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found alternativelyif the hash table size is power of two and the probe function is (ki( )/ then every slot in the table will be visited by the probe function both pseudo-random probing and quadratic probing eliminate primary clusteringwhich is the problem of keys sharing substantial segments of probe sequence if two k...
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chap searching project at the end of this asks you to improve the implementation with other hash functions and collision resolution policies analysis of closed hashing how efficient is hashingwe can measure hashing performance in terms of the number of record accesses required when performing an operation the primary o...
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/*dictionary implemented using hashing *class hashdictionaryeimplements dictionary private static final int defaultsize private hashtable /the hash table private int count/of records now in table private int maxsize/maximum size of dictionary hashdictionary(this(defaultsize)hashdictionary(int szt new hashtable(sz)count...
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chap searching if and are largethen this is approximately ( / ) the expected number of probes is one plus the sum over > of the probability of collisionswhich is approximately ( / ) /( ai= the cost for successful search (or deletionhas the same cost as originally inserting that record howeverthe expected value for the ...
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sec hashing insert delete figure growth of expected record accesses with the horizontal axis is the value for athe vertical axis is the expected number of accesses to the hash table solid lines show the cost for "randomprobing ( theoretical lower bound on the cost)while dashed lines show the cost for linear probing ( r...
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chap searching depending on the pattern of record accessesit might be possible to reduce the expected cost of access even in the face of collisions recall the / rule of the accesses will come to of the data in other wordssome records are accessed more frequently if two records hash to the same home positionwhich would ...
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occupied the slot but does so no longer if tombstone is encountered when searching through probe sequencethe search procedure is to continue with the search when tombstone is encountered during insertionthat slot can be used to store the new record howeverto avoid inserting duplicate keysit will still be necessary for ...
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chap searching bentley et al " locally adaptive data compression scheme[bstw for more on ziv-lempel codingsee data compressionmethods and theory by james storer [sto knuth covers self-organizing lists and zipf distributions in volume of the art of computer programming[knu introduction to modern information retrieval by...
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sec exercises list ordered by frequency where the frequency conforms to zipf distributioncharacterize the difference in running times assume that the values through are stored in self-organizing listinitially in ascending order consider the three self-organizing list heuristicscountmove-to-frontand transpose for counta...
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chap searching (chow many students must be in the class for the probability to be at least that there are who share birthday in the same month assume that you are hashing key to hash table of slots (indexed from to for each of the following functions ( )is the function acceptable as hash function ( would the hash progr...
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sec projects be sure to indicate how you are using and to do the hashing function rev(kreverses the decimal digits of kfor examplerev( rev( (kk mod ( (rev( mod keys write an algorithm for deletion function for hash tables that replaces the record with special value indicating tombstone modify the functions hashinsert a...
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chap searching implement database stored on disk using bucket hashing define records to be bytes long with -byte key and bytes of data the remaining bytes are available for you to store necessary information to support the hash table bucket in the hash table will be bytes longso each bucket has space for records the ha...
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indexing many large-scale computing applications are centered around data sets that are too large to fit into main memory the classic example is large database of records with multiple search keysrequiring the ability to insertdeleteand search for records hashing provides outstanding performance for such situationsbut ...
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chap indexing file is created whose records consist of key/pointer pairs hereeach key is associated with pointer to complete record in the main database file the index file could be sorted or organized using tree structurethereby imposing logical order on the records without physically rearranging them one database mig...
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sec linear indexing linear index database records figure linear indexing for variable-length records each record in the index file is of fixed length and contains pointer to the beginning of the corresponding record in the database file begins with discussion of the variant normally referred to simply as " -tree sectio...
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chap indexing second level index linear indexdisk blocks figure simple two-level linear index the linear index is stored on disk the smallersecond-level index is stored in main memory each element in the second-level index stores the first key value in the corresponding disk block of the index file in this examplethe f...
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whose records have the indicated secondary key value figure illustrates this approach now there is no duplication of secondary key valuespossibly yielding considerable space savings the cost of insertion and deletion is reducedbecause only one row of the table need be adjusted note that new row is added to the array wh...
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chap indexing secondary key primary key jones aa smith ab zukowski ab ff ax ax zx zq figure illustration of an inverted list each secondary key value is stored in the secondary key list each secondary key value on the list has pointer to list of the primary keys whose associated records have that secondary key value se...
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sec isam in-memory table of cylinder keys cylinder cylinder index cylinder cylinder index records records cylinder cylinder overflow overflow system overflow figure illustration of the isam indexing system in the index inverted lists reduce this problembut they are only suitable for secondary key indices with many fewe...
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chap indexing key value for each block in that cylindercalled the cylinder index when new records are insertedthey are placed in the correct cylinder' overflow area (in effecta cylinder acts as bucketif cylinder' overflow area fills completelythen system-wide overflow area is used search proceeds by determining the pro...
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is enough main memorythe bst is viable option for implementing both primary and secondary key indices unfortunatelythe bst can become unbalanced even under relatively good conditionsthe depth of leaf nodes can easily vary by factor of two this might not be significant concern when the tree is stored in main memory beca...
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chap indexing figure breaking the bst into blocks the bst is divided among disk blockseach with space for three nodes the path from the root to any leaf is contained on two blocks ( (bfigure an attempt to re-balance bst after insertion can be expensive (aa bst with six nodes in the shape of complete binary tree (ba nod...
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sec - trees figure - tree are greater than or equal to the value of the first key if there is right subtree (equivalentlyif the node stores two keys)then the values of all descendants in the center subtree are less than the value of the second keywhile values in the right subtree are greater than or equal to the value ...
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chap indexing /* - tree node implementation *class ttnode,eprivate lval/the left record private key lkey/the node' left key private rval/the right record private key rkey/the node' right key private ttnode left/pointer to left child private ttnode center/pointer to middle child private ttnode right/pointer to right chi...
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sec - trees figure simple insert into the - tree of figure the value is inserted into the tree at the leaf node containing because there is room in the node for second keyit is simply added to the left position with moved to the right position private findhelp(ttnode rootkey kif (root =nullreturn null/val not found if ...
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chap indexing figure simple node-splitting insert for - tree the value is added to the - tree of figure this makes the node containing values and splitpromoting value to the parent node with the middle of the three key value is passed up to the parent node along with pointer to this is called promotion the promoted key...
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sec - trees ( ( (cfigure example of inserting record that causes the - tree root to split (athe value is added to the - tree of figure this causes the node containing and to splitpromoting (bthis in turn causes the internal node containing and to splitpromoting (cfinallythe root node splitspromoting to become the left ...
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chap indexing private ttnode inserthelp(ttnode rtkey ke ettnode retvalif (rt =null/empty treecreate leaf node for root return new ttnode(kenullnullnullnullnull)if (rt isleaf()/at leaf nodeinsert here return rt add(new ttnode(kenullnullnullnullnull))/add to internal node if ( compareto(rt lkey() /insert left retval inse...
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sec -trees /*add new key/value pair to the node there might be subtree associated with the record being added this information comes in the form of - tree node with one key and (possibly nullsubtree through the center pointer field *public ttnode add(ttnode itif (rkey =null/only one keyadd here if (lkey compareto(it lk...
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chap indexing figure -tree of order four -trees guarantee that every node in the tree will be full at least to certain minimum percentage this improves space efficiency while reducing the typical number of disk fetches necessary during search or update operation -tree of order is defined to have the following shape pro...
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examining the node at level the third branch is taken to the next level to arrive at the leaf node containing record with key value -tree insertion is generalization of - tree insertion the first step is to find the leaf node that should contain the key to be insertedspace permitting if there is room in this nodethen i...
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chap indexing valuesbut these are used solely as placeholders to guide the search this means that internal nodes are significantly different in structure from leaf nodes internal nodes store keys to guide the searchassociating each key with pointer to child -tree node leaf nodes store actual recordsor else keys and poi...
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sec -trees figure example of -tree of order four internal nodes must store between two and four children for this examplethe record size is assumed to be such that leaf nodes store between three and five records are found in the second subtree from the second child of the rootthe first branch is taken to reach the leaf...
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chap indexing ( ( ( (dfigure examples of -tree insertion (aa -tree containing five records (bthe result of inserting record with key value into the tree of (athe leaf node splitscausing creation of the first internal node (cthe -tree of (bafter further insertions (dthe result of inserting record with key value into the...
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sec -trees figure simple deletion from -tree the record with key value is removed from the tree of figure note that even though is also placeholder used to direct search in the parent nodethat value need not be removed from internal nodes even if no record in the tree has key value thusthe leftmost node at level one in...
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chap indexing ( (bfigure deleting the record with key value from the -tree of figure via collapsing siblings (athe two leftmost leaf nodes merge together to form single leaf unfortunatelythe parent node now has only one child (bbecause the left subtree has spare leaf nodethat node is passed to the right subtree the pla...
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required (because there is less empty space in the disk fileand in more efficient processing (fewer blocks on average will be read into memory because the amount of information in each block is greaterbecause -trees have become so popularmany algorithm designers have tried to improve -tree performance one method for do...
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chap indexing as mentioned earliera buffer pool should be used to manage nodes of the -tree several nodes of the tree would typically be in main memory at one time the most straightforward approach is to use standard method such as lru to do node replacement howeversometimes it might be desirable to "lockcertain nodes ...
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sec exercises (aassume that linear index uses bytes to store the key and bytes to store the block id for the associated records what is the greatest number of records that can be stored in the file if linear index of size mb is used(bwhat is the greatest number of records that can be stored in the file if the linear in...
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chap indexing (bshow the improved inverted list (as illustrated by figure for this collection of records under what conditions will isam be more efficient than -tree implementation prove that the number of leaf nodes in - tree with levels is between - and - show the result of inserting the values and into the - tree of...
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projects implement two-level linear index for variable-length records as illustrated by figures and assume that disk blocks are bytes in length records in the database file should typically range between and bytesincluding -byte key value each record of the index file should store key value and the byte offset in the d...
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advanced data structures
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graphs graphs provide the ultimate in data structure flexibility graphs can model both real-world systems and abstract problemsso they are used in hundreds of applications here is small sampling of the range of problems that graphs are applied to modeling connectivity in computer and communications networks representin...
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chap graphs ( ( (cfigure examples of graphs and terminology (aa graph (ba directed graph (digraph(ca labeled (directedgraph with weights associated with the edges in this examplethere is simple path from vertex to vertex containing vertices and vertices and also form pathbut not simple path because vertex appears twice...
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sec terminology and representations figure an undirected graph with three connected components vertices and form one connected component vertices and form second connected component vertex by itself forms third connected component subgraph is formed from graph by selecting subset vs of ' vertices and subset es of ' edg...
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chap graphs ( ( (cfigure two graph representations (aa directed graph (bthe adjacency matrix for the graph of ( (cthe adjacency list for the graph of (aexample the entry for vertex in figure (cstores and because there are two edges in the graph leaving vertex with one going to vertex and one going to vertex the list fo...
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sec terminology and representations ( ( (cfigure using the graph representations for undirected graphs (aan undirected graph (bthe adjacency matrix for the graph of ( (cthe adjacency list for the graph of (awhich graph representation is more space efficient depends on the number of edges in the graph the adjacency list...
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chap graphs the adjacency matrix often requires higher asymptotic cost for an algorithm than would result if the adjacency list were used the reason is that it is common for graph algorithm to visit each neighbor of each vertex using the adjacency listonly the actual edges connecting vertex to its neighbors are examine...
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sec graph implementations /*graph adt *public interface graph /graph class adt /*initialize the graph @param the number of vertices *public void init(int )/*@return the number of vertices *public int ()/*@return the current number of edges *public int ()/*@return ' first neighbor *public int first(int )/*@return ' next...
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chap graphs edge list function next will return value of |vonce the end of the edge list for has been reached the following line appears in many graph algorithmsfor ( =>first( ) () ->next( , )this for loop gets the first neighbor of vthen works through the remaining neighbors of until value equal to -> (is returnedsign...
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sec graph implementations class graphm implements graph /graphadjacency matrix private int[][matrix/the edge matrix private int numedge/number of edges public int[mark/the mark array public graphm({public graphm(int ninit( )/constructor public void init(int nmark new int[ ]matrix new int[ ][ ]numedge public int (return...
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chap graphs public int weight(int iint /return edge weight return matrix[ ][ ]/get and set marks public void setmark(int vint valmark[vvalpublic int getmark(int vreturn mark[ ]figure (continued/edge class for adjacency list graph representation class edge private int vertwtpublic edge(int vint /constructor vert vwt wpu...
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sec graph traversals /adjacency list graph implementation class graphl implements graph private graphlist[vertexprivate int numedgepublic int[markpublic graphl({public graphl(int ninit( )/the vertex list /number of edges /the mark array /constructor public void init(int nmark new int[ ]vertex new graphlist[ ]for (int =...
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chap graphs /store edge weight public void setedge(int iint jint weightassert weight ! "may not set weight to "edge curredge new edge(jweight)if (isedge(ij)/edge already exists in graph vertex[iremove()vertex[iinsert(curredge)else /keep neighbors sorted by vertex index numedge++for (vertex[imovetostart()vertex[icurrpos...
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specified goal state by moving between states only through the connections typicallythe start and goal states are not directly connected to solve this problemthe vertices of the graph must be searched in some organized manner graph traversal algorithms typically begin with start vertex and attempt to visit the remainin...
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chap graphs ( (bfigure (aa graph (bthe depth-first search tree for the graph when starting at vertex the edges that were followed to any new (unvisitedvertex during the traversal and leaves out the edges that lead to already visited vertices dfs can be applied to directed or undirected graphs here is an implementation ...
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sec graph traversals call dfs on mark process (acprint (acand call dfs on mark process (cac process (cbprint (cband call dfs on mark process (bcc process (bfprint (bfand call dfs on mark process (fbb process (fcc process (fdprint (fdand call dfs on mark process (dcprocess (dfa pop process (feprint (feand call dfs on ma...
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chap graphs /breadth first (queue-basedsearch static void bfs(graph gint startqueue new aqueue( ()) enqueue(start) setmark(startvisited)while ( length( /process each vertex on int dequeue()previsit(gv)/take appropriate action for (int first( ) () next(vw)if ( getmark( =unvisited/put neighbors on setmark(wvisited) enque...
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sec graph traversals ( (bfigure (aa graph (bthe breadth-first search tree for the graph when starting at vertex topological sort may be found by performing dfs on the graph when vertex is visitedno action is taken ( function previsit does nothingwhen the recursion pops back to that vertexfunction postvisit prints the v...
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chap graphs initial call to bfs on mark and put on the queue dequeue process (caignore process (cbmark and enqueue print (cbprocess (cdmark and enqueue print (cdprocess (cfmark and enqueue print (cfd dequeue process (bcignore process (bfignore dequeue process (acmark and enqueue print (acprocess (aemark and enqueue pri...
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sec shortest-paths problems figure an example graph for topological sort seven tasks have dependencies as shown by the directed graph we can also implement topological sort using queue instead of recursion to do sowe first visit all edgescounting the number of edges that lead to each vertex ( count the number of prereq...
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chap graphs static void topsort(graph /topological sortqueue queue new aqueue( ())int[count new int[ ()]int vfor ( = < () ++count[ /initialize for ( = < () ++/process every edge for (int first( ) () next(vw)count[ ]++/add to ' prereq count for ( = < () ++/initialize queue if (count[ = / has no prerequisites enqueue( )w...
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the shortest paths from to every other vertex as well so there is no better algorithm (in the worst casefor finding the shortest path to single vertex than to find shortest paths to all vertices the algorithm described here will only compute the distance to every such vertexrather than recording the actual path recordi...
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chap graphs /compute shortest path distances from sstore them in static void dijkstra(graph gint sint[dfor (int = < () ++/initialize [iinteger max valued[ for (int = < () ++/process the vertices int minvertex(gd)/find next-closest vertex setmark(vvisited)if ( [ =integer max valuereturn/unreachable for (int first( ) () ...
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know both their vertex number and their distancewe create simple class for the purpose called dijkelemas follows dijkelem is quite similar to the edge class used by the adjacency list representation import java lang comparableclass dijkelem implements comparable private int vertexprivate int weightpublic dijkelem(int i...
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chap graphs /dijkstra' shortest-pathspriority queue version static void dijkstra(graph gint sint[dint /the current vertex dijkelem[ new dijkelem[ ()]/heap for edges [ new dijkelem( )/initial vertex minheap new minheap( ())for (int = < () ++/initialize distance [iinteger max valued[ for (int = < () ++/for each vertex do...
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sec minimum-cost spanning trees figure graph and its mst all edges appear in the original graph those edges drawn with heavy lines indicate the subset making up the mst note that edge (cfcould be replaced with edge (dfto form different mst with equal cost ( it connects them together)and it has minimum cost figure shows...
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chap graphs /compute minimal-cost spanning tree static void prim(graph gint sint[dint[vfor (int = < () ++/initialize [iinteger max valued[ for (int = < () ++/process the vertices int minvertex(gd) setmark(vvisited)if ( !saddedgetomst( [ ] )if ( [ =integer max valuereturn/unreachable for (int first( ) () next(vw)if ( [w...
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/prims' mst algorithmpriority queue version static void prim(graph gint sint[dint[vint /the current vertex dijkelem[ new dijkelem[ ()]/heap for edges [ new dijkelem( )/initial vertex minheap new minheap( ())for (int = < () ++/initialize [iinteger max value/distances [ for (int = < () ++/nowget distances do ( removemin(...
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chap graphs marked vertices vi vu vp unmarked vertices vi > "correctedge ej prim' edge vw vj figure prim' mst algorithm proof the left oval contains that portion of the graph where prim' mst and the "truemst agree the right oval contains the rest of the graph the two portions of the graph are connected by (at leastedge...
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example figure shows the first three steps of kruskal' algorithm for the graph of figure edge (cdhas the least costand because and are currently in separate mststhey are combined we next select edge (efto processand combine these vertices into single mst the third edge we process is (cf)which causes the mst containing ...
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chap graphs initial step process edge (cdd step process edge (efd step process edge (cfd figure illustration of the first three steps of kruskal' mst algorithm as applied to the graph of figure (bif an undirected graph has | edges and no cyclesthen the graph is connected (adraw the adjacency matrix representation for t...
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class kruskalelem implements comparable private int vwweightpublic kruskalelem(int inweightint invint inwweight inweightv invw inwpublic int (return vpublic int (return wpublic int key(return weightpublic int compareto(kruskalelem thatif (weight that key()return - else if (weight =that key()return else return static vo...
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chap graphs figure example graph for exercises explain whyin the worst casedijkstra' algorithm is (asymptoticallyas efficient as any algorithm for finding the shortest path from some vertex to another vertex show the shortest paths generated by running dijkstra' shortest-paths algorithm on the graph of figure beginning...
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mstshow the result on the equivalence array( show the array as in figure write an algorithm to find maximum cost spanning treethat isthe spanning tree with highest possible cost when can prim' and kruskal' algorithms yield different msts prove thatif the costs for the edges of graph are distinctthen only one mst exists...
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chap graphs triangular matrices the re-implement the adjacency matrix representation of figure to implement undirected graphs using triangular array while the underlying implementation (whether adjacency matrix or adjacency listis hidden behind the graph adtthese two implementations can have an impact on the efficiency...
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lists and arrays revisited simple lists and arrays are the right tool for the many applications other situations require support for operations that cannot be implemented efficiently by the standard list representations of this presents advanced implementations for lists and arrays that overcome some of the problems of...
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chap lists and arrays revisited figure example of multilist represented by tree figure example of reentrant multilist the shape of the structure is dag (all edges point downwarda list containing sublists will be written as hx hy ha iy ihz ix in this examplethe list has four elements the second element is the sublist hy...
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sec multilists figure example of cyclic list the shape of the structure is directed graph this list in bracket notationwe can duplicate nodes as necessary thusthe bracket notation for the list of figure could be written hhhabiihhabicihcdeiheii for conveniencewe will adopt convention of allowing sublists and atoms to be...
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chap lists and arrays revisited root figure linked representation for the pure list of figure the first field in each link node stores tag bit if the tag bit stores "+,then the data field stores an atom if the tag bit stores "-,then the data field stores pointer to sublist root figure lisp-like linked representation fo...
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sec matrix representations (aa (bfigure triangular matrices (aa lower triangular matrix (ban upper triangular matrix matrix representations some applications must represent largetwo-dimensional matrix where many of the elements have value of zero one example is the lower triangular matrix that results from solving syst...
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chap lists and arrays revisited cols rows figure the orthogonal list sparse matrix representation one approach to representing sparse matrix is to concatenate (or otherwise combinethe row and column coordinates into single value and use this as key in hash table thusif we want to know the value of particular position i...
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sec matrix representations element also stores pointers to its non-zero neighbors following and preceding it in the column thuseach non-zero element stores its own valueits position within the matrixand four pointers non-zero elements are found by traversing row or column list note that the first non-zero element in gi...
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chap lists and arrays revisited and solving for xwe find that the sparse matrix using this implementation is more space efficient when / that iswhen less than about of the elements are non-zero different values for the relative sizes of data valuespointersor matrix indices can lead to different break-even point for the...
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/*memory manager interface *interface memmanadt /*store record and return handle to it *public memhandle insert(byte[info)/request space /*get back copy of stored record *public byte[get(memhandle )/retrieve data /*release the space associated with record *public void release(memhandle )/release space figure simple adt...
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chap lists and arrays revisited figure dynamic storage allocation model memory is made up of series of variable-size blockssome allocated and some free in this exampleshaded areas represent memory currently allocated and unshaded areas represent unused memory available for future allocation small blockexternal fragment...
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that allocate file space in clusters another example of sacrificing space to internal fragmentation so as to simplify memory management is the buddy method described later in this section the process of searching the memory pool for block large enough to service the requestpossibly reserving the remaining space as free...
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chap lists and arrays revisited figure doubly linked list of free blocks as seen by the memory manager shaded areas represent allocated memory unshaded areas are part of the freelist tag sizellink rlink tag size size tag (atag (bfigure blocks as seen by the memory manager each block includes additional information such...
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sec memory management figure adding block to the freelist the word immediately preceding the start of in the memory pool stores the tag bit of the preceding block if is freemerge into we find the end of by using ' size field the word following the end of is the tag field for block if is freemerge it into then simply ha...
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chap lists and arrays revisited remainder of size best fit has the disadvantage that it requires that the entire list be searched another problem is that the remaining portion of the best-fit block is likely to be smalland thus useless for future requests in other wordsbest fit tends to maximize problems of external fr...
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assumes that memory is of size for some integer both free and reserved blocks will always be of size for < at any given timethere might be both free and reserved blocks of various sizes the buddy system keeps separate list for free blocks of each size there can be at most such listsbecause there can only be distinct bl...
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chap lists and arrays revisited buddies buddies buddies ( (bfigure example of the buddy system (ablocks of size (bblocks of size discussed in this section the advantage of zones is that some portions of memory can be managed more efficiently the disadvantage is that one zone might fill up while other zones have excess ...
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sec memory management handle memory block figure using handles for dynamic memory management the memory manager returns the address of the handle in response to memory request the handle stores the address of the actual memory block in this waythe memory block might be moved (with its address updated in the handlewitho...
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chap lists and arrays revisited freelist figure example of lisp list variablesincluding the system freelist int[ new int[ ]int[ new int[ ] qwhile in java this would be no problem (due to automatic garbage collection)in languages such as +this would be considered bad form because the original space allocated to is lost ...
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and which parts are garbage in particulara list is kept of all program variablesand any memory locations not reachable from one of these variables are considered to be garbage when the garbage collector executesall unused memory locations are placed in free store for future access this approach has the advantage that i...
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chap lists and arrays revisited figure garbage cycle example all memory elements in the cycle have non-zero reference counts because each element has one pointer to iteven though the entire cycle is garbage the memory objects are linked together without cyclessuch as the unix file system where files can only be organiz...
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sec further reading ( prev curr (bfigure example of the deutsch-schorr-waite garbage collection algorithm (athe initial multilist structure (bthe multilist structure of (aat the instant when link node is being processed by the garbage collection algorithm chain of pointers stretching from variable prev to the head node...
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chap lists and arrays revisited (aa (bc (cfigure some example multilists exercises for each of the following bracket notation descriptionsdraw the equivalent multilist in graphical form such as shown in figure (ahabhcdeihfhgihii (bhabhcdl eil (chl al hl bil hl ii (ashow the bracket notation for the list of figure ( (bs...
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write memory manager allocation and deallocation routines for the situation where all requests and releases follow last-requestedfirst-released (stackorder write memory manager allocation and deallocation routines for the situation where all requests and releases follow last-requestedlast-released (queueorder show the ...
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chap lists and arrays revisited add two matrices implement the memmanager adt shown at the beginning of section use separate linked list to implement the freelist your implementation should work for any of the three sequential-fit methodsfirst fitbest fitand worst fit test your system empirically to determine under wha...