•Depth: length of the alone way from root to node
•Height: length of the longest way from the node to a leaf •Keep kids in a linked list
•Preorder traverse: work at the node is done before its kids are processed •Postorder traverse: work at a node is performed after its kids are evaluated •Binary tree: no node can hold more than two kids

oAverage deepness is O ( rootN ) . O ( logN ) for binary hunt tree
oCan maintain mentions to kids cuz there’s merely 2
•Example of a binary tree: look tree
oLeaves are operands. other nodes contain operators
oInorder traverse: recursively print left kid. so parent. so right •O ( N )
oPostorder traverse: recursively print left subtree. right subtree. so operator > O ( N )
oPreorder traverse: print operator. so recursively publish the left and right subtrees
oConstructing an look tree from a postfix look: read one symbol at a clip ; if operand. make a one-node tree and push it onto a stack. If operator. dad two trees T1. T2 from stack. and organize a new tree whose root is the operator. and whose left and right kids are T2 and T1 ; push new tree onto stack •Binary hunt tree: binary tree with the belongings that for every node X. the value of all points in its left subtree are & lt ; X and the value of all points in the right subtree are & gt ; X oContains: Uses O ( logN ) stack infinite

ofindMin. findMax: crossbeam all the manner left or right from the root oinsert: track down tree as would with contains. lodge it at the terminal oremove: easy if foliage or has one kid ; if two kids ; replace informations in node with smallest informations of right subtree. and recursively cancel that node oLazy omission: if expected figure of omissions is little. merely tag the node as deleted but don’t really do anything ; little clip punishment as deepness doesn’t truly increase oRunning clip of all operations on a node is O ( deepness ) . and the mean deepness is O ( logN ) oIf input is presorted. inserts takes O ( N^2 ) since there are no left kids •AVL Trees

oBinary hunt tree with a balance status: guarantee deepness is O ( logN ) by necessitating that for every node in the tree. the tallness of the left and right subtrees can differ by at most 1 ( tallness of empty tree is -1 ) oMinimum figure of nodes S ( H ) of an AVL tree of tallness H is S ( H ) = S ( h-1 ) + S ( h-2 ) + 1 where S ( 1 ) = 2

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oAll operations O ( logN ) except perchance interpolation
•Violation after infixing into left subtree of left kid. or right subtree of right kid > individual rotary motion •Violation after infixing into right subtree of left kid or left subtree of right kid > dual rotary motion •Splay Trees

oAmortized O ( logN ) cost per operation
oMove accessed nodes to root
oZig-zag: node is a left kid and its parent is a right kid or frailty versa oZig-zig: node and its parent are both left or right kids •Level-order traverse: all nodes at deepness vitamin D processed before any node at d+1 ; non done recursively. it uses a queue alternatively of stack recursion •Set interface: alone operations are insert. take. and search oTreeset maintains order. basic operations take O ( logN ) worst instance •Map interface: aggregation of entries dwelling of keys and their values O.K.s are alone. but several keys can map to the same values oSortedMap: keys maintained in sorted order

oOperations include isEmpty. clear. size. containsKey. acquire. set oNo iterator. but:
• Set keySet ( )
•Collection values ( )
•Set entrySet ( )
oFor an object of type Map. Entry. available methods include •KeyType getKey ( )
•ValueType getValue ( )
•ValueType setValue ( ValueType newValue )
•TreeSet and TreeMap implemented with a balanced binary hunt tree

Ch. 5 Hashing

•Hashing is a technique for infixing. deleting and seeking in O ( N ) norm. so findMin. findMax and publishing the tabular array in order aren’t supported •Hash map maps a key into some figure from 0 to TableSize – 1 and topographic points it in the appropriate cell •If the input keys are whole numbers. so normally Key ( mod TableSize ) works •Want to hold TableSize be premier

•Separate chaining: keep a list of all elements that hash to the same value •Load factor = mean length of a list = figure of elements in table/size oIn an unsuccessful hunt. figure of nodes to analyze is O ( burden ) on norm ; successful hunt requires ~ 1 + ( load/2 ) links to be traversed •Instead of holding linked lists. usage H ( x ) = ( hash ( x ) + degree Fahrenheit ( I ) ) ( mod Tablesize ) where degree Fahrenheit is the hit declaration scheme oGenerally. maintain burden below. 5 for these “probing hash tables” oLinear examining: degree Fahrenheit ( I ) = I ; try cells consecutive with wraparound •Primary bunch: even if tabular array is comparatively empty. blocks of occupied cells signifier which makes hashes near them bad oQuadratic probing ; f ( I ) = i^2

•No warrant of happening an empty cell if tabular array is & gt ; ? full ( or before if size isn’t prime ) •Secondary bunch: elements hashed to same place investigation same alternate cells oDouble Hashing: degree Fahrenheit ( I ) = ihash_2 ( x ) so probe hash_2 ( x ) . 2hash_2 ( ten ) . … •Hash_2 ( x ) = R – x ( mod R ) with R premier & lt ; size is good

oRehash: construct new tabular array. twice every bit large. hash everything with new map •O ( N ) : N elements to rehash. table size about 2N. but really non that bad because it’s infrequent ( must hold been N/2 ) interpolations prior to last rehash. so it basically adds a changeless cost to infix •Can rehash when half full. after failed interpolation. or at certain burden •Standard Library has HashSet and HashMap ( they use separate chaining ) •HashTable utile for:

o1. Graph theory job where nodes have names alternatively of Numberss o2. Symbol tabular array: maintaining path of declared variables in beginning codification o3.
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o4. Online enchantment draughtss
•But they require an estimation of the figure of elements to be used

Ch. 7 Screening

•Bubble kind: O ( N^2 ) but O ( N ) if presorted
•Insertion kind: P base on ballss. at each base on balls move component P left until in right topographic point oO ( N^2 ) norm. O ( N ) on presorted
•Shellsort: increment sequence h1. h2. … . h_t
oAfter a stage. all elements spaced h_k apart are sorted
oWorst-case O ( N^2 )
oHibbard’s sequence 1. 3. 7. … . 2^k – 1 gives worst-case O ( N^3/2 ) •Heapsort: construct a minHeap in O ( N ) . deleteMin N times so O ( NlogN ) for all instances oUses an excess array so O ( N ) infinite

•Mergesort: O ( NlogN ) worst instance. but uses O ( N ) extra space/memory •Quicksort: usage median of left/right/center elements. kind elements smaller and larger than pivot. so unify oPartitioning scheme: move I right. skip over elements smaller than pivot. travel J left. skip over elements larger than pivot oWorst-case pivot is little component = O ( N^2 ) . happens on near-sorted informations oBest-case pivot is in-between = O ( NlogN )

oAverage-case O ( NlogN )

Ch. 8 The Disjoint Set Class

•The equality job is to look into for any a. B if a~b
•Find: returns the name of the equality category incorporating a given component •Add a relation a~b: perform find on a. B so brotherhood the categories •Impossible to make both operations in changeless worst-case. but can make either •Quick-find: array entry of node is name of its category ; makes brotherhood O ( N ) •We start with a wood of singleton trees ; the array representation contains the name of the parent. with -1 for no parent oUnion: merge two trees by doing the parent nexus of one tree’s root nexus to the root of the other tree. O ( 1 ) oFind is relative to depth of the node so worst-case is O ( N ) . or O ( manganese ) for m back-to-back operations oAverage instance depends on the theoretical account but is by and large O ( manganese )

•Union-by-size: do the smaller tree a subtree of the larger. interrupt ties any manner oDepth of a node is ne’er more than logN > discovery is O ( logN ) oHave the array entry of each root contain the negative of the size of its tree. so ab initio all -1. After a brotherhood. the new size is the amount of the old •Requires no excess infinite

oMost theoretical accounts show M operations is O ( M ) norm clip
•Union-by-height: maintain height alternatively of size of tree. and during brotherhoods make the shoal tree a subtree of the deeper one oAlso guarantees depth = O ( logN )
oEasy: tallness merely goes up ( by 1 ) when every bit deep trees are unioned oStore the negative of the tallness. minus an extra 1 ( once more start at -1 ) •Problem: worst instance O ( MlogN ) occurs often ; if there are many more discoveries than brotherhoods the running clip is worse than the quick-find algorithm •Path Compaction

oAfter discovery ( ten ) . every node on the way from ten to root has its parent changed to the root oM operations requires at most O ( MlogN ) clip ; unknown mean oNot compatible with brotherhood by tallness since it changes highs •When we use both union-by-size and path compaction. about additive worst instance oTheta ( M*Ackerman’s map ) . where Ackerman’s is merely somewhat faster than changeless. so it’s non rather additive oBook proves any M union/find operations is O ( Mlog*N ) where log*N = figure of times needed to use log to N until N


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