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25,300 | we are now able to express the buffer module in form that functions properly when used by individualconcurrent processesmodule bufferimport signalsconst (*buffer size*var ninoutintegernonfullsignals signal(* *nonemptysignals signal(* *bufarray of charprocedure deposit (xchar)begin if then signals wait(nonfullendinc( )b... |
25,301 | nc (*size of consumer block* (*buffer sizecommon multipe of np and nc*var nenfintegerinoutintegernonfullsignals signal(*ne > *nonemptysignals signal(*nf > *bufarray of charprocedure deposit (var xarray of char)begin ne :ne npif ne then signals wait(nonfullendfor : to np- do buf[in: [ ]inc(inendif in then in : endnf :nf... |
25,302 | dn- - dn- - (dn- dn- assume now that the sequence is to be read and transformedand the resulting numeric value to be assigned to the simple algorithm terminates with the reading of the first character that is not digit (arithmetic overflow is not consideredx : read(ch)(adens charstonumber *while (" <ch(ch <" "do : * (o... |
25,303 | "",":";"~items are separated by blanks and/or line breaks definition texts(adens _texts *const int real name char type textwriterreader record eotboolean endscanner record classintegeriintegerxrealsarray of charchcharnextchchar endprocedure openreader (var rreaderttextposinteger)procedure openwriter (var wwriterttextpo... |
25,304 | searching the task of searching is one of most frequent operations in computer programming it also provides an ideal ground for application of the data structures so far encountered there exist several basic variations of the theme of searchingand many different algorithms have been developed on this subject the basic ... |
25,305 | aarray + of integerand the linear search algorithm with sentinel is expressed by [ :xi : while [ix do inc(iend (adens _search *the resulting conditionderived from the same invariant as beforeis (ai (ak < ak xevidentlyi implies that no match (except that for the sentinelwas encountered binary search there is quite obvio... |
25,306 | as possible from further searchesno matter what the outcome of the comparison is the optimal solution is to choose the middle elementbecause this eliminates half of the array in any case as resultthe maximum number of steps is log nrounded up to the nearest integer hencethis algorithm offers drastic improvement over li... |
25,307 | lengths are most commonly used the length is implicitly specified by appending terminating character which does not otherwise occur usuallythe non-printing value (it is important for the subsequent applications that it be the least character in the character set the length is explicitly stored as the first element of t... |
25,308 | string search frequently encountered kind of search is the so-called string search it is characterized as follows given an array of elements and an array of elementswhere ndeclared as sarray of item parray of item string search is the task of finding the first occurrence of in typicallythe items are charactersthen may ... |
25,309 | while ( ( [js[ + ]do inc(jendreturn ~( mend let the result be the index iwhich points to the first occurrence of match of the pattern within the string then (ishould hold in additionr(kmust be false for all denote the latter condition as ( ) (iak < ~ (kwith the problem thus formulateda linear search suggests itself (se... |
25,310 | makes sensei preserves the invariant for the first branch it is the condition ( <nm( (si+ pjwhich guarantees (ijafter is incremented for the second branchthe last operand of this conjunction must have inequality instead of equalitywhich would imply ~ (iand guarantee (iafter is incremented taking into account that the t... |
25,311 | will now store the comparison pointthe variable will as before point to the corresponding element of the pattern see fig - string pattern - fig in the notations of the kmp algorithmthe alignment position of the pattern is now - (and not ias was the case with the simple algorithmthe central pont of the algorithm is the ... |
25,312 | evidently the condition ( -ddq( -dmust hold before assigning :dif the invariant (ijjq( -jis to hold thereafter this precondition is therefore our guideline for finding an appropriate expression for along with the condition ( -jjq( - )which is assumed to hold prior to the assignment (all subsequent reasoning concerns th... |
25,313 | parttherefore we may impose the condition pd pj when solving for this alows us to fully utilize the information from the inequality in the guard of this loop branch string pattern shifted pattern = =- (shift = =- (shift fig shifting pattern past position of last character the essential result is that the value apparent... |
25,314 | while ( ( [ip[ ]do : [ ]elsif ( ( ndo inc( )inc( )endif then : - else :- end end search analysis of kmp search the exact analysis of the performance of kmp-search islike the algorithm itselfvery intricate in [ - its inventors prove that the number of character comparisons is in the order of +nwhich suggests substantial... |
25,315 | while ( ( < ( [ - [ - ]do dec( )dec(jelsif ( ( <ndo : [ord( [ - ])] :mk :iend the indices satisfy < <mm <ii < thereforetermination with implies ( ( ) match at position - termination with demands that nhence (iimplies ( )signalling that no match exists of course we still have to convince ourselves that (iand (ijare inde... |
25,316 | the authors provide several ideas on possible further improvements one is to combine the strategy explained abovewhich provides greater shifting steps when mismatch is presentwith the knuth-morrispratt strategywhich allows larger shifts after detection of (partialmatch this method requires two precomputed tablesd is th... |
25,317 | program ci : : - repeat :( +jdiv if [kthen : else : + end until hintall programs must terminate with ak xif such an element existsor ak xif there exists no element with value company organizes poll to determine the success of its products its products are records and tapes of hitsand the most popular hits are to be bro... |
25,318 | - |
25,319 | sorting introduction the primary purpose of this is to provide an extensive set of examples illustrating the use of the data structures introduced in the preceding and to show how the choice of structure for the underlying data profoundly influences the algorithms that perform given task sorting is also good example to... |
25,320 | front of the sorter so that each card is visible and individually accessible (see fig structuring the cards as filehoweverimplies that from each pile only the card on the top is visible (see fig fig the sorting of file such restriction will evidently have serious consequences on the sorting method to be usedbut it is u... |
25,321 | total ordering relation is defined could be used just as well sorting method is called stable if the relative order if items with equal keys remains unchanged by the sorting process stability of sorting is often desirableif items are already ordered (sortedaccording to some secondary keysi properties not reflected by t... |
25,322 | sorting by straight insertion this method is widely used by card players the items (cardsare conceptually divided into destination sequence ai- and source sequence ai an- in each stepstarting with and incrementing by unitythe ith element of the source sequence is picked and transferred into the destination sequence by ... |
25,323 | moves are cmin mmin *( cave ( )/ mave ( )/ cmax ( )/ mmax ( )/ the minimal numbers occur if the items are initially in orderthe worst case occurs if the items are initially in reverse order in this sensesorting by insertion exhibits truly natural behavior it is plain that the given algorithm also describes stable sorti... |
25,324 | improvement is by no means drasticthe important term is still of the order andin factsorting the already sorted array takes more time than does straight insertion with sequential search this example demonstrates that an "obvious improvementoften has much less drastic consequences than one is first inclined to estimate ... |
25,325 | end end straightselection analysis of straight selection evidentlythe number of key comparisons is independent of the initial order of keys in this sensethis method may be said to behave less naturally than straight insertion we obtain ( )/ the number of moves is at least mmin *( in the case of initially ordered keys a... |
25,326 | ascension of bubble to its appropriate level of weight (see table this method is widely known as the bubblesort table sample of bubblesorting procedure bubblesort(adens _sorts *var ijintegerxitembegin for : to - do for : - to by - do if [ - [jthen : [ - ] [ - : [ ] [ : end end end end bubblesort this algorithm easily l... |
25,327 | if [ - [jthen : [ - ] [ - : [ ] [ :xk : end endl : + for : to by + do if [ - [jthen : [ - ] [ - : [ ] [ :xk : end endr : - until end shakersort dir table an example of shakersort analysis of bubblesort and shakersort the number of comparisons in the straight exchange algorithm is ( )/ and the minimumaverageand maximum ... |
25,328 | subsequentlythree improved methods will be discussednamelyone for each basic sorting methodinsertionselectionand exchange advanced sorting methods insertion sort by diminishing increment refinement of the straight insertion sort was proposed by shell in the method is explained and demonstrated on our standard example o... |
25,329 | [ + : else [ + : [ ] [ : end end end end shellsort analysis of shellsort the analysis of this algorithm poses some very difficult mathematical problemsmany of which have not yet been solved in particularit is not known which choice of increments yields the best results one surprising facthoweveris that they should not ... |
25,330 | selection steps requires only log(ncomparisons thereforethe total selection process requires only on the order of *log(nelementary operations in addition to the steps required by the construction of the tree this is very significant improvement over the straight methods requiring stepsand even over shellsort that requi... |
25,331 | fig heap with elements fig key sifting through the heap let us now assume that heap with elements hl+ hr is given for some values and rand that new element has to be added to form the extended heap hl hr takefor examplethe initial heap shown in fig and extend the heap to the left by an element new heap is obtained by f... |
25,332 | consequentlythe process of generating heap of elements hn- in situ is described as followsl : div while do dec( )sift(ln- end in order to obtain not only partialbut full ordering among the elementsn sift steps have to followwhereby after each step the next (leastitem may be picked off the top of the heap once morethe q... |
25,333 | while do : [ ] [ : [ ] [ :xdec( )sift(lrend end heapsort analysis of heapsort at first sight it is not evident that this method of sorting provides good results after allthe large items are first sifted to the left before finally being deposited at the far right indeedthe procedure is not recommended for small numbers ... |
25,334 | while [jdo : - endif < then : [ ] [ : [ ] [ :wi : + : - end until end partition as an exampleif the middle key is selected as comparand xthen the array of keys requires the two exchanges and to yield the partitioned array and the final index values and keys ai- are less or equal to key and keys aj+ an- are greater or e... |
25,335 | while [ix do : + endwhile [jdo : - endif < then : [ ] [ : [ ] [ :wi : + : - end until jif then sort(ljendif then sort(irend end sortprocedure quicksortbegin sort( - end quicksort procedure sort activates itself recursively such use of recursion in algorithms is very powerful tool and will be discussed further in chap i... |
25,336 | endr : (*now and delimit the left partition*until > until end nonrecursivequicksort analysis of quicksort in order to analyze the performance of quicksortwe need to investigate the behavior of the partitioning process first after having selected bound xit sweeps the entire array henceexactly comparisons are performed t... |
25,337 | what are the consequences of the worst case behavior mentioned above to the performance quicksortwe have realized that each split results in right partition of only single elementthe request to sort this partition is stacked for later execution consequentlythe maximum number of requestsand therefore the total required ... |
25,338 | fig bound too small the chosen bound was too large the splitting operation has to be repeated on the partition al aj (see fig < > fig bound too large ithe element ak splits the array into two partitions in the specified proportions and therefore is the desired quantile (see fig < >jki fig correct bound the splitting pr... |
25,339 | end find if we assume that on the average each split halves the size of the partition in which the desired quantile liesthen the number of necessary comparisons is / / it is of order this explains the power of the program find for finding medians and similar quantilesand it explains its superiority over the straightfor... |
25,340 | shakersort is still worse than straight insertion and straight selection (except in the pathological case of sorting sorted array quicksort beats heapsort by factor of to it sorts the inversely ordered array with speed practically identical to the one that is already sorted ordered random inverse straightinsertion bina... |
25,341 | sorting sequences straight merging unfortunatelythe sorting algorithms presented in the preceding are inapplicableif the amount of data to be sorted does not fit into computer' main storebut if it isfor instancerepresented on peripheral and sequential storage device such as tape or disk in this case we describe the dat... |
25,342 | howeveris scanned in strictly sequential fashion later version of merge sort will then be based on the sequence structureallowing comparison of the two programs and demonstrating the strong dependence of the form of program on the underlying representation of its data single array may easily be used in place of two seq... |
25,343 | partial merge the destination is switched from the lower to the upper end of the destination arrayor vice versato guarantee equal distribution onto both destinations if the destination of the merged items is the lower end of the destination arraythen the destination index is kand is incremented after each move of an it... |
25,344 | repetitionmust be changed to > after these modificationswe may now proceed to describe the entire algorithm in terms of procedure operating on the global array with elements procedure straightmerge(adens _mergesorts *var ijkltinteger(*index range of is * - *hmpqrintegerupbooleanbegin up :truep : repeat : :nif up then :... |
25,345 | the merge sort algorithm apparently compares well with even the advanced sorting techniques discussed in the previous howeverthe administrative overhead for the manipulation of indices is relatively highand the decisive disadvantage is the need for storage of items this is the reason sorting by merging is rarely used o... |
25,346 | table example of natural mergesort as an exampletable shows the file in its original state (line and after each pass (lines - in natural merge sort involving numbers note that only three passes are needed the sort terminates as soon as the number of runs on is (we assume that there exists at least one non-empty run on ... |
25,347 | operators are presented by the following definition definition runs(adens _runs *import filestextstype rider record (files riderfirstintegereorboolean endprocedure openrandomseq (ffiles filelengthseedinteger)procedure set (var rridervar ffiles file)procedure copy (var sourcedestinationrider)procedure listseq (var wtext... |
25,348 | if then (*konets serii*texts write( "|")inc(nendx : endtexts write( "$")texts writeint(wk )texts writeint(wn )texts writeln(wend listseqend runs we now return to the process of successive refinement of the process of natural merging procedure copyrun and the statement merge are now conveniently expressible as shown bel... |
25,349 | table incorrect result of mergesort program the example of this programming mistake is typical for many programming situations the mistake is caused by an oversight of one of the possible consequences of presumably simple operation it is also typical in the sense that serval ways of correcting the mistake are open and ... |
25,350 | if ~ eof then copyrun( end until eofruns set( )runs set( ) :files new("")files set( )(*merge from and to * : repeat repeat if first first then runs copy( )if eor then copyrun( end else runs copy( )if eor then copyrun( end end until eor eorinc(luntil eof or eofwhile ~ eof do copyrun( )inc(lendwhile ~ eof do copyrun( )in... |
25,351 | different riders obviouslyfile numbers are used to index the array of files let us then assume that the initial file is the parameter srcand that for the sorting process files are availablefgarray of files filerwarray of runs rider the algorithm can now be sketched as followsprocedure balancedmerge (srcfiles file)files... |
25,352 | if then : else : endfor : to - do runs set( [ ] [ ]end naturallystatement ( is to decrement whenever an input source ceases hencepredicate ( may easily be expressed by the relation statement ( )howeveris more difficult to refineit consists of the repeated selection of the least key among the available sources and its s... |
25,353 | repeat runs copy(rw[ ]until eorinc( )inc( )if then : end until eofrepeat if then : else : endk : for : to - do (*set input riders*runs set( [ ] [ ]endfor : to - do (*set output riders* [ :files new("")files set( [ ] [ ] end(*merge from [ [ - to [ [ - ]*for : to - do [ : endl : (*nof runs merged* : repeat (*merge on run... |
25,354 | destinations at the end of each distinct pass insteadthe notion of pass becomes diffuse the method was invented by gilstad [ - and called polyphase sort it is first illustrated by an example using three sequences at any timeitems are merged from two sources into third sequence variable whenever one of the source sequen... |
25,355 | being the number of items to be sorted and being the degree of the merge operationspolyphase promises significant improvement over balanced merging of coursethe distribution of initial runs was carefully chosen in the above examples in order to find out which initial distributions of runs lead to proper functioningwe w... |
25,356 | substituting fi for (iyields + fi fi- - fi- fi- fi for for these numbers are the fibonacci numbers of order in generalthe fibonacci numbers of order are defined as followsf + (pfi(pf - (pfi- (pf ( ( for for < note that the ordinary fibonacci numbers are those of order we have now seen that the initial numbers of runs f... |
25,357 | sequences as uniformly as possiblesince we are interested in active merges from as many sources as possible let us forget dummy runs for moment and consider the problem of distributing an unknown number of runs onto - sequences it is plain that the fibonacci numbers of order - specifying the desired numbers of runs on ... |
25,358 | we are now in position to describe the algorithm in the form of procedure called selectwhich is activated each time run has been copied and new source is selected for the next run we assume the existence of variable denoting the index of the current destination sequence ai and di denote the ideal and dummy distribution... |
25,359 | becomes necessary to retain the keys of the last item of the last run on each sequence fortunatelyour implementation of runs does exactly this in the case of output sequencesr first represents the item last written next attempt to describe the distribution algorithm could therefore be repeat selectif [jfirst < first th... |
25,360 | else ta[ : [ ]inc(kend endif then inc( [ - ]else merge one real run from [ [ - to [ - enddec(zuntil runs set( [ [ - ]] [ [ - ]])rotate sequences in map tcompute [ifor next leveldec(leveluntil level (*sorted output is [ [ ]]*the actual merge operation is almost identical with that of the -way merge sortthe only differen... |
25,361 | repeat runs copy(rr[ ]until eor end copyrunbegin runs set(rsrc)for : to - do [ : [ : [ :files new("")files set( [ ] [ ] end(*distribute initial runs*level : : [ - : [ - : repeat selectcopyrun until eof or ( - )while ~ eof do select(* [jfirst last item written on [ ]*if [jfirst < first then copyrunif eof then inc( [ ]el... |
25,362 | end until enddec(zuntil runs set( [ [ - ]] [ [ - ]])(*rotate sequences*tn : [ - ]dn : [ - ] : [ - ]for : - to by - do [ : [ - ] [ : [ - ] [ : [ - endt[ :tnd[ :dna[ :zdec(leveluntil level return [ [ ]end polyphase distribution of initial runs we were led to the sophisticated sequential sorting programsbecause the simple... |
25,363 | [ fig sifting key through heap before proceedinglet us formally declare the variables that are evidently involved in the processvar lrxintegersrcdestfiles filerwfiles riderharray of integer(*heap* is the size of the heap we use the constant mh to denote / and are indices delimiting the heap the funnelling process can t... |
25,364 | copyrun that delivers exactly one run to the selected destination the heapsort programon the other handis complex routine relying on the availability of closed procedure select which simply selects new destination there would be no problemif in one (or bothof the programs the required procedure would be called at singl... |
25,365 | dec( ) [ : [ ]sift( - ) [ :xif mh then sift(lm- endif then (*heap fullstart new run* : end endfiles readint(rxend(*step flush lower half of heap* :mrepeat dec( )files writeint(wh[ ]) [ : [ ]sift( - )dec( ) [ : [ ]if mh then sift(lr- end until (*step flush upper half of heapstart new run*while do files writeint(wh[ ]) [... |
25,366 | and find coefficients by which the factors and have to be multiplied to yield real time estimates specifty invariants for the repetitions in the three straight sorting algorithms consider the following "obviousversion of the procedure partition and find sets of values an- for which this version fails : : - : [ div ]rep... |
25,367 | the same way starting with five-way merge to and so on although this scheme seems to be inferior to polyphase because at times it chooses to leave some sequences idleand because it involves simple copy operationsit surprisingly is superior to polyphase for (verylarge files and for six or more sequences write well struc... |
25,368 | recursive algorithms introduction an object is said to be recursiveif it partially consists or is defined in terms of itself recursion is encountered not only in mathematicsbut also in daily life who has never seen an advertising picture which contains itselffig picture with recursion recursion is particularly powerful... |
25,369 | it is common to associate set of local objects with procedurei set of variablesconstantstypesand procedures which are defined locally to this procedure and have no existence or meaning outside this procedure each time such procedure is activated recursivelya new set of localbound variables is created although they have... |
25,370 | programs in which the use of algorithmic recursion is to be avoided can be characterized by schema which exhibits the pattern of their composition the equivalent schemata are shown below their characteristic is that there is only single call of either at the end (or the beginningof the composition if then sp end sif th... |
25,371 | for fibn+ fibn fibn- and fib fib directnaive transcription leads to the recursive program procedure fib (ninteger)integervar resintegerbegin if then res : elsif then res : else res :fib( - fib( - endreturn res end fib computation of fibn by call fib(ncauses this function procedure to be activated recursively how oftenw... |
25,372 | two examples of recursive programs the attractive graphic pattern shown in fig consists of superposition of five curves these curves follow regular pattern and suggest that they might be drawn by display or plotter under control of computer our goal is to discover the recursion schemaaccording to which the drawing prog... |
25,373 | main procedure determines the initial point of the curvei the initial coordinates of the pen denoted by and and the unit increment the square in which the curves are drawn is placed into the middle of the page with given width and height these parameters as well as the drawing procedure line are taken from module draw ... |
25,374 | : ( div ) : ( div )draw set( ) (iuntil end hilbert fig hilbert curves similar but slightly more complex and aesthetically more sophisticated example is shown in fig this pattern is again obtained by superimposing several curvestwo of which are shown in fig si is called the sierpinski curve of order what is its recursio... |
25,375 | fig sierpinski curves and the base pattern of the sierpinski curves is sa and the recursion patterns are (horizontal and vertical arrows denote lines of double length aa bb cc dd if we use the same primitives for drawing as in the hilbert curve examplethe above recursion scheme is transformed without difficulties into ... |
25,376 | begin if then ( - )draw line( ) ( - )draw line( * ) ( - )draw line( ) ( - end end bprocedure (kinteger)begin if then ( - )draw line( ) ( - )draw line( * ) ( - )draw line( ) ( - end end cprocedure (kinteger)begin if then ( - )draw line( ) ( - )draw line( * ) ( - )draw line( ) ( - end end dprocedure sierpinski(ninteger)c... |
25,377 | fig sierpinski curves backtracking algorithms particularly intriguing programming endeavor is the subject of so-called general problem solving the task is to determine algorithms for finding solutions to specific problems not by following fixed rule of computationbut by trial and error the common pattern is to decompos... |
25,378 | the obvious way to reduce the problem of covering fields is to consider the problem of either performing next move or finding out that none is possible let us define the corresponding algorithm first approach is to employ linear search in order to find the next move from which the tour can be completedprocedure trynext... |
25,379 | procedure trynextmove (xyiintegervar donebooleanthe function-procedure canbedone expresses the predicate tour can be completed from this move and is used within trynextmove in the linear search over possible move destinations determined according to the jump pattern of knights introduce two local variables and to stand... |
25,380 | procedure trynextmove on board that is infinite in all directionseach position xy has number candidate moves uvwhich at this point there is no need to specify (see fig the predicate to choose an acceptable move can be expressed as the logical conjunction of the conditions that the new field lies on the boardi < and < n... |
25,381 | be given the value all others are to be marked free var harray nn of integer(adens _knightstour *dxdyarray of integerprocedure canbedone (uviinteger)booleanvar donebooleanbegin [uv:itrynextmove(uvidone)if ~done then [uv: endreturn done end canbedoneprocedure trynextmove (xyiintegervar doneboolean)var eosbooleanuvintege... |
25,382 | table three knightstours what abstractions can now be made from this examplewhich pattern does it exhibit that is typical for this kind of problem-solving algorithmswhat does it teach usthe characteristic feature is that steps toward the total solution are attempted and recorded that may later be taken back and erased ... |
25,383 | note that the search condition in the while loop is modeled as procedure-function canbedone for maximal clarification of the logic of the algorithm while keeping the program easily comprehensible certainly the program can be optimized in other respects via appropriate equivalent transformations one canfor instanceelimi... |
25,384 | such representation would lead to fairly cumbersome operations for checking the availability of positions this is highly undesirable since it is the most frequently executed operation we should therefore choose data representation which makes checking as simple as possible the best recipe is to represent as directly as... |
25,385 | begin repeat inc( )until ( or ( [jb[ +jc[ - + ])eos :( end nextprocedure firstbegin eos :falsej :- next end firstbegin if then firstwhile ~eos ~canbedone(ijdo next enddone :~eos else done :true end end tryprocedure canbedone (ijinteger)boolean(*solution can be completed with -th queen in -th row*var donebooleanbegin [ ... |
25,386 | procedure body in place of its call to return logical value is no longer necessary the general schema is as followsprocedure trybegin if solution incomplete then initialize selection of candidate moves and select the first onewhile ~(no more movesdo record movetryerase moveselect next move end else print solution end e... |
25,387 | end end tryprocedure allqueens*var ijintegerbegin for : to do [ :truex[ :- endfor : to do [ :truec[ :true endm : try( )log string('no of solutions')log int( )log ln end allqueens table twelve solutions to the eight queens problem the stable marriage problem assume that two disjoint sets and of equal size are given find... |
25,388 | var rrankbegin if then for : to - do pick the -th preference of man mif acceptable then record the marriagetry(successor( ))cancel the marriage end end else record the stable set end end try the initial data are represented by two matrices that indicate the men' and women' preferences var wmrarray nn of womanmwrarray n... |
25,389 | with the meaning that singlemm implies that xm is undefinedand singleww implies that yw is undefined an inspection of the proposed algorithmhoweverquickly reveals that the marital status of man is determined by the value through the relation ~singlem[kk this suggests that the array singlem be omittedaccordinglywe will ... |
25,390 | already married becausewere anyone of them still singlem would have picked her beforehand the described process can be formulated by simple linear searchs denotes stability :- :truerepeat inc( )if then pw :wmr[ , ]if ~single[pwthen :rwm[pw,mrwm[pwy[pw]end end until ( ror ~ hunting for trouble source we must investigate... |
25,391 | return end stableprocedure try (minteger)var wrintegerbegin if then for : to - do :wmr[ , ]if single[wstable( , ,rthen [ :wy[ :msingle[ :falsetry( + )single[ :true end end else write end end tryprocedure findstablemarriages (var stexts scanner)var mwrintegerbegin for : to - do for : to - do texts scan( )wmr[ , : irmw[m... |
25,392 | rm rw number of evaluations of stability solution male optimal solutionsolution female optimal solution table result of the stable marriage problem the solution with the least value rm is called the male-optimal stable solutionthe one with the smallest rw is the female-optimal stable solution it lies in the nature of t... |
25,393 | procedure try (iinteger)begin if then if inclusion is acceptable then include -th objecttry( + )eliminate -th object endif exclusion is acceptable then try( + end else check optimality end end try from this pattern it is evident that there are possible setsclearlyappropriate acceptability criteria must be employed to r... |
25,394 | if av maxv then (*new optimumrecord it*opts :smaxv :av end the last assignment is based on the reasoning that the achievable value is the achieved valueonce all objects have been dealt with the condition exclusion is acceptable is expressed by av [ivalue maxv since it is used again thereafterthe value av [ivalue is giv... |
25,395 | table sample output from optimal selection program the asterisks mark the objects that form the optimal sets opts for the total weight limits ranging from to this backtracking scheme with limitation factor curtailing the growth of the potential search tree is also known as branch and bound algorithm exercises (towers o... |
25,396 | fig curves only of the solutions computed by the eight queens algorithm are essentially different the other ones can be derived by reflections about axes or the center point devise program that determines the principal solutions note thatfor examplethe search in column may be restricted to positions change the stable m... |
25,397 | experience no ( ) - [ wirth program development by stepwise refinement comm acm no ( ) |
25,398 | dynamic information structures recursive data types in chap the arrayrecordand set structures were introduced as fundamental data structures they are called fundamental because they constitute the building blocks out of which more complex structures are formedand because in practice they do occur most frequently the pu... |
25,399 | end type term record if tboolean then idname else subexexpression end end henceevery variable of type term consists of two componentsnamelythe tagfield andif is truethe field idor of the field subex otherwise consider nowfor examplethe following four expressions + ( ( ( ( /( ) these expressions may be visualized by the... |
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