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Elementary Data Structures

Elementary Data Structures. Stacks, Queues, Lists, Vectors, Sequences, Trees, Priority Queues, Heaps, Dictionaries & Hash Tables. The Stack ADT stores arbitrary objects Insertions and deletions follow the last-in first-out scheme Think of a spring-loaded plate dispenser

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Elementary Data Structures

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  1. Elementary Data Structures Stacks, Queues, Lists, Vectors, Sequences, Trees, Priority Queues, Heaps, Dictionaries & Hash Tables

  2. The Stack ADT stores arbitrary objects Insertions and deletions follow the last-in first-out scheme Think of a spring-loaded plate dispenser Main stack operations: push(object): inserts an element object pop(): removes and returns the last inserted element Auxiliary stack operations: object top(): returns the last inserted element without removing it integer size(): returns the number of elements stored boolean isEmpty(): indicates whether no elements are stored The Stack ADT (§2.1.1) Elementary Data Structures

  3. Applications of Stacks • Direct applications • Page-visited history in a Web browser • Undo sequence in a text editor • Chain of method calls in the Java Virtual Machine or C++ runtime environment • Indirect applications • Auxiliary data structure for algorithms • Component of other data structures Elementary Data Structures

  4. The Queue ADT stores arbitrary objects Insertions and deletions follow the first-in first-out scheme Insertions are at the rear of the queue and removals are at the front of the queue Main queue operations: enqueue(object): inserts an element at the end of the queue object dequeue(): removes and returns the element at the front of the queue Auxiliary queue operations: object front(): returns the element at the front without removing it integer size(): returns the number of elements stored boolean isEmpty(): indicates whether no elements are stored Exceptions Attempting the execution of dequeue or front on an empty queue throws an EmptyQueueException The Queue ADT (§2.1.2) Elementary Data Structures

  5. Applications of Queues • Direct applications • Waiting lines • Access to shared resources (e.g., printer) • Multiprogramming • Indirect applications • Auxiliary data structure for algorithms • Component of other data structures Elementary Data Structures

  6. The Position ADT models the notion of place within a data structure where a single object is stored It gives a unified view of diverse ways of storing data, such as a cell of an array a node of a linked list Just one method: object element(): returns the element stored at the position Position ADT Elementary Data Structures

  7. The List ADT models a sequence of positions storing arbitrary objects It allows for insertion and removal in the “middle” Query methods: isFirst(p), isLast(p) Accessor methods: first(), last() before(p), after(p) Update methods: replaceElement(p, o), swapElements(p, q) insertBefore(p, o), insertAfter(p, o), insertFirst(o), insertLast(o) remove(p) List ADT (§2.2.2) Elementary Data Structures

  8. Singly Linked List • A singly linked list is a concrete data structure consisting of a sequence of nodes • Each node stores • element • link to the next node next node elem  A B C D Elementary Data Structures

  9. Doubly Linked List prev next • A doubly linked list provides a natural implementation of the List ADT • Nodes implement Position and store: • element • link to the previous node • link to the next node • Special trailer and header nodes elem node trailer nodes/positions header elements Elementary Data Structures

  10. The Vector ADT extends the notion of array by storing a sequence of arbitrary objects An element can be accessed, inserted or removed by specifying its rank (number of elements preceding it) An exception is thrown if an incorrect rank is specified (e.g., a negative rank) Main vector operations: object elemAtRank(integer r): returns the element at rank r without removing it object replaceAtRank(integer r, object o): replace the element at rank with o and return the old element insertAtRank(integer r, object o): insert a new element o to have rank r object removeAtRank(integer r): removes and returns the element at rank r Additional operations size() and isEmpty() The Vector ADT Elementary Data Structures

  11. Applications of Vectors • Direct applications • Sorted collection of objects (elementary database) • Indirect applications • Auxiliary data structure for algorithms • Component of other data structures Elementary Data Structures

  12. The Sequence ADT is the union of the Vector and List ADTs Elements accessed by Rank, or Position Generic methods: size(), isEmpty() Vector-based methods: elemAtRank(r), replaceAtRank(r, o), insertAtRank(r, o), removeAtRank(r) List-based methods: first(), last(), before(p), after(p), replaceElement(p, o), swapElements(p, q), insertBefore(p, o), insertAfter(p, o), insertFirst(o), insertLast(o), remove(p) Bridge methods: atRank(r), rankOf(p) Sequence ADT Elementary Data Structures

  13. Applications of Sequences • The Sequence ADT is a basic, general-purpose, data structure for storing an ordered collection of elements • Direct applications: • Generic replacement for stack, queue, vector, or list • small database (e.g., address book) • Indirect applications: • Building block of more complex data structures Elementary Data Structures

  14. Computers”R”Us Sales Manufacturing R&D US International Laptops Desktops Europe Asia Canada Trees (§2.3) • In computer science, a tree is an abstract model of a hierarchical structure • A tree consists of nodes with a parent-child relation • Applications: • Organization charts • File systems • Programming environments Elementary Data Structures

  15. A C D B E G H F K I J Tree Terminology • Subtree: tree consisting of a node and its descendants • Root: node without parent (A) • Internal node: node with at least one child (A, B, C, F) • External node (a.k.a. leaf ): node without children (E, I, J, K, G, H, D) • Ancestors of a node: parent, grandparent, grand-grandparent, etc. • Depth of a node: number of ancestors • Height of a tree: maximum depth of any node (3) • Descendant of a node: child, grandchild, grand-grandchild, etc. subtree Elementary Data Structures

  16. Tree ADT (§2.3.1) • Query methods: • boolean isInternal(p) • boolean isExternal(p) • boolean isRoot(p) • Update methods: • swapElements(p, q) • object replaceElement(p, o) • Additional update methods may be defined by data structures implementing the Tree ADT • We use positions to abstract nodes • Generic methods: • integer size() • boolean isEmpty() • objectIterator elements() • positionIterator positions() • Accessor methods: • position root() • position parent(p) • positionIterator children(p) Elementary Data Structures

  17. Preorder Traversal (§2.3.2) AlgorithmpreOrder(v) visit(v) foreachchild w of v preorder (w) • A traversal visits the nodes of a tree in a systematic manner • In a preorder traversal, a node is visited before its descendants • Application: print a structured document 1 Make Money Fast! 2 5 9 1. Motivations 2. Methods References 6 7 8 3 4 2.3 BankRobbery 2.1 StockFraud 2.2 PonziScheme 1.1 Greed 1.2 Avidity Elementary Data Structures

  18. Postorder Traversal (§2.3.2) AlgorithmpostOrder(v) foreachchild w of v postOrder (w) visit(v) • In a postorder traversal, a node is visited after its descendants • Application: compute space used by files in a directory and its subdirectories 9 cs16/ 8 3 7 todo.txt1K homeworks/ programs/ 4 5 6 1 2 Robot.java20K h1c.doc3K h1nc.doc2K DDR.java10K Stocks.java25K Elementary Data Structures

  19. Inorder Traversal AlgorithminOrder(v) ifisInternal (v) inOrder (leftChild (v)) visit(v) ifisInternal (v) inOrder (rightChild (v)) • In an inorder traversal a node is visited after its left subtree and before its right subtree • Application: draw a binary tree • x(v) = inorder rank of v • y(v) = depth of v 6 2 8 1 4 7 9 3 5 Elementary Data Structures

  20. Binary Trees (§2.3.3) • Applications: • arithmetic expressions • decision processes • searching • A binary tree is a tree with the following properties: • Each internal node has two children • The children of a node are an ordered pair • We call the children of an internal node left child and right child • Alternative recursive definition: a binary tree is either • a tree consisting of a single node, or • a tree whose root has an ordered pair of children, each of which is a binary tree A C B D E F G I H Elementary Data Structures

  21. Properties of Binary Trees • Properties: • e = i +1 • n =2e -1 • h  i • h  (n -1)/2 • e 2h • h log2e • h log2 (n +1)-1 • Notation n number of nodes e number of external nodes i number of internal nodes h height Elementary Data Structures

  22. A … B D C E F J G H Array-Based Representation of Binary Trees • nodes are stored in an array 1 2 3 • let rank(node) be defined as follows: • rank(root) = 1 • if node is the left child of parent(node), rank(node) = 2*rank(parent(node)) • if node is the right child of parent(node), rank(node) = 2*rank(parent(node))+1 4 5 6 7 10 11 Elementary Data Structures

  23. A priority queue stores a collection of items An item is a pair(key, element) Main methods of the Priority Queue ADT insertItem(k, o)inserts an item with key k and element o removeMin()removes the item with smallest key and returns its element Additional methods minKey(k, o)returns, but does not remove, the smallest key of an item minElement()returns, but does not remove, the element of an item with smallest key size(), isEmpty() Applications: Standby flyers Auctions Stock market Priority Queue ADT Elementary Data Structures

  24. A heap is a binary tree storing keys at its internal nodes and satisfying the following properties: Heap-Order: for every internal node v other than the root,key(v)key(parent(v)) Complete Binary Tree: let h be the height of the heap for i = 0, … , h - 1, there are 2i nodes of depth i at depth h- 1, the internal nodes are to the left of the external nodes The last node of a heap is the rightmost internal node of depth h- 1 What is a heap (§2.4.3) 2 5 6 9 7 last node Elementary Data Structures

  25. Height of a Heap (§2.4.3) • Theorem: A heap storing nkeys has height O(log n) Proof: (we apply the complete binary tree property) • Let h be the height of a heap storing n keys • Since there are 2i keys at depth i=0, … , h - 2 and at least one key at depth h - 1, we have n1 + 2 + 4 + … + 2h-2 + 1 • Thus, n2h-1 , i.e., hlog n + 1 depth keys 0 1 1 2 h-2 2h-2 h-1 1 Elementary Data Structures

  26. The dictionary ADT models a searchable collection of key-element items The main operations of a dictionary are searching, inserting, and deleting items Multiple items with the same key are allowed Applications: address book credit card authorization mapping host names (e.g., cs16.net) to internet addresses (e.g., 128.148.34.101) Dictionary ADT methods: findElement(k): if the dictionary has an item with key k, returns its element, else, returns the special element NO_SUCH_KEY insertItem(k, o): inserts item (k, o) into the dictionary removeElement(k): if the dictionary has an item with key k, removes it from the dictionary and returns its element, else returns the special element NO_SUCH_KEY size(), isEmpty() keys(), Elements() Dictionary ADT Elementary Data Structures

  27. Binary Search • Binary search performs operation findElement(k) on a dictionary implemented by means of an array-based sequence, sorted by key • similar to the high-low game • at each step, the number of candidate items is halved • terminates after a logarithmic number of steps • Example: findElement(7) 0 1 3 4 5 7 8 9 11 14 16 18 19 m h l 0 1 3 4 5 7 8 9 11 14 16 18 19 m h l 0 1 3 4 5 7 8 9 11 14 16 18 19 m h l 0 1 3 4 5 7 8 9 11 14 16 18 19 l=m =h Elementary Data Structures

  28. A lookup table is a dictionary implemented by means of a sorted sequence We store the items of the dictionary in an array-based sequence, sorted by key We use an external comparator for the keys Performance: findElement takes O(log n) time, using binary search insertItem takes O(n) time since in the worst case we have to shift n/2 items to make room for the new item removeElement take O(n) time since in the worst case we have to shift n/2 items to compact the items after the removal The lookup table is effective only for dictionaries of small size or for dictionaries on which searches are the most common operations, while insertions and removals are rarely performed (e.g., credit card authorizations) Lookup Table Elementary Data Structures

  29. A binary search tree is a binary tree storing keys (or key-element pairs) at its internal nodes and satisfying the following property: Let u, v, and w be three nodes such that u is in the left subtree of v and w is in the right subtree of v. We have key(u)key(v) key(w) External nodes do not store items An inorder traversal of a binary search trees visits the keys in increasing order 6 2 9 1 4 8 Binary Search Tree Elementary Data Structures

  30. A hash functionh maps keys of a given type to integers in a fixed interval [0, N- 1] Example:h(x) =x mod Nis a hash function for integer keys The integer h(x) is called the hash value of key x A hash table for a given key type consists of Hash function h Array (called table) of size N When implementing a dictionary with a hash table, the goal is to store item (k, o) at index i=h(k) Hash Functions and Hash Tables (§2.5.2) Elementary Data Structures

  31. 0  1 025-612-0001 2 981-101-0002 3  4 451-229-0004 … 9997  9998 200-751-9998 9999  Example • We design a hash table for a dictionary storing items (SSN, Name), where SSN (social security number) is a nine-digit positive integer • Our hash table uses an array of sizeN= 10,000 and the hash functionh(x) = last four digits of x Elementary Data Structures

  32. A hash function is usually specified as the composition of two functions: Hash code map:h1:keysintegers Compression map:h2: integers [0, N- 1] The hash code map is applied first, and the compression map is applied next on the result, i.e., h(x) = h2(h1(x)) The goal of the hash function is to “disperse” the keys in an apparently random way Hash Functions (§ 2.5.3) Elementary Data Structures

  33. Memory address: We reinterpret the memory address of the key object as an integer (default hash code of all Java objects) Good in general, except for numeric and string keys Integer cast: We reinterpret the bits of the key as an integer Suitable for keys of length less than or equal to the number of bits of the integer type (e.g., byte, short, int and float in Java) Component sum: We partition the bits of the key into components of fixed length (e.g., 16 or 32 bits) and we sum the components (ignoring overflows) Suitable for numeric keys of fixed length greater than or equal to the number of bits of the integer type (e.g., long and double in Java) Hash Code Maps (§2.5.3) Elementary Data Structures

  34. Polynomial accumulation: We partition the bits of the key into a sequence of components of fixed length (e.g., 8, 16 or 32 bits)a0 a1 … an-1 We evaluate the polynomial p(z)= a0+a1 z+a2 z2+ … … +an-1zn-1 at a fixed value z, ignoring overflows Especially suitable for strings (e.g., the choice z =33gives at most 6 collisions on a set of 50,000 English words) Polynomial p(z) can be evaluated in O(n) time using Horner’s rule: The following polynomials are successively computed, each from the previous one in O(1) time p0(z)= an-1 pi(z)= an-i-1 +zpi-1(z) (i =1, 2, …, n -1) We have p(z) = pn-1(z) Hash Code Maps (cont.) Elementary Data Structures

  35. Division: h2 (y) = y mod N The size N of the hash table is usually chosen to be a prime The reason has to do with number theory and is beyond the scope of this course Multiply, Add and Divide (MAD): h2 (y) =(ay + b)mod N a and b are nonnegative integers such thata mod N 0 Otherwise, every integer would map to the same value b Compression Maps (§2.5.4) Elementary Data Structures

  36. Collisions occur when different elements are mapped to the same cell Chaining: let each cell in the table point to a linked list of elements that map there Chaining is simple, but requires additional memory outside the table 0  1 025-612-0001 2  3  4 451-229-0004 981-101-0004 Collision Handling (§ 2.5.5) Elementary Data Structures

  37. Open addressing: the colliding item is placed in a different cell of the table Linear probing handles collisions by placing the colliding item in the next (circularly) available table cell Each table cell inspected is referred to as a “probe” Colliding items lump together, causing future collisions to cause a longer sequence of probes Example: h(x) = x mod13 Insert keys 18, 41, 22, 44, 59, 32, 31, 73, in this order Linear Probing (§2.5.5) 0 1 2 3 4 5 6 7 8 9 10 11 12 41 18 44 59 32 22 31 73 0 1 2 3 4 5 6 7 8 9 10 11 12 Elementary Data Structures

  38. Consider a hash table A that uses linear probing findElement(k) We start at cell h(k) We probe consecutive locations until one of the following occurs An item with key k is found, or An empty cell is found, or N cells have been unsuccessfully probed Search with Linear Probing AlgorithmfindElement(k) i h(k) p0 repeat c A[i] if c= returnNO_SUCH_KEY else if c.key () =k returnc.element() else i(i+1)mod N p p+1 untilp=N returnNO_SUCH_KEY Elementary Data Structures

  39. To handle insertions and deletions, we introduce a special object, called AVAILABLE, which replaces deleted elements removeElement(k) We search for an item with key k If such an item (k, o) is found, we replace it with the special item AVAILABLE and we return element o Else, we return NO_SUCH_KEY insert Item(k, o) We throw an exception if the table is full We start at cell h(k) We probe consecutive cells until one of the following occurs A cell i is found that is either empty or stores AVAILABLE, or N cells have been unsuccessfully probed We store item (k, o) in cell i Updates with Linear Probing Elementary Data Structures

  40. Double hashing uses a secondary hash function d(k) and handles collisions by placing an item in the first available cell of the series (i+jd(k)) mod Nfor j= 0, 1, … , N - 1 The secondary hash function d(k) cannot have zero values The table size N must be a prime to allow probing of all the cells Common choice of compression map for the secondary hash function: d2(k) =q-k mod q where q<N q is a prime The possible values for d2(k) are1, 2, … , q Double Hashing Elementary Data Structures

  41. Example of Double Hashing • Consider a hash table storing integer keys that handles collision with double hashing • N= 13 • h(k) = k mod13 • d(k) =7 - k mod7 • Insert keys 18, 41, 22, 44, 59, 32, 31, 73, in this order 0 1 2 3 4 5 6 7 8 9 10 11 12 31 41 18 32 59 73 22 44 0 1 2 3 4 5 6 7 8 9 10 11 12 Elementary Data Structures

  42. In the worst case, searches, insertions and removals on a hash table take O(n) time The worst case occurs when all the keys inserted into the dictionary collide The load factor a=n/N affects the performance of a hash table Assuming that the hash values are like random numbers, it can be shown that the expected number of probes for an insertion with open addressing is1/ (1 -a) The expected running time of all the dictionary ADT operations in a hash table is O(1) In practice, hashing is very fast provided the load factor is not close to 100% Applications of hash tables: small databases compilers browser caches Performance of Hashing Elementary Data Structures

  43. A family of hash functions is universal if, for any 0<i,j<M-1, Pr(h(j)=h(k)) < 1/N. Choose p as a prime between M and 2M. Randomly select 0<a<p and 0<b<p, and define h(k)=(ak+b mod p) mod N Theorem: The set of all functions, h, as defined here, is universal. Universal Hashing (§ 2.5.6) Elementary Data Structures

  44. Proof of Universality (Part 1) • So a(j-k) is a multiple of p • But both are less than p • So a(j-k) = 0. I.e., j=k. (contradiction) • Thus, f causes no collisions. • Let f(k) = ak+b mod p • Let g(k) = k mod N • So h(k) = g(f(k)). • f causes no collisions: • Let f(k) = f(j). • Suppose k<j. Then Elementary Data Structures

  45. Proof of Universality (Part 2) • If f causes no collisions, only g can make h cause collisions. • Fix a number x. Of the p integers y=f(k), different from x, the number such that g(y)=g(x) is at most • Since there are p choices for x, the number of h’s that will cause a collision between j and k is at most • There are p(p-1) functions h. So probability of collision is at most • Therefore, the set of possible h functions is universal. Elementary Data Structures

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