Elementary Data Structures and Hash Tables
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1 Elementary Data Structures and Hash Tables Antonio Carzaniga Faculty of Informatics University of Lugano November 24, 2006
2 Outline Common concepts and notation Stacks Queues Linked lists Trees Direct-access tables Hash tables
3 Concepts A data structure is a way to organize and store information to facilitate access, or for other purposes
4 Concepts A data structure is a way to organize and store information to facilitate access, or for other purposes A data structure has an interface consisting of procedures for adding, deleting, accessing, reorganizing, etc.
5 Concepts A data structure is a way to organize and store information 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 meta-data e.g., a heap needs an array A to store the keys plus a variable heap-size(a) to remember how many elements are in the heap
6 Notation
7 Notation Data is generally represented with simple variables names e.g., A for the elements of a heap A function notation for meta-data e.g., heap-size(a) for the number of elements in a heap
8 Notation Data is generally represented with simple variables names e.g., A for the elements of a heap A function notation for meta-data e.g., heap-size(a) for the number of elements in a heap Java class Heap { int [] A; int size; } Heap h; //... h.a[0] = 2; h.size = h.size + 1; Our pseudo-code notation A heap-size(a) A[1] 2 heap-size(a) heap-size(a) + 1
9 Notation (2) Another example
10 Notation (2) Another example Java class ListElement { int key; ListElement prev; ListElement next; } class List { ListElement head; } List l; ListElement x; int k; //... k = l.head.next.key; x = l.head.next.next; Our pseudo-code notation x key(x) prev(x) next(x) L head(l) k key(next(head(l))) x next(next(head(l)))
11 Stacks A ubiquitous data structure
12 Stacks A ubiquitous data structure Interface STACK-EMPTY(S) returns TRUE if and only if S is empty PUSH(S, x) pushes the value x onto the stack S empty POP(S) extracts and returns the value on the top of the stack S
13 Stacks A ubiquitous data structure Interface STACK-EMPTY(S) returns TRUE if and only if S is empty PUSH(S, x) pushes the value x onto the stack S empty POP(S) extracts and returns the value on the top of the stack S Implementation using an array using a linked
14 A Stack Implementation
15 A Stack Implementation Array-based implementation S is an array that holds the elements of the stack top(s) is the current position of the top element of S
16 A Stack Implementation Array-based implementation S is an array that holds the elements of the stack top(s) is the current position of the top element of S STACK-EMPTY(S) 1 if top(s) = 0 2 then return TRUE 3 else return FALSE PUSH(S,X) 1 top(s) top(s) S[top(S)] x POP(S) 1 if STACK-EMPTY(S) 2 then error underflow 3 else top(s) top(s) 1 4 return S[top(S) 1]
17 Queues Interface ENQUEUE(Q, x) adds element x at the back of queue Q DEQUEUE(Q) extracts the element at the head of queue Q
18 Queues Interface ENQUEUE(Q, x) adds element x at the back of queue Q DEQUEUE(Q) extracts the element at the head of queue Q Implementation Q is an array of total length length(q) head(q) is the position of the head of the queue tail(q) is the first empty position at the tail of the queue
19 Enqueue ENQUEUE(Q,X) 1 if QUEUE-FULL(Q) 2 then error overflow 3 else Q[tail(Q)] x 4 if tail(q) < length(q) 5 then tail(q) tail(q) else tail(q) 1
20 Enqueue ENQUEUE(Q,X) 1 if QUEUE-FULL(Q) 2 then error overflow 3 else Q[tail(Q)] x 4 if tail(q) < length(q) 5 then tail(q) tail(q) else tail(q) 1 head(q) tail(q)
21 Enqueue ENQUEUE(Q,X) 1 if QUEUE-FULL(Q) 2 then error overflow 3 else Q[tail(Q)] x 4 if tail(q) < length(q) 5 then tail(q) tail(q) else tail(q) 1 head(q) tail(q)
22 Enqueue ENQUEUE(Q,X) 1 if QUEUE-FULL(Q) 2 then error overflow 3 else Q[tail(Q)] x 4 if tail(q) < length(q) 5 then tail(q) tail(q) else tail(q) 1 head(q) tail(q)
23 Enqueue ENQUEUE(Q,X) 1 if QUEUE-FULL(Q) 2 then error overflow 3 else Q[tail(Q)] x 4 if tail(q) < length(q) 5 then tail(q) tail(q) else tail(q) 1 head(q) tail(q)
24 Enqueue ENQUEUE(Q,X) 1 if QUEUE-FULL(Q) 2 then error overflow 3 else Q[tail(Q)] x 4 if tail(q) < length(q) 5 then tail(q) tail(q) else tail(q) 1 head(q) tail(q)
25 Enqueue ENQUEUE(Q,X) 1 if QUEUE-FULL(Q) 2 then error overflow 3 else Q[tail(Q)] x 4 if tail(q) < length(q) 5 then tail(q) tail(q) else tail(q) 1 head(q) tail(q)
26 Enqueue ENQUEUE(Q,X) 1 if QUEUE-FULL(Q) 2 then error overflow 3 else Q[tail(Q)] x 4 if tail(q) < length(q) 5 then tail(q) tail(q) else tail(q) 1 head(q) tail(q)
27 Dequeue DEQUEUE(Q) 1 if QUEUE-EMPTY(Q) 2 then error underflow 3 else x Q[head(Q)] 4 if head(q) < length(q) 5 then head(q) head(q) else head(q) 1 7 return x
28 Dequeue DEQUEUE(Q) 1 if QUEUE-EMPTY(Q) 2 then error underflow 3 else x Q[head(Q)] 4 if head(q) < length(q) 5 then head(q) head(q) else head(q) 1 7 return x head(q) tail(q)
29 Dequeue DEQUEUE(Q) 1 if QUEUE-EMPTY(Q) 2 then error underflow 3 else x Q[head(Q)] 4 if head(q) < length(q) 5 then head(q) head(q) else head(q) 1 7 return x head(q) tail(q)
30 Dequeue DEQUEUE(Q) 1 if QUEUE-EMPTY(Q) 2 then error underflow 3 else x Q[head(Q)] 4 if head(q) < length(q) 5 then head(q) head(q) else head(q) 1 7 return x head(q) tail(q)
31 Dequeue DEQUEUE(Q) 1 if QUEUE-EMPTY(Q) 2 then error underflow 3 else x Q[head(Q)] 4 if head(q) < length(q) 5 then head(q) head(q) else head(q) 1 7 return x head(q) tail(q)
32 Linked Lists Interface LIST-INSERT(L, x) adds element x at beginning of a list L LIST-DELETE(L, x) removes element x from a list L LIST-SEARCH(L, k) finds an element whose key is k in a list L
33 Linked Lists Interface LIST-INSERT(L, x) adds element x at beginning of a list L LIST-DELETE(L, x) removes element x from a list L LIST-SEARCH(L, k) finds an element whose key is k in a list L Implementation a doubly-linked list head(l) is the first element each element x has two links prev(x) and next(x) to the previous and next elements, respectively
34 Trees Structure fixed branching unbounded branching
35 Trees Structure fixed branching unbounded branching Implementation for each node x = root(t), parent(x) is x s parent node fixed branching: e.g., left-child(x) and right-child(x) in a binary tree unbounded branching: left-child(x) is x s first (leftmost) child right-sibling(x) is x closest sibling to the right
36 Complexity STACK-EMPTY PUSH POP ENQUEUE DEQUEUE LIST-INSERT LIST-DELETE LIST-SEARCH O(1) O(1) O(1) O(1) O(1) O(1) O(1) O(n)
37 Dictionary A dictionary is an abstract data structure that represents a set of elements (or keys) a dynamic set
38 Dictionary A dictionary is an abstract data structure that represents a set of elements (or keys) a dynamic set Interface (generic interface) INSERT(D, k) adds a key k to the dictionary D DELETE(D, k) removes key k from D SEARCH(D, x) tells whether D contains a key k
39 Dictionary A dictionary is an abstract data structure that represents a set of elements (or keys) a dynamic set Interface (generic interface) INSERT(D, k) adds a key k to the dictionary D DELETE(D, k) removes key k from D SEARCH(D, x) tells whether D contains a key k Implementation many (concrete) data structures
40 Dictionary A dictionary is an abstract data structure that represents a set of elements (or keys) a dynamic set Interface (generic interface) INSERT(D, k) adds a key k to the dictionary D DELETE(D, k) removes key k from D SEARCH(D, x) tells whether D contains a key k Implementation many (concrete) data structures hash tables
41 Direct-Address Table A direct-address table implements a dictionary
42 Direct-Address Table A direct-address table implements a dictionary The universe of keys is U = {1, 2,...,M}
43 Direct-Address Table A direct-address table implements a dictionary The universe of keys is U = {1, 2,...,M} Implementation an array T of size M each key has its own position in M
44 Direct-Address Table A direct-address table implements a dictionary The universe of keys is U = {1, 2,...,M} Implementation an array T of size M each key has its own position in M 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]
45 Direct-Address Table (2) All direct-address table operations are O(1)
46 All direct-address table operations are O(1) Direct-Address Table (2) So why isn t every set implemented with a direct-address table? The space complexity is Θ( U )... U is usually a very large number the size of the represented set is usually much smaller than U, which means that a direct-address table usually wastes a lot of space
47 All direct-address table operations are O(1) Direct-Address Table (2) So why isn t every set implemented with a direct-address table? The space complexity is Θ( U )... U is usually a very large number the size of the represented set is usually much smaller than U, which means that a direct-address table usually wastes a lot of space Can we have the benefits of direct-address tables but with much smaller tables?
48 Hash Table Idea we use a table T with length(t) U we map each key k U to a position in T through a hash function h : U {1,...,length(T)}
49 Hash Table Idea we use a table T with length(t) U we map each key k U to a position in T through a hash function h : U {1,...,length(T)} HASH-INSERT(T, k) 1 T[h(k)] TRUE HASH-DELETE(T, k) 1 T[h(k)] FALSE HASH-SEARCH(T, k) 1 return T[h(k)]
50 Hash Table Idea we use a table T with length(t) U we map each key k U to a position in T through a hash function h : U {1,...,length(T)} HASH-INSERT(T, k) 1 T[h(k)] TRUE HASH-DELETE(T, k) 1 T[h(k)] FALSE HASH-SEARCH(T, k) 1 return T[h(k)] Are these algorithms correct?
51 Hash Table Idea we use a table T with length(t) U we map each key k U to a position in T through a hash function h : U {1,...,length(T)} HASH-INSERT(T, k) 1 T[h(k)] TRUE HASH-DELETE(T, k) 1 T[h(k)] FALSE HASH-SEARCH(T, k) 1 return T[h(k)] Are these algorithms correct? No!
52 Hash Table Idea we use a table T with length(t) U we map each key k U to a position in T through a hash function h : U {1,...,length(T)} HASH-INSERT(T, k) 1 T[h(k)] TRUE HASH-DELETE(T, k) 1 T[h(k)] FALSE HASH-SEARCH(T, k) 1 return T[h(k)] Are these algorithms correct? No! What if two distinct keys k 1 = k 2 collide? (I.e., h(k 1 ) = h(k 2 ))
53 Hash Table U T
54 Hash Table U k 1 T
55 Hash Table U k 1 T k 2
56 Hash Table U k 1 T k 2 k 3
57 Hash Table U k 1 k 4 T k 2 k 3
58 Hash Table U k 1 k 4 T k1 k 3 k 2 k 4 k 2 k 3
59 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
60 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)
61 Hash Table U k 1 k 4 T CHAINED-HASH-INSERT(T, k) 1 return LIST-INSERT(T[h(k)], k) k1 k 2 k 3 k 4 k 2 load factor n length(t) α = k 3 CHAINED-HASH-SEARCH(T, k) 1 return LIST-SEARCH(T[h(k)], k)
62 Analysis We assume uniform hashing for our hash function h : U {1... T } (where T = length(t))
63 Analysis We assume uniform hashing for our hash function h : U {1... T } (where T = length(t)) Pr[h(k) = i] = 1 T for all i {1... T }
64 Analysis We assume uniform hashing for our hash function h : U {1... T } (where T = length(t)) Pr[h(k) = i] = 1 T for all i {1... T } So, given n distinct keys, the expected length n i of the linked list at position i is E[n i ] = n T = α
65 Analysis We assume uniform hashing for our hash function h : U {1... T } (where T = length(t)) Pr[h(k) = i] = 1 T for all i {1... T } So, given n distinct keys, the expected length n i of the linked list at position i is E[n i ] = n T = α We further assume that h(k) can be computed in O(1) time
66 Analysis We assume uniform hashing for our hash function h : U {1... T } (where T = length(t)) Pr[h(k) = i] = 1 T for all i {1... T } So, given n distinct keys, the expected length n i of the linked list at position i is E[n i ] = n T = α We further assume that h(k) can be computed in O(1) time Therefore, the complexity of CHAINED-HASH-SEARCH is Θ(1 +α)
67 Open-Address Hash Table U T
68 Open-Address Hash Table U k 1 T
69 Open-Address Hash Table U k 1 T k 1
70 Open-Address Hash Table U k 1 T k 1 k 2
71 Open-Address Hash Table U k 1 T k 1 k 2 k 2
72 Open-Address Hash Table U k 1 T k 1 k 2 k 2 k 3
73 Open-Address Hash Table U k 1 T k 1 k 3 k 2 k 2 k 3
74 Open-Address Hash Table U k 1 k 4 T k 1 k 3 k 2 k 2 k 3
75 Open-Address Hash Table U k 1 k 4 T k 1 k 3 k 2 k 2 k 3 k 4
76 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 for i 1 to length(t) 3 do if T[j] = NIL 4 then T[j] k 5 return j 6 elseif j < length(t) 7 then j j else j 1 9 error overflow
77 Open-Addressing (2) Idea: instead of using linked lists, we can store the all elements in the table this implies α 1
78 Open-Addressing (2) Idea: instead of using linked lists, we can store the all elements in the table this implies α 1 When a collision occurs, we simply find another free cell in T
79 Open-Addressing (2) Idea: instead of using linked lists, we can store the all elements in the table this implies α 1 When a collision occurs, we simply find another free cell in T A sequential probe may not be optimal can you figure out why?
80 Open-Addressing (3) HASH-INSERT(T, k) 1 for i 1 to length(t) 2 j h(k, i) 3 do if T[j] = NIL 4 then T[j] k 5 return j 6 error overflow
81 Open-Addressing (3) HASH-INSERT(T, k) 1 for i 1 to length(t) 2 j h(k, i) 3 do if T[j] = NIL 4 then T[j] k 5 return j 6 error overflow Notice that h(k, ) must be a permutation i.e., h(k, 1), h(k, 2),...,h(k, T ) must cover the entire table T
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