Array Lists Outline and Required Reading: Array Lists ( 7.1) CSE 2011, Winter 2017, Section Z Instructor: N. Vlajic
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1 rray Lists Outline and Required Reading: rray Lists ( 7.) CSE 20, Winter 207, Section Z Instructor: N. Vlajic
2 Lists in Everyday Life 2 List convenient way to organize data examples: score list, to-do list, address list, event list items in a list have a position : first, last, item after item rank of item K = number of elements in the list that precede K Score List: 0. lice score: 00. Bob score: Peter score: 9 3. Tim Score 87 To Do List: 0. Wake up.. Take bus to school. 2. Meet lice. 3. ttend lecture. Typical Operations on Lists add a new item at the beginning/end of the list add a new item at a particular place (e.g. rank 5) add a new item B right after item find an item remove an item remove all items of certain type
3 Lists vs. Stacks, Queues, Deques 3 Lists linear container which allows direct access to any of its elements items can be accessed either through rank / index or position relative to the position of other items in the list Stacks, Queues, Deques restricted lists with methods for accessing, inserting, and removing only the first and/or last element C B B C B C B C Stack Queue Deque List
4 List DT 4
5 E remove(int k) throws IndexOutOfBounds ; /* remove and return the element at rank k */ /* error if k<0 or k size()=n current # of elements */ /* rank of all subsequent elements will decrease! */ } List DT (cont.) 5 public interface List<E> { int size(); /* return the # of objects in this list */ boolean isempty(); /* return true if the list is empty */ 0 n-3 n-2 n- P Q R size()=n E get(int k) throws IndexOutOfBoundsException; /* return the element at rank k without removing it*/ /* error if k<0 or k size()=n current # of elements */ E set(int k, E e) throws IndexOutOfBounds ; /* replace with e elem. at rank k; return replaced element */ /* error if k<0 or k size()=n current # of elements */ last rank = size() - void add(int k, E e) throws IndexOutOfBounds ; /* insert a new element e into list at rank k */ /* error if k<0 or k>size()=n current # of elements */ /* rank of all subsequent elements will increase! */
6 List DT (cont.) 6 Example [ operations on a List ]
7 dapter Design Pattern 7 dapter Design - allows that one class be implemented by simply modifying another already existing class new class will contain an instance of the old class as a hidden instance variable each method of the new class is implemented using the methods of the hidden instance variable Deque Method List Method Deque B C List size() isempty() first() last() addfirst(e) addlast(e) removefirst() removelast() size() isempty() get(0) get(size()-) add(0,e) add(size(), e) remove(0) remove(size()-)
8 List DT Design 8 Rank (only!) an abstraction of the concept of index in arrays allow us to refer to the place where an element is stored in a sequence, regardless of the underlying implementation array or linked list! the rank of an element may change whenever the sequence is updated (e.g. an element added/removed) Rank: 0 Rank: k Rank: k+ P Q R P Q R P Q R NOTE: List/vector does NOT have to be implemented using arrays.
9 rraylist DT: Simple rray-based Implementation 9 rray-based implementation of rraylist is not the only possible, but it is the simplest! Trivial rray-based Implementation use array of size N (> expected # of elements) and store element with rank k at [k] L(i) = [i] rank in the list = index in the array Drawback very costly! O(n) - shifting of elements required on every add and remove shifting removal at rank= insertion at rank= Shifting
10 rraylist DT: Simple rray-based Implementation (cont.) 0 Run Times Method Time veragetime size isempty get set add remove O(n) O(n) n/2 n/2 Space Usage Poor! O(N), where N n General Note problems with fixed array size!!! (space may be wasted or may be insufficient)
11 rraylist: Linked List Implementation dvantages of SLL Implementation () shifting of elements during insert and remove operations avoided (2) unlimited capacity Disadvantage of get, set, add, remove will require O(n) time SLL Implementation (worst case!) as they all involve walking through the list in order to find item at rank k use of Doubly LL enables faster walk through splitting, but RT still O(n) start from head if rank < n/2 start from tail if rank < n/2 head 2 n tail
12 rraylist: rray vs. Linked List Implementation 2 Run Times in rray Implementation Method size isempty get set add remove Time O(n) O(n) veragetime n/2 n/2 Run Times in Linked List Implementation Method size isempty get set add remove Time O(n) O(n) O(n) O(n) veragetime n/4 n/4 n/4 n/4
13 rraylist DT: Extensible rray Implementation 3 Extensible rray efficient tool for dealing with array overflows (Dynamic rray) when size()=n and method add(size(), e) is called steps involved () allocate a new larger array (how much larger!?) (2) copy old elements to the new array (3) reassign old reference to the new array B B step () step (2) step (3) growing array strategies (a) (b) fixed increment: newcapacity=oldcapacity+c doubling: newcapacity=2*oldcapacity
14 rraylist DT: Extensible rray Implementation (cont.) 4 Run Times of rraylist in Case of Extensible rray Implementation Method Time veragetime size isempty get set add remove O(n) O(n)? n/2 Potentially bigger than n/2, as in some cases adding the last element will involve creating and copying elements remove average = remove(0) (+n-) + remove(size()) + 2 n/2 add average = add(0) (n+) + add(size()) +?? 2 =??
15 rraylist DT: Extensible rray Implementation (cont.) 5 Run Time of add() non-trivial to find a single method may have In Extensible rrays different run times depending on the current array occupancy if size()<n add(size(), e) runs at time if size()=n add(size(), e) runs at O(N) time (only after all N elements have been copied to the new array new element can be inserted) when array is not full insert at the end has cost when array is full insert at the end has O(N) cost The RT analysis requires that the actual array growing policy be known.
16 rraylist DT: Extensible rray Implementation (cont.) 6 Run Times in Extensible rrays with Fixed Increments Strategy Run Times in Extensible rrays with Doubling Strategy C C C n 2n On every capacity increase, next C inserts run at cost. On every capacity increase, next n inserts run at cost Which strategy would be better, overall? mortization Technique useful analysis tool when some method calls are more expensive than others, we should average out the costs over the total number of calls RT average = RT(over X operations) X
17 rraylist DT: Extensible rray Implementation (cont.) 7 Ext. rray with Fixed Increments: Cost of add(size(), e) ssume: there are n=+kc elements to be inserted at the end of a vector; initial size of the extensible array C capacity increment value ccordingly: total number of capacity increases: k total number of copy operations: + (+C) + (+2C) + + (+(k-)c) = = k + C(+2+..+(k-)) = = k + Ck(k-)/2 hence, the total run time (insert + copy) over n elements is: + kc + k + Ck(k-)/2 Θ(k 2 ) = Θ(((n-)/C) 2 ) = Θ(n 2 ) The amortized (average) time of add(size(), e) is O(n)!
18 rraylist DT: Extensible rray Implementation (cont.) 8 every k th call for inserttrank() on full array requires that +(k-)c elements be copied st call for add(size(), e) on full array additional copy operations performed copy operations 2 st call for add(size(), e) on full array additional +C copy operations performed C (+C) copy operations 3 rd call for add(size(), e) on full array additional +2C copy operations performed C C (+2C) copy operations C C C k th call for add(size(), e) on full array additional +(k-)c copy operations performed!
19 rraylist DT: Extensible rray Implementation (cont.) 9 Extensible rray with Doubling: Cost of add(size(), e) ssume: there are n=2 k elements to be inserted at the end of a vector; initial size of the extensible array ccordingly: total number of capacity increases: k total number of copy operations: k- = = 2 k - = = n - hence, the total run time (insert + copy) over n elements is: n + n - Θ(n) The amortized (average) time of add(size(), e) operation is!
20 20 initial size=, extend by doubling initial size=2, extend by C=3
21 rraylist DT: Extensible rray Implementation (cont.) 2 public class rraylist<e> implements List<E> { private E[] data; private int capacity=6; // initial size of array // private int size=0; // current # of elements // public rraylist() { data = (E[]) Object[CPCITY]; } public void add(int r, E e) throws IndexOutOfBoundsException { checkindex(r, size()); //check if 0 <= r <= data.length // if (size==data.length) { // check if enough capacity // capacity *= 2; E[] newdata = (E[]) Object[capacity]; for (int i=0; i<size; i++) { newdata[i] = data[i]; } data = newdata; }
22 rraylist DT: Extensible rray Implementation (cont.) 22 public void add(int r, E e) throws IndexOutOfBoundsException { for (int i=size-; i>=r; i--) { data[i+] = data[i]; } data[r] = e; size ++; } public E remove(int r) throws IndexOutOfBoundsException { checkindex(r, size()-); E temp = data[r]; for (int i=r; i<size-; i++) { [i] = [i+]; } size--; return temp } rank r rank size- rank r rank size-
23 mortization Technique: Example 23 Example [ clearable stack ] Consider ClearableStack DT, implemented with an array of fixed size N, and with the following interface: standard operations: push(), pop(), top(), new operation: clear() - remove everything from the stack by means of multiple pop() operations C B Method push pop top clear Time O(N) What is the worst case running time of n operations on a ClearableStack, starting with an empty stack!?
24 mortization Technique: Example (cont.) 24 pproach : In the worst case, the most costly (clear()) operation will be executed each time. RT(n operations) = n* O(N) = O(N*n) Correct, but overstated result! clear() cannot come before at least one push(). That is, clear() has cost O(N) only if N elements have been previously pushed on the stack. pproach 2: clear() comes occasionally, after a number of push()-es. push() push() clear() push() clear() k pushes k 2 pushes push() push() clear() k Q pushes n operations
25 mortization Technique: Example (cont.) 25 st set of pushes 2 nd set of pushes M th set of pushes k *push + clear + k 2 *push + clear k M *push + clear M = n operations Θ(k ) Θ(k ) Θ(k 2 ) Θ(k 2 ) Θ(k M ) Θ(k M ) # of all pushes + # of clears = [k + k k M ] + M = n, where M n/2 one clear at the end one clear after each push # of all clears does not have to be = # of all pushes RT(all clears) = RT(all pushes)!!! RT= 2* RT(all pushes) = 2*(n M) RT(n operations) [ n, 2*(n-) ]
26 mortization Technique: Example (cont.) 26 Lower bound on RT is achieved when M=n/2: every 2 nd operation is clear(). B C st push st clear 2 nd push 2 nd clear (n/2) th push (n/2) th clear Upper bound on RT is achieved when M=: one clear() after (n-) pushes. M N M P N M st push 2 nd push 3 rd push clear Θ(n-)
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