id int64 0 25.6k | text stringlengths 0 4.59k |
|---|---|
23,500 | hashing empty table after after after after after figure hash table with double hashingafter each insertion as hash (xr ( mod )with prime smaller than tablesizewill work well if we choose then figure shows the results of inserting the same keys as before the first collision occurs when is inserted hash ( so is inserted... |
23,501 | figure hash table with linear probing with input as an examplesuppose the elements and are inserted into linear probing hash table of size the hash function is (xx mod the resulting hash table appears in figure if is inserted into the tablethe resulting table in figure will be over percent full because the table is so ... |
23,502 | hashing figure hash table after rehashing rehashing can be implemented in several ways with quadratic probing one alternative is to rehash as soon as the table is half full the other extreme is to rehash only when an insertion fails thirdmiddle-of-the-road strategy is to rehash when the table reaches certain load facto... |
23,503 | /*rehashing for quadratic probing hash table *void rehashvector oldarray array/create new double-sizedempty table array resizenextprime oldarray size)forauto entry array entry info empty/copy table over currentsize forauto entry oldarray ifentry info =active insertstd::moveentry element )/*rehashing for separate chaini... |
23,504 | hashing class caseinsensitivestringhash publicsize_t operatorconst string const static hash hfreturn hftolowers )/tolower implemented elsewhere bool operatorconst string lhsconst string rhs const return equalsignorecaselhsrhs )/equalsignorecase is elsewhere }unordered_set sfigure creating case-insensitive unordered_set... |
23,505 | for separate chainingassuming load factor of this is one version of the classic balls and bins problemgiven balls placed randomly (uniformlyin binswhat is the expected number of balls in the most occupied binthe answer is well known to be th(log nlog log )meaning that on averagewe expect some queries to take nearly log... |
23,506 | hashing collisions in the entire table is (ij) < cij since there are ( )/ pairsthis sum is ( )/( mn( )/( since the expected number of collisions is below the probability that there is even one collision must also be below of courseusing lists is impractical howeverthe preceding analysis suggests the following alternati... |
23,507 | in the secondary hash tableand that this accounts for bi (bi )/ pairwise collisions which we will call ci thus the amount of spacebi used for the ith secondary hash table is ci bi the total space is then ci bi the total number of collisions is ( )/ (from the first sentence of this proof)the total number of items is o... |
23,508 | hashing randomly chosen hash functionsitem can be at either position in table or position in table item can be at either position in table or position in table and so on immediatelythis implies that search in cuckoo hash table requires at most two table accessesand remove is trivialonce the item is located (lazy deleti... |
23,509 | table table figure cuckoo hash table after insertion of table table figure cuckoo hash table after insertion of after this is donee can be easily inserted so farso goodbut can we now insert ffigures to show that this algorithm successfully inserts fby displacing ethen aand then clearlyas we mentioned beforewe cannot su... |
23,510 | hashing table table figure continuing the insertion of into the table in figure nexte displaces table table figure continuing the insertion of into the table in figure nexta displaces table table figure completing the insertion of into the table in figure miraculously (?) finds an empty position in table would try to g... |
23,511 | table table figure inserting into the table in figure displaces dwhich displaces bwhich displaces awhich displaces ewhich displaces fwhich displaces cwhich displaces it is not yet hopeless since when is displacedwe would now try the other hash tableat position howeverwhile that could be successful in generalin this cas... |
23,512 | hashing template class cuckoohashfamily publicsize_t hashconst anytype xint which constint getnumberoffunctions)void generatenewfunctions)}figure generic hashfamily interface for cuckoo hashing figure provides the class interface for cuckoo hashing we will code variant that will allow an arbitrary number of hash functi... |
23,513 | template class cuckoohashtable publicexplicit cuckoohashtableint size )void makeempty)bool containsconst anytype constbool removeconst anytype )bool insertconst anytype )bool insertanytype & )privatestruct hashentry anytype elementbool isactivehashentryconst anytype anytype)bool false elemente }isactivea hashentryanyty... |
23,514 | hashing explicit hashtableint size arraynextprimesize numhashfunctions hashfunctions getnumberoffunctions)rehashes makeempty)void makeemptycurrentsize forauto entry array entry isactive falsefigure routines to initialize and empty the cuckoo hash table /*compute the hash code for using specified function *int myhashcon... |
23,515 | /*return true if is found *bool containsconst anytype const return findposx !- figure routine to search cuckoo hash table /*remove from the hash table return true if item was found and removed *bool removeconst anytype int currentpos findposx )if!isactivecurrentpos return falsearraycurrentpos isactive false--currentsiz... |
23,516 | hashing static const int allowed_rehashes bool inserthelper const anytype xx const int count_limit anytype xxwhiletrue int lastpos - int posforint count count count_limit++count forint numhashfunctions++ pos myhashxi )if!isactivepos arraypos std::movehashentrystd::movex )true )++currentsizereturn true/none of the spots... |
23,517 | evicting the first item did not perform well in experiments evicting the last item did not perform well in experiments evicting the items in sequence ( the first eviction uses hash function the next uses hash function etc did not perform well in experiments evicting the item purely randomly did not perform well in expe... |
23,518 | hashing template class stringhashfamily publicstringhashfamilymultiplierscount generatenewfunctions)int getnumberoffunctionsconst return countvoid generatenewfunctionsforauto mult multipliers mult nextint)size_t hashconst string xint which const const int multiplier multiplierswhich ]size_t hashval forauto ch hashval m... |
23,519 | howevercuckoo hashing is extremely sensitive to the choice of hash functionsthe inventors of the cuckoo hash table reported that many of the standard hash functions that they attempted performed poorly in tests furthermorealthough the insertion time is expected to be constant time as long as the load factor is below th... |
23,520 | hashing item hop figure hopscotch hashing table the hops tell which of the positions in the block are occupied with cells containing this hash value thus hop[ indicates that only position currently contains items whose hash value is while positions and do not failsince there would now be max_dist items trying to reside... |
23,521 | item hop item hop item hop figure hopscotch hashing table attempting to insert linear probing suggests location but that is too farso we evict from position to find closer position item hop item hop item hop figure hopscotch hashing table attempting to insert linear probing suggests location but that is too farconsulti... |
23,522 | hashing item hop item hop item hop figure hopscotch hashing table insertion of continuesnextb is evictedand finallywe have spot that is close enough to the hash value and can insert universal hashing although hash tables are very efficient and have constant average cost per operationassuming appropriate load factorsthe... |
23,523 | is at most /mand when adding into table with itemsthe probability of collision at the initial point is at most /mor the load factor the use of universal hash function for separate chaining or hopscotch hashing would be sufficient to meet the assumptions used in the analysis of those data structures howeverit is not suf... |
23,524 | hashing values for sfor total of ( possible (rspairs notice that the number of (abpairs and the number of (rspairs is identicalthus each (rspair will correspond to exactly one (abpair if we can solve for (abin terms of and but that is easyas beforesubtracting equations yields ( ( (mod )which means that by multiplying b... |
23,525 | const int digs const int mersennep ( <<digs int universalhashint xint aint bint long long hashval static_casta bhashval hashval >digs hashval mersennep )ifhashval >mersennep hashval -mersennepreturn static_casthashval mfigure simple implementation of universal hashing of two since the remainder could be almost as large... |
23,526 | hashing ( ( ( ( figure extendible hashingoriginal data let us supposefor the momentthat our data consists of several -bit integers figure shows an extendible hashing scheme for these data the root of the "treecontains four pointers determined by the leading two bits of the data each leaf has up to elements it happens t... |
23,527 | ( ( ( ( ( figure extendible hashingafter insertion of and directory split ( ( ( ( ( ( figure extendible hashingafter insertion of and leaf split these possibilities suggest that it is important for the bits to be fairly random this can be accomplished by hashing the keys into reasonably long integer--hence the name we ... |
23,528 | hashing the more surprising result is that the expected size of the directory (in other words is ( + / /mif is very smallthen the directory can get unduly large in this casewe can have the leaves contain pointers to the records instead of the actual recordsthus increasing the value of this adds second disk access to ea... |
23,529 | hash table is useful for any graph theory problem where the nodes have real names instead of numbers hereas the input is readvertices are assigned integers from onward by order of appearance againthe input is likely to have large groups of alphabetized entries for examplethe vertices could be computers then if one part... |
23,530 | hashing large number of deletions in separate chaining hash table can cause the table to be fairly emptywhich wastes space in this casewe can rehash to table half as large assume that we rehash to larger table when there are twice as many elements as the table size how empty should the table be before we rehash to smal... |
23,531 | mod tablesize (section describes method of performing this calculation without overflowsbut it is unlikely that overflow matters in this case explain why this strategy tends to avoid secondary clusteringand compare this strategy with both double hashing and quadratic probing rehashing requires recomputing the hash func... |
23,532 | hashing that the first starts at number and each statement has number higher than the previous statement you may assume an upper limit of statementsbut the statement numbers in the input might be as large as -bit integer your program must run in linear time implement the word puzzle program using the algorithm describe... |
23,533 | template class pair hashedobj keyobject def/appropriate constructorsetc }template class dictionary publicdictionary)void insertconst hashedobj keyconst object definition )const object lookupconst hashedobj key constbool isemptyconstvoid makeempty)privatehashtableitems}figure dictionary skeleton for exercise implement h... |
23,534 | hashing hashing dates to at least when luhn wrote an internal ibm memorandum that used separate chaining hashing early papers on hashing are [ and [ wealth of information on the subjectincluding an analysis of hashing with linear probing under the assumption of totally random and independent hashingcan be found in [ mo... |
23,535 | dietzfelbingera karlink melhornf meyer auf der heideh rohnertand tarjan"dynamic perfect hashingupper and lower bounds,siam journal on computing ( ) - dietzfelbinger and schellbach"on risks of using cuckoo hashing with simple universal hash classes,proceedings of the twentieth annual acm-siam symposium on discrete algor... |
23,536 | hashing mckenzier harriesand bell"selecting hashing algorithm,software--practice and experience ( ) - pagh and rodler"cuckoo hashing,journal of algorithms ( ) - patrascu and thorup"on the -independence required by linear probing and minwise independence,proceedings of the th international colloquium on automatalanguage... |
23,537 | priority queues (heapsalthough jobs sent to printer are generally placed on queuethis might not always be the best thing to do for instanceone job might be particularly importantso it might be desirable to allow that job to be run as soon as the printer is available converselyifwhen the printer becomes availablethere a... |
23,538 | priority queues (heapsdeletemin priority queue insert figure basic model of priority queue as with most data structuresit is sometimes possible to add other operationsbut these are extensions and not part of the basic model depicted in figure priority queues have many applications besides operating systems in we will s... |
23,539 | binary heap the implementation we will use is known as binary heap its use is so common for priority queue implementations thatin the context of priority queueswhen the word heap is used without qualifierit is generally assumed to be referring to this implementation of the data structure in this sectionwe will refer to... |
23,540 | priority queues (heapstemplate class binaryheap publicexplicit binaryheapint capacity )explicit binaryheapconst vector items )bool isemptyconstconst comparable findminconstvoid insertconst comparable )void insertcomparable & )void deletemin)void deletemincomparable minitem )void makeempty)privateint currentsizevector a... |
23,541 | figure two complete trees (only the left tree is heapapplying this logicwe arrive at the heap-order property in heapfor every node xthe key in the parent of is smaller than (or equal tothe key in xwith the exception of the root (which has no parent in figure the tree on the left is heapbut the tree on the right is not ... |
23,542 | figure attempt to insert creating the holeand bubbling the hole up figure the remaining two steps to insert in previous heap /*insert item xallowing duplicates *void insertconst comparable ifcurrentsize =array size array resizearray size )/percolate up int hole ++currentsizecomparable copy xarray std::movecopy )forx ar... |
23,543 | we could have implemented the percolation in the insert routine by performing repeated swaps until the correct order was establishedbut swap requires three assignment statements if an element is percolated up levelsthe number of assignments performed by the swaps would be our method uses assignments if the element to b... |
23,544 | priority queues (heaps figure next two steps in deletemin figure last two steps in deletemin make sure not to assume that there are always two childrenso this usually involves an extra test in the code depicted in figure we've done this test at line one extremely tricky solution is always to ensure that your algorithm ... |
23,545 | /*remove the minimum item throws underflowexception if empty *void deleteminifisemptythrow underflowexception}array std::movearraycurrentsize-)percolatedown )/*remove the minimum item and place it in minitem throws underflowexception if empty *void deletemincomparable minitem ifisemptythrow underflowexception}minitem s... |
23,546 | priority queues (heapsfigure very large complete binary tree so there is no way to find any particular element without linear scan through the entire heap to see thisconsider the large heap structure (the elements are not shownin figure where we see that the only information known about the maximum element is that it i... |
23,547 | is terminated by user (instead of finishing normally)it must be removed from the priority queue buildheap the binary heap is sometimes constructed from an initial collection of items this constructor takes as input items and places them into heap obviouslythis can be done with successive inserts since each insert will ... |
23,548 | priority queues (heaps figure leftinitial heaprightafter percolatedown( figure leftafter percolatedown( )rightafter percolatedown( figure leftafter percolatedown( )rightafter percolatedown( theorem for the perfect binary tree of height containing + - nodesthe sum of the heights of the nodes is + ( proof it is easy to s... |
23,549 | figure leftafter percolatedown( )rightafter percolatedown( sh ( ii= ( ( ( ( - ( ( multiplying by gives the equation ( ( ( ( ( we subtract these two equations and obtain equation ( we find that certain terms almost cancel for instancewe have ( ( ( and so on the last term in equation ( ) does not appear in equation ( )th... |
23,550 | priority queues (heapsthe selection problem the first problem we will examine is the selection problem from section recall that the input is list of elementswhich can be totally orderedand an integer the selection problem is to find the kth largest element two algorithms were given in but neither is very efficient the ... |
23,551 | if the element goes into splus (log )to delete sk and insert the new element if this is necessary thusthe total time is ( ( klog ko( log kthis algorithm also gives bound of ( log nfor finding the median in we will see how to solve this problem in (naverage time in we will see an elegantalbeit impracticalalgorithm to s... |
23,552 | priority queues (heapsif the event is an arrivalwe check for an available teller if there is nonewe place the arrival on the line (queue)otherwise we give the customer tellercompute the customer' departure timeand add the departure to the set of events waiting to happen the waiting line for customers can be implemented... |
23,553 | in this casea -heap can be advantageous in much the same way as -trees finallythere is evidence suggesting that -heaps may outperform binary heaps in practice the most glaring weakness of the heap implementationaside from the inability to perform findsis that combining two heaps into one is hard operation this extra op... |
23,554 | priority queues (heaps figure null path lengths for two treesonly the left tree is leftist proof the proof is by induction if there must be at least one tree node otherwisesuppose that the theorem is true for consider leftist tree with nodes on the right path then the root has right subtree with nodes on the right path... |
23,555 | figure two leftist heaps and figure result of merging with ' right subheap reasonable to assume that the resulting tree is leftist heapbecause it was obtained via recursive step this is much like the inductive hypothesis in proof by induction since we can handle the base case (which occurs when one tree is empty)we can... |
23,556 | priority queues (heaps figure result of attaching leftist heap of previous figure as ' right child the merge notice that if the null path length is not updatedthen all null path lengths will be and the heap will not be leftist but merely random in this casethe algorithm will workbut the time bound we will claim will no... |
23,557 | template class leftistheap publicleftistheap)leftistheapconst leftistheap rhs )leftistheapleftistheap &rhs )~leftistheap)leftistheap operator=const leftistheap rhs )leftistheap operator=leftistheap &rhs )bool isemptyconstconst comparable findminconstvoid insertconst comparable )void insertcomparable & )void deletemin)v... |
23,558 | priority queues (heapsthe node referenced by the root will need to change we use the usual technique of implementing private recursive methods to do the merging the class skeleton is also shown in figure the two merge routines (fig are drivers designed to remove special cases and ensure that has the smaller root the ac... |
23,559 | /*internal method to merge two roots assumes trees are not emptyand ' root contains smallest item *leftistnode merge leftistnode * leftistnode * ifh ->left =nullptr /single node ->left /other fields in already accurate else ->right mergeh ->righth )ifh ->left->npl right->npl swapchildrenh ) ->npl ->right->npl return fi... |
23,560 | priority queues (heaps /*inserts xduplicates allowed *void insertconst comparable root mergenew leftistnodex }root )figure insertion routine for leftist heaps as mentioned abovewe can carry out insertions by making the item to be inserted one-node heap and performing merge to perform deleteminwe merely destroy the root... |
23,561 | skew heaps skew heap is self-adjusting version of leftist heap that is incredibly simple to implement the relationship of skew heaps to leftist heaps is analogous to the relation between splay trees and avl trees skew heaps are binary trees with heap orderbut there is no structural constraint on these trees unlike left... |
23,562 | priority queues (heaps figure result of merging with ' right subheap figure result of merging skew heaps and the exception of the last after few examplesit becomes clear that since all but the last node on the right path have their children swappedthe net effect is that this becomes the new left path (see the preceding... |
23,563 | the implementation of skew heaps is left as (trivialexercise note that because right path could be longa recursive implementation could fail because of lack of stack spaceeven though performance would otherwise be acceptable skew heaps have the advantage that no extra space is required to maintain path lengths and no t... |
23,564 | figure binomial trees and figure binomial queue with six elements figure two binomial queues and |
23,565 | the merge is performed by essentially adding the two queues together let be the new binomial queue since has no binomial tree of height and doeswe can just use the binomial tree of height in as part of nextwe add binomial trees of height since both and have binomial trees of height we merge them by making the larger ro... |
23,566 | figure after is inserted figure after is inserted figure after is inserted figure after is inserted figure after is inserted figure after is inserted figure after is inserted |
23,567 | with obtaining new tree of height we then merge this tree with obtaining tree of height which is the new priority queue we count this as three steps (two tree merges plus the stopping casethe next insertion after is inserted is another bad case and would require three tree merges deletemin can be performed by first fin... |
23,568 | priority queues (heaps figure result of applying deletemin to implementation of binomial queues the deletemin operation requires the ability to find all the subtrees of the root quicklyso the standard representation of general trees is requiredthe children of each node are kept in linked listand each node has pointer t... |
23,569 | template class binomialqueue publicbinomialqueue)binomialqueueconst comparable item )binomialqueueconst binomialqueue rhs )binomialqueuebinomialqueue &rhs )~binomialqueue)binomialqueue operator=const binomialqueue rhs )binomialqueue operator=binomialqueue &rhs )bool isemptyconstconst comparable findminconstvoid insertc... |
23,570 | priority queues (heaps figure merging two binomial trees in order to merge two binomial queueswe need routine to merge two binomial trees of the same size figure shows how the links change when two binomial trees are merged the code to do this is simple and is shown in figure we provide simple implementation of the mer... |
23,571 | /*merge rhs into the priority queue rhs becomes empty rhs must be different from this exercise needed to make this operation more efficient *void mergebinomialqueue rhs ifthis =&rhs /avoid aliasing problems returncurrentsize +rhs currentsizeifcurrentsize capacityint oldnumtrees thetrees size)int newnumtrees maxthetrees... |
23,572 | case /this and rhs *carry combinetreest )thetreesi rhs thetreesi nullptrbreakcase /this and carry *carry combinetreest carry )thetreesi nullptrbreakcase /rhs and carry *carry combinetreest carry )rhs thetreesi nullptrbreakcase /all three *thetreesi carrycarry combinetreest )rhs thetreesi nullptrbreakforauto root rhs th... |
23,573 | binomialnode *oldroot thetreesminindex ]binomialnode *deletedtree oldroot->leftchilddelete oldroot/construct 'binomialqueue deletedqueuedeletedqueue thetrees resizeminindex )deletedqueue currentsize <minindex forint minindex > -- deletedqueue thetreesj deletedtreedeletedtree deletedtree->nextsiblingdeletedqueue thetree... |
23,574 | priority queues (heaps#include #include #include #include #include using namespace std/empty the priority queue and print its contents template void dumpcontentsconst string msgpriorityqueue pq cout <msg <":<endlwhile!pq emptycout <pq top<endlpq pop)/do some inserts and removes (done in dumpcontentsint mainpriority_que... |
23,575 | void pushconst object )const object topconstvoid pop)bool empty)void clear)push adds to the priority queuetop returns the largest element in the priority queueand pop removes the largest element from the priority queue duplicates are allowedif there are several largest elementsonly one of them is removed the priority q... |
23,576 | priority queues (heapsa binary tree that has two extra levels (that isit is very slightly unbalancedb binary tree that has deepest node at depth log binary tree that has deepest node at depth log the worst-case binary tree rewrite the binaryheap insert routine by placing copy of the inserted item in position how many n... |
23,577 | suppose we need to perform percolateups and deletemins on -heap that initially has elements what is the total running time of all operations in terms of mnand db if what is the running time of all heap operationsc if ( )what is the total running timed what choice of minimizes the total running time suppose that binary... |
23,578 | priority queues (heaps figure input for exercises and merge the two leftist heaps in figure show the result of inserting keys to in order into an initially empty leftist heap prove or disprovea perfectly balanced tree forms if keys to are inserted in order into an initially empty leftist heap give an example of input t... |
23,579 | figure input for exercise prove that binomial tree of height has dk nodes at depth merge the two binomial queues in figure show that inserts into an initially empty binomial queue take (ntime in the worst case give an algorithm to build binomial queue of elementsusing at most comparisons between elements propose an al... |
23,580 | priority queues (heapssuppose we want to add the decreaseallkeysoperation to the heap repertoire the result of this operation is that all keys in the heap have their value decreased by an amount for the heap implementation of your choiceexplain the necessary modifications so that all other operations retain their runni... |
23,581 | atkinsonj sackn santoroand strothotte"min-max heaps and generalized priority queues,communications of the acm ( ) - bright"range restricted mergeable priority queues,information processing letters ( ) - brodal"worst-case efficient priority queues,proceedings of the seventh annual acmsiam symposium on discrete algorithm... |
23,582 | priority queues (heaps mcdiarmid and reed"building heaps fast,journal of algorithms ( ) - sleator and tarjan"self-adjusting heaps,siam journal on computing ( ) - strothottep erikssonand vallner" note on constructing min-max heaps,bit ( ) - van emde boasr kaasand zijlstra"design and implementation of an efficient priori... |
23,583 | sorting in this we discuss the problem of sorting an array of elements to simplify matterswe will assume in our examples that the array contains only integersalthough our code will once again allow more general objects for most of this we will also assume that the entire sort can be done in main memoryso that the numbe... |
23,584 | sorting the iterators must support random access the sort algorithm does not guarantee that equal items retain their original order (if that is importantuse stable_sort instead of sortas an examplein std::sortv begin) end)std::sortv begin) end)greater)std::sortv begin) beginv endv begin )the first call sorts the entire... |
23,585 | /*simple insertion sort *template void insertionsortvector forint size)++ comparable tmp std::moveap )int jforj pj &tmp aj ]-- aj std::moveaj )aj std::movetmp )figure insertion sort routine stl implementation of insertion sort in the stlinstead of having the sort routines take an array of comparable items as single par... |
23,586 | sorting /the two-parameter version calls the three-parameter versionusing ++ decltype *template void insertionsortconst iterator beginconst iterator end insertionsortbeginendless)figure two-parameter sort invokes three-parameter sort via ++ decltype template void insertionsortconst iterator beginconst iterator endcompa... |
23,587 | on the other handif the input is presortedthe running time is ( )because the test in the inner for loop always fails immediately indeedif the input is almost sorted (this term will be more rigorously defined in the next section)insertion sort will run quickly because of this wide variationit is worth analyzing the aver... |
23,588 | sorting proof the average number of inversions is initially ( - )/ ( each swap removes only one inversionso ( swaps are required this is an example of lower-bound proof it is valid not only for insertion sortwhich performs adjacent exchanges implicitlybut also for other simple algorithms such as bubble sort and selecti... |
23,589 | /*shellsortusing shell' (poorincrements *template void shellsortvector forint gap size gap gap / forint gapi size)++ comparable tmp std::moveai )int iforj >gap &tmp aj gap ] -gap aj std::moveaj gap )aj std::movetmp )figure shellsort routine using shell' increments (better increments are possibleaffect the implementatio... |
23,590 | sorting we prove the lower bound first by constructing bad case firstwe choose to be power of this makes all the increments evenexcept for the last incrementwhich is nowwe will give as input an array with the / largest numbers in the even positions and the / smallest numbers in the odd positions (for this proofthe firs... |
23,591 | previous theorem although this bound holds for the other incrementsit is too large to be useful intuitivelywe must take advantage of the fact that this increment sequence is special what we need to show is that for any element [pin position pwhen it is time to perform an hk -sortthere are only few elements to the left ... |
23,592 | sorting the performance of shellsort is quite acceptable in practiceeven for in the tens of thousands the simplicity of the code makes it the algorithm of choice for sorting up to moderately large input heapsort as mentioned in priority queues can be used to sort in ( log ntime the algorithm based on this idea is known... |
23,593 | figure (maxheap after buildheap phase figure heap after first deletemax analysis of heapsort as we saw in the first phasewhich constitutes the building of the heapuses less than comparisons in the second phasethe ith deletemax uses at most less than log ( comparisonsfor total of at most log (ncomparisons (assuming > co... |
23,594 | /*standard heapsort *template void heapsortvector forint size > -- /buildheap *percdownaia size)forint size -- std::swapa ]aj )/deletemax *percdowna )/*internal method for heapsort is the index of an item in the heap returns the index of the left child *inline int leftchildint return /*internal method for heapsort that... |
23,595 | experiments have shown that the performance of heapsort is extremely consistenton average it uses only slightly fewer comparisons than the worst-case bound suggests for many yearsnobody had been able to show nontrivial bounds on heapsort' average running time the problemit seemsis that successive deletemax operations d... |
23,596 | sorting if we choose (log log log )then the number of heaps that have cost sequence less than is at most ( / ) and the theorem follows from our earlier comments using more complex argumentit can be shown that heapsort always uses at least log (ncomparisons and that there are inputs that can achieve this bound the avera... |
23,597 | is added to cand then and are compared this proceeds until and are compared actr bctr actr cctr bctr actr cctr bctr cctr is added to cand the array is exhausted actr bctr cctr the remainder of the array is then copied to actr bctr cctr the time to merge two sorted lists is clearly linearbecause at most comparisons are ... |
23,598 | sorting /*mergesort algorithm (driver*template void mergesortvector vector tmparraya size)mergesortatmparray size )/*internal method that makes recursive calls is an array of comparable items tmparray is an array to place the merged result left is the left-most index of the subarray right is the right-most index of the... |
23,599 | /*internal method that merges two sorted halves of subarray is an array of comparable items tmparray is an array to place the merged result leftpos is the left-most index of the subarray rightpos is the index of the start of the second half rightend is the right-most index of the subarray *template void mergevector ave... |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.