Operating System- Multilevel Paging, Inverted Page Table

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Modern computer systems support a large logicaladdress space. In such a condition, the page table itself
becomes excessively large and it is clear that, we would
not want to allocate the page table contiguously in main
memory. One simple solution to this problem is to divide
the page table into smaller pieces.There are several ways
to accompalish this division.
One way is to use a two-level paging algorithm,in which
the page table itself is also paged.
Two-Level Paging
A logical address (on 32-bit machine with 4K page size)
is divided into:

1. a page number consisting of 20 bits.
2. a page offset consisting of 12 bits.
Since the page table is paged, the page number is
further divided into:
3. a 10-bit page number.
4. a 10-bit page offset.
5. Thus, a logical address is as follows:
page number

p1

p2

page offset
d

10
10
12
where pi is an index into the outer page table, and p2 is
the displacement within the page of the outer page table.

page number
p1

10

p2

10

page offset
d

12

...
Page 0
...

...

...

...
Logical
Memory

Outer Page
Table

...

Page Table

page number

p1

p2

page offset

d

p1
p2
d
outer page
table

inner page
table

desired
page

Advantages of multi-level paging:
The hierarchical system has its advantages in
that all those that live in a system understand
what extortionists they need to pay and obey.

Problem with multi-level paging:
Extra memory references to access
translation tables can slow programs down by
a factor of two or more.
Solution: Use translation look-aside buffers
(TLB) to speed up address translation

Inverted Page Table

Consider a system in which the virtual address space is 64
bits, the page size is 4KB, and the amount of physical
memory is 512MB. How much space would a simple singlelevel page table take? Such a table
contains one entry per virtual page, or 264−12 = 252
entries. Each entry would require about 4 bytes, so the
total page table size is 254 bytes, or 16 petabytes (peta- >
tera- > giga-)! And this is for each process!
Of course, a process is unlikely to use all 64 bits of address
space, so how about using multilevel page
tables? How many levels would be required to ensure that
each page table require only a single page?
Assuming an entry takes a constant 4 bytes of space, each
page table can store 1024 entries, or 10 bits of
address space. Thus d52/10e = 6 levels are required. But
this results in 6 memory accesses for each address
translation!

But notice that there are only 512MB of memory in the
system, or 229−12 = 217 = 128K physical pages.
If we can somehow manage to store only a single page table
entry per physical page, the page table size
decreases considerably, to 2MB assuming each entry takes
16 bytes. And since processes share physical
memory, we need only have a single global page table. This
is the concept of an inverted page table.

1.1 Linear Inverted Page Tables

The simplest form of an inverted page table contains one
entry per physical page in a linear array. Since
the table is shared, each entry must contain the process ID
of the page owner. And since physical pages
are now mapped to virtual, each entry contains a virtual
page number instead of a physical. The physical
page number is not stored, since the index in the table
corresponds to it.

As usual, information bits are
kept around for protection and accounting purposes.
Assuming 16 bits for a process ID, 52 bits for a virtual
page number, and 12 bits of information, each entry takes
80 bits or 10 bytes, so the total page table size is
10 · 128KB = 1.3MB.
In order to translate a virtual address, the virtual page
number and current process ID are compared
against each entry, traversing the array sequentially. When a
match is found, the index of the match replaces
the virtual page number in the address to obtain a physical
address. If no match is found, a page fault occurs.

1.2 Hashed Inverted Page Tables

A hashed inverted page table adds an extra level
before the actual page table, called a hash anchor
table.
This table is at least as large as the page table, and
maps process IDs and virtual page numbers to
page
table entries. Since collisions may occur, the page
table must do chaining. Since each member in the
chain
must map to a physical page and therefore must
have a corresponding page table entry, the chain
can be
represented as a sequence of page table entries,
with each entry pointing to the next entry in the
chain

This requires an extra 4 bytes per entry, so the page table
size is now 14 ·128KB = 1.8MB. The hash anchor table
takes an additional 4 · 128KB = 512KB, for a total of
2.3MB.
Now in order to translate a virtual address, the process ID
and virtual page number are hashed to get
an entry in the hash anchor table. The entry’s pointer to a
page table entry is followed, and the process ID

and virtual page number are compared against that stored
there. If they don’t match, the page table entry’s
next pointer is followed to get another page table entry,
and the process is repeated until a match is found.
Figure-1

Figure 1: Translation procedure using a linear inverted
page table. Info bits exist in each entry, though they
are not shown.
If a chain is exhausted by hitting an invalid next pointer, a
page fault results. The translation process is
illustrated in figure 2.
Assuming a good hash function, the average chain length
should be about 1.5, so only 2.5 memory
accesses are required on average to translate an address
[1]. This is pretty good, considering a two-level page
table requires 2 accesses. Caching can still improve the
access time considerably.

● Use hash table to limit the search to one — or
at most
a few — page-table entries.

● One entry for each real page of memory.
● Entry consists of the virtual address of the
page stored
in that real memory location, with information
about
the process that owns that page.
● Decreases memory needed to store each page
table,
but increases time needed to search the table
when a
page reference occurs.

Operating System- Multilevel Paging, Inverted Page Table

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