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0 1 2 3 4 025-612-0001 981-101-0002 451-229-0004 Chapter 9: Maps, Dictionaries, Hashing Nancy Amato Parasol Lab, Dept. CSE, Texas A&M University Acknowledgement: These slides are adapted from slides provided with Data Structures and Algorithms in C++, Goodrich, Tamassia and Mount (Wiley 2004) http://parasol.tamu.edu Outline and Reading • • • • Map ADT (§9.1) Dictionary ADT (§9.5) Ordered Maps (§9.3) Hash Tables (§9.2) Dictionaries & Hashing Dictionaries 2 4/13/2015 1:29 AM Map ADT • The map ADT models a searchable collection of keyelement items • The main operations of a map are searching, inserting, and deleting items • Multiple items with the same key are not allowed • Applications: • address book • credit card authorization • mapping host names (e.g., cs16.net) to internet addresses (e.g., 128.148.34.101) • Map ADT methods: • find(k): if M has an entry with key k, return an iterator p referring to this element, else, return special end iterator. • put(k, v): if M has no entry with key k, then add entry (k, v) to M, otherwise replace the value of the entry with v; return iterator to the inserted/modified entry • erase(k) or erase(p): remove from M entry with key k or iterator p; An error occurs if there is no such element. • size(), empty() Dictionaries & Hashing Dictionaries 3 4/13/2015 1:29 AM Map - Direct Address Table • A direct address table is a map in which • The keys are in the range {0,1,2,…,N} • Stored in an array of size N - T[0,N] • Item with key k stored in T[k] • Performance: • insertItem, find, and removeElement all take O(1) time • Space - requires space O(N), independent of n, the number of items stored in the map • The direct address table is not space efficient unless the range of the keys is close to the number of elements to be stored in the map, I.e., unless n is close to N. Dictionaries & Hashing Dictionaries 4 4/13/2015 1:29 AM Dictionary ADT • The dictionary ADT models a • Dictionary ADT methods: searchable collection of key• find(k): if the dictionary has an element items entry with key k, returns an iterator p to an arbitrary elt. • The main difference from a map • findAll(k): Return iterators (b,e) s.t. is that multiple items with the that all entries with key k are same key are allowed between them. • Any data structure that supports • insert(k, v): insert entry (k, v) into a dictionary also supports a map D, return iterator to it. • Applications: • Dictionary which has multiple definitions for the same word • erase(k), erase(p): remove arbitrary entry with key k or entry referenced by p. Error occurs if there is no such entry • Begin(), end(): return iterator to first or just beyond last entry of D • size(), isEmpty() Dictionaries & Hashing Dictionaries 5 4/13/2015 1:29 AM Dictionary - Log File (unordered sequence implementation) • A log file is a dictionary implemented by means of an unsorted sequence • We store the items of the dictionary in a sequence (based on a doubly-linked lists or a circular array), in arbitrary order • Performance: • insert takes O(1) time since we can insert the new item at the beginning or at the end of the sequence • find and erase take O(n) time since in the worst case (the item is not found) we traverse the entire sequence to look for an item with the given key • Space - can be O(n), where n is the number of elements in the dictionary • The log file is effective only for dictionaries of small size or for dictionaries on which insertions are the most common operations, while searches and removals are rarely performed (e.g., historical record of logins to a workstation) Dictionaries & Hashing Dictionaries 6 4/13/2015 1:29 AM Map/Dictionary implementations • n - #elements in map/Dictionary Vectors Insert Find Space Log File O(1) O(n) O(n) Direct Address Table (map only) O(1) O(1) O(N) 4/13/2015 1:29 AM 7 Ordered Map/Dictionary ADT • An Ordered Map/Dictionary supports the usual map/dictionary operations, but also maintains an order relation for the keys. • Naturally supports • Look-Up Tables - store dictionary in a vector by non-decreasing order of the keys • Binary Search • Ordered Map/Dictionary ADT: • In addition to the generic map/dictionary ADT, supports the following functions: • closestBefore(k): return the position of an item with the largest key less than or equal to k • closestAfter(k): return the position of an item with the smallest key greater than or equal to k 4/13/2015 1:29 8 AM Lookup Table • A lookup table is a dictionary implemented by means of a sorted sequence • We store the items of the dictionary in an array-based sequence, sorted by key • We use an external comparator for the keys • Performance: • find takes O(log n) time, using binary search • insertItem takes O(n) time since in the worst case we have to shift n/2 items to make room for the new item • removeElement take O(n) time since in the worst case we have to shift n/2 items to compact the items after the removal • The lookup table is effective only for dictionaries of small size or for dictionaries on which searches are the most common operations, while insertions and removals are rarely performed (e.g., credit card authorizations) Dictionaries 4/13/2015 1:29 9 AM Example of Ordered Map: Binary Search • Binary search performs operation find(k) on a dictionary implemented by means of an array-based sequence, sorted by key • similar to the high-low game • at each step, the number of candidate items is halved • terminates after a logarithmic number of steps • Example: find(7) 0 1 3 4 5 7 1 0 3 4 5 m l 0 9 11 14 16 18 m l 0 8 1 1 3 3 7 19 h 8 9 11 14 16 18 19 8 9 11 14 16 18 19 8 9 11 14 16 18 19 h 4 5 7 l m h 4 5 7 l=m =h Dictionaries 4/13/2015 1:29 10 AM Map/Dictionary implementations • n - #elements in map/Dictionary Insert Find Space O(1) O(n) O(n) Direct Address Table (map O(1) only) O(1) O(N) Lookup Table (ordered map/dictionary) O(logn) O(n) Log File Vectors O(n) 4/13/2015 1:29 AM 11 Hash Tables • Hashing • Hash table (an array) of size N, H[0,N] • Hash function h that maps keys to indices in H • Issues • Hash functions - need method to transform key to an index in H that will have nice properties. • Collisions - some keys will map to the same index of H (otherwise we have a Direct Address Table). Several methods to resolve the collisions • Chaining - put elts that hash to same location in a linked list • Open addressing - if a collision occurs, have a method to select another location in the table. • Probe sequences Dictionaries & Hashing Dictionaries 12 4/13/2015 1:29 AM Hash Functions and Hash Tables • A hash function h maps keys of a given type to integers in a fixed interval [0, N - 1] • Example: h(x) = x mod N is a hash function for integer keys • The integer h(x) is called the hash value of key x • A hash table for a given key type consists of • Hash function h • Array (called table) of size N • When implementing a dictionary with a hash table, the goal is to store item (k, o) at index i = h(k) Dictionaries & Hashing Dictionaries 13 4/13/2015 1:29 AM Example 0 1 2 3 4 025-612-0001 981-101-0002 451-229-0004 … • We design a hash table for a dictionary storing items (SSN, Name), where SSN (social security number) is a nine-digit positive integer • Our hash table uses an array of size N = 10,000 and the hash function h(x) = last four digits of x 9997 9998 9999 Dictionaries & Hashing Dictionaries 200-751-9998 14 4/13/2015 1:29 AM Collisions • Collisions occur when • Example with Division different elements are Method mapped to the same cell • h(k) = k mod N • collisions must be resolved • Chaining (store in list outside the table) • Open addressing (store in another cell in the table) • If N=10, then • h(k)=0 for k=0,10,20, … • h(k)= 1 for k=1, 11, 21, etc • … Dictionaries & Hashing Dictionaries 15 4/13/2015 1:29 AM Collision Resolution with Chaining • Collisions occur when 0 1 different elements are 2 025-612-0001 mapped to the same 3 4 451-229-0004 981-101-0004 cell • Chaining: let each cell in the table point to a • Chaining is simple, linked list of elements but requires that map there additional memory outside the table Dictionaries & Hashing Dictionaries 16 4/13/2015 1:29 AM Exercise: chaining • Assume you have a hash table H with N=9 slots (H[0,8]) and let the hash function be h(k)=k mod N. • Demonstrate (by picture) the insertion of the following keys into a hash table with collisions resolved by chaining. • 5, 28, 19, 15, 20, 33, 12, 17, 10 Dictionaries & Hashing Dictionaries 17 4/13/2015 1:29 AM Collision Resolution in Open Addressing - Linear Probing • Open addressing: the colliding item is placed in a different cell of the table • Linear probing handles collisions by placing the colliding item in the next (circularly) available table cell. So the i-th cell checked is: • Example: • h(x) = x mod 13 • Insert keys 18, 41, 22, 44, 59, 32, 31, 73, in this order • H(k,i) = (h(k)+i)mod N • Each table cell inspected is referred to as a “probe” • Colliding items lump together, causing future collisions to cause a longer sequence of probes 0 1 2 3 4 5 6 7 8 9 10 11 12 18 44 59 32 22 31 73 0 1 2 3 4 5 6 7 8 9 10 11 12 41 Dictionaries & Hashing Dictionaries 18 4/13/2015 1:29 AM Search with Linear Probing • Consider a hash table A that uses linear probing • find(k) • We start at cell h(k) • We probe consecutive locations until one of the following occurs • An item with key k is found, or • An empty cell is found, or • N cells have been unsuccessfully probed Algorithm find(k) i h(k) p0 repeat c A[i] if c = return Position(null) else if c.key () = k return Position(c) else i (i + 1) mod N pp+1 until p = N return Position(null) Dictionaries & Hashing Dictionaries 19 4/13/2015 1:29 AM Updates with Linear Probing • To handle insertions and deletions, we introduce a special object, called AVAILABLE, which replaces deleted elements • removeElement(k) • We search for an item with key k • If such an item (k, o) is found, we replace it with the special item AVAILABLE and we return the position of this item • Else, we return a null position • insertItem(k, o) • We throw an exception if the table is full • We start at cell h(k) • We probe consecutive cells until one of the following occurs • A cell i is found that is either empty or stores AVAILABLE, or • N cells have been unsuccessfully probed • We store item (k, o) in cell i Dictionaries & Hashing Dictionaries 20 4/13/2015 1:29 AM Exercise: Linear Probing • Assume you have a hash table H with N=11 slots (H[0,10]) and let the hash function be h(k)=k mod N. • Demonstrate (by picture) the insertion of the following keys into a hash table with collisions resolved by linear probing. • 10, 22, 31, 4, 15, 28, 17, 88, 59 Dictionaries & Hashing Dictionaries 21 4/13/2015 1:29 AM Open Addressing: Double Hashing • Double hashing uses a secondary hash function h2(k) and handles collisions by placing an item in the first available cell of the series h(k,i) =(h1(k) + ih2(k)) mod N for i = 0, 1, … , N - 1 • The secondary hash function h2(k) cannot have zero values • The table size N must be a prime to allow probing of all the cells • Common choice of compression map for the secondary hash function: h2(k) = q - k mod q where • q<N • q is a prime • The possible values for h2(k) are 1, 2, … , q Dictionaries & Hashing Dictionaries 22 4/13/2015 1:29 AM Example of Double Hashing • Consider a hash table storing integer keys that handles collision with double hashing • N = 13 • h1(k) = k mod 13 • h2(k) = 7 - k mod 7 • Insert keys 18, 41, 22, 44, 59, 32, 31, 73, in this order 0 1 2 3 4 5 6 7 8 9 10 11 12 31 41 18 32 59 73 22 44 0 1 2 3 4 5 6 7 8 9 10 11 12 Dictionaries & Hashing Dictionaries 23 4/13/2015 1:29 AM Exercise: Double Hashing • Assume you have a hash table H with N=11 slots (H[0,10]) and let the hash functions for double hashing be • h(k,i)=(h1(k) + ih2(k))mod N • h1(k)=k mod N • h2(k)=1 + (k mod (N-1)) • Demonstrate (by picture) the insertion of the following keys into H • 10, 22, 31, 4, 15, 28, 17, 88, 59 Dictionaries & Hashing Dictionaries 24 4/13/2015 1:29 AM Performance of Hashing • In the worst case, searches, insertions and removals on a hash table take O(n) time • The worst case occurs when all the keys inserted into the dictionary collide • The load factor a = n/N affects the performance of a hash table • Assuming that the hash values are like random numbers, it can be shown that the expected number of probes for an insertion with open addressing is 1 / (1 - a) • The expected running time of all the dictionary ADT operations in a hash table is O(1) • In practice, hashing is very fast provided the load factor is not close to 100% • Applications of hash tables: • small databases • compilers • browser caches Dictionaries & Hashing Dictionaries 29 4/13/2015 1:29 AM Uniform Hashing Assumption • The probe sequence of a key k is the sequence of slots that will be probed when looking for k • In open addressing, the probe sequence is h(k,0), h(k,1), h(k,2), h(k,3), … • Uniform Hashing Assumption: Each key is equally likely to have any one of the N! permutations of {0,1, 2, …, N-1} as is probe sequence • Note: Linear probing and double hashing are far from achieving Uniform Hashing • Linear probing: N distinct probe sequences • Double Hashing: N2 distinct probe sequences Dictionaries & Hashing Dictionaries 30 4/13/2015 1:29 AM Performance of Uniform Hashing • Theorem: Assuming uniform hashing and an openaddress hash table with load factor a = n/N < 1, the expected number of probes in an unsuccessful search is at most 1/(1-a). • Exercise: compute the expected number of probes in an unsuccessful search in an open address hash table with a = ½ , a=3/4, and a = 99/100. Dictionaries Dictionaries & Hashing 31 4/13/2015 1:29 AM Universal Hashing • A family of hash functions is universal if, for any 0<i,j<M-1, • Theorem: The set of all functions, h, as • Choose p as a prime defined here, is between M and 2M. universal. • Randomly select 0<a<p and • Pr(h(j)=h(i)) < 1/N. 0<b<p, and define h(k)=(ak+b mod p) mod N Dictionaries & Hashing Dictionaries 32 4/13/2015 1:29 AM Maps/Dictionaries • n = #elements in map/dictionary, • N=#possible keys (it could be N>>n) or size of hash table Vectors Insert Find Space Log File O(1) O(n) O(n) Direct Address Table (map only) O(1) O(1) O(N) Lookup Table (ordered map/dictionary) O(n) O(logn) O(n) Hashing (chaining) O(1) O(n/N) O(n+N) Hashing (open addressing) O(1/(1-n/N)) O(1/(1-n/N)) O(N) 4/13/2015 1:29 AM 35