Hash Tables. Lower bounds on sorting Hash functions. March 05, 2018 Cinda Heeren / Geoffrey Tien 1
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1 Hash Tables Lower bounds on sorting Hash functions Cinda Heeren / Geoffrey Tien 1
2 Detour back to sorting Lower bounds on worst case Selection sort, Insertion sort, Quicksort, Merge sort Worst case complexities Θ n 2, Θ n 2, Θ n 2, Θ n log n respectively All sort based on comparison of two values, a and b The algorithm performs some action based on the result a < b, a > b, a b, a b, a = b Each comparison is a decision point in the algorithm Do one thing if comparison is true, do another if false The algorithm can be expressed as a (binary) decision tree The big question: Is it possible for a comparison-based sorting algorithm to have better asymptotic worst-case performance than Merge sort? Cinda Heeren / Geoffrey Tien 2
3 Lower bounds on sorting Given some permutation of items to be sorted, in positions a, b, c, d, e The leaves represent the sorted output for some particular input permutation a < b < c < sorted Y a < c a b c d N Y a < b Cinda Heeren / Geoffrey Tien 3 N a < c b < c b < c b < c b < c Y N Y N Y N Y N sorted sorted sorted Y sorted Longest path: maximum decisions in worst case Each path from root to a leaf is the sequence of decisions made to sort some input permutation N < c < b < a sorted
4 Lower bounds on sorting How many leaves are there in this decision tree? How many starting permutations are there? The algorithm must be able to correctly sort every permutation n! input permutations, therefore n! different paths to a leaf If there were fewer than n! paths, it means the algorithm is unable to correctly sort some input permutations What is the minimum height of a tree with n! leaves? A perfect tree with n! leaves Decision tree model Cinda Heeren / Geoffrey Tien 4
5 Lower bounds on sorting A tree with h levels has 2 h+1 1 nodes Bottom level has 2 h nodes (i.e. leaves), 2 h = n! log 2 h = log n! h log n i 1 n / 2 i 1 n! logi log n / 2 n log n / 2 2 n log n Height of a perfect decision tree The longest decision path can be no shorter than this The worst case of any comparison-based sorting algorithm can be no better than Ω n log n But there are non-comparison-based sorting algorithms that can perform better (with assumptions). Stay tuned in CPSC 320 (maybe)! Cinda Heeren / Geoffrey Tien 5
6 Review: dictionary ADT Stores values associated with user-specified keys Values may be any (homogenous) type Keys may be any (homogenous) comparable type Dictionary operations Create Destroy Insert Find Remove Insert Feet Useful for something, presumably Find(Z125 Pro) Z125 Pro Fun in the sun! Stumpjumper The favourite baby of VanCity planners Z125 Pro Fun in the sun! GL1800 Quiet comfort Cinda Heeren / Geoffrey Tien 6
7 Implementing a dictionary Using some of the data structures we have seen so far Worst case complexities for Insert, Remove, Find Insert Remove Find Unordered array O 1 O n O n Ordered array O n O n O log n Unordered list O 1 O n O n Ordered list O n O n O n BST O n O n O n AVL tree O log n O log n O log n So you think AVL trees are pretty good? Let's go back to basics arrays! What if we are allowed to leave gaps in our array, and we know the index of the element we want to access? Cinda Heeren / Geoffrey Tien 7
8 A simple example Suppose a company has 300 numbered lockers in its office building, and assigns a locker to every employee it hires Suppose also, the company currently has 250 employees, and maintains information such as locker (employee) number, name, position, salary, etc. every month when computing payroll, the company system must access an employee record, given a locker number Locker number Employee name Position, etc. 103 Jorge Lorenzo Scott Redding Dani Pedrosa Katsuyuki Nakasuga Valentino Rossi Why not simply use an array with 300 (plus one) elements? We can do all insert/remove/find in O 1 time That's even better than AVL tree, but how does it scale? Cinda Heeren / Geoffrey Tien 8
9 A larger example Suppose we want to store some census data on Canadians with telephone numbers telephone number in the format (123) , can be conveniently converted to a number between 0 and 9,999,999,999, let's use this as an array index But, consider that the entire population of Canada is approximately 35,151,728 (as of 2016 census) Over 99.6% of the array will be empty And it probably will not fit in your computer's RAM anyway So the full-ranged array doesn't scale well, but let's hold onto that idea. What if the data we want to store don't have a convenient integer field? Cinda Heeren / Geoffrey Tien 9
10 Storing strings What if we had to store data by name? We would need to convert strings to integer indices Here is one way to encode strings as integers Assign a value between 1 and 26 to each letter a = 1, z = 26 (regardless of case) Sum the letter values in the string "dog" = = 26 loves to eat Milk-Bone pees on the rug Insert("dog", description) Find("god") 26 We found an entry, return true! "god" = = 26 Cinda Heeren / Geoffrey Tien 10
11 Finding unique string values Ideally we would like to have a unique integer for each possible string The "sum the letters" encoding scheme does not achieve this There is a simple method to achieve this goal As before, assign each letter a value between 1 and 26 Treat the string as a base 26 number Multiply the letter's value by 26 i, where i is the position of the letter in the word: "dog" = 4* * *26 0 = 3,101 "god" = 7* * *26 0 = 5,126 But, suppose we store strings of length 10, there are possible combinations most of which are meaningless, e.g. "achcyertxa" Cinda Heeren / Geoffrey Tien 11
12 A different approach Don't determine the array size by the maximum possible number of keys Fix the array size based on the amount of data to be stored Map the key value (phone number or name or some other data) to an array element We still need to convert the key value to an integer index using a hash function This is the basic idea behind hash tables Cinda Heeren / Geoffrey Tien 12
13 Hash tables A hash table consists of an array to store data Data often consists of complex types, or pointers to such objects One attribute of the object is designated as the table's key A hash function maps a key to an array index in 2 steps The key should be converted to an integer And then that integer mapped to an array index using some function (often the modulo function) Cinda Heeren / Geoffrey Tien 13
14 Collisions A hash function may map two different keys to the same index Referred to as a collision Consider mapping phone numbers to an array of size 1,000 where h = phone mod 1,000 Both and map to the same index (6,045,551,987 mod 1,000 = 987) A good hash function can significantly reduce the number of collisions It is still necessary to have a policy to deal with any collisions that may occur Collisions are actually unavoidable due to pigeonhole principle Cinda Heeren / Geoffrey Tien 14
15 Collisions Pigeonhole principle (informally) Try to fit k + 1 pigeons into k pigeon-sized holes Try to get 33 CPSC 221 students into a lab section with 32 seats Try to hash without collision m keys into n array indices with m > n Formally: Let X and Y be finite sets where X > Y If f: X Y, then f x 1 = f x 2 for some x 1, x 2 X, where x 1 x 2 f x 1 X Y x 2 f x 1 = f x 2 Cinda Heeren / Geoffrey Tien 15
16 Hash functions A hash function is a function that map key values to array indexes Hash functions are performed in two steps Map the key value to an integer Map the integer to a legal array index Hash functions should have the following properties Fast Deterministic Uniformity Cinda Heeren / Geoffrey Tien 16
17 Hash function speed Hash functions should be fast and easy to calculate Access to a hash table should be nearly instantaneous and in constant time Most common hash functions require a single division on the representation of the key Converting the key to a number should also be able to be performed quickly Cinda Heeren / Geoffrey Tien 17
18 Deterministic hash functions A hash function must be deterministic For a given input it must always return the same value Otherwise it will not generate the same array index And the item will not be found in the hash table Hash functions should therefore not be determined by System time Memory location Pseudo-random numbers Cinda Heeren / Geoffrey Tien 18
19 Scattering data A typical hash function usually results in some collisions Where two different search keys map to the same index A perfect hash function avoids collisions entirely Each search key value maps to a different index The goal is to reduce the number and effect of collisions To achieve this the data should be distributed evenly over the table Cinda Heeren / Geoffrey Tien 19
20 Possible values Any set of values stored in a hash table is an instance of the universe of possible values The universe of possible values may be much larger than the instance we wish to store There are many possible combinations of 10 letters But we might want a hash table to store 1,000 names Cinda Heeren / Geoffrey Tien 20
21 Uniformity A good hash function generates each value in the output range with the same probability That is, each legal hash table index has the same chance of being generated This property should hold for the universe of possible values and for the expected inputs The expected inputs should also be scattered evenly over the hash table Cinda Heeren / Geoffrey Tien 21
22 A bad hash function A hash table is to store 1,000 numeric estimates that can range from 1 to 1,000,000 Hash function h(estimate) = estimate % n Where n = array size = 1,000 Is the distribution of values from the universe of all possible values uniform? What about the distribution of expected values? Cinda Heeren / Geoffrey Tien 22
23 Another bad hash function A hash table is to store 676 names The hash function considers just the first two letters of a name Each letter is given a value where a = 1, b = 2, Function = (1 st letter * 26 + value of 2 nd letter) % 676 Is the distribution of values from the universe of all possible values uniform? What about the distribution of expected values? Cinda Heeren / Geoffrey Tien 23
24 General principles Use the entire search key in the hash function If the hash function uses modulo arithmetic make the table size a prime number A simple and (usually) effective hash function is Convert the key value to an integer, x h x = x mod tablesize Where tablesize is the first prime number larger than twice the size of the number of expected values Determining a good hash function is a complex subject for now, just use provided hash functions Cinda Heeren / Geoffrey Tien 24
25 Readings for this lesson Carrano & Henry Chapter (Hash functions) Next class: Carrano & Henry, Chapter (Collision resolution) Cinda Heeren / Geoffrey Tien 25
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