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25,400 | ted fred adam mary eva fig an example of recursive data structure the important role of the variant facility becomes clearit is the only means by which recursive data structure can be boundedand it is therefore an inevitable companion of every recursive definition the analogy between program and data structuring concep... |
25,401 | ted fred adam mary eva fig data structure linked by pointers it must be emphasized that the use of pointers to implement recursive structures is merely technique the programmer need not be aware of their existence storage may be allocated automatically the first time new component is referenced howeverif the technique ... |
25,402 | pointer has any fieldsbut only the referenced record ^we allow the abbreviated notation in place of px it was mentioned above that variant component is essential in every recursive type to ensure finite instances the example of the family predigree is of pattern that exhibits most frequently occurring constellationname... |
25,403 | adam fred ted nil nil nil mary nil eva nil nil fig data structure with nil pointers further consequence of the explicitness of pointers is that it is possible to define and manipulate cyclic data structures this additional flexibility yieldsof coursenot only increased power but also requires increased care by the progr... |
25,404 | if bboolean then tt end end where is type not involving is equivalent and replaceable by sequence of the remainder of this is devoted to the generation and manipulation of data structures whose components are linked by explicit pointers structures with specific simple patterns are emphasized in particularrecipes for ha... |
25,405 | generatedstarting with the empty lista heading element is added repeatedly the process of list generation is expressed in by the following piece of programhere the number of elements to be linked is :nil(*start with empty list*while do new( ) next :pp :qq key :ndec(nend this is the simplest way of forming list howevert... |
25,406 | fig insertion before pthe trick evidently consists of actually inserting new component after pand thereafter interchanging the values of the new element and pnextwe consider the process of list deletion deleting the successor of pis straightforward this is shown here in combination with the reinsertion of the deleted e... |
25,407 | end it follows from the definitions of the while statement and of the linking structure that is applied to all elements of the list and to no other ones very frequent operation performed is list searching for an element with given key unlike for arraysthe search must here be purely sequential the search terminates eith... |
25,408 | end end searchprocedure printlist (wword)begin (*uses global writer *while nil do texts writeint(ww key )texts writeint(ww count )texts writeln( ) : next end end printlistthe linear scan algorithm resembles the search procedure for arraysand reminds us of simple technique used to simplify the loop termination condition... |
25,409 | nil fig insertion in ordered list referring to fig we determine the condition under which the scan continues to proceed to the next elementit consists of two factorsnamely( nil( key xthe resulting search procedure isprocedure search (xintegervar rootword)(adens _list *var wordbegin (* nil* :rootw : nextwhile ( nil( key... |
25,410 | ( )variesandmoreoverwhen no information is given about their frequencies of access typical example is the symbol table in compilers of programming languages each declaration causes the addition of new symboland upon exit from its scope of validityit is deleted from the list the use of simple linked lists is appropriate... |
25,411 | procedure search (xintegervar rootword)(adens _list *var wordbegin :rootsentinel key :xif sentinel then (*first element*new(root)root key :xroot count : root next :sentinel elsif key then inc( countelse (*search*repeat : : next until key xif sentinel then (*new entry* :rootnew(root)root key :xroot count : root next : e... |
25,412 | an applicationpartial ordering (topological sortingan appropriate example of the use of flexibledynamic data structure is the process of topological sorting this is sorting process of items over which partial ordering is definedi where an ordering is given over some pairs of items but not between all of them the follow... |
25,413 | fig linear arrangement of the partially ordered set of fig how do we proceed to find one of the possible linear orderingsthe recipe is quite simple we start by choosing any item that is not preceded by another item (there must be at least oneotherwise loop would existthis object is placed at the head of the resulting l... |
25,414 | (*input phase*new(head)tail :headtexts scan( )while class texts int do : itexts scan( ) : ip :find( ) :find( )new( ) id :qt next : trailp trail :tinc( count)texts scan(send tail head key count next trail id next id next fig list structure generated by topsort program after the data structure of fig has been constructed... |
25,415 | fig list of leaders with zero count after all this preparatory establishing of convenient representation of the partially ordered set swe can finally proceed to the actual task of topological sortingi of generating the output sequence in first rough version it can be described as followsq :headwhile nil do (*output thi... |
25,416 | (*input phase*texts scan( )while class texts int do : itexts scan( ) : ip :find( ) :find( )new( ) id :qt next : trailp trail :tinc( count)texts scan(send(*search for leaders without predecessors* :headhead :nilwhile tail do :pp : nextif count then (*insert in new chain* next :headhead : end end(*output phase* :headwhil... |
25,417 | node of type with finite number of associated disjoint tree structures of base type called subtrees from the similarity of the recursive definitions of sequences and tree structures it is evident that the sequence (listis tree structure in which each node has at most one subtree the list is therefore also called degene... |
25,418 | descendant of xif is at level ithen is said to be at level + converselynode is said to be the (directancestor of the root of tree is defined to be at level the maximum level of any element of tree is said to be its depth or height fig two distinct trees if an element has no descendantsit is called terminal node or leaf... |
25,419 | fig ternary tree extended with special nodes of particular importance are the ordered trees of degree they are called binary trees we define an ordered binary tree as finite set of elements (nodeswhich either is empty or consists of root (nodewith two disjoint binary trees called the left and the right subtree of the r... |
25,420 | root nil nil nil nil nil nil nil nil nil nil nil nil fig tree of fig represented as linked data structure before investigating how trees might be used advantageously and how to perform operations on treeswe give an example of how tree may be constructed by program assume that tree is to be generated containing nodes wi... |
25,421 | keyintegerleftrightnode endvar rtexts readerwtexts writerrootnodeprocedure tree (ninteger)nodevar newnodexnlnrintegerbegin (*construct perfectly balanced tree with nodes*if then new :nil else nl : div nr : -nl- new(new)texts readint(rnew key)new key :xnew left :tree(nl)new right :tree(nrendreturn new end treeprocedure ... |
25,422 | prints the resulting treethe empty tree results in no printingthe subtree at level in first printing its own left subtreethen the nodeproperly indented by preceding it with tabsand finally in printing its right subtree basic operations on binary trees there are many tasks that may have to be perfomed on tree structurea... |
25,423 | procedure preorder (tnode)begin if nil then ( )preorder( left)preorder( rightend end preorder procedure inorder (tnode)begin if nil then inorder( left) ( )inorder( rightend end inorder procedure postorder (tnode)begin if nil then postorder( left)postorder( right) (tend end postorder note that the pointer is passed as v... |
25,424 | (fig the sentinel may be considered as commonshared representative of all external nodes by which the original tree was extended (see fig )procedure locate (xintegertnode)nodebegin key : (*sentinel*while key do if key then : right else : left end endreturn end locate note that in this case locate(xtyields the value ins... |
25,425 | the empty subtree considerfor examplethe binary tree shown in fig and the insertion of the name paul the result is shown in dotted lines in the same picture norma george ann nil er mary nil nil paul nil nil walter nil nil nil fig insertion in ordered binary tree the search process is formulated as recursive procedure n... |
25,426 | procedure search (xintegervar pnode)begin if nil then (* not in treeinsert*new( ) key :xp count : left :nilp right :nil elsif key then search(xp leftelsif key then search(xp rightelse inc( countend end search the use of sentinel again simplifies the task somewhat clearlyat the start of the program the variable root mus... |
25,427 | word is considered as any sequence of letters and digits starting with letter since words may be of widely different lengthsthe actual characters are stored in an array bufferand the tree nodes contain the index of the key' first character it is desirable that the line numbers be printed in ascending order in the cross... |
25,428 | iintegerchcharwwordbegin root :nilline : texts writeint( )texts write(wtab)texts read(rch)while ~ eot do if ch dx then (*line end*texts writeln( )inc(line)texts writeint(wline )texts write( )texts read(rchelsif (" <ch(ch <" "or (" <ch(ch <" "then : repeat if wordlen- then [ :chinc(iendtexts write(wch)texts read(rchunti... |
25,429 | tree deletion we now turn to the inverse problem of insertiondeletion our task is to define an algorithm for deletingi removing the node with key in tree with ordered keys unfortunatelyremoval of an element is not generally as simple as insertion it is straightforward if the element to be deleted is terminal node or on... |
25,430 | ab fig tree deletion analysis of tree search and insertion it is natural reaction to be suspicious of the algorithm of tree search and insertion at least one should retain some skepticism until having been given few more details about its behaviour what worries many programmers at first is the peculiar fact that genera... |
25,431 | - - fig weight distribution of branches in the tree in fig we divide the nodes into three classes the - nodes in the left subtree have an average path length ai- the root has path length of the - nodes in the right subtree have an average path length an- hencethe equation above can be expressed as sum of two terms and ... |
25,432 | equal probability expect an average improvement in the search path length of at most emphasis is to be put on the word averagefor the improvement may of course be very much greater in the unhappy case in which the generated tree had completely degenerated into listwhichhoweveris very unlikely to occur in this connectio... |
25,433 | strongly resembles that of fibonacci numbersthey are called fibonacci-trees (see fig they are defined as follows the empty tree is the fibonacci-tree of height single node is the fibonacci-tree of height if - and - are fibonacci-trees of heights - and - then is fibonacci-tree no other trees are fibonacci-trees fig fibo... |
25,434 | needing individual treatment the remaining ones can be derived by symmetry considerations from those two case is characterized by inserting keys or in the tree of fig case by inserting nodes or the two cases are generalized in fig in which rectangular boxes denote subtreesand the height added by the insertion is indica... |
25,435 | checked on that node' ancestors)we shall first adhere to this evidently correct schema because it can be implemented through pure extension of the already established search and insertion procedures this procedure describes the search operation needed at each single nodeand because of its recursive formulation it can e... |
25,436 | procedure search (xintegervar pnodevar hboolean)var node(adens _avltrees *begin (*~ *if nil then (*insert*new( ) key :xp count : left :nilp right :nilp bal : :trueelsif key then search(xp lefth)if then (*left branch has grown*if bal then bal : :false elsif bal then bal :- else (*bal - rebalance* : leftif bal - then (*s... |
25,437 | if all npermutations of keys occur with equal probabilitywhat is the expected height of the constructed balanced tree what is the probability that an insertion requires rebalancingmathematical analysis of this complicated algorithm is still an open problem empirical tests support the conjecture that the expected height... |
25,438 | gh fig deletions in balanced tree procedure balancel (var pnodevar hboolean)(adens _avltrees *var nodebegin (*hleft branch has shrunk*if bal - then bal : elsif bal then bal : :false else (*bal rebalance* : rightif bal > then (*sinlge rr rotation* right : leftp left :pif bal then bal : bal :- :false else bal : bal : end... |
25,439 | right : leftp left :pif bal + then bal :- else bal : endif bal - then bal : else bal : endp : bal : end end end balancelprocedure balancer (var pnodevar hboolean)var nodebegin (*hright branch has shrunk*if bal then bal : elsif bal then bal :- :false else (*bal - rebalance* : leftif bal < then (*single ll rotation* left... |
25,440 | else (*delete ^* :pif right nil then : lefth :true elsif left nil then : righth :true else del( lefth)if then balancel(phend end end end delete fortunatelydeletion of an element in balanced tree can also be performed with -in the worst case - (log noperations an essential difference between the behaviour of the inserti... |
25,441 | ab fig the search trees with nodes the weighted path lengths of trees (ato (eare computed according to their definition as ( / ( / ( / ( / ( / hencein this examplenot the perfectly balanced tree ( )but the degenerate tree (aturns out to be optimal the example of the compiler scanner immediately suggests that this probl... |
25,442 | instead of using the probabilities pi and qjwe will subsequently use such frequency counts and denote them by ai number of times the search argument equals ki bj number of times the search argument lies between kj and kj+ by conventionb is the number of times that is less than and bn is the frequency of being greater t... |
25,443 | we are now ready to construct the optimization algorithm in detail we recall the following definitionswhich are based on optimal trees ij consisting of nodes with keys ki+ kj aithe frequency of search for ki bjthe frequency of search argument between kj and kj+ wijthe weight of ij pijthe weighted path length of ij rijt... |
25,444 | the details of the refinement of the statement in italics can be found in the program presented below the average path length of , is now given by the quotient , / ,nand its root is the node with index , let us now describe the structure of the program to be designed its two main components are the procedures to find t... |
25,445 | procedure writetree (ijlevelinteger)var kinteger(*uses global writer *begin if then writetree(ir[ij]- level+ )for : to level do texts write(wtabendtexts writestring(wkey[ [ij]])texts writeln( )writetree( [ij]jlevel+ end end writetreeprocedure find (var stexts scanner)var ijninteger(*uses global writer *begin texts scan... |
25,446 | key ernst key peter the results of procedure find are shown in fig and demonstrate that the structures obtained for the three cases may differ significantly the total weight is the path length of the balanced tree is and that of the optimal tree is balanced tree optimal tree albert not considering key misses albert alb... |
25,447 | to the youngest (or eldestoffspring assigned to the parent possible type definition for this case is the followingand possible data structure is shown in fig type person =pointer to record namealfasiblingoffspringperson end john albert peter mary paul robert carol chris george pamela tina fig multiway tree represented ... |
25,448 | fig binary tree subdivided into pages the saving in the number of disk accesses -each page access now involves disk access -can be considerable assume that we choose to place nodes on page (this is reasonable figure)then the million item search tree will on the average require only log ( ( about page accesses instead o... |
25,449 | increasing order from left to right if the -tree is squeezed into single level by inserting the descendants in between the keys of their ancestor page this arrangement represents natural extension of binary search treesand it determines the method of searching an item with given key consider page of the form shown in f... |
25,450 | back along the search path the general structure of the program will therefore be similar to balanced tree insertionalthough the details are different first of alla definition of the page structure has to be formulated we choose to represent the items in the form of an array type page pointer to pagedescriptoritem reco... |
25,451 | as consequencethe new root page contains single item only the details can be gathered from the program presented belowand fig shows the result of using the program to construct -tree with the following insertion sequence of keys the semicolons designate the positions of the snapshots taken upon each page allocation ins... |
25,452 | number of items on the reduced pagebecauseif nthe primary characteristic of -trees would be violated some additional action has to be takenthis underflow condition is indicated by the boolean variable parameter the only recourse is to borrow or annect an item from one of the neighboring pagessay from since this involve... |
25,453 | keyintegerppage endpagerec record minteger(*no of entries on page* pageearray * of entry endvar rootpagewtexts writerprocedure search (xintegervar ppagevar kinteger)var ilrintegerfoundbooleanapagebegin :rootfound :falsewhile ( nil~found do : : (*binary search*while do :( +rdiv if < [ikey then : else : + end endif ( ( [... |
25,454 | for : to + by - do [ : [ - enda [ :uinc( melse (*overflowsplit into ab and assign the middle entry to *new( )if then (*insert in left pate * : [ - ]for : - to + by - do [ : [ - enda [ :ufor : to - do [ : [ +nend else (*insert in right page *dec(rn)if then : else : [ ]for : to - do [ : [ + + endb [ - : endfor : to - do ... |
25,455 | :( - + div (* nof items available on page *if then for : - to by - do [ + : [ienda [ - : [ ] [ - : (*move - items from to aone to *dec( mk)for : - to by - do [ : [ + + endc [ : [ ] : [spc [sp :aa : - +kh :false else (*merge pages and bdiscard * [sp : [ : [ ]for : to - do [ + + : [iendb : *ndec( ) : end end end underflo... |
25,456 | end end deleteprocedure showtree (var wtexts writerppagelevelinteger)var iintegerbegin if nil then for : to level do texts write( xendfor : to - do texts writeint(wp [ikey endtexts writeln( )if then showtree(wp level+ endfor : to - do showtree(wp [iplevel+ end end end showtreeextensive analysis of -tree performance has... |
25,457 | end fig representation of bb-tree nodes considering the problem of key insertionone must distinguish four possible situations that arise from growth of the left or right subtrees the four cases are illustrated in fig |
25,458 | fig node insertion in bb-tree remember that -trees have the characteristic of growing from the bottom toward the root and that the property of all leafs being at the same level must be maintained the simplest case ( is when the right subtree of node grows and when is the only key on its (hypotheticalpage thenthe descen... |
25,459 | (llb (lra (rra (rla fig insertion in sbb-trees it is advisable to stick no longer to the notion of pages out of which this organization had developedfor we are only interested in bounding the maximum path length to *log(nfor this we need only ensure that two horizontal pointers may never occur in succession on any sear... |
25,460 | ( ( ( ( ( ( ( ( fig insertion of keys to these pictures make the third property of -trees particularly obviousall terminal nodes appear on the same level one is therefore inclined to compare these structures with garden hedges that have been recently trimmed with hedge scissors the algorithm for the construction of sbb... |
25,461 | the subtree requires no changes of the tree structure node has obtained sibling the subtree has increased in height procedure search (var pnodexintegervar hinteger)var qrnode(adens _bbtrees *begin (* = *if nil then (*insert new node*new( ) key :xp :nilp :nilp lh :falsep rh :falseh : elsif key then search( lxh)if then (... |
25,462 | emerges as an alternative to the avl-balancing criterion performance comparison is therefore both possible and desirable we refrain from involved mathematical analysis and concentrate on some basic differences it can be proven that the avl-balanced trees are subset of the sbb-trees hencethe class of the latter is large... |
25,463 | called cartesian tree [ - we prefer the term priority search treebecause it exhibits that this structure emerged from combination of the priority tree and the search tree it is characterized by the following invariants holding for each node pp left nil implies ( left ( < left yp right nil implies ( right ( < right yit ... |
25,464 | left nil implies ( < left ( < left yp right nil implies ( < right ( < right where ( rdiv left left right right for all node pand root xminroot xmax decisive advantage of the radix scheme is that maintenance operations (preserving the invariants under insertion and deletionare confined to single spine of the treebecause... |
25,465 | found note that one side of the rectangle lies on the -axisi the lower bound for is this guarantees that enumeration requires at most (log(nsoperationswhere is the cardinality of the search space in and is the number of nodes enumerated procedure enumerate (pptrx xlxrinteger)var xminteger(adens _prioritysearchtrees *be... |
25,466 | fig circular list bidirectional list is list of elements that are linked in both ways (see fig both links are originating from header analogous to the preceding exerciseconstruct module with procedures for searchinginsertingand deleting elements fig bidirectional list does the given program for topological sorting work... |
25,467 | occupies words and that each pointer occupies one word of storage what is the gain in storage when using binary tree versus an -ary tree assume that tree is built upon the following definition of recursive data structure (see exercise formulate procedure to find an element with given key and to perform an operation on ... |
25,468 | the shown procedure to perform all four rebalancing acts (lllrrrrlat least once what is the shortest such sequence write procedure for the deletion of elements in symmetric binary -tree then find tree and short sequence of deletions causing all four rebalancing situations to occur at least once formulate data structure... |
25,469 | key transformations (hashing introduction the principal question discussed in chap at length is the followinggiven set of items characterized by key (upon which an ordering relation is defined)how is the set to be organized so that retrieval of an item with given key involves as little effort as possibleclearlyin compu... |
25,470 | (kord(kmod it has the property that the key values are spread evenly over the index rangeand it is therefore the basis of most key transformations it is also very efficiently computableif is power of but it is exactly this case that must be avoidedif the keys are sequences of letters the assumption that all keys are eq... |
25,471 | this tends to be too costlyand methods that offer compromise by being simple to compute and still superior to the linear function are preferred one of them consists of using quadratic function such that the sequence of indices for probing is (khi ( mod ni > note that computation of the next index need not involve the o... |
25,472 | var lineintegerprocedure search (ttablevar aword)var idintegerhlongintfoundbooleanbegin (*compute hash index for auses global variable line* : : while [ do :( * ord( [ ])mod ninc(iendd : found :falserepeat if [hkey then (*match*found :truet[hlno[ [hn:lineif [hn noc then inc( [hnend elsif [hkey[ then (*new entry*found :... |
25,473 | [ : (*string terminator*search(hwelse texts write(wch)texts read(rchendtexts writeln( )texts writeln( )tabulate(hend end crossref analysis of key transformation insertion and retrieval by key transformation has evidently miserable worst-case performance after allit is entirely possible that search argument may be such ... |
25,474 | an empty tablea /( + full table the expected number of probes to retrieve or insert randomly chosen key is listed in table as function of the load factor table expected number of probes as function of the load factor the numerical results are indeed surprisingand they explain the exceptionally good performance of the k... |
25,475 | exercises if the amount of information associated with each key is relatively large (compared to the key itself)this information should not be stored in the hash table explain why and propose scheme for representing such set of data consider the proposal to solve the clustering problem by constructing overflow trees in... |
25,476 | appendix the ascii character set nul dle soh dc stc etx dc dc eot dc enq nak ack syn bel etb bs can ht em lf sub vt esc ff fs cr gs so rs si us del |
25,477 | the syntax of the append(method issyntaxlistname append(elementl=[ , , , append( print("updated list=", #output:[ , , , , =[' ',' ',' ' append(' 'print("updated list=", #output:[' ',' ',' ',' ' insert():the insert(method insert one element at desired location syntaxlistname insert(index,elementl=[ , , insert( , print("... |
25,478 | remove():this method removes the first instance of element in list syntaxlist remove(elementexl=[ , , , , , remove( print(lo/ :[ , , , , reverse():this method reverses the list elements syntaxlist reverse(exl=[ , , , , , reverse(print(lo/ :[ , , , , , extend()the extend(extends the list by adding all items of list (pas... |
25,479 | iterate through rows for in range(len( ))iterate through columns for in range(len( [ ]))print( [ ][ ],end='\ 'for in resultprint )#write program to add two matrices [[ , , ][ , , ],[ , , ] [[ , , ],[ , , ],[ , , ]result [[ , , ],[ , , ][ , , ]iterate through rows for in range(len( ))iterate through columns for in range... |
25,480 | ex =( , ,'abc'print( basic tuple operations concatenation:we can use operator to combine the two tuples this is also called concatenation syntaxtuplename +tuplename ext =( , , =( , , print( + / :( , , , , , repitition:we can also repeat elements in tuple for given number of times using operator syntaxtuple*int-value =(... |
25,481 | print( [ : ]print( [ :- ]print( [: ]print( [ :]print( [:- ]print( [ :: ]print( [::- ]methodstuple does not support remove(),pop(),append(),sort(),reverse(),insert(,extend( index():this method returns the index of the element syntaxextuplename index(elementt=( , , , , , , , , print( index( ) / count()this method returns... |
25,482 | no list tuple implication of iterations is comparatively faster tuple data type is appropriate for accessing the elements tuple consume less memory as compared to the list tuple does no have must built-in methods implication of iterations is time-consuming the list is better for performing operationssuch as insertion a... |
25,483 | print( & )difference(-):difference of contain only elements that are in ,but not in ={ , , , , ={ , , , , print( - )set symmetric difference(^):symmetric difference of and is set of elements in both and except that are common in both ={ , , , , ={ , , , , print( ^ #output )membership operators:it return boolean values ... |
25,484 | syntaxset intersection(set exa={ , , , , ={ , , , , print( intersection( ) )union():it contains all the elements of set and set syntaxset union(set exa={ , , , , ={ , , , , print( union( ) )symmetric_difference():symmetric difference of and is set of elements in both and except that are common in both syntaxset symmetr... |
25,485 | print( isdisjoint( ) )issubset()this method returns true if all elements of one set is present in second set syntaxset issubset(set ={ , , ={ , , , print( issubset( )dictionariespython dictionary is an unordered collection of elements and it is mutable we can store values as pair of keys and values represented by dict(... |
25,486 | print( [ ]output: )membership operatorthis operator returns boolean valuesto check whether the keys are present in dictionary or not exs={ : , : , : , : print( in sprint( not in sdictionary methods )keys():it returns keys from the dictionary syntaxdictname keys(exd ={' ': ,' ': ,' ': print( keys() )values():it returns ... |
25,487 | exd={ :'one', :'two' clear(print( )pop():this method is used to remove the keys from dictionary syntaxdictname pop(keyexf={ :'app', :'man', :'stra' pop( print( #output{ :'man', :'stra'iterators and generators note:for clear explanation refer to iterators are containers for objects so that you can loop over the objects ... |
25,488 | list comprehension is the most popular python comprehension it allows us to create new list of elements that satisfy condition from an iterable an iterable is any python construct that can be looped over like listsstringstuplessets in list comprehensions we use square brackets general syntax of list comprehension is as... |
25,489 | introduction to functions function is block of organized and reusable program code that performs specificsingleand well-defined task function provides an interface for communication in terms of how information is transferred to it and how results are generated need for functions simplifies program development by making... |
25,490 | the function code should indented properly function definition contains two partsfunction header function body the syntax of function definition is as followsdef function_name(arg arg /no arg)"""documentation string""statement block function header ,formal parameters body of function example for defining function witho... |
25,491 | the function name and number of parameters must be same in the function call and function definition when the number parameters passed doesn' match with the parameters in the function definitionerror is generated names of arguments in the function call and function definition can be different arguments can be passed as... |
25,492 | return (expressionexample of function returning value is as followsdef cube( )return * * =int(input("enter the value of ") =cube(aprint("cube=",bdef sq( )return * =int(input("enter the value of ") =sq(xprint("square=",bpassing arguments if function definition contains argumentsthen in function call there is need to pas... |
25,493 | = add( ,bdefault arguments the formal parameters in function definition can be assigned default value such parameters to which default values are assigned are called default arguments default value can be assigned to parameter by using the assignment operator function definition can have one or more default arguments d... |
25,494 | following example demonstrates keyword arguments and their usedef ( , , )print( , , #output: ( , , def ( , , )print( , ,cf( = , = , = #output: def ( , , )print( , ,cf( = , = , = #output: def ( , , )print( , ,cf( , = , = #output: def ( , , )print( , ,cf( , , = #output: def ( , , )print( , ,cf( , = #output:error def ( , ... |
25,495 | following example demonstrates variable-length argumentdef fun(name*friendslist)print("friends of",name,"are"for in friendslistprint(xend 'fun("shaa""ayaan""avi","velan"output of the above example is as followsfriends of shaa areayaan avi velan scope of variables variables in program has two thingsscopeparts of the pro... |
25,496 | output of above code is as followslocal global global statement local variable inside function can be made global by using the global keyword if local variable which is made global using the global statement have the same name as another global variablethen changes on the variable will be reflected everywhere in the pr... |
25,497 | the arguments-list contains comma separated list of arguments expression is an arithmetic expression that uses the arguments in the arguments-list lambda function can be assigned to variable to give it name following is an example for creating lambda function and using itpower lambda * * print(power( )#output: lambda p... |
25,498 | recursion uses divide and conquer strategy to solve problems if > fact( )= *fact( - if = fact( )= !=fact( )= *fact( * *fact( * * *fact( * * * *fact( * * * * *fact( * * * * * = advantages recursive functions make the code look clean and easy complex task can be broken into sub problem using recursion sequence generation... |
25,499 | =int(input("enter the value of ") =int(input("enter the value of ")print("gcd of two numbers =",gcd( , )modulescreating modulesimport statementfrom import statementname spacing modules function allows to reuse piece of code module on the other hand contains multiple functionsvariablesand other elements which can be reu... |
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