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20,300 | chap internal sorting shellsort the next sort we consider is called shellsortnamed after its inventord shell it is also sometimes called the diminishing increment sort unlike insertion and selection sortthere is no real life intuitive equivalent to shellsort unlike the exchange sortsshellsort makes comparisons and swap... |
20,301 | figure an example of shellsort sixteen items are sorted in four passes the first pass sorts sublists of size and increment the second pass sorts sublists of size and increment the third pass sorts sublists of size and increment the fourth pass sorts list of size and increment ( regular insertion sortstatic void sort( [... |
20,302 | chap internal sorting some choices for increments will make shellsort run more efficiently than others in particularthe choice of increments described above ( - turns out to be relatively inefficient better choice is the following series based on division by three the analysis of shellsort is difficultso we must accept... |
20,303 | sec mergesort figure an illustration of mergesort the first row shows eight numbers that are to be sorted mergesort will recursively subdivide the list into sublists of one element eachthen recombine the sublists the second row shows the four sublists of size created by the first merging pass the third row shows the tw... |
20,304 | chap internal sorting subarrays of size which are in turn merged into subarrays of size and so on we need to avoid having each merge operation require new array with some difficultyan algorithm can be devised that alternates between two arrays much simpler approach is to copy the sorted sublists to the auxiliary array ... |
20,305 | static void mergesort( [ae[tempint lint rint ijkmid ( + )/ /select the midpoint if ( =rreturn/list has one element if ((mid- >thresholdmergesort(atemplmid)else inssort(almid- + )if (( -midthresholdmergesort(atempmid+ )else inssort(amid+ -mid)/do the merge operation firstcopy halves to temp for ( =li<=midi++temp[ia[ ]fo... |
20,306 | chap internal sorting sort routine such as the unix qsort function interestinglyquicksort is hampered by exceedingly poor worst-case performancethus making it inappropriate for certain applications before we get to quicksortconsider for moment the practicality of using binary search tree for sorting you could insert al... |
20,307 | static void qsort( [aint iint /quicksort int pivotindex findpivot(aij)/pick pivot dsutil swap(apivotindexj)/stick pivot at end / will be the first position in the right subarray int partition(ai- ja[ ])dsutil swap(akj)/put pivot in place if (( - qsort(aik- )/sort left partition if (( - qsort(ak+ )/sort right partition ... |
20,308 | chap internal sorting static int partition( [aint lint re pivotdo /move bounds inward until they meet while ( [++lcompareto(pivot)< )while (( != &( [--rcompareto(pivot)> ))dsutil swap(alr)/swap out-of-place values while ( )/stop when they cross dsutil swap(alr)/reverse lastwasted swap return /return first position in r... |
20,309 | sec quicksort initial pass swap pass swap pass , figure the quicksort partition step the first row shows the initial positions for collection of ten key values the pivot value is which has been swapped to the end of the array the do loop makes three iterationseach time moving counters and inwards until they meet in the... |
20,310 | chap internal sorting happens at each partition stepthen the total cost of the algorithm will be th( = in the worst casequicksort is th( this is terribleno better than bubble sort when will this worst case occuronly when each pivot yields bad partitioning of the array if the pivot values are selected at randomthen this... |
20,311 | the initial cn term is the cost of doing the findpivot and partition stepsfor some constant the closed-form solution to this recurrence relation is th( log nthusquicksort has average-case cost th( log nthis is an unusual situation that the average case cost and the worst case cost have asymptotically different growth r... |
20,312 | chap internal sorting simple improvement might then be to replace quicksort with faster sort for small numberssay insertion sort or selection sort howeverthere is an even better -and still simpler -optimization when quicksort partitions are below certain sizedo nothingthe values within that partition will be out of ord... |
20,313 | and fast the algorithm should take advantage of the fact that sorting is specialpurpose application in that all of the values to be stored are available at the start this means that we do not necessarily need to insert one value at time into the tree structure heapsort is based on the heap data structure presented in s... |
20,314 | chap internal sorting original numbers build heap remove remove remove figure an illustration of heapsort the top row shows the values in their original order the second row shows the values after building the heap the third row shows the result of the first removefirst operation on key value note that is now at the en... |
20,315 | over the time required to find the largest elements using one of the other sorting methods described earlier one situation where we are able to take advantage of this concept is in the implementation of kruskal' minimum-cost spanning tree (mstalgorithm of section that algorithm requires that edges be visited in ascendi... |
20,316 | chap internal sorting that each possible key value have corresponding bin in the extended binsort algorithm is as followsstatic void binsort(integer []list[ (llist[])new llist[maxkey]integer itemfor (int = <maxkeyi++ [inew llist()for (int = < lengthi++ [ [ ]append( [ ])for (int = <maxkeyi++for ( [imovetostart()(item [i... |
20,317 | sec binsort and radix sort initial list first pass (on right digitsecond pass (on left digit result of first passresult of second pass figure an example of radix sort for twelve two-digit numbers in base ten two passes are required to sort the list of in other wordsassign the ith record from array to bin using the form... |
20,318 | chap internal sorting as with mergesortan efficient implementation of radix sort is somewhat difficult to achieve in particularwe would prefer to sort an array of values and avoid processing linked lists if we know how many values will be in each binthen an auxiliary array of size can be used to hold the bins for examp... |
20,319 | sec binsort and radix sort initial inputarray first pass values for count rtoi count arrayindex positions for array end of pass array second pass values for count rtoi count arrayindex positions for array end of pass array figure an example showing function radix applied to the input of figure row shows the initial val... |
20,320 | chap internal sorting one could use base or base would be appropriate for sorting character strings for nowwe will treat as constant value and ignore it for the purpose of determining asymptotic complexity variable is related to the key rangeit is the maximum number of digits that key may have in base in some applicati... |
20,321 | are real numbers or arbitrary length stringsthen some care will be necessary in implementation in particularradix sort will need to be careful about deciding when the "last digithas been found to distinguish among real numbersor the last character in variable length strings implementing the concept of radix sort with t... |
20,322 | chap internal sorting sort insertion bubble selection shell shell/ merge merge/ quick quick/ heap heap/ radix/ radix/ up down figure empirical comparison of sorting algorithms run on -ghz intel pentium cpu running linux shellsortquicksortmergesortand heapsort each are shown with regular and optimized versions radix sor... |
20,323 | arraysoptimized quicksort performs well because it does one partition step before calling insertion sort compared to the other ( log nsortsunoptimized heapsort is quite slow due to the overhead of the class structure when all of this is stripped away and the algorithm is implemented to manipulate an array directlyit is... |
20,324 | chap internal sorting comparisonssecondthis proof is one of the few non-trivial lower-bounds proofs that we have for any problemthat isthis proof provides one of the relatively few instances where our lower bound is tighter than simply measuring the size of the input and output as suchit provides useful model for provi... |
20,325 | sec lower bounds for sorting xyz xyz yzx xzy zxy yxz zyx yes yxz yxz yzx zyx [ ]< [ ]( < ?no xyz xyz xzy zxy yes no yes no [ ]< [ ] [ ]< [ ]yzx ( < ?yxz xzy ( < ?xyz yzx xzy zyx zxy yes no yes no [ ]< [ ] [ ]< [ ]zxy ( < ?xzy zyx ( < ?yzx figure decision tree for insertion sort when processing three values labeled xyan... |
20,326 | chap internal sorting compared with xagainthere are two possibilities if is less than xthen these items should be swapped (the left branchif is not less than xthen insertion sort is complete (the right branchnote that the right branch reaches leaf nodeand that this leaf node contains only permutation yxz this means tha... |
20,327 | binary tree of height can store at most nodes equivalentlya tree with nodes requires at least dlog( ) levels what is the minimum number of nodes that must be in the decision tree for any comparison-based sorting algorithm for valuesbecause sorting algorithms are in the business of determining which unique permutation o... |
20,328 | chap internal sorting exercises using inductionprove that insertion sort will always produce sorted array write an insertion sort algorithm for integer key values howeverhere' the catchthe input is stack (not an array)and the only variables that your algorithm may use are fixed number of integers and fixed number of st... |
20,329 | recall that sorting algorithm is said to be stable if the original ordering for duplicate keys is preserved we can make any algorithm stable if we alter the input keys so that (potentiallyduplicate key values are made unique in way that the first occurrence of the original duplicate value is less than the second occurr... |
20,330 | chap internal sorting graph (nn log nf (nn and (nn in the range < < to visually compare their growth rates typicallythe constant factor in the running-time expression for an implementation of insertion sort will be less than the constant factors for shellsort or quicksort how many times greater can the constant factor ... |
20,331 | (adevise an algorithm to sort three numbers it should make as few comparisons as possible how many comparisons and swaps are required in the bestworstand average cases(bdevise an algorithm to sort five numbers it should make as few comparisons as possible how many comparisons and swaps are required in the bestworstand ... |
20,332 | chap internal sorting can stop early this makes the best case performance become ( (because if the list is already sortedthen no iterations will take place on the first passand the sort will stop right theremodify the bubble sort implementation to add this flag and test compare the modified implementation on range of i... |
20,333 | classmateswhat does this say about the difficulty of doing empirical timing studies perform study of shellsortusing different increments compare the version shown in section where each increment is half the previous onewith others in particulartry implementing "division by where the increments on list of length will be... |
20,334 | file processing and external sorting earlier presented basic data structures and algorithms that operate on data stored in main memory some applications require that large amounts of information be stored and processed -so much information that it cannot all fit into main memory in that casethe information must reside ... |
20,335 | chap file processing and external sorting medium ram disk flash drive floppy tape $ figure price comparison table for some writeable electronic data storage media in common use prices are in us dollars/mb primary versus secondary storage computer storage devices are typically classified into primary or main memory and ... |
20,336 | they are not erased from disk and tape when the power is turned off in contrastram used for main memory is usually volatile -all information is lost with the power second advantage is that floppy diskscd-romsand "flashdrives can easily be transferred between computers this provides convenient way to take information fr... |
20,337 | chap file processing and external sorting there are generally two approaches to minimizing disk accesses the first is to arrange information so that if you do access data from secondary memoryyou will get what you need in as few accesses as possibleand preferably on the first access file structure is the term used for ... |
20,338 | of the data on disk likewisewhen writing to particular logical byte position with respect to the beginning of the filethis position must be converted by the file manager into the corresponding physical location on the disk to gain some appreciation for the the approximate time costs for these operationsyou need to unde... |
20,339 | chap file processing and external sorting boom (armplatters read/write heads spindle (atrack (bfigure (aa typical disk drive arranged as stack of platters (bone track on disk drive platter intersector gaps bits of data sectors ( (bfigure the organization of disk platter dots indicate density of information (anominal ar... |
20,340 | the information flow at constant rate along the spiralthe drive must speed up the rate of disk spin as the / head moves toward the center of the disk this makes for more complicated and slower mechanism three separate steps take place when reading particular byte or series of bytes of data from hard disk firstthe / hea... |
20,341 | chap file processing and external sorting growsthere might not be free space physically adjacent thusa file might consist of several extents widely spaced on the disk the fuller the diskand the more that files on the disk changethe worse this file fragmentation (and the resulting seek timebecomes file fragmentation lea... |
20,342 | sec disk drives intersector gap sector header sector data sector header sector data intrasector gap figure an illustration of sector gaps within track each sector begins with sector header containing the sector address and an error detection code for the contents of that sector the sector header is followed by small in... |
20,343 | chap file processing and external sorting how much time is required to read the trackon averageit will require half rotation to bring the first sector of the track under the / headand then one complete rotation to read the track how long will it take to read file of mb divided into sectorsized ( byterecordsthis file wi... |
20,344 | buffers and buffer pools given the specifications of the disk drive from example we find that it takes about + ms to read one track of data on average it takes about ++( / ) ms on average to read single sector of data this is good savings (slightly over half the time)but less than of the data on the track are read if w... |
20,345 | chap file processing and external sorting takes maximum advantage of this micro-parallelism is double buffering imagine that file is being processed sequentially while the first sector is being readthe cpu cannot process that information and so must wait or find something else to do in the meantime once the first secto... |
20,346 | tion was first accessed typically it is more important to know how many times the information has been accessedor how recently the information was last accessed another approach is called "least frequently used(lfulfu tracks the number of accesses to each buffer in the buffer pool when buffer must be reusedthe buffer t... |
20,347 | chap file processing and external sorting reading blocks stored on slowersecondary memory (such as on the disk drivethe disk stores the complete contents of the virtual memory blocks are read into main memory as demanded by memory accesses naturallyprograms using virtual memory techniques are slower than programs whose... |
20,348 | sec buffers and buffer pools secondary storage (on diskmain memory (in ram figure an illustration of virtual memory the complete collection of information resides in the slowersecondary storage (on diskthose sectors recently accessed are held in the fast main memory (in ramin this examplecopies of sectors and from seco... |
20,349 | chap file processing and external sorting buffer pool' logical storage space physicallyit will actually be copied to the appropriate byte position in some buffer in the buffer pool example assume each sector of the disk file (and thus each block in the buffer poolstores bytes assume that the buffer pool is in the state... |
20,350 | pointer to the buffer containing sector would be returned by the call to getblock the client would then copy bytes to positions - of the bufferand call dirtyblock to warn the buffer pool that the contents of this block have been modified further problem with the second adt is the risk of stale pointers when the buffer ... |
20,351 | chap file processing and external sorting tents must then issue readblock request to read the data from disk into the bufferand then getdatapointer request to gain direct access to the buffer' data contents /*improved adt for buffer pools using the buffer-passing style most user functionality is in the buffer classnot ... |
20,352 | where disk / is the bottleneck for the programeven the time to copy lots of information between the buffer pool user and the buffer might be inconsequential another advantage to buffer passing is the reduction in unnecessary read operations for data that will be overwritten anyway you should note that the implementatio... |
20,353 | chap file processing and external sorting randomaccessfile(string namestring mode)class constructoropens disk file for processing read(byte[ )read some bytes from the current position in the file the current position moves forward as the bytes are read write(byte[ )write some bytes at the current position in the file (... |
20,354 | block contains the same number of fixed-size data records depending on the applicationa record might be only few bytes -composed of little or nothing more than the key -or might be hundreds of bytes with relatively small key field records are assumed not to cross block boundaries these assumptions can be relaxed for sp... |
20,355 | chap file processing and external sorting of single bank of read/write heads that move together over stack of platters if the sorting process involves reading from an input filealternated with writing to an output filethen the / head will continuously seek between the input file and the output file similarlyif two inpu... |
20,356 | simple approaches to external sorting if your operating system supports virtual memorythe simplest "externalsort is to read the entire file into virtual memory and run an internal sorting method such as quicksort this approach allows the virtual memory manager to use its normal buffer pool mechanism to control disk acc... |
20,357 | chap file processing and external sorting runs of length runs of length runs of length figure simple external mergesort algorithm input records are divided equally between two input files the first runs from each input file are merged and placed into the first output file the second runs from each input file are merged... |
20,358 | quentially and write the output run files sequentially for sequential processing and double buffering to be effectivehoweverit is necessary that there be separate / head available for each file this typically means that each of the input and output files must be on separate disk drivesrequiring total of four disk drive... |
20,359 | chap file processing and external sorting merge the runs together to form single sorted file replacement selection this section treats the problem of creating initial runs as large as possible from disk fileassuming fixed amount of ram is available for processing as mentioned previouslya simple approach is to allocate ... |
20,360 | sec external sorting input file input buffer ram output buffer output run file figure overview of replacement selection input records are processed sequentially initially ram is filled with records as records are processedthey are written to an output buffer when this buffer becomes fullit is written to disk meanwhilea... |
20,361 | chap file processing and external sorting input memory output figure replacement selection example after building the heaproot value is output and incoming value replaces it value is output nextreplaced with incoming value the heap is reorderedwith rising to the root value is output next incoming value is too small for... |
20,362 | sec external sorting falling snow future snow existing snow snowplow movement start time figure the snowplow analogy showing the action during one revolution of the snowplow circular track is laid out straight for purposes of illustrationand is shown in cross section at any time the most snow is directly in front of th... |
20,363 | chap file processing and external sorting input runs output buffer figure illustration of multiway merge the first value in each input run is examined and the smallest sent to the output this value is removed from the input and the process repeated in this examplevalues and are compared first value is removed from the ... |
20,364 | replaces several (potentiallysequential passes with single random access pass if the processing would not be sequential anyway (such as when all processing is on single disk drive)no time is lost by doing so multiway merging can greatly reduce the number of passes required if there is room in memory to store one block ... |
20,365 | chap file processing and external sorting file size sort , , , , sort memory size (in blocks , , , , , , , , , , , , sort memory size (in blocks , , , , , , , , , figure comparison of three external sorts on collection of small records for files of various sizes each entry in the table shows time in seconds and total n... |
20,366 | great discussion on external sorting methods can be found in salzberg' book the presentation in this is similar in spirit to salzberg' for details on disk drive modeling and measurementsee the article by ruemmler and wilkes"an introduction to disk drive modeling[rw see andrew tanenbaum' structured computer organization... |
20,367 | chap file processing and external sorting show your calculations assume that disk drive is configured as follows the total storage is approximately mb divided among surfaces each surface has tracksthere are sectors/track bytes/sectorand sectors/cluster the disk turns at rpm the track-to-track seek time is msand the ave... |
20,368 | (athe file is stored on series of contiguous tracksas few tracks as possible (bthe file is spread randomly across the disk in kb clusters show your calculations at the end of the fastest disk drive could find specifications for was the maxtor atlas this drive had nominal capacity of gb using platters ( surfacesor gb/su... |
20,369 | chap file processing and external sorting assume that virtual memory is managed using buffer pool the buffer pool contains five buffers and each buffer stores one block of data memory accesses are by block id assume the following series of memory accesses takes place for each of the following buffer pool replacement st... |
20,370 | processing is sorting operation on all of the employee recordsand that an external sorting algorithm is used the company' payroll program is so good that it plans to hire out its services to do payroll processing for other companies the president has an offer from second company with times as many employees she realize... |
20,371 | chap file processing and external sorting implement disk-based buffer pool class based on the lru buffer pool replacement strategy disk blocks are numbered consecutively from the beginning of the file with the first block numbered as assume that blocks are bytes in sizewith the first bytes used to store the block id co... |
20,372 | searching organizing and retrieving information is at the heart of most computer applicationsand searching is surely the most frequently performed of all computing tasks search can be viewed abstractly as process to determine if an element with particular value is member of particular set the more common view of search... |
20,373 | chap searching this and the following treat these three approaches in turn any of these approaches are potentially suitable for implementing the dictionary adt introduced in section howevereach has different performance characteristics that make it the method of choice in particular circumstances the current considers ... |
20,374 | sec searching unsorted and sorted arrays let pi be the probability that is in position of when is not in lsequential search will require comparisons let be the probability that is not in then the average cost (nwill be (nnp ipi = what happens to the equation if we assume all the pi ' are equal (except ) (np ip = = ( ( ... |
20,375 | chap searching element in lthat iswe check elements [ ] [ ]and so on so long as is greater than the values we are checkingwe continue on but when we reach value in greater than kwe do linear search on the piece of length that we know brackets if it is in the list if mj < ( )jthen the total cost of this algorithm is at ... |
20,376 | looking for the search generally does not start at the middle of the dictionary person looking for word starting with 'sgenerally assumes that entries beginning with 'sstart about three quarters of the way through the dictionary thushe or she will first open the dictionary about three quarters of the way through and th... |
20,377 | chap searching this is equal tonpn (need at least probesi= ( ( pn ( pn ( pn ( pn npn we require at least two probes to set the boundsso the cost is (need at least probesi= we now make take advantage of useful fact known as cebysev' inequality cebysev' inequality states that (need probes)or pi is pi < ( ) <( ) ( ) becau... |
20,378 | sec searching unsorted and sorted arrays it is not always practical to reduce an algorithm' growth rate there is practicality window for every problemin that we have practical limit to how big an input we wish to solve for if our problem size never grows too bigit might not matter if we can reduce the cost by an extra ... |
20,379 | chap searching self-organizing lists while lists are most commonly ordered by key valuethis is not the only viable option another approach to organizing lists to speed search is to order the records by expected frequency of access while the benefits might not be as great as when organized by key valuethe cost to organi... |
20,380 | sec self-organizing lists geometric probability distribution can yield quite different results example calculate the expected cost for searching list ordered by frequency when the probabilities are defined as / if < < pi / - if thencn ( / = for this examplethe expected number of accesses is constant this is because the... |
20,381 | chap searching when frequency distribution follows the / rulethe average search looks at about one tenth of the records in table ordered by frequency in most applicationswe have no means of knowing in advance the frequencies of access for the data records to complicate matters furthercertain records might be accessed f... |
20,382 | sec self-organizing lists searches are performed in other wordsif we had known the series of (at least nsearches in advance and had stored the records in order of frequency so as to minimize the total cost for these accessesthis cost would be at least half the cost required by the move-to-front heuristic (this will be ... |
20,383 | chap searching and the total number of comparisons required is finallyif the list is organized by the transpose heuristicthen the final list will be hand the total number of comparisons required is while self-organizing lists do not generally perform as well as search trees or sorted listboth of which require (log nsea... |
20,384 | sec bit vectors for representing sets figure the bit table for the set of primes in the range to the bit at position is set to if and only if is prime the car on left hit this approach to compression is similar in spirit to ziv-lempel codingwhich is class of coding algorithms commonly used in file compression utilities... |
20,385 | chap searching the set difference can be implemented in java using the expression &~ (is the symbol for bit-wise negationfor larger sets that do not fit into single computer wordthe equivalent operations can be performed in turn on the series of words making up the entire bit vector this method of computing sets from b... |
20,386 | are many approaches to hashing and it is easy to devise an inefficient implementation hashing is suitable for both in-memory and disk-based searching and is one of the two most widely used methods for organizing large databases stored on disk (the other is the -treewhich is covered in as simple (though unrealisticexamp... |
20,387 | chap searching probability that some pair of students shares the same birthday ( the same day of the yearnot necessarily the same year)if there are studentsthen the odds are about even that two will share birthday this is despite the fact that there are days in which students can have birthdays (ignoring leap years)on ... |
20,388 | natural distributions are geometric for exampleconsider the populations of the largest cities in the united states if you plot these populations on number linemost of them will be clustered toward the low sidewith few outliers on the high side this is an example of zipf distribution (see section viewed the other waythe... |
20,389 | chap searching example good hash function for numerical values is the mid-square method the mid-square method squares the key valueand then takes the middle bits of the resultgiving value in the range to this works well because most or all bits of the key value contribute to the result for exampleconsider records whose... |
20,390 | for examplebecause the ascii value for "ais and "zis sum will always be in the range to for string of ten upper case letters for hash table of size or lessa good distribution results with all slots in the table accepting either two or three of the values in the key range for hash table of size the distribution is terri... |
20,391 | chap searching and the next four bytes ("bbbb"will be interpreted as the integer value , , , their sum is , , , (when treated as an unsigned integerif the table size is then the modulus function will cause this key to hash to slot in the table note that for any sufficiently long stringthe sum for the integer quantities... |
20,392 | sec hashing figure an illustration of open hashing for seven numbers stored in ten-slot hash table using the hash function (kk mod the numbers are inserted in the order and two of the values hash to slot one value hashes to slot three of the values hash to slot and one value hashes to slot access record will be much hi... |
20,393 | chap searching hash table overflow figure an illustration of bucket hashing for seven numbers stored in fivebucket hash table using the hash function (kk mod each bucket contains two slots the numbers are inserted in the order and two of the values hash to bucket three values hash to bucket one value hashes to bucket a... |
20,394 | the desired key value is not found and the bucket still has free slotsthen the search is complete if the bucket is fullthen it is possible that the desired record is stored in the overflow bucket in this casethe overflow bucket must be searched until the record is found or all records in the overflow bucket have been c... |
20,395 | chap searching /*insert record with key into ht *void hashinsert(key ke rint home/home position for int pos home ( )/initial position for (int = ht[pos!nulli++pos (home (ki) /next pobe slot assert ht[poskey(compareto( ! "duplicates not allowed"ht[posnew kvpair(kr)/insert figure insertion method for dictionary implement... |
20,396 | sec hashing number of records storedand refuse to insert into table that has only one free slot the discussion on bucket hashing presented simple method of collision resolution if the home position for the record is occupiedthen move down the bucket until free slot is found this is an example of technique for collision... |
20,397 | chap searching ( (bfigure example of problems with linear probing (afour values are inserted in the order and using hash function (kk mod (bthe value is added to the hash table in slot with probability / this is illustrated by figure (bthis tendency of linear probing to cluster items together is known as primary cluste... |
20,398 | sec hashing sequence that cycles through only the even slots likewisethe probe sequence for key whose home position is in an odd slot will cycle through the odd slots thusthis combination of table size and linear probing constant effectively divides the records into two sets stored in two disjoint sections of the hash ... |
20,399 | chap searching example consider table of size with perm[ perm[ and perm[ assume that we have two keys and where ( and ( the probe sequence for is then then then the probe sequence for is then then then thuswhile will probe to ' home position as its second choicethe two keysprobe sequences diverge immediately thereafter... |
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