Two dimensional array

Jul 4, 2015Download as ppt, pdf5 likes5,915 views

Two dimensional arrays in C can be declared and initialized similarly to one dimensional arrays, with the first subscript specifying the row and second subscript specifying the column. Elements are stored in contiguous memory locations in row-major order. The document then presents a sample problem of reading a 2D array of integers from a file, finding the largest element, and printing it out. It also discusses using typedef to define custom data types like matrices and strings.

- Two dimensional arrays are declared
similarly to one dimensional arrays:
char t[20][10]; (and referred to as a
matrix, or as a "table")
The first subscript gives the row number, and
the second subscript specifies column number.

We can also initialize a two-dimensional array
in a declaration statement; E.g.,
int m[2][3] = {{1, 2, 3}, {4, 5, 6}};
which specifies the elements in each row of the
array (the interior braces around each row of
values could be omitted). The matrix for this
example is






=
654
321
m

Specification of the number of rows could be
omitted. But other subscript specifications can
not be omitted. So, we could also write the
initialization as
int m[ ][3] = {{1, 2, 3}, {4, 5, 6}};

int a[2][3]= {1,2,3,4,5,6};
specifies 6 integer locations.
Storage for array elements are in contiguous locations in
memory in row major order (unlike column major order in
Fortran), referenced by subscripts(or index) values starting at 0
for both rows and columns.
a[0][0] a[0][1] a[0][2] a[1][0] a[1][1] a[1][2]
RAM
1 2 3 4 5 6

1. Problem Definition
Assume we have a file “in.dat” (located in the pwd) consisting
of 10 rows of 10 integers per row (100 integers total). Write a
C program which reads from the file and stores the numbers
in a two dimensional array named ‘mat’, computes the largest
of all of these numbers and prints the largest value to the
screen.

2. Refine, Generalize, Decompose the problem definition
(i.e., identify sub-problems, I/O, etc.)
Input = from file “in.dat”
Output= Largest of the 100 integers printed to screen.
3. Develop Algorithm
Use Unix redirection to read file “in.dat”.
Use nested for loop to go through two dimensional array to

#include <stdio.h>
void main(void)
{
int row, col, maximum;
/* Declare a 2-D array called ‘mat’ which can hold a 10-by-10 */
/* matrix of integers */
int mat[10][10];
/* Write code here to read from the file ‘in.dat’ */
for(row=0;row<10;++row)
for(col=0;col<10;++col)
scanf("%i",&mat[row][col]);
/* Write code to find the largest value in the array ‘mat’ */
maximum = mat[0][0]; /* why ??? */
for(row=0;row<10;++row) /* How does this work??? */
for(col=0;col<10;++col)
if (mat[row][col] > maximum)
maximum = mat[row][col];
/* Print the largest value to the screen */
printf(" max value in the array = %in",maximum);
}

The typedef mechanism in C allows us to define our own
data types. For example,
typedef double matrix[10][20];
In this example we create a new data-type named matrix.
We can declare a variable of data-type matrix by
matrix a;
and this declaration is equivalent to
double a[10][20];

Other examples of the use of typedef are,
typedef int blah; /* don’t do this !!!!! */
typedef char string[5];
In the first example we give another name or alias to the
data type int .
In the second example we create a new data-type named
string.

To declare a variable of data-type blah ,
blah x;
this declaration is equivalent to
int x;
To declare a variable of data-type string ,
string var ="init"; /* declare and initialize */
scanf("%s",var); /* read in a string of chars */
strcpy(var , "next");
The declaration
string morse[26];
declares morse as an array of 26 elements. Each element has
data-type string.

The declaration and initialization:
string morse[26] =
{".-", "-...", "-.-.", "-..", ".", "..-.",
"--.", "....", "..", ".---", "-.-", ".-..",
"--", "-.", "---", ".--.", "--.-", ".-.", "...",
"-", "..-", "...-", ".--", "-..-", "-.--", "--.. " };
declares morse as an array of 26 elements with
morse[0] = ".-"
morse[1] = "-..." and so on...

Standard (ANSI) C allows up to 12 dimension.
But more computation is required by the system
to locate elements in multi-subscripted arrays.
Therefore, in problems involving intensive
numerical calculations, we should use arrays
with fewer dimension.
We can also use 3, 4, … dimensional arrays,
e.g.,
threeDArray [i][j][k];

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The document discusses lower bounds for sorting algorithms. It shows that the worst case and average case lower bounds for sorting are both Ω(n log n). This lower bound is derived by analyzing the minimum depth of balanced binary decision trees required to represent the n! possible permutations of n elements. Several algorithms, including heapsort, are shown to achieve the optimal time complexity of O(n log n), matching this lower bound. Problem transformations are also introduced as a method for proving lower bounds.

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The document discusses red-black trees, which are binary search trees augmented with node colors to guarantee a height of O(log n). It first defines the properties of red-black trees, then proves their height is O(log n), and finally describes insertion and deletion operations. The key points are that nodes can be red or black, insertion can violate properties so recoloring is needed, and rotations are used to restructure the tree during insertion and deletion while maintaining the red-black properties.

Hash tableRajendran

This document discusses hashing techniques for implementing symbol tables. It begins by reviewing the motivation for symbol tables in compilers and describing the basic operations of search, insertion and deletion that a hash table aims to support efficiently. It then discusses direct addressing and its limitations when key ranges are large. The concept of a hash function is introduced to map keys to a smaller range to enable direct addressing. Collision resolution techniques of chaining and open addressing are covered. Analysis of expected costs for different operations on chaining hash tables is provided. Various hash functions are described including division and multiplication methods, and the importance of choosing a hash function to distribute keys uniformly is discussed. The document concludes by mentioning universal hashing as a technique to randomize the hash function

Medians and order statisticsRajendran

The document discusses algorithms for finding order statistics like the median in a list of numbers. It presents a randomized selection algorithm that finds the ith smallest element in expected linear time O(n) by using a randomized partition like in quicksort. It also describes a deterministic selection algorithm that finds the ith element in worst-case linear time O(n) by partitioning the list around the median of medians. Finding the median can then be used to sort in linear time by recursively finding order statistics.

Proof master theoremRajendran

1) The document provides the proof of the Master Theorem for analyzing divide-and-conquer recurrences. 2) It introduces key lemmas used in the proof, including bounding the contribution of subproblem sizes. 3) The proof considers three cases depending on how the work at each level relates to the decrease in problem size, showing the recurrence has a runtime of O(nlogba), O(nklogn), or O(nk) respectively.

Recursion tree methodRajendran

- Recurrences describe functions in terms of their values on smaller inputs and arise when algorithms contain recursive calls to themselves. - To analyze the running time of recursive algorithms, the recurrence must be solved to find an explicit formula or bound the expression in terms of n. - Examples of recurrences and their solutions are given, including binary search (O(log n)), dividing the input in half at each step (O(n)), and dividing the input in half but examining all items (O(n)). - Methods for solving recurrences include iteration, substitution, and using recursion trees to "guess" the solution.

Recurrence theoremRajendran

The document discusses different methods for analyzing recursive functions, including: 1. The recursion-tree method, which represents each subproblem as a node with costs summed at each level and total. This is suitable for divide-and-conquer recurrences. 2. An example of using the recursion-tree method to solve T(n)=3T(n/4)+Θ(n^2), showing the tree has height log4n, Θ(nlog43) leaf nodes each costing T(1), and total cost O(n^2). 3. Another example of T(n)=T(n/3)+T(2n/3)+O(n

Master method Rajendran

This document discusses recurrences and algorithms analysis. It covers: 1. Recurrences arise when an algorithm contains recursive calls to itself. The running time is described by a recurrence relation. 2. Examples of recurrence relations are given for different types of recursive algorithms. 3. The binary search algorithm is presented as an example recursive algorithm and its recurrence relation is derived.

Master method theoremRajendran

Hash tablesRajendran

The document discusses hash tables and how they can be used to implement dictionaries. Hash tables map keys to table slots using a hash function in order to store and retrieve items efficiently. Collisions may occur when multiple keys hash to the same slot. Chaining is described as a method to handle collisions by storing colliding items in linked lists attached to table slots. Analysis shows that with simple uniform hashing, dictionary operations like search, insert and delete take expected O(1) time on average.

Lower boundRajendran

This chapter discusses limitations on algorithmic power and methods for establishing lower bounds on algorithms. It introduces lower bounds as estimates of the minimum amount of work needed to solve a problem. Several methods are presented for establishing lower bounds, including trivial lower bounds based on input/output sizes, decision trees to model comparisons, adversary arguments, and reducing one problem to another with a known lower bound. Examples are given for sorting, searching, and matrix multiplication.

Master method theoremRajendran

Greedy algorithmsRajendran

This document discusses greedy algorithms and dynamic programming techniques for solving optimization problems. It covers the activity selection problem, which can be solved greedily by always selecting the shortest remaining activity. It also discusses the knapsack problem and how the fractional version can be solved greedily while the 0-1 version requires dynamic programming due to its optimal substructure but non-greedy nature. Dynamic programming builds up solutions by combining optimal solutions to overlapping subproblems.

Longest common subsequences in Algorithm AnalysisRajendran

The document discusses dynamic programming and the longest common subsequence (LCS) problem. It provides an example of using dynamic programming to find the LCS of two strings "ABCB" and "BDCAB" in multiple steps. The algorithm builds up the solution by filling a 2D array c[m,n] where m and n are the lengths of the two strings. Each entry c[i,j] represents the length of the LCS of the prefixes of length i and j of the two strings. The running time is O(mn).

Dynamic programming in Algorithm AnalysisRajendran

The document discusses dynamic programming and amortized analysis. It covers: 1) An example of amortized analysis of dynamic tables, where the worst case cost of an insert is O(n) but the amortized cost is O(1). 2) Dynamic programming can be used when a problem breaks into recurring subproblems. Longest common subsequence is given as an example that is solved using dynamic programming in O(mn) time rather than a brute force O(2^m*n) approach. 3) The dynamic programming algorithm for longest common subsequence works by defining a 2D array c where c[i,j] represents the length of the LCS of the first i elements

Average case Analysis of QuicksortRajendran

Np completenessRajendran

1) NP-Completeness refers to problems that are in NP (can be verified in polynomial time) and are as hard as any problem in NP. 2) The first problem proven to be NP-Complete was the Circuit Satisfiability problem, which asks whether there exists an input assignment that makes a Boolean circuit output 1. 3) To prove a problem P is NP-Complete, it must be shown that P is in NP and that any problem in NP can be reduced to P in polynomial time. This establishes P as at least as hard as any problem in NP.

computer languagesRajendran

The document discusses the differences between low-level machine code used by CPUs and high-level computer languages used by programmers. It explains that programmers write source code in high-level languages, which are then compiled into machine code through a compiler or interpreted line-by-line using an interpreter. Compiled code tends to run faster while interpreted languages typically have more programmer-friendly features. Modern dynamic languages often use a just-in-time compiler to bridge performance gaps.

Element distinctness lower boundsRajendran

Scheduling with Startup and Holding CostsRajendran

Divide and conquer surfing lower boundsRajendran

Red black treeRajendran

Hash tableRajendran

Medians and order statisticsRajendran

Proof master theoremRajendran

Recursion tree methodRajendran

Recurrence theoremRajendran

Master method Rajendran

Master method theoremRajendran

Hash tablesRajendran

Lower boundRajendran

Master method theoremRajendran

Greedy algorithmsRajendran

Longest common subsequences in Algorithm AnalysisRajendran

Dynamic programming in Algorithm AnalysisRajendran

Average case Analysis of QuicksortRajendran

Np completenessRajendran

computer languagesRajendran

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Two dimensional array

1. Two dimensional

2. - Two dimensional arrays are declared similarly to one dimensional arrays: char t[20][10]; (and referred to as a matrix, or as a "table") The first subscript gives the row number, and the second subscript specifies column number.

3. We can also initialize a two-dimensional array in a declaration statement; E.g., int m[2][3] = {{1, 2, 3}, {4, 5, 6}}; which specifies the elements in each row of the array (the interior braces around each row of values could be omitted). The matrix for this example is       = 654 321 m

4. Specification of the number of rows could be omitted. But other subscript specifications can not be omitted. So, we could also write the initialization as int m[ ][3] = {{1, 2, 3}, {4, 5, 6}};

5. int a[2][3]= {1,2,3,4,5,6}; specifies 6 integer locations. Storage for array elements are in contiguous locations in memory in row major order (unlike column major order in Fortran), referenced by subscripts(or index) values starting at 0 for both rows and columns. a[0][0] a[0][1] a[0][2] a[1][0] a[1][1] a[1][2] RAM 1 2 3 4 5 6

6. 1. Problem Definition Assume we have a file “in.dat” (located in the pwd) consisting of 10 rows of 10 integers per row (100 integers total). Write a C program which reads from the file and stores the numbers in a two dimensional array named ‘mat’, computes the largest of all of these numbers and prints the largest value to the screen. 2. Refine, Generalize, Decompose the problem definition (i.e., identify sub-problems, I/O, etc.) Input = from file “in.dat” Output= Largest of the 100 integers printed to screen. 3. Develop Algorithm Use Unix redirection to read file “in.dat”. Use nested for loop to go through two dimensional array to

7. #include <stdio.h> void main(void) { int row, col, maximum; /* Declare a 2-D array called ‘mat’ which can hold a 10-by-10 */ /* matrix of integers */ int mat[10][10]; /* Write code here to read from the file ‘in.dat’ */ for(row=0;row<10;++row) for(col=0;col<10;++col) scanf("%i",&mat[row][col]); /* Write code to find the largest value in the array ‘mat’ */ maximum = mat[0][0]; /* why ??? */ for(row=0;row<10;++row) /* How does this work??? */ for(col=0;col<10;++col) if (mat[row][col] > maximum) maximum = mat[row][col]; /* Print the largest value to the screen */ printf(" max value in the array = %in",maximum); }

8. The typedef mechanism in C allows us to define our own data types. For example, typedef double matrix[10][20]; In this example we create a new data-type named matrix. We can declare a variable of data-type matrix by matrix a; and this declaration is equivalent to double a[10][20];

9. Other examples of the use of typedef are, typedef int blah; /* don’t do this !!!!! */ typedef char string[5]; In the first example we give another name or alias to the data type int . In the second example we create a new data-type named string.

10. To declare a variable of data-type blah , blah x; this declaration is equivalent to int x; To declare a variable of data-type string , string var ="init"; /* declare and initialize */ scanf("%s",var); /* read in a string of chars */ strcpy(var , "next"); The declaration string morse[26]; declares morse as an array of 26 elements. Each element has data-type string.

11. The declaration and initialization: string morse[26] = {".-", "-...", "-.-.", "-..", ".", "..-.", "--.", "....", "..", ".---", "-.-", ".-..", "--", "-.", "---", ".--.", "--.-", ".-.", "...", "-", "..-", "...-", ".--", "-..-", "-.--", "--.. " }; declares morse as an array of 26 elements with morse[0] = ".-" morse[1] = "-..." and so on...

12. Standard (ANSI) C allows up to 12 dimension. But more computation is required by the system to locate elements in multi-subscripted arrays. Therefore, in problems involving intensive numerical calculations, we should use arrays with fewer dimension. We can also use 3, 4, … dimensional arrays, e.g., threeDArray [i][j][k];

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