Elementary Data Structures and Hash Tables
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1 Elementary Data Structures and Hash Tables Antonio Carzaniga Faculty of Informatics Università della Svizzera italiana March 23, 2016
2 Outline Common concepts and notation Stacks Queues Linked lists Trees Direct-access tables Hash tables
3 Concepts Adatastructureisawaytoorganizeandstoreinformation to facilitate access, or for other purposes
4 Concepts Adatastructureisawaytoorganizeandstoreinformation to facilitate access, or for other purposes A data structure has an interface consisting of procedures for adding, deleting, accessing, reorganizing, etc.
5 Concepts Adatastructureisawaytoorganizeandstoreinformation to facilitate access, or for other purposes A data structure has an interface consisting of procedures for adding, deleting, accessing, reorganizing, etc. A data structure stores data and possibly meta-data
6 Concepts Adatastructureisawaytoorganizeandstoreinformation to facilitate access, or for other purposes A data structure has an interface consisting of procedures for adding, deleting, accessing, reorganizing, etc. A data structure stores data and possibly meta-data e.g.,aheapneedsanarrayatostorethekeys,plusavariable A.heap-sizetorememberhowmanyelementsareintheheap
7 The ubiquitous last-in first-out container(lifo) Stack
8 Stack The ubiquitous last-in first-out container(lifo) Interface STACK-EMPTY(S)returnsTRUEifandonlyifSisempty PUSH(S,x)pushesthevaluexontothestackS POP(S)extractsandreturnsthevalueonthetopofthestackS
9 Stack The ubiquitous last-in first-out container(lifo) Interface STACK-EMPTY(S)returnsTRUEifandonlyifSisempty PUSH(S,x)pushesthevaluexontothestackS POP(S)extractsandreturnsthevalueonthetopofthestackS Implementation usinganarray usingalinkedlist...
10 A Stack Implementation
11 Array-based implementation A Stack Implementation
12 A Stack Implementation Array-based implementation Sisanarraythatholdstheelementsofthestack S.topisthecurrentpositionofthetopelementofS
13 A Stack Implementation Array-based implementation Sisanarraythatholdstheelementsofthestack S.topisthecurrentpositionofthetopelementofS STACK-EMPTY(S) 1 ifs.top==0 2 return TRUE 3 else return FALSE
14 A Stack Implementation Array-based implementation Sisanarraythatholdstheelementsofthestack S.topisthecurrentpositionofthetopelementofS STACK-EMPTY(S) 1 ifs.top==0 2 return TRUE 3 else return FALSE PUSH(S,X) 1 S.top =S.top+1 2 S[S.top] =x POP(S) 1 if STACK-EMPTY(S) 2 error underflow 3 elses.top =S.top 1 4 return S[S.top + 1]
15 The ubiquitous first-in first-out container(fifo) Queue
16 Queue The ubiquitous first-in first-out container(fifo) Interface ENQUEUE(Q,x)addselementxatthebackofqueueQ DEQUEUE(Q)extractstheelementattheheadofqueueQ
17 Queue The ubiquitous first-in first-out container(fifo) Interface ENQUEUE(Q,x)addselementxatthebackofqueueQ DEQUEUE(Q)extractstheelementattheheadofqueueQ Implementation QisanarrayoffixedlengthQ.length i.e.,qholdsatmostq.lengthelements enqueueing more than Q elements causes an overflow error Q.headisthepositionofthe head ofthequeue Q.tailisthefirstemptypositionatthetailofthequeue
18 Enqueue ENQUEUE(Q,X) 1 if Q.queue-full 2 error overflow 3 elseq[q.tail] =x 4 if Q.tail < Q.length 5 Q.tail =Q.tail+1 6 elseq.tail =1 7 if Q.tail == Q.head 8 Q.queue-full = TRUE 9 Q.queue-empty = FALSE
19 Enqueue ENQUEUE(Q,X) 1 if Q.queue-full 2 error overflow 3 elseq[q.tail] =x 4 if Q.tail < Q.length 5 Q.tail =Q.tail+1 6 elseq.tail =1 7 if Q.tail == Q.head 8 Q.queue-full = TRUE 9 Q.queue-empty = FALSE Q.head Q.tail
20 Enqueue ENQUEUE(Q,X) 1 if Q.queue-full 2 error overflow 3 elseq[q.tail] =x 4 if Q.tail < Q.length 5 Q.tail =Q.tail+1 6 elseq.tail =1 7 if Q.tail == Q.head 8 Q.queue-full = TRUE 9 Q.queue-empty = FALSE Q.head Q.tail
21 Enqueue ENQUEUE(Q,X) 1 if Q.queue-full 2 error overflow 3 elseq[q.tail] =x 4 if Q.tail < Q.length 5 Q.tail =Q.tail+1 6 elseq.tail =1 7 if Q.tail == Q.head 8 Q.queue-full = TRUE 9 Q.queue-empty = FALSE Q.head Q.tail
22 Enqueue ENQUEUE(Q,X) 1 if Q.queue-full 2 error overflow 3 elseq[q.tail] =x 4 if Q.tail < Q.length 5 Q.tail =Q.tail+1 6 elseq.tail =1 7 if Q.tail == Q.head 8 Q.queue-full = TRUE 9 Q.queue-empty = FALSE Q.head Q.tail
23 Enqueue ENQUEUE(Q,X) 1 if Q.queue-full 2 error overflow 3 elseq[q.tail] =x 4 if Q.tail < Q.length 5 Q.tail =Q.tail+1 6 elseq.tail =1 7 if Q.tail == Q.head 8 Q.queue-full = TRUE 9 Q.queue-empty = FALSE Q.head Q.tail
24 Enqueue ENQUEUE(Q,X) 1 if Q.queue-full 2 error overflow 3 elseq[q.tail] =x 4 if Q.tail < Q.length 5 Q.tail =Q.tail+1 6 elseq.tail =1 7 if Q.tail == Q.head 8 Q.queue-full = TRUE 9 Q.queue-empty = FALSE Q.head Q.tail
25 Enqueue ENQUEUE(Q,X) 1 if Q.queue-full 2 error overflow 3 elseq[q.tail] =x 4 if Q.tail < Q.length 5 Q.tail =Q.tail+1 6 elseq.tail =1 7 if Q.tail == Q.head 8 Q.queue-full = TRUE 9 Q.queue-empty = FALSE Q.head Q.tail
26 Dequeue DEQUEUE(Q) 1 if Q.queue-empty 2 error underflow 3 elsex =Q[Q.head] 4 if Q.head < Q.length 5 Q.head =Q.head+1 6 elseq.head =1 7 if Q.tail == Q.head 8 Q.queue-empty = TRUE 9 Q.queue-full = FALSE 10 return x
27 Dequeue DEQUEUE(Q) 1 if Q.queue-empty 2 error underflow 3 elsex =Q[Q.head] 4 if Q.head < Q.length 5 Q.head =Q.head+1 6 elseq.head =1 7 if Q.tail == Q.head 8 Q.queue-empty = TRUE 9 Q.queue-full = FALSE 10 return x Q.head Q.tail
28 Dequeue DEQUEUE(Q) 1 if Q.queue-empty 2 error underflow 3 elsex =Q[Q.head] 4 if Q.head < Q.length 5 Q.head =Q.head+1 6 elseq.head =1 7 if Q.tail == Q.head 8 Q.queue-empty = TRUE 9 Q.queue-full = FALSE 10 return x Q.head Q.tail
29 Dequeue DEQUEUE(Q) 1 if Q.queue-empty 2 error underflow 3 elsex =Q[Q.head] 4 if Q.head < Q.length 5 Q.head =Q.head+1 6 elseq.head =1 7 if Q.tail == Q.head 8 Q.queue-empty = TRUE 9 Q.queue-full = FALSE 10 return x Q.head Q.tail
30 Dequeue DEQUEUE(Q) 1 if Q.queue-empty 2 error underflow 3 elsex =Q[Q.head] 4 if Q.head < Q.length 5 Q.head =Q.head+1 6 elseq.head =1 7 if Q.tail == Q.head 8 Q.queue-empty = TRUE 9 Q.queue-full = FALSE 10 return x Q.head Q.tail
31 Linked List Interface LIST-INSERT(L,x)addselementxatbeginningofalistL LIST-DELETE(L,x)removeselementxfromalistL LIST-SEARCH(L,k)findsanelementwhosekeyiskinalistL
32 Linked List Interface LIST-INSERT(L,x)addselementxatbeginningofalistL LIST-DELETE(L,x)removeselementxfromalistL LIST-SEARCH(L,k)findsanelementwhosekeyiskinalistL Implementation a doubly-linked list eachelementxhastwo links x.prevandx.nexttothe previous and next elements, respectively eachelementxholdsakeyx.key itisconvenienttohaveadummy sentinel elementl.nil
33 Linked List With a Sentinel LIST-INIT(L) 1 L.nil.prev = L.nil 2 L.nil.next = L.nil LIST-INSERT(L, x) 1 x.next = L.nil.next 2 L.nil.next.prev = x 3 L.nil.next =x 4 x.prev =L.nil LIST-SEARCH(L, k) 1 x =L.nil.next 2 whilex L.nil x.key k 3 x =x.next 4 returnx
34 Trees Structure fixed branching unbounded branching
35 Trees Structure fixed branching unbounded branching Implementation foreachnodex T.root,x.parentisx sparentnode fixed branching: e.g., x. left-child and x. right-child in a binary tree unbounded branching: x. left-child is x s first(leftmost) child x. right-sibling is x closest sibling to the right
36 Complexity
37 Algorithm Complexity Complexity
38 Complexity Algorithm Complexity STACK-EMPTY
39 Complexity Algorithm Complexity STACK-EMPTY O(1) PUSH
40 Complexity Algorithm Complexity STACK-EMPTY O(1) PUSH POP ENQUEUE DEQUEUE O(1) O(1) O(1) O(1) LIST-INSERT
41 Complexity Algorithm Complexity STACK-EMPTY O(1) PUSH POP ENQUEUE DEQUEUE LIST-INSERT O(1) O(1) O(1) O(1) O(1) LIST-DELETE
42 Complexity Algorithm Complexity STACK-EMPTY O(1) PUSH POP ENQUEUE DEQUEUE LIST-INSERT LIST-DELETE O(1) O(1) O(1) O(1) O(1) O(1) LIST-SEARCH
43 Complexity Algorithm Complexity STACK-EMPTY O(1) PUSH POP ENQUEUE DEQUEUE LIST-INSERT LIST-DELETE O(1) O(1) O(1) O(1) O(1) O(1) LIST-SEARCH Θ(n)
44 Dictionary A dictionary is an abstract data structure that represents a set of elements(or keys) adynamicset
45 Dictionary A dictionary is an abstract data structure that represents a set of elements(or keys) adynamicset Interface(generic interface) INSERT(D,k)addsakeyktothedictionaryD DELETE(D,k)removeskeykfromD SEARCH(D,k)tellswhetherDcontainsakeyk
46 Dictionary A dictionary is an abstract data structure that represents a set of elements(or keys) adynamicset Interface(generic interface) INSERT(D,k)addsakeyktothedictionaryD DELETE(D,k)removeskeykfromD SEARCH(D,k)tellswhetherDcontainsakeyk Implementation many(concrete) data structures
47 Dictionary A dictionary is an abstract data structure that represents a set of elements(or keys) adynamicset Interface(generic interface) INSERT(D,k)addsakeyktothedictionaryD DELETE(D,k)removeskeykfromD SEARCH(D,k)tellswhetherDcontainsakeyk Implementation many(concrete) data structures hash tables
48 A direct-address table implements a dictionary Direct-Address Table
49 Direct-Address Table A direct-address table implements a dictionary TheuniverseofkeysisU = {1,2,...,M}
50 Direct-Address Table A direct-address table implements a dictionary TheuniverseofkeysisU = {1,2,...,M} Implementation anarraytofsizem eachkeyhasitsownpositionint
51 Direct-Address Table A direct-address table implements a dictionary TheuniverseofkeysisU = {1,2,...,M} Implementation anarraytofsizem eachkeyhasitsownpositionint DIRECT-ADDRESS-INSERT(T, k) 1 T[k] =TRUE DIRECT-ADDRESS-DELETE(T, k) 1 T[k] =FALSE DIRECT-ADDRESS-SEARCH(T, k) 1 return T[k]
52 Complexity Direct-Address Table (2)
53 Direct-Address Table (2) Complexity All direct-address table operations are O(1)!
54 Direct-Address Table (2) Complexity All direct-address table operations are O(1)! So why isn t every set implemented with a direct-address table?
55 Direct-Address Table (2) Complexity All direct-address table operations are O(1)! So why isn t every set implemented with a direct-address table? ThespacecomplexityisΘ( U ) U istypicallyaverylargenumber Uistheuniverseofkeys! therepresentedsetistypicallymuchsmallerthan U i.e.,adirect-addresstableusuallywastesalotofspace
56 Direct-Address Table (2) Complexity All direct-address table operations are O(1)! So why isn t every set implemented with a direct-address table? ThespacecomplexityisΘ( U ) U istypicallyaverylargenumber Uistheuniverseofkeys! therepresentedsetistypicallymuchsmallerthan U i.e.,adirect-addresstableusuallywastesalotofspace Canwehavethebenefitsofadirect-addresstablebutwitha table of reasonable size?
57 Hash Table Idea useatabletwith T U mapeachkeyk UtoapositioninT,usingahashfunction h :U {1,..., T }
58 Hash Table Idea useatabletwith T U mapeachkeyk UtoapositioninT,usingahashfunction h :U {1,..., T } HASH-INSERT(T, k) 1 T[h(k)] =TRUE HASH-DELETE(T, k) 1 T[h(k)] =FALSE HASH-SEARCH(T, k) 1 returnt[h(k)]
59 Hash Table Idea useatabletwith T U mapeachkeyk UtoapositioninT,usingahashfunction h :U {1,..., T } HASH-INSERT(T, k) 1 T[h(k)] =TRUE HASH-DELETE(T, k) 1 T[h(k)] =FALSE HASH-SEARCH(T, k) 1 returnt[h(k)] Are these algorithms correct?
60 Hash Table Idea useatabletwith T U mapeachkeyk UtoapositioninT,usingahashfunction h :U {1,..., T } HASH-INSERT(T, k) 1 T[h(k)] =TRUE HASH-DELETE(T, k) 1 T[h(k)] =FALSE HASH-SEARCH(T, k) 1 returnt[h(k)] Are these algorithms correct? No!
61 Hash Table Idea useatabletwith T U mapeachkeyk UtoapositioninT,usingahashfunction h :U {1,..., T } HASH-INSERT(T, k) 1 T[h(k)] =TRUE HASH-DELETE(T, k) 1 T[h(k)] =FALSE HASH-SEARCH(T, k) 1 returnt[h(k)] Are these algorithms correct? No! Whatiftwodistinctkeysk 1 k 2 collide?(i.e.,h(k 1 ) =h(k 2 ))
62 Hash Table U T
63 Hash Table U k 1 T
64 Hash Table U k 1 T k 2
65 Hash Table U k 1 T k 2 k 3
66 Hash Table U k 1 k 4 T k 2 k 3
67 Hash Table U k 1 k 4 T k1 k 3 k 2 k 4 k 2 k 3
68 Hash Table U k 1 k 4 T CHAINED-HASH-INSERT(T, k) 1 return LIST-INSERT(T[h(k)], k) k1 k 3 k 2 k 4 k 2 k 3
69 Hash Table U k 1 k 4 T CHAINED-HASH-INSERT(T, k) 1 return LIST-INSERT(T[h(k)], k) k1 k 3 k 2 k 4 k 2 k 3 CHAINED-HASH-SEARCH(T, k) 1 return LIST-SEARCH(T[h(k)], k)
70 Hash Table U k 1 k 4 T CHAINED-HASH-INSERT(T, k) 1 return LIST-INSERT(T[h(k)], k) k 2 k1 k 3 k 4 k 2 load factor α = n T k 3 CHAINED-HASH-SEARCH(T, k) 1 return LIST-SEARCH(T[h(k)], k)
71 We assume uniform hashing for our hash function h :U {1... T }(where T =T.length) Analysis
72 Analysis We assume uniform hashing for our hash function h :U {1... T }(where T =T.length) Pr[h(k) =i] = 1 T foralli {1... T } (The formalism is actually a bit more complicated.)
73 Analysis We assume uniform hashing for our hash function h :U {1... T }(where T =T.length) Pr[h(k) =i] = 1 T foralli {1... T } (The formalism is actually a bit more complicated.) So,givenndistinctkeys,theexpectedlengthn i ofthelinked listatpositioniis E[n i ] = n T = α
74 Analysis We assume uniform hashing for our hash function h :U {1... T }(where T =T.length) Pr[h(k) =i] = 1 T foralli {1... T } (The formalism is actually a bit more complicated.) So,givenndistinctkeys,theexpectedlengthn i ofthelinked listatpositioniis E[n i ] = n T = α Wefurtherassumethath(k)canbecomputedinO(1)time
75 Analysis We assume uniform hashing for our hash function h :U {1... T }(where T =T.length) Pr[h(k) =i] = 1 T foralli {1... T } (The formalism is actually a bit more complicated.) So,givenndistinctkeys,theexpectedlengthn i ofthelinked listatpositioniis E[n i ] = n T = α Wefurtherassumethath(k)canbecomputedinO(1)time Therefore, the complexity of CHAINED-HASH-SEARCH is Θ(1+α)
76 Open-Address Hash Table U T
77 Open-Address Hash Table U k 1 T
78 Open-Address Hash Table U k 1 T k 1
79 Open-Address Hash Table U k 1 T k 1 k 2
80 Open-Address Hash Table U k 1 T k 1 k 2 k 2
81 Open-Address Hash Table U k 1 T k 1 k 2 k 2 k 3
82 Open-Address Hash Table U k 1 T k 1 k 3 k 2 k 2 k 3
83 Open-Address Hash Table U k 1 k 4 T k 1 k 3 k 2 k 2 k 3
84 Open-Address Hash Table U k 1 k 4 T k 1 k 3 k 2 k 2 k 3 k 4
85 Open-Address Hash Table U k 1 k 4 T k 1 k 3 k 2 k 2 k 4 k 3 HASH-INSERT(T, k) 1 j =h(k) 2 fori =1toT.length 3 ift[j]==nil 4 T[j] =k 5 return j 6 elseifj<t.length 7 j =j+1 8 elsej =1 9 error overflow
86 Open-Addressing (2) Idea:insteadofusinglinkedlists,wecanstoreallthe elements in the table thisimpliesα 1
87 Open-Addressing (2) Idea:insteadofusinglinkedlists,wecanstoreallthe elements in the table thisimpliesα 1 Whenacollisionoccurs,wesimplyfindanotherfreecellinT
88 Open-Addressing (2) Idea:insteadofusinglinkedlists,wecanstoreallthe elements in the table thisimpliesα 1 Whenacollisionoccurs,wesimplyfindanotherfreecellinT A sequential probe may not be optimal canyoufigureoutwhy?
89 Open-Addressing (3) HASH-INSERT(T, k) 1 fori =1toT.length 2 j =h(k,i) 3 ift[j]==nil 4 T[j] =k 5 return j 6 error overflow
90 Open-Addressing (3) HASH-INSERT(T, k) 1 fori =1toT.length 2 j =h(k,i) 3 ift[j]==nil 4 T[j] =k 5 return j 6 error overflow Noticethath(k, )mustbeapermutation i.e.,h(k,1),h(k,2),...,h(k, T )mustcovertheentiretablet
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