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23,800 | algorithm design techniques at this pointwe have and now the largest distance is so either or but if then which is impossiblesince is no longer in on the other handif then and this is also impossiblesince only appears once in thusthis line of reasoning leaves no solutionso we backtrack since failed to produce solutionw... |
23,801 | = = = = * = = = = = = figure decision tree for the worked turnpike reconstruction example bool turnpikevector xdistset dint deletemaxxn ) deletemaxxn )ifxn xn removexn xn )return placexdn )else return falsefigure turnpike reconstruction algorithmdriver routine (pseudocodepass the same arguments plus the boundaries left... |
23,802 | backtracking algorithm to place the points [leftx[rightx[ [left- and [right+ [nalready tentatively placed if place returns truethen [leftx[rightwill have values *bool placevector xdistset dint nint leftint right int dmaxbool found false ifd isemptyreturn true dmax findmax) /check if setting [rightdmax is feasible ifx[j... |
23,803 | the analysis of the algorithm involves two factors suppose lines to and to are never executed we can maintain as balanced binary search (or splaytree (this would require code modificationof courseif we never backtrackthere are at most ( operations involving dsuch as deletion and the contains implied at lines and this c... |
23,804 | /*recursive function to find best move for computer returns the evaluation and sets bestmovewhich ranges from to and indicates the best square to occupy possible evaluations satisfy comp_loss draw comp_win complementary function findhumanmove is figure *int tictactoe::findcompmoveint bestmove int iresponsevalueint dc/d... |
23,805 | positionsline initializes it to the smallest possible valueand the loop in lines to searches for improvements each successor position is recursively evaluated in turn by lines to this is recursivebecauseas we will seefindhumanmove calls findcompmove if the human' response to move leaves the computer with more favorable... |
23,806 | algorithm design techniques passing an extra variablewhich indicates whose turn it is to move this does make the code somewhat less readableso we have stayed with separate routines we leave supporting routines as an exercise the most costly computation is the case where the computer is asked to pick the opening move si... |
23,807 | figure two searches that arrive at identical position tree is actually constructed by the algorithm the game tree is just an abstract concept the value of the game tree is figure shows the evaluation of the same game tree with several (but not all possibleunevaluated nodes almost half of the terminal nodes have not bee... |
23,808 | algorithm design techniques max min min max max figure pruned game tree >= max <= min dfigure the node marked is unimportant <= min >= max cfigure the node marked is unimportant levelwhich has called findcompmovehas already determined that it can force value of at most (recall that low values are good for the human sid... |
23,809 | /*same as beforebut perform alpha-beta pruning the main routine should make the call with alpha comp_loss and beta comp_win *int tictactoe::findcompmoveint bestmoveint alphaint beta int iresponsevalueint dc/dc means don' careits value is unused int valueiffullboardvalue drawelse ifimmediatecompwinbestmove return comp_w... |
23,810 | algorithm design techniques search the result is even more pruning than one would expect from random ordering of the nodes other techniquessuch as searching deeper in more active lines of playare also employed in practicea- pruning limits the searching to only onnodeswhere is the size of the full game tree this is huge... |
23,811 | show that any online bin packing algorithm can be forced to use at least the optimal number of binsby considering the following sequence of itemsn items of size items of size items of size give simple analysis to show the performance bound for first fit decreasing bin packing when the smallest item size is larger than ... |
23,812 | algorithm design techniques this gives an ( algorithm to multiply -bit numbers compare this method to the solution in the text show how to multiply two numbers by solving five problems that are roughly one-third of the original size generalize this problem to obtain an ( algorithm for any constant is the algorithm in... |
23,813 | show that is nondecreasing along each row and column use this to show that all entries in can be computed in ( time which of the dynamic programming algorithms can be solved in ( using these techniqueswrite routine to reconstruct the shortest paths from the algorithm in section the binomial coefficients (nkcan be defin... |
23,814 | algorithm design techniques max min max min max figure game treewhich can be pruned write the remaining procedures for tic-tac-toe the one-dimensional circle packing problem is as followsyou have circles of radii rn these circles are packed in box such that each circle is tangent to the bottom of the box and are arrang... |
23,815 | figure voronoi diagram give an ( log nalgorithm to find one such arrangement your algorithm may serve as proof for part ( we are given set pn of points in plane voronoi diagram is partition of the plane into regions ri such that all points in ri are closer to pi than any other point in figure shows sample voronoi diagr... |
23,816 | algorithm design techniques consider the problem of right-justifying paragraph the paragraph contains sequence of words wn of length an which we wish to break into lines of length words are separated by blanks whose ideal length is (millimeters)but blanks can stretch or shrink as necessary (but must be > )so that line ... |
23,817 | as an exampleif we are searching for the pattern "textbookwith at most three mismatches in the string "data structures txtborpk"we find match (insert an echange an to an odelete pgive an (mnalgorithm to solve the approximate string matching problemwhere |pand |sone form of the knapsack problem is as followswe are given... |
23,818 | algorithm design techniques distance graph::shortestst distance dt tempifs = return dt for each vertex adjacent to tmp shortestvt )ifcs, tmp dt dt cs, tmpreturn dt figure recursive shortest-path algorithm (pseudocode *repeat part (aif is allowed to be rectangleinstead of square largest is measured by area even if the c... |
23,819 | as the initial current position two players alternate turns at each turna player must select grid element in the current row or column the value of the selected position is added to the player' scoreand that position becomes the current position and cannot be selected again players alternate until all grid elements in ... |
23,820 | algorithm design techniques automatically be improved by factor of or more this is discussed in exercise [ ]and [ the discussion of random-number generators is based on [ park and miller attribute the portable implementation to schrage [ the mersenne twister generator was proposed in [ ]the subtract with-carry generato... |
23,821 | bloom" counterexample to the theorem of piccard,journal of combinatorial theory ( ) - blumr floydv prattr rivestand tarjan"time bounds for selection,journal of computer and system sciences ( ) - borodin and munrothe computational complexity of algebraic and numerical problemsamerican elseviernew york chang and korsh"ca... |
23,822 | algorithm design techniques knuththe art of computer programmingvol seminumerical algorithms ed addison-wesleyreadingmass knuth"optimum binary search trees,acta informatica ( ) - knuth"an analysis of alpha-beta cutoffs,artificial intelligence ( ) - knuthtex and metafontnew directions in typesettingdigital pressbedfordm... |
23,823 | preparata and shamoscomputational geometryan introductionspringer-verlagnew york pugh"skip listsa probabilistic alternative to balanced trees,communications of the acm ( ) - rabin"probabilistic algorithms,in algorithms and complexityrecent results and new directions ( traubed )academic pressnew york - rabin"probabilist... |
23,824 | |
23,825 | amortized analysis in this we will analyze the running times for several of the advanced data structures that have been presented in and in particularwe will consider the worstcase running time for any sequence of operations this contrasts with the more typical analysisin which worst-case bound is given for any single ... |
23,826 | amortized analysis an unrelated puzzle consider the following puzzletwo kittens are placed on opposite ends of football field yards apart they walk toward each other at the speed of yards per minute at the same timetheir mother is at one end of the field she can run at yards per minute the mother runs from one kitten t... |
23,827 | figure binomial trees and figure two binomial queues and figure binomial queue the result of merging and |
23,828 | amortized analysis minimum and splitting the original binomial queue into two binomial queueswhich are then merged less terse explanation of these operations is given in we consider very simple problem first suppose we want to build binomial queue of elements we know that building binary heap of elements can be done in... |
23,829 | if we add all these equationsmost of the ti terms cancelleaving ci tn = or equivalentlyn ci (tn = recall that and tn the number of trees after the insertionsis certainly not negativeso (tn is not negative thus ci < = which proves the claim during the buildbinomialqueue routineeach insertion had worstcase time of (log )... |
23,830 | amortized analysis total time total potential figure sequence of inserts that works is found neverthelessthe discussion above suggests few ruleswhich tell us the properties that good potential functions have the potential function should always assume its minimum at the start of the sequence popular method of choosing ... |
23,831 | insert cost - - potential change - - - figure the insertion cost and potential change for each operation in sequence skew heaps the analysis of binomial queues is fairly easy example of an amortized analysis we now look at skew heaps as is common with many of our examplesonce the right potential function is foundthe an... |
23,832 | amortized analysis will drop the big-oh notation and charge one unit of time for each node on the paths since the heaps have no structureit is possible that all the nodes in both heaps lie on the right pathand this would give (nworst-case bound to merge the heaps (exercise asks you to construct an examplewe will show ... |
23,833 | figure change in heavy/light status after merge proof let and be the two heapswith and nodesrespectively suppose the right path of has light nodes and heavy nodesfor total of likewiseh has light and heavy nodes on its right pathfor total of nodes if we adopt the convention that the cost of merging two skew heaps is the... |
23,834 | amortized analysis in order to lower this time boundthe time required to perform the decreasekey operation must be improved -heapswhich were described in section give an (logd | |time bound for the decreasekey operation as well as for insertbut an ( logd | |bound for deletemin by choosing to balance the costs of |edecr... |
23,835 | - - - - figure decreasing to via percolate up would take (ntime figure sample leftist heap figure decreasing to creates heap-order violation |
23,836 | amortized analysis figure the two trees after the cut we do not want to percolate the to the rootbecauseas we have seenthere are cases where this could be expensive the solution is to cut the heap along the dashed linethus creating two treesand then merge the two trees back into one let be the node to which the decreas... |
23,837 | figure converted to the leftist heap figure decreasekey( completed by merging and the same sizeso it violates the binomial queue property we will call this lazy binomial queue in order to maintain consistency this is fast operation that always takes constant (worst-casetime as beforean insertion is done by creating one... |
23,838 | amortized analysis figure lazy binomial queue figure lazy binomial queue after removing the smallest element ( forr <log ++ while|lr > remove two trees from lr merge the two trees into new treeadd the new tree to lr+ figure procedure to reinstate binomial queue be able to perform the merge in time proportional to the n... |
23,839 | figure combining the binomial trees into binomial queue theorem the amortized running times of merge and insert are both ( for lazy binomial queues the amortized running time of deletemin is (log nproof the potential function is the number of trees in the collection of binomial queues the initial potential is and the p... |
23,840 | amortized analysis if we ignore the constant in the big-oh notationby the argument above once this is donethere can be at most log trees remainingso the potential function can increase by at most (log nt adding the actual time and the change in potential gives an amortized bound of log since all the trees are binomial ... |
23,841 | figure the resulting segment of the fibonacci heap after the decreasekey operation lost its second childso it is cut from the process stops heresince was unmarked the node is now marked the result is shown in figure notice that and which used to be marked nodesare no longer markedbecause they are now root nodes this wi... |
23,842 | amortized analysis because it is well known that the fibonacci numbers grow exponentiallyit immediately follows that any node with descendants has rank at most (log sthuswe have lemma the rank of any node in fibonacci heap is (log nproof immediate from the discussion above if all we were concerned about were the time b... |
23,843 | increasethe potential change is at most (log nt adding the actual time and potential change gives the (log namortized bound for deletemin finallyfor the decreasekey operationlet be the number of cascading cuts the actual cost of decreasekey is which is the total number of cuts performed the first (noncascadingcut creat... |
23,844 | amortized analysis rotation and each zig-zig or zig-zag as two rotationsthen the cost of any access is equal to plus the number of rotations in order to show an (log namortized bound for the splaying stepwe need potential function that can increase by at most (log nover the entire splaying step but that will also cance... |
23,845 | squaring both sides gives ab < / taking logarithms of both sides proves the lemma with the preliminaries taken care ofwe are ready to prove the main theorem theorem the amortized time to splay tree with root at node is at most ( ( )- ( ))+ (log nproof the potential function is the sum of the ranks of the nodes in if is... |
23,846 | amortized analysis we also see that si ( >si (xconsequentlyri ( <ri (pmaking this substitution gives atzig-zag < rf (prf ( ri (xfrom figure we see that sf (psf ( <sf (xif we apply lemma we obtain log sf (plog sf ( < log sf ( by the definition of rankthis becomes rf (prf ( < rf ( substituting thiswe obtain atzig-zag < r... |
23,847 | in generalby adding up the amortized costs of all the rotationsof which at most one can be zigwe see that the total amortized cost to splay at node is at most (rf (xri ( ) where ri (xis the rank of before the first splaying step and rf (xis the rank of after the last splaying step since the last splaying step leaves at... |
23,848 | amortized analysis the keys to choosing potential function are to guarantee that the minimum potential occurs at the beginning of the algorithmand to have the potential increase for cheap operations and decrease for expensive operations it is important that the excess or saved time be measured by an opposite change in ... |
23,849 | in we described rehashingwhen table becomes more than half fulla new table twice as large is constructedand the entire old table is rehashed give formal amortized analysiswith potential functionto show that the amortized cost of an insertion is still ( what is the maximum depth of fibonacci heapa deque with heap order ... |
23,850 | amortized analysis factor of the best static search trees exercise (bshows that splay trees are optimalto within constant factor of the best optimal search trees theseas well as two other strong resultsare proved in the original splay tree paper [ amortization is used in [ to merge balanced search tree efficiently the ... |
23,851 | advanced data structures and implementation in this we discuss six data structures with an emphasis on practicality we begin by examining alternatives to the avl tree discussed in these include an optimized version of the splay treethe red-black treeand the treap we also examine the suffix treewhich allows searching fo... |
23,852 | advanced data structures and implementation figure top-down splay rotationszigzig-zigand zig-zag xthat is the root of its subtreethis is represented in our diagrams as the "middletree tree stores nodes in the tree that are less than xbut not in ' subtreesimilarly tree stores nodes in the tree that are larger than xbut ... |
23,853 | figure simplified top-down zig-zag identical to the zig case this would seem advantageous because testing for host of cases is time-consuming the disadvantage is that by descending only one levelwe have more iterations in the splaying procedure once we have performed the final splaying stepfigure shows how lrand the mi... |
23,854 | empty simplified zig-zag empty zig-zig zig reassemble figure steps in top-down splay (access in top tree |
23,855 | template class splaytree publicsplaytreenullnode new binarynodenullnode->left nullnode->right nullnoderoot nullnode~splaytreemakeempty)delete nullnode/same methods as for binarysearchtree (omittedsplaytreeconst splaytree rhs )splaytreesplaytree &rhs )splaytree operator=const splaytree rhs )splaytree operator=splaytree ... |
23,856 | /*internal method to perform top-down splay the last accessed node becomes the new root this method may be overridden to use different splaying algorithmhoweverthe splay tree code depends on the accessed item going to the root is the target item to splay around is the root of the subtree to splay *void splayconst compa... |
23,857 | as we mentioned abovebefore the reassembly at the end of the splayheader left and header right point to the roots of and lrespectively (this is not typo--follow the linksexcept for this detailthe code is relatively straightforward figure shows the method to insert an item into tree new node is allocated (if necessary)a... |
23,858 | advanced data structures and implementation void removeconst comparable if!containsx return/item not founddo nothing /if is foundit will be splayed to the root by contains binarynode *newtreeifroot->left =nullnode newtree root->rightelse /find the maximum in the left subtree /splay it to the rootand then attach right c... |
23,859 | red-black tree is binary search tree with the following coloring properties every node is colored either red or black the root is black if node is redits children must be black every path from node to null pointer must contain the same number of black nodes consequence of the coloring rules is that the height of red-bl... |
23,860 | advanced data structures and implementation figure zig rotation and zig-zag rotation work if is black chain or zig-zag chain (in either of two directionsfigure shows how we can rotate the tree for the case where is left child (note there is symmetric caseeven though is leafwe have drawn more general case that allows to... |
23,861 | figure color fliponly if ' parent is red do we continue with rotation turns out that we can apply top-down procedure to red-black trees that guarantees that won' be red the procedure is conceptually easy on the way downwhen we see node that has two red childrenwe make red and the two children black (if is the rootafter... |
23,862 | advanced data structures and implementation of red-black trees is the relatively low overhead required to perform insertionand the fact thatin practicerotations occur relatively infrequently an actual implementation is complicated not only by the host of possible rotations but also by the possibility that some subtrees... |
23,863 | void printtreeconst ifheader->right =nullnode cout <"empty tree<endlelse printtreeheader->right )void printtreeredblacknode * const ift ! ->left printtreet->left )cout element <endlprinttreet->right )redblacktreeconst redblacktree rhs nullnode new redblacknodenullnode->left nullnode->right nullnodeheader header->left h... |
23,864 | template class redblacktree publicexplicit redblacktreeconst comparable neginf )redblacktreeconst redblacktree rhs )redblacktreeredblacktree &rhs )~redblacktree)const comparable findminconstconst comparable findmaxconstbool containsconst comparable constbool isemptyconstvoid printtreeconstvoid makeempty)void insertcons... |
23,865 | redblacknode *parentredblacknode *grandredblacknode *great/usual recursive stuff void reclaimmemoryredblacknode * )void printtreeredblacknode * constredblacknode cloneredblacknode const/red-black tree manipulations void handlereorientconst comparable item )redblacknode rotateconst comparable itemredblacknode *theparent... |
23,866 | /*internal routine that performs single or double rotation because the result is attached to the parentthere are four cases called by handlereorient item is the item in handlereorient theparent is the parent of the root of the rotated subtree return the root of the rotated subtree *redblacknode rotateconst comparable i... |
23,867 | grand->color redifitem element !item element parent rotateitemgrand )/start dbl rotate current rotateitemgreat )current->color blackheader->right->color black/make root black void insertconst comparable current parent grand headernullnode->element xwhilecurrent->element ! great grandgrand parentparent currentcurrent el... |
23,868 | advanced data structures and implementation figure three cases when is left child and has two black children treaps our last type of binary search treeknown as the treapis probably the simplest of all like the skip listit uses random numbers and gives (log nexpected time behavior for any input searching time is identic... |
23,869 | template class treap publictreapnullnode new treapnodenullnode->left nullnode->right nullnodenullnode->priority int_maxroot nullnodetreapconst treap rhs )treaptreap &rhs )~treap)treap operator=const treap rhs )treap operator=treap &rhs )/additional public member functions (not shownprivatestruct treapnode comparable el... |
23,870 | advanced data structures and implementation insertion into the treap is simpleafter an item is added as leafwe rotate it up the treap until its priority satisfies heap order it can be shown that the expected number of rotations is less than after the item to be deleted has been foundit can be deleted by increasing its ... |
23,871 | /*internal method to remove from subtree is the item to remove is the node that roots the tree set the new root of the subtree *void removeconst comparable xtreapnode ift !nullnode ifx element removext->left )else ift->element removext->right )else /match found ift->left->priority right->priority rotatewithleftchildt )... |
23,872 | advanced data structures and implementation assuming that the size of is less than (and usually it is significantly less)then we would reasonably expect that the time to solve this problem for given and would be at least linear in the length of tand in fact there are several ot algorithms howeverwe are interested in mo... |
23,873 | index substring being represented ana anana banana na nana figure suffix array that stores only indices (full substrings shown for referencewe can also use the suffix array to find the number of occurrences of pthey will be stored sequentially in the suffix arraythus two binary searches suffice to find range of suffixe... |
23,874 | /returns the lcp for any two strings *int computelcpconst string const string int whilei length& length& = ++ireturn /fill in the suffix array and lcp information for string str str is the input string sa is an existing array to place the suffix array lcp is an existing array to place the lcp information *void createsu... |
23,875 | the running time of the suffix array computation is dominated by the sorting stepwhich uses on log comparisons in many circumstances this can be reasonably acceptable performance for instancea suffix array for , , -character english-language novel can be built in just few seconds howeverthe on log costbased on the numb... |
23,876 | advanced data structures and implementation eed deed ed ed eed ed eed deed figure lefttrie representing the suffixes for deed{ddeededeed}rightcompressed trie that collapses single-node branches trie in figure is exactly one less than the number of internal branching nodes in the original trie in figure and of course wr... |
23,877 | node with letter depth of is representing the common string "anain all nodes that are below it and the internal node with letter depth of is representing the common string "nain all nodes that are below it in factthis analysis makes clear that suffix tree is equivalent to suffix array plus an lcp array if we have suffi... |
23,878 | advanced data structures and implementation if there is no such nodethe answer is either zero or one and is found by checking the suffix at which the search terminates this takes timeproportional to the length of the pattern pand is independent of the size of | find the most common substring of specified length return ... |
23,879 | step sort the characters in the stringassigning them numbers sequentially starting at then use those numbers for the remainder of the algorithm note that the numbers that are assigned depend on the text soif the text contains dna characters acgand onlythen there will be only four numbers then pad the array with three t... |
23,880 | advanced data structures and implementation array of length three and thus have fourfourand three suffixesrespectively ' suffixes are [abr][aca][dab][ra$][aca][dab][ra$][dab][ra$][ra$]which clearly correspond to the suffixes abracadabraacadabradabraand ra in the original string in the original string sthese suffixes ar... |
23,881 | [bra[cad[abr[ $$[rac[ada[braintegers sa[ index figure mapping of tri-charactersthe resulting array that is formed for and the resulting suffix array that is computed recursively step compute the suffix array for this is easy to do because [ - [ - [ is[ - [ ] [ - [ ] [ -since our recursive call has already sorted all [ ... |
23,882 | advanced data structures and implementation [abrindex index of second element radix pass ordering radix pass ordering rank in group sausing ' indices [aca[dab[ra$ add one to above last line of figure stably radix sort by first char using results of previous line using results of previous line figure computing suffix ar... |
23,883 | sa for sa for and final sa input index again the first characters matchso we compare indices and and this is already computedwith index having the smaller string so that means that now goes into the final suffix arrayand we advance the second groupobtaining sa for sa for and final sa input index once againthe first cha... |
23,884 | advanced data structures and implementation the same situation occurs on the next comparison between pair of 'sthe second comparison is between index ( cand index ( )so the element from the first group advances sa for sa for and final sa input index at this pointthere are no ties for whileso we quickly advance to the l... |
23,885 | sa for sa for and final sa input index /fill in the suffix array information for string str str is the input string sa is an existing array to place the suffix array lcp is an existing array to place the lcp information *void createsuffixarrayconst string strvector savector lcp ifsa size!str length|lcp size!str lengtht... |
23,886 | advanced data structures and implementation since this is standard mergewith at most two comparisons per suffix pairthis step takes linear time the entire algorithm thus satisfies (nt( / on and takes linear time although we have only computed the suffix arraythe lcp information can also be computed as the algorithm run... |
23,887 | /assigns the new supercharacter names /at end of routinesa will have indices into sin sorted order /and will have new character names /returns the number of names assignednote that if /this value is the same as then sa is suffix array for int assignnamesconst vector svector vector sa int int int /radix sort the new cha... |
23,888 | advanced data structures and implementation /stably sort in[ - with indices into that has keys in /into out[ - ]sort is relative to offset into /uses counting radix sort void radixpassconst vector invector outconst vector sint offsetint nint vector countk )/counter array forint ++ ++countsini offset ]forint < ++ counti... |
23,889 | /compute the suffix array for placing result into sa void computes vector vector sa int int ifk = /if unique namesdon' need recursion forint ++ sa [ ielse makesuffixarrays sa )/store unique names in using the suffix array forint ++ sa void computes const vector svector vector sa const vector sa int int int forint ++ if... |
23,890 | advanced data structures and implementation /merge sorted sa suffixes and sorted sa suffixes void mergeconst vector sconst vector vector saconst vector sa const vector sa int nint int int int whilet ! & ! int getindexintossa tn )/ int sa ]/ ifsuffix issmallerss sa ijt sak+ ++telse sak+ ++pwhilep sak+sa +]whilet sak+get... |
23,891 | int getindexintosconst vector sa int tint ifsa return sa else return sa /true if [ <[ bool leqint int int int return | = & < /true if [ <[ bool leqint int int int int int return | = &leqa )bool suffix issmallerconst vector sconst vector const vector sa int int iint jint ifsa / vs can break tie after character return le... |
23,892 | advanced data structures and implementation publicvoid insertconst vector insertxroot )privatevoid insertconst vector xkdnode tint level ift =nullptr new kdnodex }else ifxlevel datalevel insertxt->left level )else insertxt->right level )figure insertion into - trees solved by recursive tree traversalas shown in figure ... |
23,893 | public/*print items satisfying low < <high and low < <high *void printrangeconst vector lowconst vector high const printrangelowhighroot )privatevoid printrangeconst vector lowconst vector highkdnode *tint level const ift !nullptr iflow data &high > ->data &low data &high > ->data cout data <",data <")<endliflowlevel d... |
23,894 | advanced data structures and implementation pairing heaps the last data structure we examine is the pairing heap the analysis of the pairing heap is still openbut when decreasekey operations are neededit seems to outperform other heap structures the most likely reason for its efficiency is its simplicity the pairing he... |
23,895 | figure compareandlink merges two subheaps don' know if this violates the heap order thus we cut the adjusted node from its parent and complete the decreasekey by merging the two heaps that result to perform deleteminwe remove the rootcreating collection of heaps if there are children of the rootthen calls to the merge ... |
23,896 | advanced data structures and implementation /*internal method that is the basic operation to maintain order links first and second together to satisfy heap order first is root of tree which may not be nullptr first->nextsibling must be nullptr on entry second is root of tree which may be nullptr first becomes the resul... |
23,897 | struct pairnodetypedef pairnode position/*insert item into the priority queuemaintaining heap order return the position ( pointer to the nodecontaining the new item *position insertconst comparable pairnode *newnode new pairnodex }ifroot =nullptr root newnodeelse compareandlinkrootnewnode )return newnode/*change the va... |
23,898 | advanced data structures and implementation void deleteminifisemptythrow underflowexception}pairnode *oldroot rootifroot->leftchild =nullptr root nullptrelse root combinesiblingsroot->leftchild )delete oldrootfigure pairing heap deletemin scan with the current merged result as an exampleif we have eight childrenc throu... |
23,899 | /*internal method that implements two-pass merging firstsibling the root of the conglomerate and is assumed not nullptr *pairnode combinesiblingspairnode *firstsibling iffirstsibling->nextsibling =nullptr return firstsibling/allocate the array static vector treearray )/store the subtrees in an array int numsiblings for... |
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