Stacks and Linked Lists

Abstract Data Types (ADTs) An ADT is an abstraction of a data structure that specifies Data stored Operations on the data Error conditions associated with operations

Abstract Data Types (ADTs) An ADT is an abstraction of a data structure that specifies Data stored Operations on the data Error conditions associated with operations Example: Registering for classes The data stored are the courses in your schedule The operations supported are Register(course) Unregister(course) ForceRequest(course) Error conditions: Registering for multiple classes meeting at the same time

Stacks

Stacks Stacks store arbitrary objects (Pez in this case)

Stacks Stacks store arbitrary objects (Pez in this case) Operations push(e): inserts an element to the top of the stack

Stacks Stacks store arbitrary objects (Pez in this case) Operations push(e): inserts an element to the top of the stack

Stacks Stacks store arbitrary objects (Pez in this case) Operations push(e): inserts an element to the top of the stack pop(): removes and returns the top element of the stack

Stacks Stacks store arbitrary objects (Pez in this case) Operations push(e): inserts an element to the top of the stack pop(): removes and returns the top element of the stack

Stacks Stacks store arbitrary objects (Pez in this case) Operations push(e): inserts an element to the top of the stack pop(): removes and returns the top element of the stack

Stacks Stacks store arbitrary objects (Pez in this case) Operations push(e): inserts an element to the top of the stack pop(): removes and returns the top element of the stack top(): returns a reference to the top element of the stack, but doesn’t remove it

Stacks Stacks store arbitrary objects (Pez in this case) Operations push(e): inserts an element to the top of the stack pop(): removes and returns the top element of the stack top(): returns a reference to the top element of the stack, but doesn’t remove it Optional operations size(): returns the number of elements in the stack empty(): returns a bool indicating if the stack contains any objects

Stack Exceptions Attempting to execute an operation of ADT may cause an error condition called an exception Exceptions are said to be “thrown” by an operation that cannot be executed In the Stack ADT, pop and top cannot be performed if the stack is empty Attempting to execute pop or top on an empty stack throws an EmptyStackException

Exercise: Stacks Describe the output and final structure of the stack after the following operations: Push(8) Push(3) Pop() Push(2) Push(5) Push(9) Push(1)

Applications of Stacks Direct applications Page-visited history in a Web browser Undo sequence in a text editor Saving local variables when one function calls another, and this one calls another, and so on. Indirect applications Auxiliary data structure for algorithms Component of other data structures

C++ Run-time Stack main() { int i; i = 5; foo(i); } foo(int j) { int k; k = j+1; bar(k); } bar(int m) { … } The C++ run-time system keeps track of the chain of active functions with a stack When a function is called, the run-time system pushes on the stack a frame containing Local variables and return value Program counter, keeping track of the statement being executed When a function returns, its frame is popped from the stack and control is passed to the method on top of the stack bar PC = 1 m = 6 foo PC = 3 j = 5 k = 6 main PC = 2 i = 5

Array-based Stack Algorithm size() return t + 1 Algorithm empty() return size () == 0 Algorithm pop() if empty() then throw EmptyStackException else t  t  1 return S[t + 1] A simple way of implementing the Stack ADT uses an array We add elements from left to right A variable keeps track of the index of the top element … S 1 2 t

Array-based Stack (cont.) The array storing the stack elements may become full A push operation will then throw a FullStackException Limitation of the array-based implementation Not intrinsic to the Stack ADT Algorithm push(e) if t = S.length  1 then throw FullStackException else t  t + 1 S[t]  e S 1 2 t …

Performance and Limitations (array-based implementation of stack ADT) Let n be the number of elements in the stack The space used is O(n) Each operation runs in time O(1) Limitations The maximum size of the stack must be defined a priori , and cannot be changed Trying to push a new element into a full stack causes an implementation-specific exception

Growable Array-based Stack In a push operation, when the array is full, instead of throwing an exception, we can replace the array with a larger one How large should the new array be? incremental strategy: increase the size by a constant c doubling strategy: double the size Algorithm push(o) if t = S.length  1 then A  new array of size … for i  0 to t do A[i]  S[i] S  A t  t + 1 S[t]  o

Comparison We compare the incremental strategy and the doubling strategy by analyzing the total time T(n) needed to perform a series of n push operations Assume that we start with an empty stack represented by an array of size 1 We call amortized time of a push operation the average time taken by a push over the series of operations, i.e., T(n)/n

Incremental Strategy Analysis We replace the array k = n/c times The total time T(n) of a series of n push operations is proportional to n + c + 2c + 3c + 4c + … + kc = n + c(1 + 2 + 3 + … + k) = n + ck(k + 1)/2 Since c is a constant, T(n) is O(n + k2) = O(n2) The amortized time of a push operation is O(n)

Doubling Strategy Analysis We replace the array k = log2 n times The total time T(n) of a series of n push operations is proportional to n + 1 + 2 + 4 + 8 + …+ 2k = n + 2k + 1 -1 = 3n -1 T(n) is O(n) The amortized time of a push operation is O(1)

Stack Interface in C++ template <class Type> class Stack { public: int size(); bool isEmpty(); Type& top() throw(EmptyStackException); void push(Type e); Type pop() throw(EmptyStackException); }; Requires the definition of class EmptyStackException Most similar STL construct is vector

Array-based Stack in C++ template <class Type> class ArrayStack { private: int capacity; // stack capacity Type *S; // stack array int t; // top of stack public: ArrayStack(int c) : capacity(c) { S = new Type [ capacity ]; t = -1; } bool isEmpty() { return t < 0; } Type pop() throw(EmptyStackException) { if ( isEmpty ( ) ) throw EmptyStackException(“Popping from empty stack”); return S [ t-- ]; //… (other functions omitted)

Singly Linked List A singly linked list is a structure consisting of a sequence of nodes A singly linked list stores a pointer to the first node (head) and last (tail) Each node stores element link to the next node next node elem tail head  Leonard Sheldon Howard Raj

Singly Linked List Node in C++ template <class Type> class SLinkedListNode { public: Type elem; SLinkedListNode<Type> *next; }; next node elem  Leonard Sheldon Howard Raj

Singly Linked List A singly linked list is a structure consisting of a sequence of nodes Operations insertFront(e): inserts an element on the front of the list removeFront(): returns and removes the element at the front of the list insertBack(e): inserts an element on the back of the list removeBack(): returns and removes the element at the end of the list

Inserting at the Front Allocate a new node Have new node point to old head Update head to point to new node head tail  Leonard Sheldon Howard Raj

Inserting at the Front Allocate a new node Have new node point to old head Update head to point to new node head tail   Penny Leonard Sheldon Howard Raj

Inserting at the Front Allocate a new node Have new node point to old head Update head to point to new node head tail  Penny Leonard Sheldon Howard Raj

Inserting at the Front Allocate a new node Have new node point to old head Update head to point to new node tail head  Penny Leonard Sheldon Howard Raj

Inserting at the Front Allocate a new node Have new node point to old head Update head to point to new node head tail  

Inserting at the Front Allocate a new node Have new node point to old head Update head to point to new node head tail    Raj

Inserting at the Front Allocate a new node Have new node point to old head Update head to point to new node If tail is NULL, update tail to point to the head node head tail  Raj

Removing at the Front Update head to point to next node in the list Return elem of previous head and delete the node head tail  Leonard Sheldon Howard Raj

Removing at the Front Update head to point to next node in the list Return elem of previous head and delete the node head tail  Leonard Sheldon Howard Raj

Removing at the Front Update head to point to next node in the list Return elem of previous head and delete the node head tail  Leonard Sheldon Howard Raj

Removing at the Front Update head to point to next node in the list Return elem of previous head and delete the node head tail  Leonard Sheldon Howard Raj

Removing at the Front Update head to point to next node in the list Return elem of previous head and delete the node head tail  Sheldon Howard Raj

Removing at the Front Update head to point to next node in the list Return elem of previous head and delete the node head tail  Sheldon

Removing at the Front Update head to point to next node in the list Return elem of previous head and delete the node head tail   Sheldon

Removing at the Front Update head to point to next node in the list Return elem of previous head and delete the node head tail   Sheldon

Removing at the Front Update head to point to next node in the list Return elem of previous head and delete the node head tail   Sheldon

Removing at the Front Update head to point to next node in the list Return elem of previous head and delete the node If head is NULL, update tail to NULL head tail  

Inserting at the Back Allocate a new node If tail is NULL, update head and tail to point to the new node; otherwise Have the old tail point to the new node Update tail to point to new node head tail  Leonard Sheldon Howard

Inserting at the Back Allocate a new node If tail is NULL, update head and tail to point to the new node; otherwise Have the old tail point to the new node Update tail to point to new node head tail   Leonard Sheldon Howard Raj

Inserting at the Back Allocate a new node If tail is NULL, update head and tail to point to the new node; otherwise Have the old tail point to the new node Update tail to point to new node head tail  Leonard Sheldon Howard Raj

Inserting at the Back Allocate a new node If tail is NULL, update head and tail to point to the new node; otherwise Have the old tail point to the new node Update tail to point to new node head tail  Leonard Sheldon Howard Raj

Removing at the Back No efficient way of doing so (O(n)) Typically would not use a singly linked-list if this operation is commonly used head tail  Leonard Sheldon Howard Raj

Stack with a Singly Linked List We can implement a stack with a singly linked list The top element of the stack is the first node of the list The space used is O(n) and each operation of the Stack ADT takes O(1) time nodes t top  elements

Stack Summary Stack Operation Complexity for Different Implementations Array Fixed-Size Expandable (doubling strategy) Singly Linked List Pop() O(1) Push(o) O(n) Worst Case O(1) Best Case O(1) Average Case Top() Size(), isEmpty()

Queues

Queues Auxiliary queue operations: Exceptions Queues store arbitrary objects Insertions are at the end of the queue and removals are at the front of the queue Main queue operations: enqueue(e): inserts an element at the end of the queue dequeue(): removes and returns the element at the front of the queue Auxiliary queue operations: front(): returns the element at the front without removing it size(): returns the number of elements stored isEmpty(): returns a boolean value indicating if there are no elements in the queue Exceptions Attempting to execute dequeue or front on an empty queue throws an EmptyQueueException

Exercise: Queues Describe the output and final structure of the queue after the following operations: enqueue(8) enqueue(3) dequeue() enqueue(2) enqueue(5) enqueue(9) enqueue(1)

Applications of Queues Direct applications Waiting lines Access to shared resources (e.g., printer) User input in a game Indirect applications Auxiliary data structure for algorithms Component of other data structures

wrapped-around configuration Array-based Queue Use an array of size N in a circular fashion Two variables keep track of the front and rear f index of the front element r index immediately past the rear element Array location r is kept empty normal configuration Q 1 2 r f wrapped-around configuration Q 1 2 f r

Queue Operations We use the modulo operator (remainder of division) Algorithm size() return (N - f + r) mod N Algorithm isEmpty() return (f = r) Q 1 2 r f Q 1 2 f r

Queue Operations (cont.) Algorithm enqueue(o) if size() = N  1 then throw FullQueueException else Q[r]  o r  (r + 1) mod N Operation enqueue throws an exception if the array is full This exception is implementation-dependent Q 1 2 r f Q 1 2 f r

Queue Operations (cont.) Algorithm dequeue() if isEmpty() then throw EmptyQueueException else o  Q[f] f  (f + 1) mod N return o Operation dequeue throws an exception if the queue is empty This exception is specified in the queue ADT Q 1 2 r f Q 1 2 f r

Performance and Limitations - array-based implementation of queue ADT Let n be the number of elements in the queue The space used is O(n) Each operation runs in time O(1) Limitations The maximum size of the queue must be defined a priori , and cannot be changed Trying to enqueue a new element into a full queue causes an implementation-specific exception

Growable Array-based Queue In an enqueue operation, when the array is full, instead of throwing an exception, we can replace the array with a larger one Similar to what we did for an array-based stack The enqueue operation has amortized running time O(n) with the incremental strategy O(1) with the doubling strategy

Exercise Describe how to implement a queue using a singly-linked list Queue operations: enqueue(x), dequeue(), size(), isEmpty() For each operation, give the running time

Queue with a Singly Linked List We can implement a queue with a singly linked list The front element is stored at the head of the list The rear element is stored at the tail of the list The space used is O(n) and each operation of the Queue ADT takes O(1) time NOTE: we do not have the limitation of the array based implementation on the size of the stack b/c the size of the linked list is not fixed, I.e., the queue is NEVER full. head tail  Leonard Sheldon Howard Raj

Informal C++ Queue Interface template <class Type> class Queue { public: int size(); bool isEmpty(); Type& front() throw(EmptyQueueException); void enqueue(Type e); Type dequeue() throw(EmptyQueueException); }; Informal C++ interface for our Queue ADT Requires the definition of class EmptyQueueException No corresponding built-in STL class

Queue Summary Queue Operation Complexity for Different Implementations Array Fixed-Size Expandable (doubling strategy) List Singly-Linked dequeue() O(1) enqueue(o) O(n) Worst Case O(1) Best Case O(1) Average Case front() Size(), isEmpty()

Double-Ended Queues Auxiliary deque operations: Exceptions The Double-Ended Queue, or Deque, ADT stores arbitrary objects. (Pronounced ‘deck’) Richer than stack or queue ADTs. Supports insertions and deletions at both the front and the end. Main deque operations: insertFirst(object o): inserts element o at the beginning of the deque insertLast(object o): inserts element o at the end of the deque removeFirst(): removes and returns the element at the front of the deque removeLast(): removes and returns the element at the end of the deque Auxiliary deque operations: first(): returns the element at the front without removing it last(): returns the element at the front without removing it size(): returns the number of elements stored isEmpty(): returns a Boolean value indicating whether no elements are stored Exceptions Attempting to execute removeFirst,removeLast, front, or last on an empty deque throws an EmptyDequeException

Doubly Linked List A doubly linked list is a structure consisting of a sequence of nodes A doubly linked list stores a pointer to a special head/tail node Each node stores element link to the prev, next node next prev elem node head tail Leonard Sheldon Howard Raj

Doubly Linked List A doubly linked list is a structure consisting of a sequence of nodes A doubly linked list stores a pointer to a special head/tail node Each node stores element link to the prev, next node next prev elem node head tail

Doubly Linked List Node in C++ template <class Type> class DLinkedListNode { public: Type elem; DLinkedListNode<Type> *prev, *next; }; next prev elem node head tail Leonard Sheldon Howard Raj

Doubly Linked List A doubly linked list is a structure consisting of a sequence of nodes Operations insertFront(e): inserts an element on the front of the list removeFront(): returns and removes the element at the front of the list insertBack(e): inserts an element on the back of the list removeBack(): returns and removes the element at the end of the list Private operations add(n, e): inserts the element after the node n remove(n): returns and removes the element stored in the node n

Adding a Node Allocate a new node Have new node point to the previous and next nodes Update the previous and next nodes to point to the new node head tail Leonard Sheldon Howard Raj

Adding a Node Allocate a new node Have new node point to the previous and next nodes Update the previous and next nodes to point to the new node Bernadette head tail Leonard Sheldon Howard Raj

Adding a Node Allocate a new node Have new node point to the previous and next nodes Update the previous and next nodes to point to the new node Bernadette head tail Leonard Sheldon Howard Raj

Adding a Node Allocate a new node Have new node point to the previous and next nodes Update the previous and next nodes to point to the new node Bernadette head tail Leonard Sheldon Howard Raj

Adding a Node Allocate a new node Have new node point to the previous and next nodes Update the previous and next nodes to point to the new node head tail Leonard Sheldon Howard Bernadette Raj

Adding a Node Allocate a new node Have new node point to the previous and next nodes Update the previous and next nodes to point to the new node Sheldon head tail

Adding a Node Allocate a new node Have new node point to the previous and next nodes Update the previous and next nodes to point to the new node Sheldon head tail

Adding a Node Allocate a new node Have new node point to the previous and next nodes Update the previous and next nodes to point to the new node Sheldon head tail

Adding a Node Allocate a new node Have new node point to the previous and next nodes Update the previous and next nodes to point to the new node head tail Sheldon

Removing a Node Have the prev node’s next point to the next of the current node Have the next node’s prev point to the prev of the current node Delete the current node head tail Leonard Sheldon Howard Raj

Removing a Node Have the prev node’s next point to the next of the current node Have the next node’s prev point to the prev of the current node Delete the current node head tail Leonard Sheldon Howard Raj

Removing a Node Have the prev node’s next point to the next of the current node Have the next node’s prev point to the prev of the current node Delete the current node head tail Leonard Sheldon Howard Raj

Removing a Node Have the prev node’s next point to the next of the current node Have the next node’s prev point to the prev of the current node Delete the current node head tail Leonard Sheldon Howard Raj

Removing a Node Have the prev node’s next point to the next of the current node Have the next node’s prev point to the prev of the current node Delete the current node head tail Leonard Sheldon Raj

Deque with a Doubly Linked List We can implement a deque with a doubly linked list The front element is pointed to by head The rear element is pointed to by tail The space used is O(n) and each operation of the Deque ADT takes O(1) time head tail Leonard Sheldon Howard Raj

Performance and Limitations - doubly linked list implementation of deque ADT Let n be the number of elements in the deque The space used is O(n) Each operation runs in time O(1) Limitations NOTE: we do not have the limitation of the array based implementation on the size of the deque b/c the size of the linked list is not fixed, I.e., the deque is NEVER full.

Deque Summary Deque Operation Complexity for Different Implementations Array Fixed-Size Expandable (doubling strategy) List Singly-Linked Doubly-Linked removeFirst(), removeLast() O(1) O(1) – removeFirst, O(n) – removeLast insertFirst(o), InsertLast(o) O(n) Worst Case O(1) Best Case O(1) Average Case first(), last size(), isEmpty()