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Find the Longest Increasing Subsequence of a Given Sequence

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/* Find the Longest Increasing Subsequence of a Given Sequence This is a C++ Program to implement LCS. The longest common subsequence (LCS) problem is to find the longest subsequence common to all sequences in a set of sequences (often just two). (Note that a subsequence is different from a substring, for the terms of the former need not be consecutive terms of the original sequence.) It is a classic computer science problem, the basis of data comparison programs such as the diff utility, and has applications in bioinformatics. */ #include <iostream> #include <string.h> #include <stdio.h> using namespace std; #define ARRAY_SIZE(A) sizeof(A)/sizeof(A[0]) // Binary search (note boundaries in the caller) // A[] is ceilIndex in the caller int CeilIndex(int A[], int l, int r, int key) { int m; while (r - l > 1) { m = l + (r - l) / 2; (A[m] >= key ? r : l) = m; // ternary expression returns an l-value } return r; } int LongestIncreasingSubsequenceLength(int A[], int size) { // Add boundary case, when array size is one int *tailTable = new int[size]; int len; // always points empty slot memset(tailTable, 0, sizeof(tailTable[0]) * size); tailTable[0] = A[0]; len = 1; for (int i = 1; i < size; i++) { if (A[i] < tailTable[0]) // new smallest value tailTable[0] = A[i]; else if (A[i] > tailTable[len - 1]) // A[i] wants to extend largest subsequence tailTable[len++] = A[i]; else // A[i] wants to be current end candidate of an existing subsequence // It will replace ceil value in tailTable tailTable[CeilIndex(tailTable, -1, len - 1, A[i])] = A[i]; } delete[] tailTable; return len; } int main() { int A[] = { 2, 5, 3, 7, 11, 8, 10, 13, 6 }; int n = ARRAY_SIZE(A); printf("Length of Longest Increasing Subsequence is %d\n", LongestIncreasingSubsequenceLength(A, n)); return 0; }

In computer programming, loops are used to repeat a block of code. For example, when you are displaying number from 1 to 100 you may want set the value of a variable to 1 and display it 100 times, increasing its value by 1 on each loop iteration. When you know exactly how many times you want to loop through a block of code, use the for loop instead of a while loop. A for loop is a repetition control structure that allows you to efficiently write a loop that needs to execute a specific number of times.

An array is defined as the collection of similar type of data items stored at contiguous memory locations. Arrays are the derived data type in C++ programming language which can store the primitive type of data such as int, char, double, float, etc. It also has the capability to store the collection of derived data types, such as pointers, structure, etc. The array is the simplest data structure where each data element can be randomly accessed by using its index number. C++ array is beneficial if you have to store similar elements. For example, if we want to store the marks of a student in 6 subjects, then we don't need to define different variables for the marks in the different subject. Instead of that, we can define an array which can store the marks in each subject at the contiguous memory locations.

The sizeof() is an operator that evaluates the size of data type, constants, variable. It is a compile-time operator as it returns the size of any variable or a constant at the compilation time. The size, which is calculated by the sizeof() operator, is the amount of RAM occupied in the computer. The sizeof is a keyword, but it is a compile-time operator that determines the size, in bytes, of a variable or data type. The sizeof operator can be used to get the size of classes, structures, unions and any other user defined data type. The data_type can be the data type of the data, variables, constants, unions, structures, or any other user-defined data type.

#include is a way of including a standard or user-defined file in the program and is mostly written at the beginning of any C/C++ program. This directive is read by the preprocessor and orders it to insert the content of a user-defined or system header file into the following program. These files are mainly imported from an outside source into the current program. The process of importing such files that might be system-defined or user-defined is known as File Inclusion. This type of preprocessor directive tells the compiler to include a file in the source code program.

Return length of string. Returns the length of the string, in terms of bytes. This function is used to find the length of the string in terms of bytes. This is the actual number of bytes that conform the contents of the string , which is not necessarily equal to the storage capacity. This is the number of actual bytes that conform the contents of the string, which is not necessarily equal to its storage capacity. Note that string objects handle bytes without knowledge of the encoding that may eventually be used to encode the characters it contains. Therefore, the value returned may not correspond to the actual number of encoded characters in sequences of multi-byte or variable-length characters (such as UTF-8).

The if...else statement executes two different codes depending upon whether the test expression is true or false. Sometimes, a choice has to be made from more than 2 possibilities. The if...else ladder allows you to check between multiple test expressions and execute different statements. In C/C++ if-else-if ladder helps user decide from among multiple options. The C/C++ if statements are executed from the top down. As soon as one of the conditions controlling the if is true, the statement associated with that if is executed, and the rest of the C else-if ladder is bypassed. If none of the conditions is true, then the final else statement will be executed.

In while loop, condition is evaluated first and if it returns true then the statements inside while loop execute, this happens repeatedly until the condition returns false. When condition returns false, the control comes out of loop and jumps to the next statement in the program after while loop. The important point to note when using while loop is that we need to use increment or decrement statement inside while loop so that the loop variable gets changed on each iteration, and at some point condition returns false. This way we can end the execution of while loop otherwise the loop would execute indefinitely. A while loop that never stops is said to be the infinite while loop, when we give the condition in such a way so that it never returns false, then the loops becomes infinite and repeats itself indefinitely.

In computer programming, we use the if statement to run a block code only when a certain condition is met. An if statement can be followed by an optional else statement, which executes when the boolean expression is false. There are three forms of if...else statements in C++: • if statement, • if...else statement, • if...else if...else statement, The if statement evaluates the condition inside the parentheses ( ). If the condition evaluates to true, the code inside the body of if is executed. If the condition evaluates to false, the code inside the body of if is skipped.

Search range for subsequence. Searches the range [first1,last1) for the first occurrence of the sequence defined by [first2,last2), and returns an iterator to its first element, or last1 if no occurrences are found. The elements in both ranges are compared sequentially using operator== (or pred, in version (2)): A subsequence of [first1,last1) is considered a match only when this is true for all the elements of [first2,last2). This function returns the first of such occurrences. For an algorithm that returns the last instead, see find_end. The function shall not modify any of its arguments.

Strings are objects that represent sequences of characters. The standard string class provides support for such objects with an interface similar to that of a standard container of bytes, but adding features specifically designed to operate with strings of single-byte characters. The string class is an instantiation of the basic_string class template that uses char (i.e., bytes) as its character type, with its default char_traits and allocator types. Note that this class handles bytes independently of the encoding used: If used to handle sequences of multi-byte or variable-length characters (such as UTF-8), all members of this class (such as length or size), as well as its iterators, will still operate in terms of bytes (not actual encoded characters).

Check whether eofbit is set. Returns true if the eofbit error state flag is set for the stream. This flag is set by all standard input operations when the End-of-File is reached in the sequence associated with the stream. Note that the value returned by this function depends on the last operation performed on the stream (and not on the next). Operations that attempt to read at the End-of-File fail, and thus both the eofbit and the failbit end up set. This function can be used to check whether the failure is due to reaching the End-of-File or to some other reason.

Deallocate storage space. Default deallocation functions (single-object form). A delete operator is used to deallocate memory space that is dynamically created using the new operator, calloc and malloc() function, etc., at the run time of a program in C++ language. In other words, a delete operator is used to release array and non-array (pointer) objects from the heap, which the new operator dynamically allocates to put variables on heap memory. We can use either the delete operator or delete [ ] operator in our program to delete the deallocated space. A delete operator has a void return type, and hence, it does not return a value.

Fill block of memory. Sets the first num bytes of the block of memory pointed by ptr to the specified value (interpreted as an unsigned char). This function converts the value of a character to unsigned character and copies it into each of first num character of the object pointed by the given str[]. If the num is larger than string size, it will be undefined.

In the C++ Programming Language, the #define directive allows the definition of macros within your source code. These macro definitions allow constant values to be declared for use throughout your code. Macro definitions are not variables and cannot be changed by your program code like variables. You generally use this syntax when creating constants that represent numbers, strings or expressions. The syntax for creating a constant using #define in the C++ is: #define token value

A program shall contain a global function named main, which is the designated start of the program in hosted environment. main() function is the entry point of any C++ program. It is the point at which execution of program is started. When a C++ program is executed, the execution control goes directly to the main() function. Every C++ program have a main() function.

Consider a situation, when we have two persons with the same name, jhon, in the same class. Whenever we need to differentiate them definitely we would have to use some additional information along with their name, like either the area, if they live in different area or their mother's or father's name, etc. Same situation can arise in your C++ applications. For example, you might be writing some code that has a function called xyz() and there is another library available which is also having same function xyz(). Now the compiler has no way of knowing which version of xyz() function you are referring to within your code.

Allocate storage space. Default allocation functions (single-object form). A new operator is used to create the object while a delete operator is used to delete the object. When the object is created by using the new operator, then the object will exist until we explicitly use the delete operator to delete the object. Therefore, we can say that the lifetime of the object is not related to the block structure of the program.