Lec4
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The document discusses different data structures for implementing dictionaries, including arrays, linked lists, hash tables, binary trees, and B-trees. It focuses on hashing as a technique for implementing dictionaries. Hashing maps keys to table slots using a hash function to achieve fast average-case search, insertion, and deletion times of O(1). Collisions are resolved using chaining, where each slot contains a linked list. The load factor affects performance, with lower load factors resulting in faster operations.
![A recursive procedure
Algorithm BinarySearch(A, k, low, high)
iflow > high thenreturnNil
elsemid ←(low+high) / 2
ifk = A[mid] then returnmid
elseifk < A[mid] then
returnBinarySearch(A, k, low, mid-1)
else returnBinarySearch(A, k, mid+1, high)](https://image.slidesharecdn.com/lec4-140928015804-phpapp01/85/Lec4-8-320.jpg)
![An iterative procedure
INPUT: A[1..n] –a sorted (non-decreasing) array of integers, key–an integer.
OUTPUT: an index j such that A[j] = k.
NIL, if ∀j (1 ≤ j ≤ n): A[j] ≠ k
low←1
high← n
do
mid←(low+high)/2
if A[mid] = kthenreturn mid
else if A[mid] > kthen high← mid-1
else low← mid+1
while low<= high
return NIL](https://image.slidesharecdn.com/lec4-140928015804-phpapp01/85/Lec4-9-320.jpg)
![Searching in an unsorted array
INPUT: A[1..n] –an array of integers, q–an integer.
OUTPUT: an index jsuch that A[j] = q. NIL, if ∀j(1≤j≤n): A[j] ≠ q
j ← 1
while j ≤ n and A[j]≠ q
doj++
if j≤ n thenreturnj
else return NIL
…Worst-case running time: O(n), average-case: O(n)
…
We can’t do better. This is a lower bound for the problem of searching in an arbitrary sequence.](https://image.slidesharecdn.com/lec4-140928015804-phpapp01/85/Lec4-11-320.jpg)
![Analysis of Hashing
…An element with key k is stored in slot h(k) (instead of slot kwithout hashing)
…
The hash function hmaps the universe Uof keys into the slots of hash table T[0...m-1]
…
Assume time to compute h(k) is Θ(1)
{}:0,1,...,1hUm→−](https://image.slidesharecdn.com/lec4-140928015804-phpapp01/85/Lec4-20-320.jpg)
Lec4
- 1. Dictionaries … the dictionary ADT … binary search … Hashing
- 2. Dictionaries … Dictionaries store elements so that they can be located quickly using keys … Adictionary may hold bank accounts … each account is an object that is identified by an account number … each account stores a wealth of additional information … including the current balance, … the name and address of the account holder, and … the history of deposits and withdrawals performed … an application wishing to operate on an account would have to provide the account number as a search key
- 3. The Dictionary ADT … A dictionary is an abstract model of a database … A dictionary stores key-element pairs … The main operation supported by a dictionary is searching by key … simple container methods: size(), isEmpty(), elements() … query methods: findElem(k), findAllElem(k) … update methods: insertItem(k,e), removeElem(k), removeAllElem(k) … special element: NIL, returned by an unsuccessful search
- 4. The Dictionary ADT …Supporting order (methods min, max, successor, predecessor ) is not required, thus it is enough that keys are comparable for equality
- 5. The Dictionary ADT …Different data structures to realize dictionaries … arrays, linked lists (inefficient) … Hash table (used in Java...) … Binary trees … Red/Black trees … AVL trees … B-trees … In Java: … java.util.Dictionary –abstract class … java.util.Map –interface
- 6. Searching INPUT •sequence of numbers (database) • a single number (query) OUTPUT •index of the found number or NIL a1, a2, a3,….,an; q 2 5 4 10 7; 5 2 5 4 10 7; 9 j 2 NIL
- 7. BinarySearch …Idea: Divide and conquer, a key design technique … narrow down the search range in stages … findElement(22)
- 8. A recursive procedure Algorithm BinarySearch(A, k, low, high) iflow > high thenreturnNil elsemid ←(low+high) / 2 ifk = A[mid] then returnmid elseifk < A[mid] then returnBinarySearch(A, k, low, mid-1) else returnBinarySearch(A, k, mid+1, high)
- 9. An iterative procedure INPUT: A[1..n] –a sorted (non-decreasing) array of integers, key–an integer. OUTPUT: an index j such that A[j] = k. NIL, if ∀j (1 ≤ j ≤ n): A[j] ≠ k low←1 high← n do mid←(low+high)/2 if A[mid] = kthenreturn mid else if A[mid] > kthen high← mid-1 else low← mid+1 while low<= high return NIL
- 10. Running Time of Binary Search …The range of candidate items to be searched is halved after each comparison … In the array-based implementation, access by rank takes O(1) time, thus binary search runs in O(log n) time
- 11. Searching in an unsorted array INPUT: A[1..n] –an array of integers, q–an integer. OUTPUT: an index jsuch that A[j] = q. NIL, if ∀j(1≤j≤n): A[j] ≠ q j ← 1 while j ≤ n and A[j]≠ q doj++ if j≤ n thenreturnj else return NIL …Worst-case running time: O(n), average-case: O(n) … We can’t do better. This is a lower bound for the problem of searching in an arbitrary sequence.
- 12. The Problem T&T is a large phone company, and they want toprovide caller ID capability: … given a phone number, return the caller’s name … phone numbers range from 0 to r = 108-1 … There are n phone numbers, n << r. … want to do this as efficiently as possible
- 13. Using an unordered sequence …searching and removing takes O(n) time … inserting takes O(1) time … applications to log files (frequent insertions, rare searches and removals)
- 14. Using an ordered sequence …searching takes O(log n) time (binary search) … inserting and removing takes O(n) time … application to look-up tables (frequent searches, rare insertions and removals)
- 15. Other Suboptimal ways direct addressing: an array indexed by key: … takes O(1) time, … O(r)spacewhere r is the range of numbers (108) … huge amount of wasted space (null) (null) Ankur (null) (null) 0000-0000 0000-0000 9635-8904 0000-0000 0000-0000
- 16. Another Solution …Can do better, with a Hash table--O(1) expected time, O(n+m) space, where mis table size … Like an array, but come up with a function to map the large range into one which we can manage … e.g., take the original key, modulo the (relatively small) size of the array, and use that as an index … Insert (9635-8904, Ankur) into a hashed array with, say, five slots. 96358904 mod 5 = 4 (null) (null) (null) (null) Ankur 0 1 2 3 4
- 17. An Example „Let keys be entry no’s of students in CSL201. eg. 2004CS10110. „ There are 100 students in the class. We create a hash table of size, say 100. „ Hash function is, say, last two digits. „ Then 2004CS10110 goes to location 10. „ Where does 2004CS50310 go? „ Also to location 10. We have a collision!!
- 18. Collision Resolution …How to deal with two keys which hash to the same spot in the array? … Use chaining … Set up an array of links (a table), indexed by the keys, to lists of items with the same key … Most efficient (time-wise) collision resolution scheme
- 19. Collision resolution (2) …To find/insert/delete an element … using h, look up its position in table T … Search/insert/delete the element in the linked list of the hashed slot
- 20. Analysis of Hashing …An element with key k is stored in slot h(k) (instead of slot kwithout hashing) … The hash function hmaps the universe Uof keys into the slots of hash table T[0...m-1] … Assume time to compute h(k) is Θ(1) {}:0,1,...,1hUm→−
- 21. Analysis of Hashing(2) …An good hash function is one which distributes keys evenly amongst the slots. … An ideal hash function would pick a slot, uniformly at random and hash the key to it. … However, this is not a hash function since we would not know which slot to look into when searching for a key. … For our analysis we will use this simple uniformhash function … Given hash table Twith mslots holding nelements, the load factoris defined as α=n/m
- 22. Analysis of Hashing(3) Unsuccessful search … element is not in the linked list … Simple uniformhashing yields an average list length α= n/m … expected number of elements to be examined α … search time O(1+α) (includes computing the hash value)
- 23. Analysis of Hashing (4) Successful search … assume that a new element is inserted at the end of the linked list … upon insertion of the i-th element, the expected length of the list is (i-1)/m … in case of a successful search, the expected number of elements examined is 1 more that the number of elements examined when the sought- for element was inserted!
- 24. Analysis of Hashing (5) …The expected number of elements examined is thus … Considering the time for computing the hash function, we obtain () () 11111111111211211221122nniiiinmnmnnnmnmnmmm α == −⎛⎞+=+−⎜⎟ ⎝⎠ − =+⋅ − =+ =+− +− ΣΣ(2/21/2)(1)mααΘ+−=Θ+
- 25. Analysis of Hashing (6) Assuming the number of hash table slots is proportional to the number of elements in the table … n=O(m) … α= n/m = O(m)/m = O(1) … searching takes constant time on average … insertion takes O(1) worst-case time … deletion takes O(1) worst-case time when the lists are doubly-linked