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22,400 | writing this short book has been fun and rewarding experience we would like to thankin no particular order the following people who have helped us during the writing of this book sonu kapoor generously hosted our book which when we released the first draft received over thirteen thousand downloadswithout his generosity... |
22,401 | granville barnett granville is currently ph candidate at queensland university of technology (qutworking on parallelism at the microsoft qut eresearch centre he also holds degree in computer scienceand is microsoft mvp his main interests are in programming languages and compilers granville can be contacted via one of t... |
22,402 | |
22,403 | introduction what this book isand what it isn' this book provides implementations of common and uncommon algorithms in pseudocode which is language independent and provides for easy porting to most imperative programming languages it is not definitive book on the theory of data structures and algorithms for the most pa... |
22,404 | figure algorithmic run time expansion figure shows some of the run times to demonstrate how important it is to choose an efficient algorithm for the sanity of our graph we have omitted cubic ( )and exponential ( run times cubic and exponential algorithms should only ever be used for very small problems (if ever!)avoid ... |
22,405 | and recursive calls--so that you can get the most efficient run times for your algorithms the biggest asset that big oh notation gives us is that it allows us to essentially discard things like hardware if you have two sorting algorithmsone with quadratic run timeand the other with logarithmic run time then the logarit... |
22,406 | the reason that we are explicit in this requirement is simple--all our implementations are based on an imperative thinking style if you are functional programmer you will need to apply various aspects from the functional paradigm to produce efficient solutions with respect to your functional language whether it be hask... |
22,407 | the type of parameters is inferred all primitive language constructs are explicitly begun and ended if an algorithm has return type it will often be presented in the postconditionbut where the return type is sufficiently obvious it may be omitted for the sake of brevity most algorithms in this book require parametersan... |
22,408 | tips for working through the examples as with most books you get out what you put in and so we recommend that in order to get the most out of this book you work through each algorithm with pen and paper to track things like variable namesrecursive calls etc the best way to work through algorithms is to set up tableand ... |
22,409 | testing all the data structures and algorithms have been tested using minimised test driven development style on paper to flesh out the pseudocode algorithm we then transcribe these tests into unit tests satisfying them one by one when all the test cases have been progressively satisfied we consider that algorithm suit... |
22,410 | data structures |
22,411 | linked lists linked lists can be thought of from high level perspective as being series of nodes each node has at least single pointer to the next nodeand in the last node' case null pointer representing that there are no more nodes in the linked list in dsa our implementations of linked lists always maintain head and ... |
22,412 | figure singly linked list node figure singly linked list populated with integers insertion in general when people talk about insertion with respect to linked lists of any form they implicitly refer to the adding of node to the tail of the list when you use an api like that of dsa and you see general purpose method that... |
22,413 | algorithm contains(headvalue prehead is the head node in the list value is the value to search for postthe item is either in the linked listtrueotherwise false head while and value value next end while if return false end if return true end contains deletion deleting node from linked list is straightforward but there a... |
22,414 | algorithm remove(headvalue prehead is the head node in the list value is the value to remove from the list postvalue is removed from the listtrueotherwise false if head /case return false end if head if value value if head tail /case head tail else /case head head next end if return true end if while next and next valu... |
22,415 | algorithm traverse(head prehead is the head node in the list postthe items in the list have been traversed head while yield value next end while end traverse traversing the list in reverse order traversing singly linked list in forward manner ( left to rightis simple as demonstrated in ss howeverwhat if we wanted to tr... |
22,416 | figure reverse traveral of singly linked list figure doubly linked list node |
22,417 | the following algorithms for the doubly linked list are exactly the same as those listed previously for the singly linked list searching (defined in ss traversal (defined in ssinsertion the only major difference between the algorithm in ss is that we need to remember to bind the previous pointer of to the previous tail... |
22,418 | algorithm remove(headvalue prehead is the head node in the list value is the value to remove from the list postvalue is removed from the listtrueotherwise false if head return false end if if value head value if head tail head tail else head head next head previous end if return true end if head next while and value va... |
22,419 | figure doubly linked list reverse traversal algorithm reversetraversal(tail pretail is the tail node of the list to traverse postthe list has been traversed in reverse order tail while yield value previous end while end reversetraversal summary linked lists are good to use when you have an unknown number of items to st... |
22,420 | list |
22,421 | binary search tree binary search trees (bstsare very simple to understand we start with root node with value xwhere the left subtree of contains nodes with values and the right subtree contains nodes whose values are > each node follows the same rules with respect to nodes in their left and right subtrees bsts are of i... |
22,422 | insertion as mentioned previously insertion is an (log noperation provided that the tree is moderately balanced algorithm insert(value prevalue has passed custom type checks for type postvalue has been placed in the correct location in the tree if root root node(value else insertnode(rootvalue end if end insert algorit... |
22,423 | searching searching bst is even simpler than insertion the pseudocode is self-explanatory but we will look briefly at the premise of the algorithm nonetheless we have talked previously about insertionwe go either left or right with the right subtree containing values that are > where is the value of the node we are ins... |
22,424 | deletion removing node from bst is fairly straightforwardwith four cases to consider the value to remove is leaf nodeor the value to remove has right subtreebut no left subtreeor the value to remove has left subtreebut no right subtreeor the value to remove has both left and right subtree in which case we promote the l... |
22,425 | algorithm remove(value prevalue is the value of the node to removeroot is the root node of the bst count is the number of items in the bst postnode with value is removed if found in which case yields trueotherwise false nodet oremove findnode(value if nodet oremove return false /value not in bst end if parent findparen... |
22,426 | finding the parent of given node the purpose of this algorithm is simple to return reference (or pointerto the parent node of the one with the given value we have found that such an algorithm is very usefulespecially when performing extensive tree transformations algorithm findparent(valueroot prevalue is the value of ... |
22,427 | algorithm findnode(rootvalue prevalue is the value of the node we want to find the parent of root is the root node of the bst posta reference to the node of value if foundotherwise if root return end if if root value value return root else if value root value return findnode(root leftvalue else return findnode(root rig... |
22,428 | algorithm findmax(root preroot is the root node of the bst root postthe largest value in the bst is located if root right return root value end if findmax(root right end findmax tree traversals there are various strategies which can be employed to traverse the items in treethe choice of strategy depends on which node v... |
22,429 | ( ( ( ( (efigure preorder visit binary search tree example ( |
22,430 | ( ( ( ( (efigure postorder visit binary search tree example ( |
22,431 | inorder another variation of the algorithms defined in ss and ss is that of inorder traversal where the value of the current node is yielded in between traversing the left subtree and the right subtree an example of inorder traversal is shown in figure ( ( ( ( ( (ffigure inorder visit binary search tree example algorit... |
22,432 | breadth first traversing tree in breadth first order yields the values of all nodes of particular depth in the tree before any deeper ones in other wordsgiven depth we would visit the values of all nodes at in left to right fashionthen we would proceed to and so on until we hade no more nodes to visit an example of bre... |
22,433 | algorithm breadthfirst(root preroot is the root node of the bst postthe nodes in the bst have been visited in breadth first order queue while root yield root value if root left enqueue(root left end if if root right enqueue(root right end if if ! isempty( root dequeue( else root end if end while end breadthfirst summar... |
22,434 | heap heap can be thought of as simple tree data structurehowever heap usually employs one of two strategies min heapor max heap each strategy determines the properties of the tree and its values if you were to choose the min heap strategy then each parent node would have value that is <than its children for examplethe ... |
22,435 | figure array representation of simple tree data structure figure direct children of the nodes in an array representation of tree data structure vector arraylist list figure does not specify how we would handle adding null references to the heap this varies from case to casesometimes null values are prohibited entirelyi... |
22,436 | figure converting tree data structure to its array counterpart |
22,437 | figure calculating node properties the run time efficiency for heap insertion is (log nthe run time is by product of verifying heap order as the first part of the algorithm (the actual insertion into the arrayis ( figure shows the steps of inserting the values and into min-heap |
22,438 | figure inserting values into min-heap |
22,439 | algorithm add(value prevalue is the value to add to the heap count is the number of items in the heap postthe value has been added to the heap heap[countvalue count count + minheapify( end add algorithm minheapify( precount is the number of items in the heap heap is the array used to store the heap items postthe heap h... |
22,440 | algorithm remove(value prevalue is the value to remove from the heap lef tand right are updated aliasfor index and index respectively count is the number of items in the heap heap is the array used to store the heap items postvalue is located in the heap and removedtrueotherwise false /step index findindex(heapvalue if... |
22,441 | figure deleting an item from heap |
22,442 | algorithm contains(value prevalue is the value to search the heap for count is the number of items in the heap heap is the array used to store the heap items postvalue is located in the heapin which case trueotherwise false - while count and heap[ value - + end while if count return true else return false end if end co... |
22,443 | algorithm contains(value prevalue is the value to search the heap for count is the number of items in the heap heap is the array used to store the heap items postvalue is located in the heapin which case trueotherwise false start nodes while start count start nodes end nodes start count while start count and start end ... |
22,444 | figure determining is not in the heap after inspecting the nodes of level figure living and dead space in the heap backing array if you have followed the advice we gave in the deletion algorithm then heap that has been mutated several times will contain some form of default value for items no longer in the heap potenti... |
22,445 | and max heap the former strategy enforces that the value of parent node is less than that of each of its childrenthe latter enforces that the value of the parent is greater than that of each of its children when you come across heap and you are not told what strategy it enforces you should assume that it uses the min-h... |
22,446 | sets set contains number of valuesin no particular order the values within the set are distinct from one another generally set implementations tend to check that value is not in the set before adding itavoiding the issue of repeated values from ever occurring this section does not cover set theory in depthrather it dem... |
22,447 | figure aa bba these algorithms most of the algorithms defined in system linq enumerable deal mainly with sequences rather than sets exclusively set union can be implemented as simple traversal of both sets adding each item of the two sets to new union set algorithm union(set set preset and set union is set posta union ... |
22,448 | algorithm intersection(set set preset and set intersectionand smallerset are sets postan intersection of set and set has been created if set count set count smallerset set else smallerset set end if foreach item in smallerset if set contains(itemand set contains(item intersection add(item end if end foreach return inte... |
22,449 | ordered an ordered set is similar to an unordered set in the sense that its members are distinctbut an ordered set enforces some predefined comparison on each of its members to produce set whose members are ordered appropriately in dsa and earlier we used binary search tree (defined in ss as the internal backing data s... |
22,450 | queues queues are an essential data structure that are found in vast amounts of software from user mode to kernel mode applications that are core to the system fundamentally they honour first in first out (fifostrategythat is the item first put into the queue will be the first servedthe second item added to the queue w... |
22,451 | enqueue( peek( dequeue( standard queue queue is implicitly like that described prior to this section in dsa we don' provide standard queue because queues are so popular and such core data structure that you will find pretty much every mainstream library provides queue data structure that you can use with your language ... |
22,452 | figure queue mutations |
22,453 | deque' provide front and back specific versions of common queue operationse you may want to enqueue an item to the front of the queue rather than the back in which case you would use method with name along the lines of enqueuefront the following list identifies operations that are commonly supported by deque'senqueuefr... |
22,454 | figure deque data structure after several mutations |
22,455 | to look at array minimization techniques as wellit could be that after several invocations of the resizing algorithm and various mutations on the deque later that we have an array taking up considerable amount of memory yet we are only using few small percentage of that memory an algorithm described would also be (nyet... |
22,456 | avl tree in the early ' adelson-velsky and landis invented the first selfbalancing binary search tree data structurecalling it avl tree an avl tree is binary search tree (bstdefined in ss with self-balancing condition stating that the difference between the height of the left and right subtrees cannot be no more than o... |
22,457 | figure unbalanced binary search tree bfigure avl treesinsertion order- ) , , , , - ) , , , , |
22,458 | tree rotations tree rotation is constant time operation on binary search tree that changes the shape of tree while preserving standard bst properties there are left and right rotations both of them decrease the height of bst by moving smaller subtrees down and larger subtrees up right rotation left rotation figure tree... |
22,459 | algorithm leftrotation(node prenode right postnode right is the new root of the subtree node has become node right' left child and bst properties are preserved rightn ode node right node right rightn ode left rightn ode left node end leftrotation algorithm rightrotation(node prenode left postnode left is the new root o... |
22,460 | algorithm checkbalance(current precurrent is the node to start from balancing postcurrent height has been updated while tree balance is if needed restored through rotations if current left and current right current height - else current height max(height(current left),height(current right) end if if height(current left... |
22,461 | algorithm insert(value prevalue has passed custom type checks for type postvalue has been placed in the correct location in the tree if root root node(value else insertnode(rootvalue end if end insert algorithm insertnode(currentvalue precurrent is the node to start from postvalue has been placed in the correct locatio... |
22,462 | algorithm remove(value prevalue is the value of the node to removeroot is the root node of the avl postnode with value is removed and tree rebalanced if found in which case yields trueotherwise false nodet oremove root parent stackpath root while nodet oremove and nodet oremove alue alue parent nodet oremove if value n... |
22,463 | end while /set the parentsright pointer of largestv alue to findparent(largestv alue valueright nodet oremove value largestv alue value end if while path count checkbalance(path pop()/we trackback to the root node check balance end while count count return true end remove summary the avl tree is sophisticated self bala... |
22,464 | algorithms |
22,465 | sorting all the sorting algorithms in this use data structures of specific type to demonstrate sortinge bit integer is often used as its associated operations ( etcare clear in their behaviour the algorithms discussed can easily be translated into generic sorting algorithms within your respective language of choice bub... |
22,466 | figure bubble sort iterations algorithm mergesort(list prelist postlist has been sorted into values of ascending order if list count /already sorted return list end if list count lef list( right list(list count for to lef count- lef [ilist[ end for for to right count- right[ilist[ end for lef mergesort(lef right merges... |
22,467 | divide impera (mergefigure merge sort divide et impera approach quick sort quick sort is one of the most popular sorting algorithms based on divide et impera strategyresulting in an ( log ncomplexity the algorithm starts by picking an itemcalled pivotand moving all smaller items before itwhile all greater elements afte... |
22,468 | pivot pivot pivot pivot pivot pivot pivot pivot pivot figure quick sort example (pivot median strategy algorithm quicksort(list prelist postlist has been sorted into values of ascending order if list count /already sorted return list end if pivot -medianvalue(list for to list count- if list[ipivot equal insert(list[ ] ... |
22,469 | insertion sort insertion sort is somewhat interesting algorithm with an expensive runtime of ( it can be best thought of as sorting scheme similar to that of sorting hand of playing cardsi you take one card and then look at the rest with the intent of building up an ordered set of cards in your hand figure insertion so... |
22,470 | shell sort put simply shell sort can be thought of as more efficient variation of insertion sort as described in ss it achieves this mainly by comparing items of varying distances apart resulting in run time complexity of ( log nshell sort is fairly straight forward but may seem somewhat confusing at first as it differ... |
22,471 | figure shell sort |
22,472 | ones tens hundreds for further clarification what if we wanted to determine how many thousands the number hasclearly there are nonebut often looking at number as final like we often do it is not so obvious so when asked the question how many thousands does have you should simply pad the number with zero in that locatio... |
22,473 | figure radix sort base algorithm bubble sort defined in ss selecting the correct sorting algorithm is usually denoted purely by efficiencye you would always choose merge sort over shell sort and so on there are also other factors to look at though and these are based on the actual implementation some algorithms are ver... |
22,474 | numeric unless stated otherwise the alias denotes standard bit integer primality test simple algorithm that determines whether or not given integer is prime numbere and are all prime numbershowever is not as it can be the result of the product of two numbers that are in an attempt to slow down the inner loop the is use... |
22,475 | algorithm tobinary( pren > postn has been converted into its base representation while list add( / end while return reverse(list end tobinary list { table algorithm trace of tobinary attaining the greatest common denominator of two numbers fairly routine problem in mathematics is that of finding the greatest common den... |
22,476 | computing the maximum value for number of specific base consisting of digits this algorithm computes the maximum value of number for given number of digitse using the base system the maximum number we can have made up of digits is the number similarly the maximum number that consists of digits for base number is which ... |
22,477 | algorithm factorial( pren > is the number to compute the factorial of postthe factorial of is computed if return end if actorial for to actorial actorial end for return actorial end factorial summary in this we have presented several numeric algorithmsmost of which are simply here because they were fun to design perhap... |
22,478 | searching sequential search simple algorithm that search for specific item inside list it operates looping on each element (nuntil match occurs or the end is reached algorithm sequentialsearch(listitem prelist postreturn index of item if foundotherwise - index while index list count and list[index item index index end ... |
22,479 | figure asearch( )bsearch( algorithm probabilitysearch(listitem prelist posta boolean indicating where the item is found or notin the former case swap founded item with its predecessor index while index list count and list[index item index index end while if index >list count or list[index item return false end if if in... |
22,480 | the reader has already seen such an algorithm in ss searching algorithms and their efficiency largely depends on the underlying data structure being used to store the data for instance it is quicker to determine whether an item is in hash table than it is an arraysimilarly it is quicker to search bst than it is linked ... |
22,481 | strings strings have their own in this text purely because string operations and transformations are incredibly frequent within programs the algorithms presented are based on problems the authors have come across previouslyor were formulated to satisfy curiosity reversing the order of words in sentence defining algorit... |
22,482 | algorithm reversewords(value prevalue sb is string buffer postthe words in value have been reversed last value length start last while last > /skip whitespace while start > and value[startwhitespace start start end while last start /march down to the index before the beginning of the word while start > and start whites... |
22,483 | figure lef and right pointers marching in towards one another algorithm ispalindrome(value prevalue postvalue is determined to be palindrome or not word value strip(touppercase( lef right word length - while word[lef tword[rightand lef right lef lef right right end while return word[lef tword[right end ispalindrome in ... |
22,484 | figure string with three words figure string with varying number of white space delimiting the words of the previously listed index keeps track of the current index we are at in the stringwordcount is an integer that keeps track of the number of words we have encounteredand finally inw ord is boolean flag that denotes ... |
22,485 | algorithm wordcount(value prevalue postthe number of words contained within value is determined inw ord true wordcount index /skip initial white space while value[indexwhitespace and index value length - index index end while /was the string just whitespace if index value length and value[indexwhitespace return end if ... |
22,486 | figure aundesired uniques setbdesired uniques set algorithm repeatedwordcount(value prevalue postthe number of repeated words in value is returned words value split(' uniques set foreach word in words uniques add(word strip() end foreach return words length -uniques count end repeatedwordcount you will notice in the re... |
22,487 | word match index index index bfigure afirst stepbsecond step cmatch occurred algorithm any(word,match prewordmatch postindex representing match location if occured- otherwise for to word length while word[iwhitespace - + end while for index to match length while match[indexwhitespace index index end while if match[inde... |
22,488 | algorithm walkthrough learning how to design good algorithms can be assisted greatly by using structured approach to tracing its behaviour in most cases tracing an algorithm only requires single table in most cases tracing is not enoughyou will also want to use diagram of the data structure your algorithm operates on t... |
22,489 | figure visualising the data structure we are operating on value word lef right table column for each variable we wish to track the ispalindrome algorithm uses the following list of variables in some form throughout its execution value word lef right having identified the values of the variables we need to keep track of... |
22,490 | we cannot stress enough how important such traces are when designing your algorithm you can use these trace tables to verify algorithm correctness at the cost of simple tableand quick sketch of the data structure you are operating on you can devise correct algorithms quicker visualising the problem domain and keeping t... |
22,491 | figure call chain for fibonacci algorithm figure return chain for fibonacci algorithm |
22,492 | the caller and continue execution of that method execution in the caller is contiued at the next statementor expression after the recursive call was made in the fibonacci algorithmsrecursive case we make two recursive calls when the first recursive call (fibonacci( )returns to the caller we then execute the the second ... |
22,493 | translation walkthrough the conversion from pseudo to an actual imperative language is usually very straight forwardto clarify an example is provided in this example we will convert the algorithm in ss to the clanguage public static bool isprime(int number if (number return false int innerloopbound (int)math floor(math... |
22,494 | summary as you can see from the example used in this we have tried to make the translation of our pseudo code algorithms to mainstream imperative languages as simple as possible whenever you encounter keyword within our pseudo code examples that you are unfamiliar with just browse to appendix which descirbes each keywo... |
22,495 | recursive vs iterative solutions one of the most succinct properties of modern programming languages like ++ #and java (as well as many othersis that these languages allow you to define methods that reference themselvessuch methods are said to be recursive one of the biggest advantages recursive methods bring to the ta... |
22,496 | ( if you use recursion for algorithms with any of the above run time efficiency' you are inviting trouble the growth rate of these algorithms is high and in most cases such algorithms will lean very heavily on techniques like divide and conquer while constantly splitting problems into smaller problems is good practicei... |
22,497 | while activation records are an efficient way to support method calls they can build up very quickly recursive algorithms can exhaust the stack size allocated to the thread fairly fast given the chance just about now we should be dusting the cobwebs off the age old example of an iterative vs recursive solution in the f... |
22,498 | do though is somewhat limited by the fact that you are still using recursion youas the developer have to accept certain accountability' for performance |
22,499 | no pointers overloaded operators primitive variable types input/output java library data structures summary questions arrays the array workshop applet insertion searching deletion the duplicates issue not too swift the basics of arrays in java creating an array accessing array elements initialization an array example d... |
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