Is partitioning column required in the clustered index in SQL Server - sql-server

I have a table with these indexes:
pk_id_sales PRIMARY KEY (id) -> Clustered unique index
uk_sales_id UNIQUE(sales_id -> Non clustered unique index
uk_sales_date_party_name (sales_date, party_name) -> Non clustered, non unique index
I want to partition this table on the column sales_date.
Should I include sales_date into the clustered index to get the benefits of partitioning? Is this an optional one? What should be the factors to be considered to make this decision if it is an optional one?
What should be the order of columns in the clustered index If I add sales_date? Should it be (id, sales_date) or (sales_date, id)? What is the role of order here?
Will the order of columns in the index make any performance impact in this case?
If we include the partition column in the query, will partition elimination always happen regardless of the indexes we have? (Eg: I already have a unique non-clustered index on the sales_id (it doesn't contain sales_date). If I make a query with sales_id and sales_date in the where clause, will the partition elimination happen?)
Please share if there is a comprehensive write-up or video that will help to gain a fair understanding of the above-given concepts.
Any response will be appreciated. I can share more details if required.
I tried the following scenarios on an existing empty table. In both cases, the new records are getting inserted into the respective partitions and partition elimination is happening correctly (Found it based on the actual execution plan in azure data studio)
SCENARIO 1
I followed the below-given steps based on a tutorial. I don't know we are performing the 4th step.
Drop the existing clustered index on ID
Create a new non-clustered index on ID
Create a clustered index on sales_date
Drop the clustered index on `sales_date'
SCENARIO 2
Based on another tutorial, I tried the following.
I followed the below-given steps based on a tutorial. I don't know we are performing the 4th step.
Drop the existing clustered index on ID
Create a new non-clustered index on ID
Create a clustered index on sales_date

For your first question, the partitioning column is required to be specified explicitly as a key column for all unique indexes. Furthermore, SQL Server will automatically add the partitioning column to clustered index keys if not already specified.
The partitioning column is automatically added as an included column in non-unique non-clustered indexes when not already a key or included column.
EDIT:
For this question asked in comment:
The existing clustered index on my table is id (It is IDentical and
auto incremented). I want to partition the table based on sales_date.
My understanding is that we need to add sales_date to the clustered
index. In the examples I saw on web, they are adding it as a second
part of the clustered index, ie, (id, sales_date). But for me, it
looks like (sales_date,id) will be more helpful as id is unique and
it will not help to improve performance.
It depends on your queries. The partitioning column must be specified to eliminate parttions and the leftmost key column must be specified to perform an index seek.
With unique clustered index key (id,sales_date) and no other indexes:
WHERE id = 1 will perform an index seek against every partition to
find the single row.
WHERE sales_date = '20221114' will perform a full scan of single
partition containing the date and return only rows matching the date.
WHERE id = 1 AND sales_date = '20221114' will perform a seek
against only the single partition containing the date and touch the
single row.
With unique clustered index key (sales_date,id):
WHERE id = 1 will full scan every partition to find the single
row.
WHERE sales_date = '20221114' will perform an index seek on only
the partition containing the date and touch only rows that qualify.
WHERE id = 1 AND sales_date = '20221114' will perform and index
seek only the partition containing the date and touch only the single
row.

Related

Non clustered index on the same column as the clustered primary key should be unique?

In a large People's table which has a PersonID column as primary key (int identity) and is clustered indexed by that (PK_People), with a whole bunch of additional columns, I had the need to add a non clustered index also on the same column with only one additional included column (Name).
The reason for that would apparently be because this table is most of the time inner joined with other tables via it's primary key (PersonID) just to get the Person Name and not all the other columns. It seems that SQL Server would prefer to use this index because it's much smaller than the whole clustered index (actual entire table).
I saw that need because SQL Server was actually using another non clustered index that existed in the table for the Name column alone, to retrieve the Name for the joins and thus having to scan the whole index to find the matching PersonID for the person it was looking for. Still it was preferring this over doing a key lookup using the actual clustered index. (Maybe because of the size of it?)
Then, the moment I created an index like this:
CREATE UNIQUE NONCLUSTERED INDEX UX_People_PersonID ON dbo.People (PersonID ASC)
INCLUDE (Name);
SQL was happily using this new index for the table joins, doing now an index seek instead of index scan, using the PersonID as key to output the Name column (included in the index).
Now... to the actual question, or questions:
Should I make this index UNIQUE as I did? (It kind of made sense to me at the time, one reason being the fact that the PersonID column would never have duplicate values anyway since it's the primary key)
Does it help SQL Server in any way by defining it UNIQUE or should it be a simple index?
Do I still need the clustered index to be the PersonID as well?
Is it a problem to have two indexes on the same column like this?
Since SQL benefits from having this non clustered index like I described for most of the case scenarios that I use, should I make the cluster index on some other column instead of PersonID?
Please let me know what your thoughts are... if I'm on the right path here or there are better approaches to solve this (duplication and uniqueness)...
Should I make this index UNIQUE as I did?
Yes
Does it help SQL Server in any way by defining it UNIQUE or should it be a simple index?
No.
It's unique, so you should declare it as unique. It makes no difference to the actual implementation however. PersonID is the "row locator" for this table as it's the clustered index key, so it would be added to any non-unique non-clustered index as a key column. But since it's already there, physical storage doesn't change whether or not you declare the index as unique.
Do I still need the clustered index to be the PersonID as well?
You still need some clustered index, and PersonID is probably a good choice. Note that if you change the clustered index to something else, then you must make the index on PersonID unique.
Is it a problem to have two indexes on the same column like this?
It's not a problem. The clustered index has all the columns in it, so scanning it can be expensive. The seperate index on PersonID is useful for counting the rows, for paging, and (as here) for covering queries that need only the PersonID and Name.

Filter on the uniquifier column in a non-unique clustered index

I have a table with a non-unique clustered index. I want to move these records to a table with a unique clustered index (I don't care which single record I get out of the table). I can do this via a GROUP BY, but I suspect it could be faster to select off of the hidden uniquifier column.
I have found some examples of how to access the value of the uniquifier column, but none that show how it might be uses in a query, e.g.
SELECT *
FROM NonUniqueClusteredIndexTable
WHERE uniquifier = 0
Any ideas how to access this value, or otherwise how to quickly de-duplicate such a table?

Which index is better non-clustered vs clustered in this case?

I have a table which has 4 columns (region_id, product_id, cate_id, month_id) as a primary key.
This primary key was created as default, so a clustered index were created for PK.
This table contains more than 10 millions rows.
If I delete existing pk and create a new pk with non-clustered index type, is it better than clustered index for the following query?
select region_id, product_id, cate_id, month_id, a, b, c
from fact_a
where month_id > 100
Thanks in advance.
A simple nonclustered index on month_id will certainly improve the average performance for that query (assuming month_id for most of the rows is less than 100, so that the where clause excludes most of the rows). However, if you're creating the index specifically for that query (or any queries with month_id in the where clause and a, b, c, month_id or a subset of those in the select), you will get even better results by including the selected values in the index, like this:
CREATE INDEX index_fact_a_month_id ON fact_a (month_id) INCLUDE (a,b,c)
The quick answer, yes, removing the primary key (moreso, replacing the current multi-column Primary Key with a single identity column) and then creating your NCI on Month_ID will be better/faster/more efficient.
Clustered Index - it IS the data. It contains every column of every row in the table. There can only be one CI because the table data only needs to exist once. Each row has a key...
Primary Key - it is the key to identify a row in a Clustered Index.
Non-Clustered Index - it acts as a table of a subset of columns from the rows in the Clustered Index.
Keeping it simple, a Non-Clustered Index contains less data than the Clustered Index, and it orders the data in a way (Month_id ASC) that makes queries against it much more efficient than querying against the CI (A, B, C, Month_ID). SQL Server has no way to "dip" into the CI Primary Key or row data and say, "Hey, I'm filtering by Month_ID, so I'll just go right to that column." By nature of Clustered Indexes, SQL Server "reads" all CI rows (index scan), every column, every byte of data. Very inefficient and wasteful since your WHERE clause will be filtering out a lot of these rows.
The Non Clustered Index only contains a subset of columns, so it is much more efficient in that it can say, "Hey, I'm filtering by Month_ID, and I only contain Month_ID, aaannnd Month_ID is in ascending order, so I can just jump right to the rows that I want!" (index seek). Much more efficient since only the rows you want to return will be "read" by SQL Server.
Getting a little more advanced, since the Non Clustered Index is only Month_ID, but you are querying for all the columns in the Clustered Index, SQL Server needs to be able to go back to the CI from the NCI to get rest of the columns. To do that, the Primary Key of the CI is stored in the NCI, along with the column subset. So the NCI is really like a two column table of (Month_ID, CI Primary Key).
If your Primary Key is monstrous, your NCIs will also be monstrous, and therefore less efficient (more disk reads, more buffer pool consumption, bad database stuff).
Disclaimer: there can be specific scenarios where you want every column to be the clustered index key/pk. I don't sense that is applicable here, but it is possible. If you have a heavily used query that refers to every column of the table in where clauses or joins, than a coverage clustered index may be beneficial.

Covering indexes when extra columns uniquely determined by clustered index

Suppose I need to update myTab from luTab as follows
update myTab
set LookupVale = (select LookupValue from luTab B
where B.idLookup = myTab.idLookup)
luTab consists of 2 columns (idLookup(unique), LookupValue)
Which is preferable : a unique clustered index on idLookup, or one on idLookup and Lookupvalue combined? Is a covering index going to make any difference in this situation?
(I'm mostly interested in SQL server)
Epilogue :
I followed up Krips tests below with 27M rows in myTab, 1.5M rows in luTab.
The crucial part seems to be the uniqueness of the index.
If the index is specified as unique, the update uses a hash table.
If it is not specified as unique, then the update first aggreates luTab by idLookup (the Stream Aggegate) and then uses a nested loop. This is much slower.
When I use the extended index, SQL is now no longer assued that that LookupValue is unique so its forced down the much slower, stream aggregate-nested loop route
Firstly:
A covering index is always non-clustered
You should always have a PK and a clustered index (there are the same by default on SQL Server)
The 2 concepts are separate
So:
Your PK (clustered) would be idLookup if this uniquely identifies a row
The covering index would be (idLookup) INCLUDE (LookupValue)
However:
idLookup is the PK (clustered), so you don't need a covering index
the clustered index (PK) is implicitly "covering" by the nature of a clustered index (simply, index is data at the lowest level)
I've created your tables and loaded just a few records (50 or so lookup, and 15 in myTab).
Then I've tried various index options. The Index Seek on luTab always has a cost of 29%.
The interesting bit is that if you add in the LookupValue column to the index on luTab the execution plan shows two extra steps after the Index Seek: Stream Aggregate and Assert. While cost is 0%, that may go up with more data.
I've also tried a nonclustered index on just idLookup, and including LookupValue as an 'Included Column'. That way the data pages don't need to be accessed to retrieve that that column. That may be an option for you although the execution plan doesn't show anything different (but they don't have the Stream Aggregate / Assert either).
-Krip

What do Clustered and Non-Clustered index actually mean?

I have a limited exposure to DB and have only used DB as an application programmer. I want to know about Clustered and Non clustered indexes.
I googled and what I found was :
A clustered index is a special type of index that reorders the way
records in the table are physically
stored. Therefore table can have only
one clustered index. The leaf nodes
of a clustered index contain the data
pages. A nonclustered index is a
special type of index in which the
logical order of the index does not
match the physical stored order of
the rows on disk. The leaf node of a
nonclustered index does not consist of
the data pages. Instead, the leaf
nodes contain index rows.
What I found in SO was What are the differences between a clustered and a non-clustered index?.
Can someone explain this in plain English?
With a clustered index the rows are stored physically on the disk in the same order as the index. Therefore, there can be only one clustered index.
With a non clustered index there is a second list that has pointers to the physical rows. You can have many non clustered indices, although each new index will increase the time it takes to write new records.
It is generally faster to read from a clustered index if you want to get back all the columns. You do not have to go first to the index and then to the table.
Writing to a table with a clustered index can be slower, if there is a need to rearrange the data.
A clustered index means you are telling the database to store close values actually close to one another on the disk. This has the benefit of rapid scan / retrieval of records falling into some range of clustered index values.
For example, you have two tables, Customer and Order:
Customer
----------
ID
Name
Address
Order
----------
ID
CustomerID
Price
If you wish to quickly retrieve all orders of one particular customer, you may wish to create a clustered index on the "CustomerID" column of the Order table. This way the records with the same CustomerID will be physically stored close to each other on disk (clustered) which speeds up their retrieval.
P.S. The index on CustomerID will obviously be not unique, so you either need to add a second field to "uniquify" the index or let the database handle that for you but that's another story.
Regarding multiple indexes. You can have only one clustered index per table because this defines how the data is physically arranged. If you wish an analogy, imagine a big room with many tables in it. You can either put these tables to form several rows or pull them all together to form a big conference table, but not both ways at the same time. A table can have other indexes, they will then point to the entries in the clustered index which in its turn will finally say where to find the actual data.
In SQL Server, row-oriented storage both clustered and nonclustered indexes are organized as B trees.
(Image Source)
The key difference between clustered indexes and non clustered indexes is that the leaf level of the clustered index is the table. This has two implications.
The rows on the clustered index leaf pages always contain something for each of the (non-sparse) columns in the table (either the value or a pointer to the actual value).
The clustered index is the primary copy of a table.
Non clustered indexes can also do point 1 by using the INCLUDE clause (Since SQL Server 2005) to explicitly include all non-key columns but they are secondary representations and there is always another copy of the data around (the table itself).
CREATE TABLE T
(
A INT,
B INT,
C INT,
D INT
)
CREATE UNIQUE CLUSTERED INDEX ci ON T(A, B)
CREATE UNIQUE NONCLUSTERED INDEX nci ON T(A, B) INCLUDE (C, D)
The two indexes above will be nearly identical. With the upper-level index pages containing values for the key columns A, B and the leaf level pages containing A, B, C, D
There can be only one clustered index per table, because the data rows
themselves can be sorted in only one order.
The above quote from SQL Server books online causes much confusion
In my opinion, it would be much better phrased as.
There can be only one clustered index per table because the leaf level rows of the clustered index are the table rows.
The book's online quote is not incorrect but you should be clear that the "sorting" of both non clustered and clustered indices is logical, not physical. If you read the pages at leaf level by following the linked list and read the rows on the page in slot array order then you will read the index rows in sorted order but physically the pages may not be sorted. The commonly held belief that with a clustered index the rows are always stored physically on the disk in the same order as the index key is false.
This would be an absurd implementation. For example, if a row is inserted into the middle of a 4GB table SQL Server does not have to copy 2GB of data up in the file to make room for the newly inserted row.
Instead, a page split occurs. Each page at the leaf level of both clustered and non clustered indexes has the address (File: Page) of the next and previous page in logical key order. These pages need not be either contiguous or in key order.
e.g. the linked page chain might be 1:2000 <-> 1:157 <-> 1:7053
When a page split happens a new page is allocated from anywhere in the filegroup (from either a mixed extent, for small tables or a non-empty uniform extent belonging to that object or a newly allocated uniform extent). This might not even be in the same file if the filegroup contains more than one.
The degree to which the logical order and contiguity differ from the idealized physical version is the degree of logical fragmentation.
In a newly created database with a single file, I ran the following.
CREATE TABLE T
(
X TINYINT NOT NULL,
Y CHAR(3000) NULL
);
CREATE CLUSTERED INDEX ix
ON T(X);
GO
--Insert 100 rows with values 1 - 100 in random order
DECLARE #C1 AS CURSOR,
#X AS INT
SET #C1 = CURSOR FAST_FORWARD
FOR SELECT number
FROM master..spt_values
WHERE type = 'P'
AND number BETWEEN 1 AND 100
ORDER BY CRYPT_GEN_RANDOM(4)
OPEN #C1;
FETCH NEXT FROM #C1 INTO #X;
WHILE ##FETCH_STATUS = 0
BEGIN
INSERT INTO T (X)
VALUES (#X);
FETCH NEXT FROM #C1 INTO #X;
END
Then checked the page layout with
SELECT page_id,
X,
geometry::Point(page_id, X, 0).STBuffer(1)
FROM T
CROSS APPLY sys.fn_PhysLocCracker( %% physloc %% )
ORDER BY page_id
The results were all over the place. The first row in key order (with value 1 - highlighted with an arrow below) was on nearly the last physical page.
Fragmentation can be reduced or removed by rebuilding or reorganizing an index to increase the correlation between logical order and physical order.
After running
ALTER INDEX ix ON T REBUILD;
I got the following
If the table has no clustered index it is called a heap.
Non clustered indexes can be built on either a heap or a clustered index. They always contain a row locator back to the base table. In the case of a heap, this is a physical row identifier (rid) and consists of three components (File:Page: Slot). In the case of a Clustered index, the row locator is logical (the clustered index key).
For the latter case if the non clustered index already naturally includes the CI key column(s) either as NCI key columns or INCLUDE-d columns then nothing is added. Otherwise, the missing CI key column(s) silently gets added to the NCI.
SQL Server always ensures that the key columns are unique for both types of indexes. The mechanism in which this is enforced for indexes not declared as unique differs between the two index types, however.
Clustered indexes get a uniquifier added for any rows with key values that duplicate an existing row. This is just an ascending integer.
For non clustered indexes not declared as unique SQL Server silently adds the row locator into the non clustered index key. This applies to all rows, not just those that are actually duplicates.
The clustered vs non clustered nomenclature is also used for column store indexes. The paper Enhancements to SQL Server Column Stores states
Although column store data is not really "clustered" on any key, we
decided to retain the traditional SQL Server convention of referring
to the primary index as a clustered index.
I realize this is a very old question, but I thought I would offer an analogy to help illustrate the fine answers above.
CLUSTERED INDEX
If you walk into a public library, you will find that the books are all arranged in a particular order (most likely the Dewey Decimal System, or DDS). This corresponds to the "clustered index" of the books. If the DDS# for the book you want was 005.7565 F736s, you would start by locating the row of bookshelves that is labeled 001-099 or something like that. (This endcap sign at the end of the stack corresponds to an "intermediate node" in the index.) Eventually you would drill down to the specific shelf labelled 005.7450 - 005.7600, then you would scan until you found the book with the specified DDS#, and at that point you have found your book.
NON-CLUSTERED INDEX
But if you didn't come into the library with the DDS# of your book memorized, then you would need a second index to assist you. In the olden days you would find at the front of the library a wonderful bureau of drawers known as the "Card Catalog". In it were thousands of 3x5 cards -- one for each book, sorted in alphabetical order (by title, perhaps). This corresponds to the "non-clustered index". These card catalogs were organized in a hierarchical structure, so that each drawer would be labeled with the range of cards it contained (Ka - Kl, for example; i.e., the "intermediate node"). Once again, you would drill in until you found your book, but in this case, once you have found it (i.e, the "leaf node"), you don't have the book itself, but just a card with an index number (the DDS#) with which you could find the actual book in the clustered index.
Of course, nothing would stop the librarian from photocopying all the cards and sorting them in a different order in a separate card catalog. (Typically there were at least two such catalogs: one sorted by author name, and one by title.) In principle, you could have as many of these "non-clustered" indexes as you want.
Find below some characteristics of clustered and non-clustered indexes:
Clustered Indexes
Clustered indexes are indexes that uniquely identify the rows in an SQL table.
Every table can have exactly one clustered index.
You can create a clustered index that covers more than one column. For example: create Index index_name(col1, col2, col.....).
By default, a column with a primary key already has a clustered index.
Non-clustered Indexes
Non-clustered indexes are like simple indexes. They are just used for fast retrieval of data. Not sure to have unique data.
Clustered Index
A clustered index determines the physical order of DATA in a table. For this reason, a table has only one clustered index(Primary key/composite key).
"Dictionary" No need of any other Index, its already Index according to words
Nonclustered Index
A non-clustered index is analogous to an index in a Book. The data is stored in one place. The index is stored in another place and the index has pointers to the storage location. this help in the fast search of data. For this reason, a table has more than 1 Nonclustered index.
"Biology Book" at starting there is a separate index to point Chapter location and At the "END" there is another Index pointing the common WORDS location
A very simple, non-technical rule-of-thumb would be that clustered indexes are usually used for your primary key (or, at least, a unique column) and that non-clustered are used for other situations (maybe a foreign key). Indeed, SQL Server will by default create a clustered index on your primary key column(s). As you will have learnt, the clustered index relates to the way data is physically sorted on disk, which means it's a good all-round choice for most situations.
Clustered Index
A Clustered Index is basically a tree-organized table. Instead of storing the records in an unsorted Heap table space, the clustered index is actually B+Tree index having the Leaf Nodes, which are ordered by the clusters key column value, store the actual table records, as illustrated by the following diagram.
The Clustered Index is the default table structure in SQL Server and MySQL. While MySQL adds a hidden clusters index even if a table doesn't have a Primary Key, SQL Server always builds a Clustered Index if a table has a Primary Key column. Otherwise, the SQL Server is stored as a Heap Table.
The Clustered Index can speed up queries that filter records by the clustered index key, like the usual CRUD statements. Since the records are located in the Leaf Nodes, there's no additional lookup for extra column values when locating records by their Primary Key values.
For example, when executing the following SQL query on SQL Server:
SELECT PostId, Title
FROM Post
WHERE PostId = ?
You can see that the Execution Plan uses a Clustered Index Seek operation to locate the Leaf Node containing the Post record, and there are only two logical reads required to scan the Clustered Index nodes:
|StmtText |
|-------------------------------------------------------------------------------------|
|SELECT PostId, Title FROM Post WHERE PostId = #P0 |
| |--Clustered Index Seek(OBJECT:([high_performance_sql].[dbo].[Post].[PK_Post_Id]), |
| SEEK:([high_performance_sql].[dbo].[Post].[PostID]=[#P0]) ORDERED FORWARD) |
Table 'Post'. Scan count 0, logical reads 2, physical reads 0
Non-Clustered Index
Since the Clustered Index is usually built using the Primary Key column values, if you want to speed up queries that use some other column, then you'll have to add a Secondary Non-Clustered Index.
The Secondary Index is going to store the Primary Key value in its Leaf Nodes, as illustrated by the following diagram:
So, if we create a Secondary Index on the Title column of the Post table:
CREATE INDEX IDX_Post_Title on Post (Title)
And we execute the following SQL query:
SELECT PostId, Title
FROM Post
WHERE Title = ?
We can see that an Index Seek operation is used to locate the Leaf Node in the IDX_Post_Title index that can provide the SQL query projection we are interested in:
|StmtText |
|------------------------------------------------------------------------------|
|SELECT PostId, Title FROM Post WHERE Title = #P0 |
| |--Index Seek(OBJECT:([high_performance_sql].[dbo].[Post].[IDX_Post_Title]),|
| SEEK:([high_performance_sql].[dbo].[Post].[Title]=[#P0]) ORDERED FORWARD)|
Table 'Post'. Scan count 1, logical reads 2, physical reads 0
Since the associated PostId Primary Key column value is stored in the IDX_Post_Title Leaf Node, this query doesn't need an extra lookup to locate the Post row in the Clustered Index.
Clustered Index
Clustered indexes sort and store the data rows in the table or view based on their key values. These are the columns included in the index definition. There can be only one clustered index per table, because the data rows themselves can be sorted in only one order.
The only time the data rows in a table are stored in sorted order is when the table contains a clustered index. When a table has a clustered index, the table is called a clustered table. If a table has no clustered index, its data rows are stored in an unordered structure called a heap.
Nonclustered
Nonclustered indexes have a structure separate from the data rows. A nonclustered index contains the nonclustered index key values and each key value entry has a pointer to the data row that contains the key value.
The pointer from an index row in a nonclustered index to a data row is called a row locator. The structure of the row locator depends on whether the data pages are stored in a heap or a clustered table. For a heap, a row locator is a pointer to the row. For a clustered table, the row locator is the clustered index key.
You can add nonkey columns to the leaf level of the nonclustered index to by-pass existing index key limits, and execute fully covered, indexed, queries. For more information, see Create Indexes with Included Columns. For details about index key limits see Maximum Capacity Specifications for SQL Server.
Reference: https://learn.microsoft.com/en-us/sql/relational-databases/indexes/clustered-and-nonclustered-indexes-described
Let me offer a textbook definition on "clustering index", which is taken from 15.6.1 from Database Systems: The Complete Book:
We may also speak of clustering indexes, which are indexes on an attribute or attributes such that all of tuples with a fixed value for the search key of this index appear on roughly as few blocks as can hold them.
To understand the definition, let's take a look at Example 15.10 provided by the textbook:
A relation R(a,b) that is sorted on attribute a and stored in that
order, packed into blocks, is surely clusterd. An index on a is a
clustering index, since for a given a-value a1, all the tuples with
that value for a are consecutive. They thus appear packed into
blocks, execept possibly for the first and last blocks that contain
a-value a1, as suggested in Fig.15.14. However, an index on b is
unlikely to be clustering, since the tuples with a fixed b-value
will be spread all over the file unless the values of a and b are
very closely correlated.
Note that the definition does not enforce the data blocks have to be contiguous on the disk; it only says tuples with the search key are packed into as few data blocks as possible.
A related concept is clustered relation. A relation is "clustered" if its tuples are packed into roughly as few blocks as can possibly hold those tuples. In other words, from a disk block perspective, if it contains tuples from different relations, then those relations cannot be clustered (i.e., there is a more packed way to store such relation by swapping the tuples of that relation from other disk blocks with the tuples the doesn't belong to the relation in the current disk block). Clearly, R(a,b) in example above is clustered.
To connect two concepts together, a clustered relation can have a clustering index and nonclustering index. However, for non-clustered relation, clustering index is not possible unless the index is built on top of the primary key of the relation.
"Cluster" as a word is spammed across all abstraction levels of database storage side (three levels of abstraction: tuples, blocks, file). A concept called "clustered file", which describes whether a file (an abstraction for a group of blocks (one or more disk blocks)) contains tuples from one relation or different relations. It doesn't relate to the clustering index concept as it is on file level.
However, some teaching material likes to define clustering index based on the clustered file definition. Those two types of definitions are the same on clustered relation level, no matter whether they define clustered relation in terms of data disk block or file. From the link in this paragraph,
An index on attribute(s) A on a file is a clustering index when: All tuples with attribute value A = a are stored sequentially (= consecutively) in the data file
Storing tuples consecutively is the same as saying "tuples are packed into roughly as few blocks as can possibly hold those tuples" (with minor difference on one talking about file, the other talking about disk). It's because storing tuple consecutively is the way to achieve "packed into roughly as few blocks as can possibly hold those tuples".
Clustered Index:
Primary Key constraint creates clustered Index automatically if no clustered Index already exists on the table. Actual data of clustered index can be stored at leaf level of Index.
Non Clustered Index:
Actual data of non clustered index is not directly found at leaf node, instead it has to take an additional step to find because it has only values of row locators pointing towards actual data.
Non clustered Index can't be sorted as clustered index. There can be multiple non clustered indexes per table, actually it depends on the sql server version we are using. Basically Sql server 2005 allows 249 Non Clustered Indexes and for above versions like 2008, 2016 it allows 999 Non Clustered Indexes per table.
Clustered Index - A clustered index defines the order in which data is physically stored in a table. Table data can be sorted in only way, therefore, there can be only one clustered index per table. In SQL Server, the primary key constraint automatically creates a clustered index on that particular column.
Non-Clustered Index - A non-clustered index doesn’t sort the physical data inside the table. In fact, a non-clustered index is stored at one place and table data is stored in another place. This is similar to a textbook where the book content is located in one place and the index is located in another. This allows for more than one non-clustered index per table.It is important to mention here that inside the table the data will be sorted by a clustered index. However, inside the non-clustered index data is stored in the specified order. The index contains column values on which the index is created and the address of the record that the column value belongs to.When a query is issued against a column on which the index is created, the database will first go to the index and look for the address of the corresponding row in the table. It will then go to that row address and fetch other column values. It is due to this additional step that non-clustered indexes are slower than clustered indexes
Differences between clustered and Non-clustered index
There can be only one clustered index per table. However, you can
create multiple non-clustered indexes on a single table.
Clustered indexes only sort tables. Therefore, they do not consume
extra storage. Non-clustered indexes are stored in a separate place
from the actual table claiming more storage space.
Clustered indexes are faster than non-clustered indexes since they
don’t involve any extra lookup step.
For more information refer to this article.

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