Data Structure & Algorithm Lecture 2 – Basic Data Structure JJCAO Steal some from Prof. Yoram Moses

The Sorting Problem Example: Input: A sequence of n numbers < 𝑎 1 , 𝑎 2 , …, 𝑎 n > Output: A permutation (reordering) < 𝑏 1 , 𝑏 2 , …, 𝑏 n > of the input sequence such that 𝑏 1 ≤ 𝑏 2 …≤ 𝑏 n

Recitation Selection Sort 𝑂( 𝑛 2 ) Insertion Sort 𝑂( 𝑛 2 ) k=i for i = 1:n, k = i //invariant: a[k] smallest of a[i..n] for j = i+1:n if a[j] < a[k] then k = j //invariant: a[1..i] in final position swap a[i,k] end key = A[j]

How is Data Represented? Sort a sequences: Array? List? Heap? There are many options Efficiency of the algorithm also depends on the data structure used

Elementary Data Structures Arrays Lists Stacks Queues Trees “Mankind’s progress is measured by the number of things we can do without thinking.” In some languages or libs, some of them are the “off-the-shelf” components.

Components of Data Structure There are two aspects to any data structure: The abstract operations which it supports. (Abstract Data Types, ADT) The implementation of these operations. (Data Structure) Data structures such as array, list, stacks, …

Data Type vs. Data Structure Data Structure is the way an ADT is implemented. Must be distinguished from the ADT itself!

C++ Implementation of Stack

Data Structure vs. Algorithm Data Structures Represent objects of the ADT Algorithms Manipulate the data structures to implement a mission using the operations of the ADT

Contiguous vs. Linked Data Structures Data structures can be neatly classified as either contiguous or linked depending upon whether they are based on arrays or pointers: Contiguously-allocated structures are composed of single slabs of memory, and include arrays, matrices, heaps, and hash tables. Linked data structures are composed of multiple distinct chunks of memory bound together by pointers, and include lists, trees, and graph adjacency lists.

Array Array of integer Array of images An array is a number of data items of the same type arranged contiguously in memory. Fixed-size 𝑚_𝑛𝑆𝑖𝑧𝑒

Advantages of contiguously-allocated arrays Constant-time access given the index. Space efficiency - Arrays consist purely of data, so no space is wasted with links or other formatting information. Memory locality - Physical continuity (memory locality) between successive data accesses helps exploit the high-speed cache memory on modern computer architectures.

Dynamic Array Unfortunately we cannot adjust the size of simple arrays in the middle of a program’s execution. Start with an array of size 1, and double its size from m to 2m each time we run out of space. How many times will we double for n elements? Only log⁡(𝑛) 1 + 2 + 4 + 8 + … + 2^i =2^{i+1}=n => i = log(n)

How much total work The apparent waste in this procedure involves the recopying of the old contents on each expansion. If half the elements move once, a quarter of the elements twice, and so on, the total number of movements M is given by Thus each of the n elements move an average of only twice, and the total work of managing the dynamic array is the same O(n) as a simple array.

Remove an Element

Notes Practical implementations usually include more operations Initialization/Destruction “Luxury” operations: size() print() operators

Bubble Sort Bubble sort: beginning of first pass: 1. Compare two players. 2. If the one on the left is taller, swap them. 3. Move one position right. End of first pass

Bubble Sort for i = 1:n, swapped = false for j = n:i+1, if a[j] < a[j-1], swap a[j,j-1] swapped = true break if not swapped end Is there any problem?

Several Sort Algorithms http://www.sorting-algorithms.com http://www.sorting-algorithms.com

The Linked List Structures typedef struct list { item type item; struct list *next; } list; Pointers represent the address of a location in memory. A cell-phone number can be thought of as a pointer to its owner as they move about the planet.

Searching a List Searching in a linked list can be done iteratively or recursively. list *search_list(list *l, item type x) { if (l == NULL) return(NULL); if (l->item == x) return(l); else return( search_list(l->next, x) ); }

Insertion into a List Since we have no need to maintain the list in any particular order, we might as well insert each new item at the head. void insert_list(list **l, item_type x) { list *p = new list; p->item = x; p->next = *l; *l = p; } Note the **l, since the head element of the list changes.

Deleting from a List

Deleting from a List

Advantages of Linked Lists Overflow on linked structures can never occur unless the memory is actually full. 2. Insertions and deletions are simpler than for contiguous (array) lists. 3. With large records, moving pointers is easier and faster than moving the items themselves.

Various List

Stack last-in, first-out (LIFO) http://www.algolist.net/Data_structures/Stack

Stack ADT & std::vector comments create() constructor of vector creates empty stack bool isEmpty() empty() tells whether the stack s is empty push(Item e) push_back(Item e) put e on top of the stack s Item peek() Item back() returns topmost element in stack s pop() pop_back() removes topmost element from the stack s destroy() Deconstructor of vector destroys stack s reserve() push_back() pop_back() back() size() empty()

Stack Implementation Using a Linked List

Stack Example 1: Reversing a Word part trap

Stack Example 2: Delimiter Matching When compilers compile your code When you want to write a program to parse a math formula

Stack Example 2: Delimiter Matching A successful example: b(c[d]e) Character Read Stack Contents b ( c [ [( d ] e )

Queue first-in, first-out

Implementation of Queue

A Circular Queue

Homework 1 Basic Dynamic Array & Selection Sort Deadline: 22:00, Sep. 10, 2011