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24,200 | notes there are bunch of junctions (nodes in the graphand bunch of pipes (edges in the graphconnecting the junction the pipe will only allow water to flow one way (the graph is directedeach pipe has also has capacity (the weight of the edge)representing the maximum amount of water that can flow through the pipe finally... |
24,201 | the first path we found was if we push as much flow as possiblethen we end up with the followingnotes figure we have pushed flow along the path / / / / / now we have run into problemour only options left are to push unit of water along the path and unit of water along after thatwe won' be able to find any more path fro... |
24,202 | notes an edge (vuwith capacity in the residual graph figure shows the residual graph after finding our first pathfigure the dashed edges are the edges we added in the residual graph / / / / / the capacity of the dashed edge is the same as the amount of water carried by the solid edge in the opposite direction so here i... |
24,203 | which we define by gx (vex)where ex is set of residual edges corresponding to the feasible solution notes what is exwe replace each arc ( ,jin by two arcs ( , )( , )the arc ( ,jhas (residualcapacity rij uij xijand the arc ( ,ihas (residualcapacity rji=xij then we construct the set ex from the new edges with positive re... |
24,204 | notes consider that in this case and soone can verify that / in zadeh presents examples of networks that require / and / augmentationsbut these examples are dependent on choice of the shortest path we made worst-case tests with and vertexes and compared the running times of the improved version of the algorithm against... |
24,205 | figure an example of the transportation network in this we have supply vertexes (with supply values and ) demand vertexes (with demand values and )and transshipment node each edge has two numberscapacity and costdivided by comma bi bj uij,cij notes - - - representing the flow on arc (ije by xijwe can obtain the optimiz... |
24,206 | notes we can easily avoid this situationhoweverif we add special node with the supply/demand value br - now we have two optionsif (supply dominatesthen for each node with bi we add an arc (irwith infinite capacity and zero costotherwise (demand dominates)for each node with bi we add an arc (riwith the same properties n... |
24,207 | function will be unbounded howeverin some taskswe are able to assign finite capacity to each uncapacitated edge escaping such situation notes minimum spanning tree minimum spanning tree or minimum weight spanning tree is then spanning tree with weight less than or equal to the weight of every other spanning tree more g... |
24,208 | notes figure sixteen spanning trees of complete graph on four vertices now suppose the edges of the graph have weights or lengths the weight of tree is just the sum of weights of its edges obviouslydifferent trees have different lengths the problemhow to find the minimum length spanning treethis problem can be solved b... |
24,209 | algorithm notes let (vewhich is represented by an adjacency list adj some support data structures is the forest - set of all (partialtrees mst is the minimum spanning treerepresented by set of edges is priority queue of edges kruskal( let be set of singleton set of all vertices in mst <{ < while not empty do (uv<extrac... |
24,210 | notes in the case where the weight of is less than the weight of we can conclude that is not minimum spanning treeand the assumption that there exist edges , with (ew(fis incorrect and therefore is minimum spanning tree (equal to or with different edge setbut with same weightfigure step by step representation of kruska... |
24,211 | prim' algorithm notes prim' algorithm is an algorithm in graph theory that finds minimum spanning tree for connected weighted graph this means it finds subset of the edges that forms tree that includes every vertexwhere the total weight of all the edges in the tree is minimized the algorithm was discovered in by mathem... |
24,212 | data structure is the logical or mathematical arrangement of data in memory it considers not only the physical layout of the data items in the memory but also the relationships between these data items and the operations that can be performed on these items the choice of appropriate data structures and algorithms forms... |
24,213 | preface to the second edition --new programs on parenthesis checking and evaluation of prefix expressions have been added recursionwhich is one of the most common applications of stackshas been moved to this --new programs on priority queues and multiple queues have been included --this now includes sections on general... |
24,214 | stacks focuses on the concept of last-infirst-out (lifodata structure called stacks the also shows the practical implementation of these data structures using arrays as well as linked lists it also shows how stacks can be used for the evaluation of arithmetic expressions queues deals with the concept of first-infirst-o... |
24,215 | preface to the second edition preface to the first edition viii introduction to introduction to data structures and algorithms arrays strings structures and unions linked lists stacks queues trees efficient binary trees multi-way search trees heaps graphs searching and sorting hashing and collision files and their orga... |
24,216 | detailed contents preface to the second edition preface to the first edition viii introduction to introduction identifiers and keywords basic data types variables and constants writing the first program input and output functions operators and expressions type conversion and typecasting control statements decision cont... |
24,217 | inserting an element in an array deleting an element from an array merging two arrays passing arrays to functions passing individual elements passing the entire array pointers and arrays arrays of pointers two-dimensional arrays declaring two-dimensional arrays initializing two-dimensional arrays accessing the elements... |
24,218 | deleting node from doubly linked list circular doubly linked lists inserting new node in circular doubly linked list deleting node from circular doubly linked list header linked lists multi-linked lists applications of linked lists polynomial representation stacks introduction to stacks array representation of stacks o... |
24,219 | avl trees operations on avl trees searching for node in an avl tree red-black trees properties of red-black trees operations on red-black trees applications of red-black trees splay trees operations on splay trees advantages and disadvantages of splay trees multi-way search trees introduction to -way search trees trees... |
24,220 | modified warshall' algorithm applications of graphs searching and sorting introduction to searching linear search binary search interpolation search jump search introduction to sorting sorting on multiple keys practical considerations for internal sorting bubble sort insertion sort selection sort merge sort quick sort ... |
24,221 | introduction to learning objective this book deals with the study of data structures through before going into detailed analysis of data structuresit would be useful to familiarize ourselves with the basic knowledge of programming in thereforein this we will learn about the various constructs of such as identifiers and... |
24,222 | data structures using main(statement statement statement nfunction (statement statement statement nfunction (statement statement statement nfunctionn(statement statement statement nfigure structure of program of statements arranged according to specific meaningful sequence note that programmers can choose any name for ... |
24,223 | might have been surprised to see that the range of char is given as - to char is supposed to store characters not numbersso why this rangethe answer is that in the memorycharacters are stored in their ascii codes for examplethe character 'ahas the ascii code of in memory we will not store 'abut (in binary number format... |
24,224 | data structures using numeric variables numeric variables can be used to store either integer values or floating point values modifiers like shortlongsignedand unsigned can also be used with numeric variables the difference between signed and unsigned numeric variables is that signed variables can be either negative or... |
24,225 | writing the first program to write programwe first need to write the code for thatopen text editor if you are windows useryou may use notepad and if you prefer working on unix/linuxyou can use emac or vi once the text editor is opened on your screentype the following statements#include int main(printf("\ welcome to the... |
24,226 | data structures using this command is used to compile your program if there are any mistakes in the programthen the compiler will tell you what mistake(syou have made and on which line the error has occurred in case of errorsyou need to re-open your file and correct the mistakes howeverif everything is rightthen no err... |
24,227 | the programprintf function is used that sends results to terminal like printf and scanfthere are different functions in that can carry out the input-output operations these functions are collectively known as standard input/output library program that uses standard input/output functions must contain the following stat... |
24,228 | data structures using note in case of reading stringswe do not use the sign in the scanf function printf(the printf function is used to display information required by the user and also prints the values of the variables its syntax can be given asprintf ("control string"arg ,arg ,arg ,argn)after the control stringthe f... |
24,229 | printf("% " )printf("% " ) operators and expressions language supports different types of operatorswhich can be used with variables and constants to form expressions these operators can be categorized into the following major groupsarithmetic operators relational operators equality operators logical operators unary ope... |
24,230 | data structures using note it is not possible to divide any number by zero this is an illegal operation that results in run-time division-by-zero exception thereby terminating the program except for modulus operatorall other arithmetic operators can accept mix of integer and floating point numbers if both operands are ... |
24,231 | for example( cthe whole expression is true only if both expressions are truei if is greater than and table truth table of logical or | table truth table of logical not logical or (||logical or returns false value if both the operands are false otherwise it returns true value the truth table of logical or operator is gi... |
24,232 | data structures using int yy ++is equivalent to writing xx whereas ++xis equivalent to writing xthe same principle applies to unary decrement operators note that unary operators have higher precedence than the binary operators and if in an expression we have more than one unary operator then they are evaluated from rig... |
24,233 | table truth table of bitwise xor bit in the second operand the truth table of bitwise xor operator is shown in table the bitwise xor operator compares each bit of its first operand with the corresponding bit of its second operand if one of the bits is the corresponding bit in the result is and otherwise for example bit... |
24,234 | data structures using table assignment operators operator example /float = float = / \int int \ *int int * +int int + -int int - &int int & ^int int ^ <<int int << >>int int >>ba is evaluated as ( ( ( )))first is assigned to cthen the value of is assigned to finallythe value of is assigned to table contains list of oth... |
24,235 | table operators precedence chart operator associativity (left-to-right [--++(postfixright-to-left --(postfix++(prefixright-to-left --(prefix+(unary(unary(type*(indirection&(addresssizeof left-to-right left-to-right left-to-right <left-to-right >=!left-to-right left-to-right left-to-right left-to-right &left-to-right |l... |
24,236 | data structures using type conversion and typecasting type conversion or typecasting of variables refers to changing variable of one data type into another while type conversion is done implicitlycasting has to be done explicitly by the programmer we will discuss both these concepts here type conversion type conversion... |
24,237 | control statements till now we know that the code in the program is executed sequentially from the first line of the program to its last line that isthe second statement is executed after the firstthe third statement is executed after the secondso on and so forth although this is truein some cases we want only selected... |
24,238 | data structures using #include int main(int = if ( > ++printf("\ % " )return in the above codewe take variable and initialize it to in the test expressionwe check if the value of is greater than as the test expression evaluates to trueand the value of is incremented after thatthe value of is printed on the screen the o... |
24,239 | programming example write program to find whether number is even or odd #include int main(int aprintf("\ enter the value of ")scanf("% "& )if( % == printf("\ % is even" )else printf("\ % is odd" )return output enter the value of is even if-else-if statement language supports if-else-if statements to test additional con... |
24,240 | data structures using int numprintf("\ enter any number ")scanf("% "&num)if(num== printf("\ the value is equal to zero")else if(num> printf("\ the number is positive")else printf("\ the number is negative")return note that if the first test expression evaluates to true valuei num= then the rest of the statements in the... |
24,241 | when there are many conditions to testusing the if and else-if constructs becomes complicated and confusing thereforeswitch case statements are often used as an alternative to long if statements that compare variable to several 'integralvalues (integral values are those values that can be expressed as an integersuch as... |
24,242 | data structures using printf("\ % is vowel"ch)breakcase ' 'case ' 'printf("\ % is vowel"ch)breakcase ' 'case ' 'printf("\ % is vowel"ch)breakdefaultprintf("\ % is not vowel"ch)return output enter any character is not vowel note that there is no break statement after case aso if the character is entered then control wil... |
24,243 | note that in the while loopthe condition is tested before any of the statements in the statement block is executed if the condition is trueonly then the statements will be executedotherwise if the condition is falsethe control will jump to statement ythat is the immediate statement outside the while loop block in the f... |
24,244 | data structures using figure do-while construct note that the test condition is enclosed in parentheses and followed by semi-colon the statements in the statement block are enclosed within curly brackets the curly brackets are optional if there is only one statement in the body of the do-while loop the do-while loop co... |
24,245 | printf("\ enter the value of ")scanf("% "& )do sum sum ii while( <= )avg (float)sum/nprintf("\ the sum of first % numbers % ",nsum)printf("\ the average of first % numbers "navg)return output enter the value of the sum of first numbers the average of first numbers for loop like the while and do-while loopsthe for loop ... |
24,246 | data structures using should continue to repeat itself every iteration of the loop must make the condition to exit the loop approachable sowith every iterationthe loop variable must be updated updating the loop variable may include incrementing the loop variabledecrementing the loop variable or setting it to some other... |
24,247 | break and continue statements break statement in cthe break statement is used to terminate the execution of the nearest enclosing loop in which it appears we have already seen its use in the switch statement the break statement is widely used with forwhileand do-while loops when the compiler encounters break statementt... |
24,248 | data structures using continueprintf("\ % " )return output note that the code is meant to print numbers from to but as soon as becomes equal to the continue statement is encounteredso the printf(statement is skipped and the control passes to the expression that increments the value of hencewe conclude that the continue... |
24,249 | why are functions neededlet us analyse the reasons why segmenting program into manageable chunks is an important aspect of programming dividing the program into separate well-defined functions facilitates each function to be written and tested separately this simplifies the process of getting the total program to work ... |
24,250 | data structures using return_data_type specifies the data type of the value that will be returned to the calling function as result of the processing performed by the called function (data_type variable data_type variable is list of variables of specified data types these variables are passed from the calling function ... |
24,251 | programming example write program to find whether number is even or odd using functions #include int evenodd(int)//function declaration int main(int numflagprintf("\ enter the number ")scanf("% "&num)flag evenodd(num)//function call if (flag = printf("\ % is even"num)else printf("\ % is odd"num)return int evenodd(int /... |
24,252 | data structures using printf("\ the value of num after calling the function % "num)return void add(int nn printf("\ the value of num in the called function % " )output the value of num before calling the function the value of num in the called function the value of num after calling the function following are the point... |
24,253 | * * printf("\ the value of num in the called function % "* )output the value of num before calling the function the value of num in the called function the value of num after calling the function advantages the advantages of using the call-by-reference technique of passing arguments includesince arguments are not copie... |
24,254 | data structures using output in main() and in function (call by value methoda and in main() and in main() and in function (call by reference methodc and in main() and pointers every variable in has name and value associated with it when variable is declareda specific block of memory within the computer is allocated to ... |
24,255 | pointers are used for the dynamic memory allocation of variable (refer appendix on memory allocation in programsdeclaring pointer variables the general syntax of declaring pointer variables can be given as below data_type *ptr_nameheredata_type is the data type of the value that the pointer will point to for exampleint... |
24,256 | data structures using pointer expressions and pointer arithmetic like other variablespointer variables can also be used in expressions for exampleif ptr and ptr are pointersthen the following statements are validint num num sum mul div int *ptr *ptr ptr &num ptr &num sum *ptr *ptr mul sum (*ptr )*ptr + div (*ptr )/(*pt... |
24,257 | void *ptrin csince you cannot have variable of type voidthe void pointer will therefore not point to any data andthuscannot be dereferenced you need to cast void pointer to another kind of pointer before using it generic pointers are often used when you want pointer to point to data of different types at different time... |
24,258 | data structures using for exampleconsider the following codeint = int *px**ppxpx &xppx &pxfigure pointer to pointer now if we writelet us assumethe memory locations of these variables are as shown in fig printf("\ % "**ppx)thenit would print the value of drawbacks of pointers although pointers are very useful in cthey ... |
24,259 | modulus operator (%can only be applied on integer operandsand not on float or double operands equality operators have lower precedence than relational operators like arithmetic expressionslogical expressions are evaluated from left to right both +and ++ increment the value of xbut in the former casethe value of is retu... |
24,260 | data structures using exercises review questions discuss the structure of program differentiate between declaration and definition how is memory reserved using declaration statement what do you understand by identifiers and keywords explain the terms variables and constants how many types of variables are supported by ... |
24,261 | (ccorrect to eight decimal places write program to find the smallest of three integers using functions write program to calculate area of triangle using function write program to find whether number is divisible by two or not using functions write program to print 'programming in is funusing pointers write program to r... |
24,262 | data structures using equality operators have higher precedence than the relational operators shifting once to the left multiplies the number by decision control statements are used to repeat the execution of list of statements printf("% "scanf("% "&num))is valid statement , is valid integer constant printf statement c... |
24,263 | introduction to data structures and algorithms learning objective in this we are going to discuss common data structures and algorithms which serve as building blocks for creating efficient programs we will also discuss different approaches to designing algorithms and different notations for evaluating the performance ... |
24,264 | data structures using data structures are used in almost every program or software system some common examples of data structures are arrayslinked listsqueuesstacksbinary treesand hash tables data structures are widely applied in the following areascompiler design statistical analysis package numerical analysis artific... |
24,265 | elementary data structure organization data structures are building blocks of program program built using improper data structures may not work as expected so as programmer it is mandatory to choose most appropriate data structures for program the term data means value or set of values it specifies either the value of ... |
24,266 | data structures using howeverif the elements of data structure are not stored in sequential orderthen it is non-linear data structure the relationship of adjacency is not maintained between elements of non-linear data structure examples include trees and graphs supports variety of data structures we will now introduce ... |
24,267 | the last node in the list contains null pointer to indicate that it is the end or tail of the list since the memory for node is dynamically allocated when it is added to the listthe total number of nodes that may be added to list is limited only by the amount of memory available figure shows linked list of seven nodes ... |
24,268 | data structures using front rear figure array representation of queue herefront and rear if we want to add one more value to the listsayif we want to add another element with the value then the rear would be incremented by and the value would be stored at the position pointed by the rear the queueafter the additionwoul... |
24,269 | in fig node is the left child and node is the right child of the root node note that the left sub-tree of the root node consists of the nodes and similarlythe right sub-tree of the root node consists of the nodes and note advantageprovides quick searchinsertand delete operations disadvantagecomplicated deletion algorit... |
24,270 | data structures using many timetwo or more operations are applied simultaneously in given situation for exampleif we want to delete the details of student whose name is xthen we first have to search the list of students to find whether the record of exists or not and if it exists then at which locationso that the detai... |
24,271 | and such language is known as programming language in general termsan algorithm provides blueprint to write program to solve particular problem it is considered to be an effective procedure for solving problem in finite number of steps that isa well-defined algorithm always provides an answer and is guaranteed to termi... |
24,272 | data structures using top-down vs bottom-up approach whether the top-down strategy should be followed or bottom-up is question that can be answered depending on the application at hand while top-down approach follows stepwise refinement by decomposing the algorithm into manageable modulesthe bottom-up approach on the o... |
24,273 | figure algorithm to test for equality of two numbers figure algorithm to print the first natural of programming examples write an algorithm for swapping two values step input first number as step input second number as step set temp step set step set temp step print ab step end write an algorithm to find the larger of ... |
24,274 | data structures using [end of ifstep end write an algorithm to find the sum of first natural numbers step input step set sum step repeat step while < step set sum sum set [end of loopstep print sum step end time and space complexity analysing an algorithm means determining the amount of resources (such as time and memo... |
24,275 | time-space trade-off the best algorithm to solve particular problem at hand is no doubt the one that requires less memory space and takes less time to complete its execution but practicallydesigning such an ideal algorithm is not trivial task there can be more than one algorithm to solve particular problem one may requ... |
24,276 | data structures using logarithmic loops we have seen that in linear loopsthe loop updation statement either adds or subtracts the loop-controlling variable howeverin logarithmic loopsthe loop-controlling variable is either multiplied or divided during each iteration of the loop for examplelook at the loops given belowf... |
24,277 | if we calculate the average of this loop ( / )we will observe that it is equal to the number of iterations in the outer loop ( plus divided by in general termsthe inner loop iterates ( )/ times thereforethe efficiency of such code can be given as (nn ( )/ big notation in today' era of massive advancement in computer te... |
24,278 | data structures using examples of functions not in ( includen to summarizebest case describes an upper bound for all combinations of input it is possibly lower than the worst case for examplewhen sorting an array the best case is when the array is already correctly sorted worst case describes lower bound for worst case... |
24,279 | < < this means = therefore = example show that ( solution by definitionwe have < ( <cg(nsubstituting as (nand as ( )we get < <cn dividing by / < / / <cn / < / < note that / as and / is maximum when therefore < / < this meansc to determine the value of < / < - < / < - < / < - < / < - < < this implies hence = example sho... |
24,280 | data structures using limitations of big notation there are certain limitations with the big notation of expressing the complexity of algorithms these limitations are as followsmany algorithms are simply too hard to analyse mathematically there may not be sufficient information to calculate the behaviour of the algorit... |
24,281 | - < < / so as lim / naehence ( for = and >no= example show that ( solution by the definitionwe can write <cg( < ( <cn < dividing by we get / <cn / < / < < / thusfrom the above statementwe see that the value of depends on the value of there does not exist value of that satisfies the condition as increases this could fai... |
24,282 | data structures using > you may verify this by substituting the values as shown below < / < / / and / ( < / / < / / < / < / thusin generalwe can write / other useful notations there are other notations like little notation and little notation which have been discussed below little notation this notation provides non-as... |
24,283 | < / this is contradictory value as for any value of as it cannot be assured to be less than / or / an imprecise analogy between the asymptotic comparison of functions (nand (nand the relation between their values can be given asf(no( ( ) ( > (nf(no( ( ) (ng(npoints to remember data structure is particular way of storin... |
24,284 | data structures using define data structures give some examples in how many ways can you categorize data structuresexplain each of them discuss the applications of data structures write short note on different operations that can be performed on data structures compare linked list with an array write short note on abst... |
24,285 | which notation provides strict upper bound for ( )(aomega notation (bbig notation (csmall notation (dtheta notation which notation comprises set of all functions (nthat are greater than or equal to cg(nfor all values of > (aomega notation (bbig notation (csmall notation (dtheta notation function in ( notation is ( (bn ... |
24,286 | arrays learning objective in this we will discuss arrays an array is user-defined data type that stores related information together all the information stored in an array belongs to the same data type soin this we will learn how arrays are defineddeclaredinitializedand accessed we will also discuss the different opera... |
24,287 | an array is collection of similar data elements these data elements have the same data type the elements of the array are stored in consecutive memory locations and are referenced by an index (also known as the subscriptthe subscript is an ordinal number which is used to identify an element of the array marks marks mar... |
24,288 | data structures using data type [ int marks [ ][ [ [ [ [ [ [ [ [ array name [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ char name [ ]array size [ float salary [ ][ [ [ [ figure declaring arrays of different data types and sizes accessing the elements of an array storing related data items in single array enables the programmers to d... |
24,289 | other data elements can simply be calculated using the base address the formula to perform this calculation isaddress of data elementa[kba(aw( lower_boundherea is the arrayk is the index of the element of which we have to calculate the addressba is the base address of the array aand is the size of one element in memory... |
24,290 | data structures using initialize the elements during declaration storing values in an array input values for the elements from the keyboard assign values to individual elements figure storing values in an array initializing arrays during declaration the elements of an array can be initialized at the time of declaration... |
24,291 | inputting values from the keyboard an array can be initialized by inputting values from the keyboard in this methoda while/do-while or for loop is executed to input the value for each element of the array for examplelook at the figure code for inputting each code shown in fig element of the array in the codewe start at... |
24,292 | data structures using traversing the data elements of an array can include printing every elementcounting the total number of elementsor performing any process on these elements sincearray is linear data structure (because all its elements form sequence)traversing its elements is very simple and straightforward the alg... |
24,293 | int inarr[ ]sum = float mean clrscr()printf("\ enter the number of elements in the array ")scanf("% "& )for( = ; < ; ++printf("\ arr[% " )scanf("% ",&arr[ ])for( = ; < ; ++sum +arr[ ]mean (float)sum/nprintf("\ the sum of the array elements % "sum)printf("\ the mean of the array elements "mean)return output enter the nu... |
24,294 | data structures using the smallest element is the position of the smallest element in the array is write program to find the second largest of numbers using an array #include #include int main(int inarr[ ]largesecond_largeclrscr()printf("\ enter the number of elements in the array ")scanf("% "& )printf("\ enter the ele... |
24,295 | scanf("% "&digit[ ]) = while( <numofdigitsnumber number digit[ipow( , ) ++printf("\ the number is % "number)return output enter the number of digits enter the digit at position enter the digit at position enter the digit at position enter the digit at position the number is write program to find whether the array of in... |
24,296 | data structures using inserting an element in an array if an element has to be inserted at the end of an existing arraythen the task of insertion is quite simple we just have to add to the upper_ figure algorithm to append new element to an bound and assign the value herewe assume that existing array the memory space a... |
24,297 | ( datadata[ data[ data[ data[ data[ data[ data[ data[ data[ data[ algorithm to insert an element in the middle of an array the algorithm insert will be declared as insert (anposvalthe arguments are (aathe array in which the element has to be inserted (bnthe number of elements in the array (cposthe position at which the... |
24,298 | data structures using #include int main(int innumposarr[ ]clrscr()printf("\ enter the number of elements in the array ")scanf("% "& )for( = ; < ; ++printf("\ arr[% " )scanf("% "&arr[ ])printf("\ enter the number to be inserted ")scanf("% "&num)printf("\ enter the position at which the number has to be added ")scanf("% ... |
24,299 | scanf("% "&num)for( = ; < ; ++if(arr[inumfor( - >=ij--arr[ + arr[ ]arr[inumbreakn + printf("\ the array after insertion of % is "num)for( = ; < ; ++printf("\ arr[% % "iarr[ ])getch()return output enter the number of elements in the array arr[ arr[ arr[ arr[ arr[ enter the number to be inserted the array after insertion... |
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