Download presentation

Presentation is loading. Please wait.

Published byRiley Williams Modified over 4 years ago

1
1 11. Hash Tables Heejin Park College of Information and Communications Hanyang University

2
2 Contents Direct-address tables Hash tables Hash functions

3
3 Direct address tables U (universe of keys) 9 4 0 7 6 1 / / / / / / 2 3 5 8 2 3 5 8 Key Satellite data Slot Generate a table T with |U| slots. Store k into slot k. Assume that no two elements have the same key. T K (actual keys) 0 1 2 3 4 5 6 7 8 9

4
4 Direct-address tables Running time : O(1) DIRECT-ADDRESS-SEARCH(T, k) return T[k] DIRECT-ADDRESS-INSERT(T, x) T[key[x]] x DIRECT-ADDRESS-DELETE(T, x) T[key[x]] NIL

5
5 Hash tables Space consumption of direct addressing Θ(|U|) If the universe |K|/|U| is small, most of the space allocated for T would be wasted. Is it possible to reduce the space requirement to Θ(|K|) while the running time is still O(1) ?

6
6 Hash tables Hashing The element with key k is stored in slot h(k). instead of slot k. h is called a hash function. A hash function computes the slot from the key k. h : U {0, 1,..., m - 1}. We say that an element with key k hashes to slot h(k). We also say that h(k) is the hash value of key k.

7
7 Hash tables U (universe of keys) k1k1 k2k2 k3k3 k4k4 0 h(k 1 ) h(k 3 ) h(k 2 )=h(k 5 ) h(k 4 ) m-1 k5k5 K (actual keys)

8
8 Hash tables Collision : two keys may hash to the same slot. Avoiding collisions To make h appear to be random. avoiding collisions or at least minimizing collisions. Avoiding collisions is impossible because |U| > m.

9
9 Hash tables Collision resolution by chaining Put the elements hashed to the same slot in a linked list. U (universe of keys) / / / / / / k5k5 k7k7 k4k4 k8k8 k3k3 k2k2 k1k1 k6k6 k5k5 K (actual keys) k1k1 k4k4 k8k8 k3k3 k6k6 k7k7 k2k2

10
10 Hash tables CHAINED-HASH-INSERT(T, x) insert x at the head of list T[h(key[x])] CHAINED-HASH-DELETE(T, x) delete x from the list T[h(key[x])] CHAINED-HASH-SEARCH(T, k) search for an element with key k in list T[h(k)]

11
11 Hash tables The worst-case running time for insertion is O(1). Assuming that the element x being inserted is not already present in the table. Deletion of an element x can be done in O(1) time. Assuming that the lists are doubly linked. For searching, the worst-case running time is proportional to the length of the list.

12
12 Hash tables Searching time for hashing with chaining Given a hash table T with m slots that stores n elements, we define the load factor α for T. The average number of elements stored in a chain. α = n/m.

13
13 Hash tables Worst-case All n keys hash to the same slot, creating a list of length n. Searching time : Θ(n) plus the time to compute the hash function. No better than a linked list for all the elements.

14
14 Hash tables Average case Depends on how well the hash function h distributes the set of keys to be stored among the m slots. Simple uniform hashing An element is equally likely to hash into each slot. An element hashes independently of where any other element has hashed to.

15
15 Hash tables Let n j denote the length of the list T[j] for j = 0, 1,..., m – 1. n = n 0 + n 1 + … + n m-1 average value of n j E[n j ] = α = n/m. Assume that the hash value h(k) can be computed in O(1) time, the time required to search for an element with key k depends linearly on the length n h(k) of the list T[h(k)].

16
16 Hash tables Consider two cases of search. When search is unsuccessful. No element in the table has key k. When search is successful. An element with key k is found.

17
17 Hash tables An unsuccessful search takes Θ(1 + α) expected time. A successful search takes Θ(1 + α) expected time.

Similar presentations

OK

Data Structures Using C++ 2E Chapter 9 Searching and Hashing Algorithms.

Data Structures Using C++ 2E Chapter 9 Searching and Hashing Algorithms.

© 2017 SlidePlayer.com Inc.

All rights reserved.

Ads by Google

Ppt on earth hour 2015 Ppt on course management system Ppt on business etiquette in other countries Ppt on world bank and imf Ppt on preservation of public property Download ppt on electricity Ppt on case study of leadership Ppt on home automation using gsm Ppt on ideal gas law calculator Ppt on idiopathic thrombocytopenia purpura in pregnancy