.insert(0, c) return "".join(res) Complexity: Θ(n 2 ) n list inserts Each insert moves all characters one position up in the list Alberto Montresor (UniTN) SP - Data structure 2017/11/21 65 / 69
106 Some code examples Reverse string def reverse(s): n = len(s)-1 res = [] while n >= 0: res.append(s[n]) n -= 1 return "".join(res) Alberto Montresor (UniTN) SP - Data structure 2017/11/21 66 / 69
107 Some code examples Reverse string def reverse(s): n = len(s)-1 res = [] while n >= 0: res.append(s[n]) n -= 1 return "".join(res) Complexity: Θ(n) n list append Each append has an amortized cost of O(1) Alberto Montresor (UniTN) SP - Data structure 2017/11/21 66 / 69
108 Some code examples Reverse string def reverse(s): n = len(s)-1 res = [] while n >= 0: res.append(s[n]) n -= 1 return "".join(res) Complexity: Θ(n) n list append Each append has an amortized cost of O(1) Better solution def reverse(s): return s[::-1] Alberto Montresor (UniTN) SP - Data structure 2017/11/21 66 / 69
109 Some code examples Remove duplicates def deduplicate(l): res=[] for item in L: if item not in res: res.append(item) return res Alberto Montresor (UniTN) SP - Data structure 2017/11/21 67 / 69
110 Some code examples Remove duplicates def deduplicate(l): res=[] for item in L: if item not in res: res.append(item) return res Complexity: Θ(n 2 ) n list append n checks whether an element is already present Each check costs O(n) Alberto Montresor (UniTN) SP - Data structure 2017/11/21 67 / 69
111 Some code examples Remove duplicates def deduplicate(l): res=[] present=set() for item in L: if item not in present: res.append(item) present.add(item) return res Alberto Montresor (UniTN) SP - Data structure 2017/11/21 68 / 69
112 Some code examples Remove duplicates def deduplicate(l): res=[] present=set() for item in L: if item not in present: res.append(item) present.add(item) return res Complexity: Θ(n) n list append n checks whether an element is already present Each check costs O(1) Alberto Montresor (UniTN) SP - Data structure 2017/11/21 68 / 69
113 Some code examples Remove duplicates def deduplicate(l): res=[] present=set() for item in L: if item not in present: res.append(item) present.add(item) return res Complexity: Θ(n) n list append n checks whether an element is already present Each check costs O(1) Other possibility destroy original order def deduplicate(l): return list(set(l)) Alberto Montresor (UniTN) SP - Data structure 2017/11/21 68 / 69
114 Some code examples Exercise Queues and Priority Queues are data structures which are known to most computer scientists. The Italian Queue, however, is not so well known, though it occurs often in everyday life. At lunch time the queue in front of the cafeteria is an italian queue, for example. In an italian queue each element belongs to a group. If an element enters the queue, it first searches the queue from head to tail to check if some of its group members (elements of the same group) are already in the queue. If yes, it enters the queue right behind them. If not, it enters the queue at the tail and becomes the new last element (bad luck). Dequeuing is done like in normal queues: elements are processed from head to tail in the order they appear in the italian queue. Alberto Montresor (UniTN) SP - Data structure 2017/11/21 69 / 69
Scientific Programming. Data structures
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