Data Structures: Hash Tables and Skip Lists, Slides of Computer Science

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Algorithms

Hash Tables

Homework 4

● Programming assignment:

■ Implement Skip Lists

■ Evaluate performance

■ Extra credit: compare performance to randomly

built BST

■ Due Mar 9 (Fri before Spring Break)

■ Will post later today

Summary: Skip Lists

● O(1) expected time for most operations

● O(lg n) expected time for insert

● O(n^2 ) time worst case

■ But random, so no particular order of insertion

evokes worst-case behavior

● O(n) expected storage requirements

● Easy to code

Review: Hash Tables

● Hash table:

■ Given a table T and a record x, with key (=

symbol) and satellite data, we need to support:

○ Insert (T, x)

○ Delete (T, x)

○ Search(T, x)

■ We want these to be fast, but don’t care about

sorting the records

■ In this discussion we consider all keys to be

(possibly large) natural numbers

Review: The Problem With

Direct Addressing

● Direct addressing works well when the range

m of keys is relatively small

● But what if the keys are 32-bit integers?

■ Problem 1: direct-address table will have

2 32 entries, more than 4 billion

■ Problem 2: even if memory is not an issue, the

time to initialize the elements to NULL may be

● Solution: map keys to smaller range 0..m-

● This mapping is called a hash function

Hash Functions

● Next problem: collision

T

m - 1

h(k 1 ) h(k 4 )

h(k 2 ) = h(k 5 )

h(k 3 )

k (^4)

k 2 k (^3)

k (^1)

k (^5)

U

(universe of keys)

K

(actual keys)

Open Addressing

● Basic idea (details in Section 12.4):

■ To insert: if slot is full, try another slot, …, until an

open slot is found (probing)

■ To search, follow same sequence of probes as

would be used when inserting the element

○ If reach element with correct key, return it

○ If reach a NULL pointer, element is not in table

● Good for fixed sets (adding but no deletion)

■ Example: spell checking

● Table needn’t be much bigger than n

Chaining

● Chaining puts elements that hash to the same

slot in a linked list:

T

k 4

k 2 k^3

k (^1)

k (^5)

U

(universe of keys)

K

(actual keys)

k (^6) k (^8)

k (^7)

k 1 k 4 ——

k 5 k 2

k 3 k 8 k 6 ——

k 7 ——

Chaining

T

k 4

k 2 k^3

k (^1)

k (^5)

U

(universe of keys)

K

(actual keys)

k (^6) k (^8)

k (^7)

k 1 k 4 ——

k 5 k 2

k 3 k 8 k 6 ——

k 7 ——

● How do we delete an element?

■ Do we need a doubly-linked list for efficient delete?

Chaining

● How do we search for a element with a

given key?

T

k 4

k 2 k^3

k (^1)

k (^5)

U

(universe of keys)

K

(actual keys)

k (^6) k (^8)

k (^7)

k 1 k 4 ——

k 5 k 2

k 3 k 8 k 6 ——

k 7 ——

Analysis of Chaining

● Assume simple uniform hashing: each key in

table is equally likely to be hashed to any slot

● Given n keys and m slots in the table, the

load factor α = n/m = average # keys per slot

● What will be the average cost of an

unsuccessful search for a key? A: O(1+ α)

Analysis of Chaining

● Assume simple uniform hashing: each key in

table is equally likely to be hashed to any slot

● Given n keys and m slots in the table, the

load factor α = n/m = average # keys per slot

● What will be the average cost of an

unsuccessful search for a key? A: O(1+ α)

● What will be the average cost of a successful

search?

Analysis of Chaining Continued

● So the cost of searching = O(1 + α)

● If the number of keys n is proportional to the

number of slots in the table, what is α?

● A: α = O(1)

■ In other words, we can make the expected cost of

searching constant if we make α constant

Choosing A Hash Function

● Clearly choosing the hash function well is

crucial

■ What will a worst-case hash function do?

■ What will be the time to search in this case?

● What are desirable features of the hash

function?

■ Should distribute keys uniformly into slots

■ Should not depend on patterns in the data