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24,100 | notes nextdelete although is in leafthis leaf does not have an extra keythe deletion results in node with only one keywhich is not acceptable for -tree of order if the sibling node to the immediate left or right has an extra keywe can then borrow key from the parent and move key up from this sibling in our specific cas... |
24,101 | notes of courseyou immediately see that the parent node now contains only one keyg this is not acceptable if this problem node had sibling to its immediate left or right that had spare keythen we would again "borrowa key suppose for the moment that the right sibling (the node with xhad one more key in it somewhere to t... |
24,102 | notes examplehere is portion of -tree with order (nodes have at least keys and pointersnodes are delimited with [square bracketsthe keys are city namesand are kept sorted in each node on either side of every key are pointers linking the key to subsequent nodesstart here chicago hoboken ++ aptos boston denver detroit sa... |
24,103 | to insert the key " "we first simply search for that key if is foundthe key is already in the tree and the insertion is superfluous otherwisewe must end up at leaf node at the bottom level of the tree where would be stored in the above casethe leaf node contains and room for fourth keyso is simply inserted in the leaf ... |
24,104 | notes if node underflowswe may be able to "redistributekeys by borrowing some from neighboring node for examplein the order -tree belowthe key is being deletedwhich causes node to underflow since it only has keys and left so keys from the neighbor on the left are "shifted throughthe parent node and redistributed so bot... |
24,105 | notes task consider the following binary search treea splay this tree at each of the following keys in turndbgfadbd applications databasesa database is collection of data organized in fashion that facilitates updatingretrievingand managing the data the data can consist of anythingincludingbut not limited to namesaddres... |
24,106 | notes concurrent access to -treesdatabases typically run in multiuser environments where many users can concurrently perform operations on the database unfortunatelythis common scenario introduces complications exampleimagine database storing bank account balances now assume that someone attempts to withdraw ` from an ... |
24,107 | -treesb-trees are balanced trees that are optimized for situations when part or the entire tree must be maintained in secondary storage such as magnetic disk notes self assessment choose the appropriate answers -tree has minimum number of allowable children for each node known as the (aminimization factor (bmaximizatio... |
24,108 | notes you have -tree of order in figure given below explain to delete from it there are possible ordered sequences of the four keys but only distinct binary trees with four nodes thereforethese binary trees are not equally likely to occur as search trees find which one of the binary search trees corresponds to each of ... |
24,109 | shi-kuo changdata structures and algorithmsworld scientific notes shi-kuo changdata structures and algorithmsworld scientific sorenson and tremblayan introduction to data structure with algorithms thomas cormencharles eleiserson ronald rivestintroduction to algorithmsprentice-hall of india pvt limitednew delhi timothy ... |
24,110 | anil sharmalovely professional university unit hashing notes contents objectives introduction hashing linear probing or linear open addressing rehashing overflow chaining hash functions open hashing closed hashing rehashing summary keywords self assessment review questions further readings objectives after studying thi... |
24,111 | hashing notes in many applications we require to use data object called symbol table symbol table is nothing but set of pairs (namevaluewhere value represents collection of attributes associated with the nameand this collection of attributes depends upon the program element identified by the name for exampleif name is ... |
24,112 | notes and place in the ith location of the table there are various methods available to obtain an alternative location to handle the collision they differ from each other in the way search is made for an alternative location the following are the commonly used collision handling techniqueslinear probing or linear open ... |
24,113 | figure hash table implementation using overflow chaining for collision handling notes consider using division method of hashing store the following values in the hash table of size use sequential method for resolving the collisions since division method of hashing is to be used the hash function hish(keykey mode where ... |
24,114 | notes the hash table therefore is the one shown below task " hash function must be deterministic and stateless discuss hash functions some of the methods of defining hash function are discussed below modular arithmeticin this methodfirst the key is converted to integerthen it is divided by the size of index rangeand th... |
24,115 | of the square are used to get the hash value since the middle bits of the square usually depend on all the characters in the identifierit is expected that different identifiers will result in different values the number of middle bits that we select depends on table size therefore if is the number of middle bits used t... |
24,116 | notes the set of hash values (also called the buckets or binslet { }where is positive integer hash function hu associates buckets (hash valuesto keys two main issuescollisions if and are two different keysit is possible that ( ( this is called collision collision resolution is the most important issue in hash table imp... |
24,117 | notes task "open hashing is most appropriate when the hash table is kept in main memorywith the lists implemented by standard in-memory linked list explain rehashing this is another method of collision handling in this method you find an alternative empty location by modifying the hash functionand applying the modified... |
24,118 | notes keywords foldingin folding the identifier is partitioned into several parts all but the last part being of the same length hash functiona hash function is unary function that is used by hashed associative containers hashinghashing is the transformation of string of characters into usually shorter fixed-length val... |
24,119 | (bdetermine the hash addresses and find how many collisions occur when these keys are first folded by adding their digits together (in ordinary decimal representationand then applying hash_size (cfind hash function that will produce no collisions for these keys ( hash function that has no collisions for fixed set of ke... |
24,120 | notes further readings books brian kernighan and dennis ritchiethe programming languageprentice hall burkhard moniendata structures and efficient algorithmsthomas ottmannspringer krusedata structure program designprentice hall of indianew delhi mark allen welesdata structure algorithm analysis in csecond ed addisonwesl... |
24,121 | unit heaps unit heaps notes contents objectives introduction heaps binary heaps complete trees implementation putting items into binary heap removing items from binary heap applications of heaps discrete event simulation implementation -heaps summary keywords self assessment review questions further readings objectives... |
24,122 | notes variables are allocated in the space of initialized static variablesthe local variables of procedure are allocated in the procedure' activation recordwhich is typically found in the processor stackand dynamically allocated variables are allocated in the heap in this unitthe term heap is taken to mean the storage ... |
24,123 | of height two similarlythe left subtree of node is perfect binary tree of height twoand the right subtree is complete binary tree of height two notes figure complete binary tree does there exist an complete binary with exactly nodes for every integer > the following theorem addresses this question indirectly by definin... |
24,124 | notes there are exactly + nodes in the perfect left subtree from the inductive hypothesisthere are + nodes in the right subtree thusmk + + ( + )+ thereforeby induction mh + for all > which proves the upper bound it follows from theorem that there exists exactly one complete binary tree that contains exactly internal no... |
24,125 | notes figure object class hierarchy comparable abstract object abstract container container binaryheap priorityqueue mergeablepriorityqueue tree binomialqueue abstracttree binarytree leflistheap generaltree binomialtree program- introduces the binaryheap class the binaryheap class extends the abstractcontainer class in... |
24,126 | notes in this examplethe new item to be inserted has the key note that you cannot simply drop the new item into the next position in the complete tree because the resulting tree is no longer heap ordered insteadthe hole in the heap is moved toward the root by moving items down in the heap as shown in figure (band (cthe... |
24,127 | removing items from binary heap notes the dequeuemin method removes from priority queue the item having the smallest key in order to remove the smallest itemit needs first to be located thereforethe dequeuemin operation is closely related to findmin the smallest item is always at the root of min heap thereforethe findm... |
24,128 | notes figure (aillustrates the problem the dequeuemin operation removes the key from the heapbut it is the node containing key that must be removed from the tree to make it into complete tree again when key is removed from the roota hole is created in the tree as shown in figure (bthe trick is to move the hole down in ... |
24,129 | in the worst casethe hole must be pushed from the root to leaf node each iteration of the loop makes at most two object comparisons and moves the hole down one level notes thereforethe running time of the dequeuemin operation is log ( (log )where count is the number of items in the heap if ( and ( )the dequeuemin opera... |
24,130 | notes when the server finishes serving the customerthat customer departs if the queue is not emptythe server immediately commences serving the next customer otherwisethe server becomes idle how do you keep track of which event to simulate nexteach event (arrival or departureoccurs at discrete point in simulation time i... |
24,131 | notes boolean serverbusy falseint numberinqueue randomvariable servicetime new exponentialrv ( )randomvariable interarrivaltime new exponentialrv ( )eventlist enqueue (new event arrival ))while (!eventlist isempty ()event event (eventeventlist dequeuemin ()double event gettime ()if ( timelimit{eventlist purge ()breaksw... |
24,132 | notes the state of the system being simulated is represented by the two variables serverbusy and numberinqueue the first is boolean value which indicates whether the server is busy the second keeps track of the number of customers in the queue in addition to the state variablesthere are two instances of the class expon... |
24,133 | that is at least as large as its parentin max-heapeach item has priority that is no larger than its parent notes the minimum priority item in min-heap (or the maximum priority item in max-heapmay always be found at position of the array to remove this item from the priority queuethe last item in the array is moved into... |
24,134 | notes implementing -heaps as arrays the nodes of -heap can be stored in an array in breadth-first order (aallows indices for parents and children to calculated directlyeliminating the need for pointers key if is the index of an item xthen [( - )/dis the index of (xand the indices of the children of are in the range [ (... |
24,135 | notes odreturn mincendprocedure siftdown(item iinteger xmodifies heap )integer cc :minchild( , )do and key( ( ) ( : ( ) :cc :minchild( , )odh( :iendprocedure delete(item imodified heap )item jj : (| |) (| |:nullif and key(jsiftup( , - ( ), ) and key(jkey( =>siftdown( , - ( ), )fienditem function deletemin(modifies heap... |
24,136 | notes best example of -heap is "djikstra' algorithmdjikstra' algorithm (named after its discoverdutch computer scientist dijkstrasolves the problem of finding the shortest path from point in graph (the sourceto destination with non-negative weight edge it turns out that one can find the shortest paths from given source... |
24,137 | notes from all the adjacent verticeschoose the closest vertex to the source as we initialized [sto it' (shown in bold circleadd it to relax all vertices adjacent to si and update vertices and by and as the distance from choose the nearest vertexx relax all vertices adjacent to update predecessors for uv and predecessor... |
24,138 | notes finallyadd to the predecessor list now defines the shortest path from each node to dijkstra' algorithm initialise and pifor each vertex in vg [ :infinity pi[ :nil [ : set to empty : : (gwhile ( -sis not nullwhile not empty( sort the vertices in - according to the current best estimate of their distance from the s... |
24,139 | additionallya heap is "complete tree- complete tree is one in which there are no gaps between leaves notes for instancea tree with root node that has only one child must have its child as the left node more preciselya complete tree is one that has every level filled in before adding node to the next leveland one that h... |
24,140 | notes write method and the corresponding recursive function to traverse binary tree (in whatever order you find convenientand dispose of all its nodes use this method to implement binary_tree destructor consider heap of keyswith xk being the key in position (in the contiguous representationfor < prove that the height o... |
24,141 | unit leftist heaps and binomial queues unit leftist heaps and binomial queues notes contents objectives introduction leftist heaps leftist trees implementation merging leftist heaps putting items into leftist heap removing items from leftist heap skew heaps binomial queues summary keywords self assessment review questi... |
24,142 | notes leftist trees leftist tree is tree which tends to ``lean'to the left the tendency to lean to the left is defined in terms of the shortest path from the root to an external node in leftist treethe shortest path to an external node is always found on the right every node in binary tree has associated with it quanti... |
24,143 | figure leftist heap notes the reason for our interest in leftist trees is illustrated by the following theoremstheorem- consider leftist tree which contains internal nodes the path leading from the root of downwards to the rightmost external node contains at most log ( nodes extbfproof assume that has null path length ... |
24,144 | notes public class leftistheap extends binarytree implements mergeablepriorityqueue protected int nullpathlength/program leftistheap fields merging leftist heaps in order to merge two leftist heapssay and declared as follows mergeablepriorityqueue new leftistheap ()mergeablepriorityqueue new leftistheap () invoke the m... |
24,145 | figure merging leftist heaps notes program gives the code for the merge method of the leftistheap class the merge method makes use of two other methodsswapcontents and swapsubtrees the swapcontents method takes as its argument leftist heapand exchanges all the contents (key and subtreesof this heap with the given one t... |
24,146 | notes where (isgtis time required to compare two keys if you assume that the time to compare two keys is constantthen you get (log log )where and are the number of internal nodes in trees and respectively public class leftistheap extends binarytree implements mergeablepriorityqueue protected int nullpathlengthpublic vo... |
24,147 | the expression for the running time for the insert operation follows directly from that of the merge operation that isthe time required for the insert operation in the worst case is notes ( ) (isgto( )where is the null path length of the heap into which the item is inserted if you assume that two keys can be compared i... |
24,148 | notes throw new containeremptyexception ()comparable result (comparablegetkey ()leftistheap oldleft getleftheap ()leftistheap oldright getrightheap ()purge ()swapcontents (oldleft)merge (oldright)return result/program leftistheap class dequeuemin method the running time of program is determined by the time required to ... |
24,149 | worst-case time bound for each of the heap operationswhere is the number of nodes in the heap or heaps involved notes figure union of two skew heaps (amerge of the right paths (bswapping of children along the path formed by the merge ( ( in our self-adjusting version of this data structurei perform the union operation ... |
24,150 | notes has very special shape called binomial tree binomial trees are general trees the maximum degree of node is not fixed the remarkable characteristic of binomial queues is that the merge operation is similar in structure to binary additioni the collection of binomial trees that make up the binomial queue is like the... |
24,151 | the number of nodes in binomial tree of order is function of knotes theorem- the binomial tree of order kbkcontains nodes extbfproof (by inductionlet nk be the number of nodes in bka binomial tree of order base caseby definitionb consists single node therefore inductive hypothesisassume that nk for lfor some > consider... |
24,152 | notes figure two views of binomial tree (ab (bb alternativelyyou can think of bk as being comprised of two binomial trees of order - examplefigure (bshows that is made up of two instances of in generalsuppose you have two trees of order - say - and - where - { - then you can construct binomial tree of order by combinin... |
24,153 | summary notes power-of- heap is left-heap-ordered tree consisting of root node with an empty right subtree and complete left subtree the tree corresponding to power-of- heap by the left-childright-sibling correspondence is called binomial tree binomial trees and power-of- heaps are equivalent work with both representat... |
24,154 | notes write methods to implement queues in circular array with one unused entry in the array that iswe consider that the array is full when the rear is two positions before the frontwhen the rear is one position beforeit will always indicate an empty queue prove that "the binomial tree of order kbkcontains nodes extbfp... |
24,155 | mandeep kaurlovely professional university unit sorting notes contents objectives introduction internal sorting insertion sort algorithm of insertion sort complexity analysis shell sort heap sort merge sort merging of two sorted lists quick sort bucket sort external sorting summary keywords self assessment review quest... |
24,156 | notes the difference lies in the fact that the first method moves data only over small distances in the process of sortingwhereas the second method moves data over large distancesso that items settle into the proper order soonerthus resulting in fewer comparisons performance of sorting algorithm can also depend on the ... |
24,157 | any sorting algorithm that uses comparisons of keys needs at least ( log ntime to accomplish the sorting notes sorting methods internal (in memoryquick sort heap sort bubble sort insertion sort selection sort shell sort external (appropriate for secondary storagemerge sort radix sort poly-phase sort insertion sort this... |
24,158 | notes unsorted part and inserts it to the right place of the sorted one when unsorted part becomes emptyalgorithm stops sketchyinsertion sort algorithm step looks like thissorted partitial result < > unsorted data becomes sorted partitial result < unsorted data > examplesort { - using insertion sort - unsorted - - to b... |
24,159 | swap swap swap sifting is done notes this approach writes sifted element to temporary position many times next implementation eliminates those unnecessary writes shifting instead of swapping we can modify previous algorithmso it will write sifted element only to the final correct position let us see an illustration swa... |
24,160 | notes shell sort improves the efficiency and decreases the run time complexity to ( donald shell discovered the shell sort and thence it is known as shell sort algorithm the algorithms for shell sort can be defined in two stepsstep divide the original list into smaller lists step sort individual sub lists using any kno... |
24,161 | second_list [ ] [ ] [ ] [ [ third_list [ ] [ ] [ ] [ [ forth_list [ ] [ ] [ ] [ [ fifth_list [ ] [ ] [ ] [ [ notes so the sub list will contain every kth element of the original list starting from index - th according to the algorithm mentioned abovefor each iterationthe list is divided and then sorted if we use the sa... |
24,162 | notes after first iteration = when the increment reduces to you will get the following lists ( ( ( now if we analyse the first sub lists where increment was and then second sub lists where increment was we can see that in the second set of listswe are comparing and sorting again which we had already sorted before in th... |
24,163 | notes after sorting the sub liststhe resulting list we get is as follows - you nowreduce the increment value to and get the following sub lists after sorting these sub liststhe resulting list is as follows - you further reduce increment value to which is our last increment value now there is only one listwhich after so... |
24,164 | notes the hk form geometric seriesthe sum of all hk with is in (htonthus (nnsorting steps are needed for this part where now let when the sequence is arranged as an array with hk columns there are /hk elements in each column thuso(( /hk) sorting steps are needed to sort each columnsince insertion sort has quadratic com... |
24,165 | notes list[ ]flag truej iwhile( < &flagif(( &(list[jlist[ ]) if( >list[ ]flag falseelse list[ div list[ ] list[ div kvoid build_initial_heap(list xint nint ifor( div > --adjust(xin)void heapsort(list xint nint ibuild_initial_heap(xn)for( -li > --exchange(list[ ]list[ + ])adjust(xli)void exchange(int &aint &bint tt * * ... |
24,166 | notes void adjust(int []int iint nint jkint flagk [ ]flag iwhile( < &flag = if(( &( [jx[ ]) ++if( > [ ]flag else [ [ ] [ kvoid build_initial_heap(int []int nint ifor( > --adjust(xin)void heapsort(int []int nint ibuild_initial_heap(xn)for( -li > --exchange( [ ] [ ])adjust( , , )analysis of heapsort in each pass of while... |
24,167 | ithe procedure build-initial-heap calls the adjust procedure for values of ranging from / down to hence the total number of iterations will benotes / lognlog( / log( / / log( / lognlog( / = this comes out to be some constant times hence the computation time of build_initial_heap is (nthe heapsort procedure calls adjust... |
24,168 | notes each of size and merge the first list with the second list to form single sorted list of size similarly we merge the third and the fourth lists to form second single sorted list of size and so on this completes the one pass we then consider the first sorted list of size and second sorted list of size and merge th... |
24,169 | notes [ = [ ] + ++else [kx[ ] ++; ++while ( <my[ = [ ] ++ ++while ( <ny[kx[ ] ++ ++void mpass(int [],int [],int int nint ,ji whi ( < merge(xyii+ - - ) if(( + - nmerge(xyii+ - )else while ( < [ = [ ] ++lovely professional university |
24,170 | notes void msort(int []int nint int [] while( nmpass(xyln) mpass(yxln) the merging of two sub-liststhe first running from the index to mand the second running from the index ( + to requires no more than ( - + iterations hence if then no more than iterationswhere is the size of the list to be sorted therefore if is the ... |
24,171 | when only one active run remains the algorithm finishes up as shown in lines and of twoway-merge-it just copies all the remaining elements to the end of file figure visualizes multiway merging notes figure main-memory organization for multiway merging input buffersone for each unfinished run select smallest unchosen el... |
24,172 | notes th( logmnis lower bound for the problem of external-memory sorting that ismultiway merge sort is an asymptotically optimal algorithm merging of two sorted lists assume that two lists to be merged are sorted in descending order compare the first element of the first list with the first element of the second list i... |
24,173 | notes sort( , )printf("the sorted list is:\ ")dis( , )sort( , )printf("the sorted list is:\ ")dis( , )merge( , , , )printf("the elements of merged list are \ ")dis( , )/*display the merged list*getch()void read(int [],int iint jfor( = ; < ; ++scanf("% ",& [ ])fflush(stdin)void dis(int [],int iint jfor( = ; < ; ++printf... |
24,174 | notes void merge(int [],int [],int [],int kint ptra= ,ptrb= ,ptrc= while(ptra< &ptrb<kif( [ptrab[ptrb] [ptrc]= [ptra]ptra++else [ptrc]= [ptrb]ptrb++ptrc++while(ptra<kc[ptrc]= [ptra]ptra++;ptrc++while(ptrb<kc[ptrc]= [ptrb]ptrb++ptrc++inputenter the elements of the first list outputthe elements of first list are inputent... |
24,175 | notes the sorted list is the elements of the merged list are task write program for merge sort quick sort in this method an array [ ], [nis sorted by picking some value in the array as key elementwe then swap the first element of the list with the key element so that the key will come in the first positionwe then find ... |
24,176 | notes ++ifi jinterchange(& [ ]& [ ])interchange(& [ ]& [ ])qsort(xmj- )qsort(xj+ )void interchange(int *iint *jint temptemp * * * * tempchoice of the key we can choose any entry in the list as the key the choice of first entry is often poor choice for keysince if the list is already sortedthen there will be no element ... |
24,177 | when qsort is activated first time key and = and = is incremented till it becomes because at position the value is greater than keyj is not decrementedbecause at position the value that we have is less than the key since <jwe interchange the rd element and th element then is incremented till it becomes and is decrement... |
24,178 | notes (ncomes out to bet(nc* + ( - * *( - ( - * * ( - * * *( - ( - * * * ( - * * - * ( -nc+ therefore (nn hence the order is ( space complexity the average case space complexity is log nbecause the space complexity depends on the maximum number of activations that can exist we find that if we assume that every time the... |
24,179 | notes examplesort { using quicksort swap pivot value pivot value swap and swap and swap and stop partition run quick sort recursively sorted noticethat we show here only the first recursion stepin order not to make example too long butin fact{ and { are sorted then recursively why does it workon the partition step algo... |
24,180 | notes < swap ( , , to get [ [ 'ovthere forel [ 'mvthere forer < swap ( , , to get [ [ 'uvl [ 'mv; break swap ( , , to get [ at this point 'nis in its correct place [ ] [ to [ constitutes sub list [ to [ constitutes sublist now quick sort ( quick sort ( the quick sort algorithm uses the ( log ncomparisons on average the... |
24,181 | bucket_sort (anotes length [ for to do insert [iinto list [na[ ] for to - do sort list with insertion sort concatenate the lists [ ] [ ] [ - together in order examplegiven input array [ the array [ of sorted lists or buckets after line bucket holds values in the interval [ / ( + )/ the sorted output consists of concate... |
24,182 | notes thereforethe probability follows the binomial distributionwhich has meane[ninp variancevar[ninp( / for any random variablewe have [ nivar[nie [ni / th( putting this value in equation above(do some tweakingand we have expected time for insertion_sorto(nnow back to our original problem in the above bucket sort algo... |
24,183 | summary notes arranging objects in specified order is called sorting bubble sortinsertion sortselection sortquick sortheap sortradix sort are some of very common search algorithms the comparison starts with the first element (or at the last elementand continues sequentially till either we find match or the end of the l... |
24,184 | notes ( bubble sort heap sort proceeds in (afive phase (bthree phases (cone phase (dtwo phase fill in the blanks arranging objects in specified order is called the average time complexity of the quick sort algorithm is the computing time of heapsort is the time complexity of the algorithm is both average and worst case... |
24,185 | notes consider an unsorted array [nof integer elements that may have many elements present more than once it is required to store only the distinct elements of the array in separate array the information about the number of times each element is replicated is maintained in third array for examplec[ would indicate the n... |
24,186 | mandeep kaurlovely professional university unit graphs notes contents objectives introduction defining graph basic graph terminology representations of graphs adjacent matrix adjacency list representation shortest path algorithms summary keywords self assessment review questions further readings objectives after studyi... |
24,187 | defining graph notes graph consists of set of vertice (nodesand set of edges (arcswe write =( ,ev is finite and non empty set of vertices is set of pairs of verticesthese pairs are called edges therefore ( )read as of gis set of verticesand ( )read as of gis set of edges an edge ( , )is pair of vertices and wand is sai... |
24,188 | notes figure directed graph the direction is indicated by an arrow the set of vertices for this graph remains the same as that of the graph in the earlier examplei ( ( , , , , however the set of edges would be ( {( , )( , )( , )( , )( , )( , )( , )do you notice the differencenote arrow is always from tail vertex to hea... |
24,189 | notes find out the vertices adjacent to remaining nodes of the graph patha path from vertex to vertex is sequence of verticeseach adjacent to the next consider the above example again , , is path , , is path , , is pathis , , pathhow many paths are there from vertex to vertex you may notice that there is path existing ... |
24,190 | notes in figure , is directed path is directed path is not directed path there is no directed cycle in the above graph you may verify the above statement digraph is called strongly connected if there is directed path from any vertex to any other vertex consider the digraph given in figure figure weakly connected graph ... |
24,191 | notes an alternative could be as given in figure figure strongly connected graph there may exist more alternate structures make at least one more alternate structure for the same diagraph you must have observed that there is no limitation of number of edges incident on one vertex it could be noneoneor more the number o... |
24,192 | notes graph depicted in figure is tree and so are the ones depicted in figure (ato (efigure tree structure ( ( ( ( (ebecause of their special structure and propertiestrees occur in many different applications in computer science did knowin figure how many outdegree present in vertex representations of graphs graph is m... |
24,193 | notes table (ashows the adjacency matrix for the graph given in figure table adjacency matrix for the graph in figure vertice you may observe that the adjacency matrix for an undirected graph is symmetricas the lower and upper triangles are same also all the diagonal elements are zero since we consider graphs without a... |
24,194 | notes therefore we shall havetable adjacency list structure for graph in figure note that adjacent vertices may appear in the adjacency list in arbitrary order also an arrow from to in the list linked to does not mean that and are adjacent the adjacency list representation is better for sparse graphs because the space ... |
24,195 | notes here is the algorithm let be the set of all the vertices of the graph and be the set of all the vertices considered for the determination of the minimal path set ={ while there are still vertices in (asort the vertices in according to the current best estimate of their distance from the source (badd uthe closest ... |
24,196 | notes summary graphs provide in excellent way to describe the essential features of many applications graphs are mathematical structures and are found to be useful in problem solving they may be implemented in many ways by the use of different kinds of data structures graph traversalsdepth first as well as breadth firs... |
24,197 | test the program for obtaining the depth first spanning tree for the following graph " graph may have many spanning treesfor instance the complete graph on four vertices has sixteen spanning treesexplain define vertices of graph graph is regular if every vertex has the same valence (that isif it is adjacent to the same... |
24,198 | notes further readings books brian kernighan and dennis ritchiethe programming languageprentice hall burkhard moniendata structures and efficient algorithmsthomas ottmannspringer krusedata structure program designprentice hall of indianew delhi mark allen welesdata structure algorithm analysis in second ed addisonwesle... |
24,199 | unit network flows unit network flows notes contents objectives introduction network flow ford fulkerson method comparison networks network flow problem minimum spanning tree kruskal' algorithm prim' algorithm summary keywords self assessment review questions further readings objectives after studying this unityou will... |
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