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20,100 | chap mathematical preliminaries thought processwhich in turn clarifies your explanations secondif you use one of the standard proof structures such as proof by contradiction or an induction proofthen both you and your reader are working from shared understanding of that structure that makes for less complexity to your ... |
20,101 | proofproof by contradiction step contrary assumptionassume that there is largest integer call it (for "biggest"step show this assumption leads to contradictionconsider is an integer because it is the sum of two integers alsoc bwhich means that is not the largest integer after all thuswe have reached contradiction the o... |
20,102 | chap mathematical preliminaries proving either variant of the induction step (in conjunction with verifying the base caseyields satisfactory proof by mathematical induction the two conditions that make up the induction proof combine to demonstrate that thrm holds for as an extension of the fact that thrm holds for this... |
20,103 | sec mathematical proof techniques that ( ( )( )/ and because (ns( nwe can substitute for ( to get - = +ni= ( )( + ( thusby mathematical inductions(nn ( )/ = note carefully what took place in this example first we cast (nin terms of smaller occurrence of the problems(ns( this is important because once ( comes into the p... |
20,104 | chap mathematical preliminaries takes care of the rest " - ( ( = = ( ) thusby mathematical inductionpn = ( example this example shows how we can use induction to prove that proposed closed-form solution for recurrence relation is correct theorem the recurrence relation (nt( - )+ ( has closed-form solution (nn proofto p... |
20,105 | then replace cstamp with three cstamps if notthen the makeup must have included at least two cstamps (because it is at least of size and contains only cstampsin this casereplace two of the cstamps with single cstamp in either casewe now have value of made up of cand cstamps thusby mathematical inductionthe theorem is c... |
20,106 | chap mathematical preliminaries figure two-coloring for the regions formed by three lines in the plane lines unfortunatelythe regions newly split by the nth line violate the rule for two-coloring take all regions on one side of the nth line and reverse their coloring (after doing sothis half-plane is still two-coloredt... |
20,107 | example we would like to prove that function fact does indeed compute the factorial function there are two distinct steps to such proof the first is to prove that the function always terminates the second is to prove that the function returns the correct value theorem function fact will terminate for any value of proof... |
20,108 | chap mathematical preliminaries derive an equation that relates the parameters to the problem select values for the parametersand apply the equation to yield an estimated solution when doing estimationsa good way to reassure yourself that the estimate is reasonable is to do it in two different ways in generalif you wan... |
20,109 | typical car is driven about , miles per year if gasoline costs $ /gallonthen the yearly gas bill is $ for the less efficient car and $ for the more efficient car if we ignore issues such as the payback that would be received if we invested $ in bankit would take years to make up the difference in price at this pointthe... |
20,110 | chap mathematical preliminaries also discusses the three proof techniques presented in section and the roles of investigation and argument in problem solving for more about estimating techniquessee two programming pearls by john louis bentley entitled the back of the envelope and the envelope is back [ben ben ben ben g... |
20,111 | ( { ih ih ion the set { how many total orderings can be defined on set with elementsexplain your answer define an adt for set of integers (remember that set has no concept of duplicate elementsand has no concept of orderyour adt should consist of the functions that can be performed on set to control its membershipcheck... |
20,112 | chap mathematical preliminaries /iterative fibonacci generator static long fibi(int /fibr( is the largest value that fits in long assert ( &( < " out of range"long currprevpastif (( = |( = )return curr prev /curr holds current fib value for (int = <=ni++/compute next value past prev/past holds fibi( - prev curr/prev ho... |
20,113 | sec exercises prove by contradiction that the number of primes is infinite (ause induction to show that is always even (bgive direct proof in one or two sentences that is always even (cshow that is always divisible by three (dis aways divisible by explain your answer prove that is irrational explain why - ( ( ii= = = p... |
20,114 | chap mathematical preliminaries (bgive the summation that results from expanding your recurrence (cgive closed-form solution for the summation prove (using inductionthat the recurrence (nt( nt( has as its closed-form solution (nn( )/ expand the following recurrence to help you find closed-form solutionand then use indu... |
20,115 | when buying home mortgageyou often have the option of paying some money in advance (called "discount points"to get lower interest rate assume that you have the choice between two -year mortgagesone at %and the other at with an up-front charge of of the mortgage value how long would it take to recover the charge when yo... |
20,116 | chap mathematical preliminaries down to miles per hour after traveling another milehe again slows to miles per hour this continuesprogressively slowing by mile per hour for each mile traveled until the trip is complete (ahow long does it take the man to reach his in-laws(bhow long would the trip take in the continuous ... |
20,117 | algorithm analysis how long will it take to process the company payroll once we complete our planned mergershould buy new payroll program from vendor or vendor yif particular program is slowis it badly implemented or is it solving hard problemquestions like these ask us to consider the difficulty of problemor the relat... |
20,118 | chap algorithm analysis run them on suitable range of inputsmeasuring how much of the resources in question each program uses this approach is often unsatisfactory for four reasons firstthere is the effort involved in programming and testing two algorithms when at best you want to keep only one secondwhen empirically c... |
20,119 | compiled with the same compiler and run on the same computer under the same conditions as much as possiblethe same amount of care should be taken in the programming effort devoted to each program to make the implementations "equally efficient in this senseall of the factors mentioned above should cancel out of the comp... |
20,120 | chap algorithm analysis seen so far it is reasonable to assume that it takes fixed amount of time to do one such comparisonregardless of the value of the two integers or their positions in the array because the most important factor affecting running time is normally size of the inputfor given input size we often expre... |
20,121 | example consider the following codesum for ( = <=ni++for ( = <=nj++sum++what is the running time for this code fragmentclearly it takes longer to run when is larger the basic operation in this example is the increment operation for variable sum we can assume that incrementing takes constant timecall this time (we can i... |
20,122 | chap algorithm analysis log log input size figure two views of graph illustrating the growth rates for six equations the bottom view shows in detail the lower-left portion of the top view the horizontal axis represents input size the vertical axis can represent timespaceor any other measure of cost |
20,123 | sec bestworstand average cases log log log log figure costs for growth rates representative of most computer algorithms we can get some further insight into relative growth rates for various algorithms from figure most of the growth rates that appear in typical algorithms are shownalong with some representative input s... |
20,124 | chap algorithm analysis values if we implement sequential search as program and run it many times on many different arrays of size nor search for many different values of within the same arraywe expect the algorithm on average to go halfway through the array before finding the value we seek on averagethe algorithm exam... |
20,125 | astrous consequences on program' space or time performance unusual data distributions can also be used to advantageas shown in section in summaryfor real-time applications we are likely to prefer worst-case analysis of an algorithm otherwisewe often desire an average-case analysis if we know enough about the distributi... |
20,126 | chap algorithm analysis (nn change / log nn -figure the increase in problem size that can be run in fixed period of time on computer that is ten times faster the first column lists the right-hand sides for each of the five growth rate equations of figure for the purpose of this examplearbitrarily assume that the old ma... |
20,127 | computer ten times faster yetthen the new computer ( times faster than the original computerwill only run problem of size if you had second program whose growth rate is and for which the original computer could run problem of size in an hourthan machine ten times faster can run problem only of size in an hourthusan exp... |
20,128 | chap algorithm analysis when we want an estimate of the running time or other resource requirements of an algorithm this simplifies the analysis and keeps us thinking about the most important aspectthe growth rate this is called asymptotic algorithm analysis to be preciseasymptotic analysis refers to the study of an al... |
20,129 | fast as ( (the running time of our algorithmfor the worst-case inputwe would say the algorithm is "in ( in the worst case the following is precise definition for an upper bound (nrepresents the true running time of the algorithm (nis some expression for the upper bound for (na non-negatively valued functiont(nis in set... |
20,130 | chap algorithm analysis sequential search is practical for large nin way that is not true for some other algorithms in ( we always seek to define the running time of an algorithm with the tightest (lowestpossible upper bound thuswe prefer to say that sequential search is in (nthis also explains why the phrase "is in ( ... |
20,131 | sec asymptotic analysis example assume (nc for and thenc > for all sot( >cn for and thereforet(nis in ohm( by the definition it is also true that the equation of example is in ohm(nhoweveras with big-oh notationwe wish to get the "tightest(for ohm notationthe largestbound possible thuswe prefer to say that this running... |
20,132 | chap algorithm analysis many algorithms (or their instantiations as programs)it is easy to come up with the equation that defines their runtime behavior most algorithms presented in this book are well understood and we can almost always give th analysis for them however discusses whole class of algorithms for which we ... |
20,133 | sec asymptotic analysis rule ( says that given two parts of program run in sequence (whether two statements or two sections of code)you need consider only the more expensive part this rule applies to ohm and th notations as wellfor bothyou need consider only the more expensive part rule ( is used to analyze simple loop... |
20,134 | chap algorithm analysis because grows faster than log thusn is in ohm( log calculating the running time for program this section presents the analysis for several simple code fragments example we begin with an analysis of simple assignment statement to an integer variable bbecause the assignment statement takes constan... |
20,135 | but each time the cost of the inner loop is different because it costs with changing each time you should see that for the first execution of the outer loopi is for the second execution of the outer loopi is each time through the outer loopi becomes one greateruntil the last time through the loop when thusthe total cos... |
20,136 | chap algorithm analysis sum for ( = <=nk*= for ( = <=nj++sum ++/do log times /do times sum for ( = <=nk*= for ( = <=kj++sum ++/do log times /do times when analyzing these two code fragmentswe will assume that is power of two the first code fragment has its outer for loop executed log times because on each iteration is ... |
20,137 | then multiplied by the input valuewhich takes constant time thusthe cost of the factorial functionif we wish to measure cost in terms of the number of multiplication operationsis one more than the number of multiplications made by the recursive call on the smaller input because the base case does no multiplicationsits ... |
20,138 | chap algorithm analysis position key figure an illustration of binary search on sorted array of positions consider search for the position with value binary search first checks the value at position because kthe desired value cannot appear in any position below in the array nextbinary search checks the value at positio... |
20,139 | binary search requires that the array values be ordered from lowest to highest depending on the context in which binary search is to be usedthis requirement for sorted array could be detrimental to the running time of complete programbecause maintaining the values in sorted order requires to greater cost when inserting... |
20,140 | chap algorithm analysis bubble sort and insertion sort algorithms typically given as examples in first year programming course have worst case running times in ( thusthe problem of sorting can be said to have an upper bound in ( how do we close the gap between ohm(nand ( )can there be better sorting algorithmif you can... |
20,141 | clearreread section we use th-notation to indicate that there is no meaningful difference between what we know about the growth rates of the upper and lower bound (which is usually the case for simple algorithmsit is common mistake to confuse the concepts of upper bound or lower bound on the one handand worst case or b... |
20,142 | chap algorithm analysis going up to the rightas increases (you might want to sketch this graph for yourself before reading furthernowimagine the graph showing the cost for each instance of the problem of finding the maximum value among (say elements in an array the first position along the axis of the graph might corre... |
20,143 | pixel assume that each pixel can take any integer value in the range to the problem is to find the number of pixels of each color value and then sort the color values with respect to the number of times each value appears in the picture assume that the picture is rectangle with pixels pseudocode algorithm to solve the ... |
20,144 | chap algorithm analysis the analysis techniques used to measure space requirements are similar to those used to measure time requirements howeverwhile time requirements are normally measured for an algorithm that manipulates particular data structurespace requirements are normally determined for the data structure itse... |
20,145 | classic example of space/time tradeoff is the lookup table lookup table pre-stores the value of function that would otherwise be computed each time it is needed for example is the greatest value for the factorial function that can be stored in -bit int variable if you are writing program that often computes factorialsi... |
20,146 | chap algorithm analysis of its correct position once it has been placed thereand each swap operation places at least one integer in its correct position thusthis code fragment has cost th(nhoweverit requires more time to run than the first code fragment on my computer the second version takes nearly twice as long to ru... |
20,147 | had modified the program it ran in only few hoursand he finished his thesis on time while not nearly so important as changing an algorithm to reduce its growth rate"code tuningcan also lead to dramatic improvements in running time code tuning is the art of hand-optimizing program to run faster or require less storage f... |
20,148 | chap algorithm analysis to have sides perpendicular to the and axesthat contains the object if the point is not in the bounding boxthen it cannot be in the object if the point is in the bounding boxonly then would we conduct the full comparison of the object versus the point note that if the point is outside the boundi... |
20,149 | empirical analysis this has focused on asymptotic analysis this is an analytic toolwhereby we model the key aspects of an algorithm to determine the growth rate of the algorithm as the input size grows as pointed out previouslythere are many limitations to this approach these include the effects at small problem sizede... |
20,150 | chap algorithm analysis further reading pioneering works on algorithm analysis include the art of computer programming by donald knuth [knu knu ]and the design and analysis of computer algorithms by ahohopcroftand ullman [ahu the alternate definition for ohm comes from [ahu the use of the notation " (nis in ( ( ))rathe... |
20,151 | sec exercises arrange the following expressions by growth rate from slowest to fastest log log / see stirling' approximation in section for help in classifying (asuppose that particular algorithm has time complexity ( and that executing an implementation of it on particular machine takes seconds for inputs now suppose ... |
20,152 | chap algorithm analysis (bwhat is the smallest integer such that log (nk ) (ais th( )explain why or why not (bis th( )explain why or why not for each of the following pairs of functionseither (nis in ( ( )) (nis in ohm( ( ))or (nth( ( )for each pairdetermine which relationship is correct justify your answerusing the me... |
20,153 | (fsum for ( = <=ni*= for ( = <=nj++sum++(gassume that array contains valuesrandom takes constant timeand sort takes log steps for ( = <ni++for ( = <nj++ [idsutil random( )sort( )(hassume array contains random permutation of the values from to sum for ( = <ni++for ( = [ ]!=ij++sum++(isum if (even( )for ( = <ni++sum++els... |
20,154 | chap algorithm analysis given an array storing integers ordered by valuemodify the binary search routine to return the position of the first integer with value in the situation where can appear multiple times in the array be sure that your algorithm is th(log )that isdo not resort to sequential search once an occurrenc... |
20,155 | summation for the running time of your algorithm on piecesand then derive closed-form solution for the summation can the average case cost for an algorithm be worse than the worst case costcan it be better than the best case costexplain why or why not prove that if an algorithm is th( ( )in the average casethen it is o... |
20,156 | fundamental data structures |
20,157 | listsstacksand queues if your program needs to store few things -numberspayroll recordsor job descriptions for example -the simplest and most effective approach might be to put them in list only when you have to organize or search through large number of things do more sophisticated data structures usually become neces... |
20,158 | chap listsstacksand queues lists we all have an intuitive understanding of what we mean by "list,so our first step is to define precisely what is meant so that this intuitive understanding can eventually be converted into concrete data structure and its operations perhaps the most important concept related to lists is ... |
20,159 | the next step is to define the adt for list object in terms of set of operations on that object we will use the java notation of an interface to formally define the list adt interface list defines the member functions that any list implementation inheriting from it must supportalong with their parameters and return typ... |
20,160 | chap listsstacksand queues /*list adt *public interface list /*remove all contents from the listso it is once again empty client is responsible for reclaiming storage used by the list elements *public void clear()/*insert an element at the current location the client must ensure that the list' capacity is not exceeded ... |
20,161 | mentationsassertions are used to enforce such preconditions in commercial implementationsuch violations would be best implemented by the java exception mechanism list can be iterated through as shown in the following code fragment for ( movetostart() currpos()< length() next()it getvalue()dosomething(it)in this example... |
20,162 | chap listsstacksand queues array-based list implementation there are two standard approaches to implementing liststhe array-based listand the linked list this section discusses the array-based approach the linked list is presented in section time and space efficiency comparisons for the two are discussed in section fig... |
20,163 | sec lists /*array-based list implementation *class alist implements list private static final int defaultsize /default size private int maxsize/maximum size of list private int listsize/number of list items now private int curr/position of current element private [listarray/array holding list elements /*constructors */... |
20,164 | chap listsstacksand queues public void movetostart(curr /reset position public void movetoend(curr listsize/reset public void prev(if (curr ! curr--/back up public void next(if (curr listsizecurr++/next /return list size public int length(return listsize/return current position public int currpos(return curr/set curren... |
20,165 | sec lists insert ( ( (cfigure inserting an element at the head of an array-based list requires shifting all existing elements in the array by one position toward the tail (aa list containing five elements before inserting an element with value (bthe list after shifting all existing elements one position to the right (c... |
20,166 | chap listsstacksand queues class link private elementprivate link next/singly linked list node /value for this node /pointer to next node in list /constructors link( itlink nextvalelement itnext nextvallink(link nextvalnext nextvallink next(return nextlink setnext(link nextvalreturn next nextvale element(return element... |
20,167 | sec lists curr tail head figure initial state of linked list when using header node key design decision for the linked list implementation is how to represent the current position the most reasonable choices appear to be pointer to the current element but there is big advantage to making curr point to the element prece... |
20,168 | chap listsstacksand queues head curr tail (ahead curr tail (bfigure insertion using header nodewith curr pointing one node head of the current element (alinked list before insertion the current node contains (blinked list after inserting the node containing figure shows the definition for the linked list classnamed lli... |
20,169 | sec lists /linked list implementation class llist implements list private link head/pointer to list header private link tail/pointer to last element protected link curr/access to current element int cnt/size of list //constructors llist(int sizethis()/constructor -ignore size llist(curr tail head new link(null)/create ... |
20,170 | chap listsstacksand queues public void movetostart(curr headpublic void movetoend(curr tail/set curr at list start /set curr at list end /move curr one step leftno change if already at front public void prev(if (curr =headreturn/no previous element link temp head/march down list until we find the previous element while... |
20,171 | sec lists curr insert (acurr (bfigure the linked list insertion process (athe linked list before insertion (bthe linked list after insertion marks the element field of the new link node marks the next field of the new link nodewhich is set to point to what used to be the current node (the node with value marks the next... |
20,172 | chap listsstacksand queues figure illustrates the remove method removing an element requires th( time method next simply moves curr one position toward the tail of the listwhich takes th( time method prev moves curr one position toward the head of the listbut its implementation is more difficult in singly linked listth... |
20,173 | sec lists /singly linked list node with freelist support class link private element/value for this node private link next/point to next node in list /constructors link( itlink nextvalelement itnext nextvallink(link nextvalnext nextvallink next(return nextlink setnext(link nextvalreturn next nextvale element(return elem... |
20,174 | chap listsstacksand queues in figure you should note the use of the static definition for the freelist header the purpose of the keyword static is to create single variable shared among all instances of the link nodes we want only single freelist for all link nodes of given type program might create multiple lists if t... |
20,175 | sec lists determined size whenever the list contains only few elementsa substantial amount of space might be tied up in largely empty array linked lists have the advantage that they only need space for the objects actually on the list there is no limit to the number of elements on linked listas long as there is free-st... |
20,176 | chap listsstacksand queues array-based lists are faster for random access by position positions can easily be adjusted forwards or backwards by the next and prev methods these operations always take th( time in contrastsingly linked lists have no explicit access to the previous elementand access by position requires th... |
20,177 | pointer to each element is that the pointer requires space of its own if elements are never duplicatedthen this additional space adds unnecessary overhead java most naturally stores references to objectsmeaning that only single copy of an object such as payroll record will be maintainedeven if it is on multiple lists w... |
20,178 | chap listsstacksand queues head curr tail figure doubly linked list class dlink private elementprivate dlink nextprivate dlink prev/doubly linked list node /value for this node /pointer to next node in list /pointer to previous node /constructors dlink( itdlink pdlink nelement itprev pnext ndlink(dlink pdlink nprev pne... |
20,179 | /insert "itat current position public void insert( itcurr setnext(new dlink(itcurrcurr next()))curr next(next(setprev(curr next())cnt++public void append( it/append "itto list tail setprev(new dlink(ittail prev()tail))tail prev(prev(setnext(tail prev())cnt++/remove and return first element in right partition public rem... |
20,180 | chap listsstacksand queues curr insert (acurr (bfigure insertion for doubly linked lists the labels and correspond to assignments done by the linked list node constructor marks the assignment to curr->next marks the assignment to the prev pointer of the node following the newly inserted node the three parameters to the... |
20,181 | sec lists curr (acurr it (bfigure doubly linked list removal element it stores the element of the node being removed then the nodes to either side have their pointers adjusted the only disadvantage of the doubly linked list as compared to the singly linked list is the additional space used the doubly linked list requir... |
20,182 | chap listsstacksand queues bb /now contains original value of /now contains original value of similar effect can be had by using the exclusive-or operator this fact is widely used in computer graphics region of the computer screen can be highlighted by xoring the outline of box around it xoring the box outline second t... |
20,183 | sec stacks /*stack adt *public interface stack /*reinitialize the stack the user is responsible for reclaiming the storage used by the stack elements *public void clear()/*push an element onto the top of the stack @param it the element being pushed onto the stack *public void push( it)/*remove and return the element at... |
20,184 | chap listsstacksand queues /*array-based stack implementation *class astack implements stack private static final int defaultsize private int maxsizeprivate int topprivate [listarray/maximum size of stack /index for top object /array holding stack /constructors astack(this(defaultsize)@suppresswarnings("unchecked"/gene... |
20,185 | sec stacks /linked stack implementation class lstack implements stack private link top/pointer to first element private int size/number of elements //constructors public lstack(top nullsize public lstack(int sizetop nullsize /reinitialize stack public void clear(top nullsize public void push( it/put "iton stack top new... |
20,186 | chap listsstacksand queues top top figure two stacks implemented within in single arrayboth growing toward the middle is then returned to free store (or the freelist)and the element value is returned as the value of the pop method comparison of array-based and linked stacks all operations for the array-based and linked... |
20,187 | sec stacks currptr currptr currptr call fact( currptr currptr currptr currptr currptr currptr call fact( call fact( call fact( currptr currptr return currptr currptr currptr currptr return return currptr return figure implementing recursion with stack values indicate the address of the program instruction to return to ... |
20,188 | chap listsstacksand queues ter is maderequiring that the stack store the calling address (say and current value (which is at this pointwe have reached the base case for factand so the recursion begins to unwind each return from fact involves popping the stored value for from the stackalong with the return address from ... |
20,189 | sec stacks example the toh function shown in figure makes two recursive callsone to move rings off the bottom ringand another to move these rings back to the goal pole we can eliminate the recursion by using stack to store representation of the three operations that toh must performtwo recursive calls and move operatio... |
20,190 | chap listsstacksand queues static void toh(int npole startpole goalpole temp/make stack just big enough stack new astack( * + ) push(new tohobj(operation tohnstartgoaltemp))while ( length( tohobj it pop()/get next task if (it op =operation move/do move move(it startit goal)else if (it num /imitate toh recursive /soluti... |
20,191 | sec queues /*queue adt *public interface queue /*reinitialize the queue the user is responsible for reclaiming the storage used by the queue elements *public void clear()/*place an element at the rear of the queue @param it the element being enqueued *public void enqueue( it)/*remove and return element at the front of ... |
20,192 | chap listsstacksand queues front rear (afront rear (bfigure after repeated useelements in the array-based queue will drift to the back of the array (athe queue after the initial four numbers and have been inserted (bthe queue after elements and are deletedfollowing which and are inserted enqueue operations will require... |
20,193 | sec queues front front rear ( (brear figure the circular queue with array positions increasing in the clockwise direction (athe queue after the initial four numbers and have been inserted (bthe queue after elements and are deletedfollowing which and are inserted through use of the modulus operator (denoted by in javain... |
20,194 | chap listsstacksand queues be no elements in the queueone elementtwoand so on at most there can be elements in the queue if there are array positions this means that there are different states for the queue ( through elements are possibleif the value of front is fixedthen different values for rear are needed to disting... |
20,195 | sec queues /array-based queue implementation class aqueue implements queue private static final int defaultsize private int maxsize/maximum size of queue private int front/index of front element private int rear/index of rear element private [listarray/array holding queue elements /constructors aqueue(this(defaultsize)... |
20,196 | chap listsstacksand queues /linked queue implementation class lqueue implements queue private link front/pointer to front queue node private link rear/pointer to rear queuenode int size/number of elements in queue /constructors public lqueue(init()public lqueue(int sizeinit()/ignore size private void init(/initialize q... |
20,197 | enqueue places the new element in link node at the end of the linked list ( the node that rear points toand then advances rear to point to the new link node method dequeue grabs the first element of the list removes it comparison of array-based and linked queues all member functions for both the array-based and linked ... |
20,198 | chap listsstacksand queues /*the dictionary abstract class *public interface dictionary /*reinitialize dictionary *public void clear()/*insert record @param the key for the record being inserted @param the record being inserted *public void insert(key ke )/*remove and return record @param the key of the record to be re... |
20,199 | method clear simply re-initializes the dictionary the remove method is similar to findexcept that it also deletes the record returned from the dictionary once againif there are multiple records in the dictionary that match the desired keythere is no requirement as to which one actually is removed and returned method si... |
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