How to solve memory address problems - memory

Can anyone explain how to solve these problems step by step
Assume a 2^24 byte memory.
Assume the memory is byte addressable. What is the lowest address and highest address? How many bits are needed for the address?
Assume the memory is word addressable, with a 16 bit word. What is the lowest address and highest address? How many bits are needed for the address?
Assume the memory is word addressable, with a 32 bit word. What is the lowest address and highest address? How many bits are needed for the address?

A byte is 8 bits. If it's byteaddressable, you can't reference an address by anything other than the start of some 8 bits. That is, in a 2^2 byte memory, you have 4 bytes. The lowest address starts at 0 bytes, and the highest address starts at 3 bytes. (0, 1, 2, 3 = 4 bytes total)
If the bytes are contiguous (they are juxtaposed- touching each other rather than spread out) then you can fit all 4 bytes into a 4 byte memory perfectly.
a)
If you have 2^24 bytes then you have 2^(24 + 3) bits because you're doing (2^24 * 2^3) = 2^(24+3). Thus you have 134,217,728 total bits.
The highest address would be one byte before the end, so the address at 2^24 - 1. Note that it's 2^24 - 1 and not 2^27 - 1 because you are addressing it by bytes and not bits. Lowest address would be 0.
Lowest address = 0
Highest address = 2^24 - 1
b)
A word just means a grouping of bytes. A 1-byte word is literally the same thing as a byte, it just implies that the word is some meaningful piece of data, whereas a byte is not necessarily a meaningful piece of data.
A 16-bit word == a 2-byte word because 8 bits are in a byte, thus if you have 2^24 bytes available, you only have a total of 2^23 words.
Lowest address = 0
Highest address = max number of words - 1 = 2^23 - 1.
c)
Same thing as with a 4-byte word instead of 2. Thus:
2^22 bytes available to store words.
Lowest address = 0
Highest address = max number of words - 1 = 2^22 - 1.
Feel free to correct me if you see any errors. Hope I helped.

Related

Length and size of strings in Elixir / Erlang needs explanation

Can someone explain why s is a string with 4096 chars
iex(9)> s = String.duplicate("x", 4096)
... lots of "x"
iex(10)> String.length(s)
4096
but its memory size are a few 6 words?
iex(11)> :erts_debug.size(s)
6 # WHAT?!
And why s2 is a much shorter string than s
iex(13)> s2 = "1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20"
"1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20"
iex(14)> String.length(s)
50
but its size has more 3 words than s?
iex(15)> :erts_debug.size(s2)
9 # WHAT!?
And why does the size of these strings does not match their lengths?
Thanks
First clue why this is showing that values can be found in this question. Quoting size/1 docs:
%% size(Term)
%% Returns the size of Term in actual heap words. Shared subterms are
%% counted once. Example: If A = [a,b], B =[A,A] then size(B) returns 8,
%% while flat_size(B) returns 12.
Second clue can be found in Erlang documentation about bitstrings implementation.
So in the first case the string is too big to fit on heap alone, so it uses refc binaries which are stored on stack and on heap there is only pointer to given binary.
In second case string is shorter than 64 bytes and it uses heap binaries which is just array of bytes stored directly in on the heap, so that gives us 8 bytes per word (64-bit) * 9 = 72 and when we check documentation about exact memory overhead in VM we see that Erlang uses 3..6 words per binary + data, where data can be shared.

translate virtual address to physical address

The following page table is for a system with 16-bit virtual and physical addresses and with 4,096-byte pages. The reference bit is set to 1 when the page has been referenced. Periodically, a thread zeroes out all values of the reference bit.All numbers are provided in decimal.
I want to convert the following virtual addresses (in hexadecimal) to the equivalent physical addresses. Also I want to set the reference bit for the appropriate entry in the page table.
• 0xE12C
• 0x3A9D
• 0xA9D9
• 0x7001
• 0xACA1
I know the answers are but I want to know how can I achieve these answers:
0xE12C → 0x312C
0x3A9D → 0xAA9D
0xA9D9 → 0x59D9
0x7001 → 0xF001
0xACA1 → 0x5CA1
I found and tried This but it did not help me much.
It is given that virtual address is 16 bit long.Hence, there are 2^16 addresses in the virtual address space.
Page Size is given to be 4 KB ( there are 4K (4 * (2 ^ 10) )addresses in a page), so the number of pages will be ( 2^16 ) / ( 2 ^ 12 ) = 2 ^ 4.
To address each page 4 bits are required.
The most significant 4 bits in the virtual address will denote the page number being referred and the remaining 12 bits will be the page offset.
One thing to remember is page size (in the virtual address space ) is always same as the frame size in the main memory. Hence the last 12 bits will remain same in the physical address as that of the virtual address.
To get the frame address in the main memory just use the first 4 bits.
Example: Consider the virtual address 0xACA1
Here A in ACA1 denotes the page number ( 10 ) and corresponding frame no is 5 ( 0101) hence the resulting physical address will be → 0x5CA1.
To translate a virtual address to a physical address (applies ONLY to this homework question), we need to know 2 things:
Page Size
Number of bits for virtual address
In this example: 16-bit system, 4KB page size and physical memory size is 64KB.
First of all we need to determine the number of needed bits to act as offset inside page. log2(Page-Size) = log2(4096) = 12 bits for offset
Out of the 16 bits for virtual address, 12 are for offset, that means each process has 2^4 = 16 virtual pages. Each entry in page table stores the corresponding frame accommodating the page. For example:
Now lets translate!
First of all for ease of work lets convert 0xE12C to binary.
0xE12C = (1110 0001 0010 1100) in base 2
1110 = 14 in decimal
Entry 14 in P.T => Page frame 3.
Lets concatenate it to the 12 offset bits
Answer: (0011 0001 0010 1100) = 0x312C
Another example: 0x3A9D
0x3A9D = 0011 1010 1001 1101
0011 = 3
PageTable[3] = 10
10 in decimal = 1010 in binary
1010 1010 1001 1101 in binary = 0xAA9D
To help you solve this question, we need to get our details right:
16 bit of virtual address space = 2^16 = 65,536 address space
16 bit of physical address space = 2^16 = 65,536 address space
4096 Byte page size determines the offset, which is Log(4096) / Log (2) = 12 bit. This means, 2^12 for Page size
As per #Akash Mahapatra, the offset from virtual address is directly mapped to the offset onto physical address
As such, we now have:
2^16 (16bit) for virtual address - 2^12 (12bit) for offset = 4-bit for pages, or rather total number of pages available.
I won't repeat the calculation for physical since it's the same numbers.
2^4 (4bit) for pages = 16, which correlates to the number of table entries above!
We're getting there... be patient! :)
Memory Address 0xE12C in hex notation is also known to be holding 16-bit of address. (Since it's stated in the question.)
Let's butcher the address now...
We first remove '0x' from the info.
We can convert E12C to binary notation like #Tony Tannous, but I am going to apply a little short-cut.
I simply use a ratio. Well, the address is notated in 4 characters above, and since 16/4 = 4, I can define the first letter as virtual address, while the other 3 are offset address.
With the information, 'E' in hexadecimal format, I need to convert to Decimal = 14. Then I look at your table provided, and I found page frame '3'. Page frame 3 is noted in decimal format, which then need to be converted back to Hexadecimal format... Duh!... which is 3!
So, the Physical address mapping of the virtual memory location of 0xE12C can be found at 0x312C on the physical memory.
You will then go back to the table, and refer to the reference bit column and put a '1' to the row 14.
Apply the same concept for these -
0x3A9D → 0xAA9D
0xA9D9 → 0x59D9
0x7001 → 0xF001
0xACA1 → 0x5CA1
If you notice, the last 3 digits are the same (which determines the offset).
And the 1st of the 4-digits are mapped according to the table:
table entry 3 -> page frame 10 -> hex notation A
table entry A (10) -> page frame 5 -> hex notation 5
table entry 7 -> page frame 15 -> hex notation F
table entry A (10) -> page frame 5 -> hex notation 5
Hope this explanation helps you and others like me! :)

What is the relation between address lines and memory?

These are my assignments:
Write a program to find the number of address lines in an n Kbytes of memory. Assume that n is always to the power of 2.
Sample input: 2
Sample output: 11
I don't need specific coding help, but I don't know the relation between address lines and memory.
To express in very easy terms, without any bus-multiplexing, the number of bits required to address a memory is the number of lines (address or data) required to access that memory.
Quoting from the Wikipedia article,
a system with a 32-bit address bus can address 232 (4,294,967,296) memory locations.
for a simple example, consider this, you have 3 address lines (A, B, C), so the values which can be formed using 3 bits are
A B C
0 0 0
0 0 1
0 1 0
0 1 1
1 0 0
1 0 1
1 1 0
1 1 1
Total 8 values. So using ABC, you can access any of those eight values, i.e., you can reach any of those memory addresses.
So, TL;DR, the simple relationship is, with n number of lines, we can represent 2n number of addresses.
An address line usually refers to a physical connection between a CPU/chipset and memory. They specify which address to access in the memory. So the task is to find out how many bits are required to pass the input number as an address.
In your example, the input is 2 kilobytes = 2048 = 2^11, hence the answer 11. If your input is 64 kilobytes, the answer is 16 (65536 = 2^16).

Direct Mapped Cache of Blocks Example

So i have this question in my homework assignment that i have struggling a bit with. I looked over my lecture content/notes and have been able to utilize those to answer the questions, however, i am not 100% sure that i did everything correctly. There are two parts (part C and D) in the question that i was not able to figure out even after consulting my notes and online sources. I am not looking for a solution for those two parts by any means, but it would be greatly appreciated if i could get, at least, a nudge in the right direction in how i can go about solving it.
I know this is a rather large question, however, i hope someone could possibly check my answers and tell me if all my work and methods of looking at this problem is correct. As always, thank you for any help :)
Alright, so now that we have the formalities out of the way,
--------------------------Here is the Question:--------------------------
Suppose a small direct-mapped cache of blocks with 32 blocks is constructed. Each cache block stores
eight 32-bit words. The main memory—which is byte addressable1—is 16,384 bytes in size. 32-bit words are stored
word aligned in memory, i.e., at an address that is divisible by 4.
(a) How many 32-bit words can the memory store (in decimal)?
(b) How many address bits would be required to address each byte of memory?
(c) What is the range of memory addresses, in hex? That is, what are the addresses of the first and last bytes of
memory? I'll give you a hint: memory addresses are numbered starting at 0.
(d) What would be the address of the last word in memory?
(e) Using the cache mapping scheme discussed in the Chapter 5 lecture notes, how many and which address bits
would be used to form the block offset?
(f) How many and which memory address bits would be used to form the cache index?
(g) How many and which address bits would be used to form the tag field for each cache block?
(h) To which cache block (in decimal) would memory address 0x2A5C map to?
(i) What would be the block offset (in decimal) for 0x2A5C?
(j) How many other main memory words would map to the same block as 0x2A5C?
(k) When the word at 0x2A5C is moved into a cache block, what are the memory addresses (in hex) of the other
words which will also be moved into this block? Express your answer as a range, e.g., [0x0000, 0x0200].
(l) The first word of a main memory block that is mapped to a cache block will always be at an address that is
divisible by __ (in decimal)?
(m) Including the V and tag bits of each cache block, what would be the total size of the cache (in bytes)
(n) what would be the size allocated for the data bits (in bytes)?
----------------------My answers and work-----------------------------------
a) memory = 16384 bytes. 16384 bytes into bits = 131072 bits. 131072/32 = 4096 32-bit words
b) 2^14 (main memory) * 2^2 (4 bits/word) = 2^16. take log(base2)(2^16) = 16 bits
c) couldnt figure this part out (would appreciate some input (NOT A SOLUTION) on how i can go about looking at this problem
d)could not figure this part out either :(
e)8 words in each cache line. 8 * 4(2^2 bits/word) = 32 bits in each cache line. log(base2)(2^5) = 5 bits used for block offset.
f) # of blocks = 2^5 = 32 blocks. log(base2)(2^5) = 5 bits for cache index
g) tag = 16 - 5 - 5 - 2(word alignment) = 4 bits
h) 0x2A5C
0010 10100 10111 00
tag index offset word aligned bits
maps to cache block index = 10100 = 0x14
i) maps to block offset = 10111 = 0x17
j) 4 tag bits, 5 block offset = 2^9 other main memory words
k) it is a permutation of the block offsets. so it maps the memory addresses with the same tag and cache index bits and block offsets of 0x00 0x01 0x02 0x04 0x08 0x10 0x11 0x12 0x14 0x18 0x1C 0x1E 0x1F
l)divisible by 4
m) 2(V+tag+data) = 2(1+4+2^3*2^5) = 522 bits = 65.25 bytes
n)data bits = 2^5 blocks * 2^3 words per block = 256 bits = 32 bytes
Part C:
If a memory has M bytes, and the memory is byte addressable, the the memory addresses range from 0 to M - 1.
For your question, this means that memory addresses range from 0 to 16383, or in hex 0x0 to 0x3FFF.
Part D:
Words are 4 bytes long. So given your answer to C, the last word is at:
(0x3FFFF - 3) -> 0x3FFC.
You can see that this is correct because the lowest 2 bits of the address are 0, which must be true of any 4 byte aligned address.

Cache Addressing: Length of Index, Block offset, Byte offset & Tag?

Let's say I know the following values:
W = Word length (= 32 bits)
S = Cache size in words
B = Block size in words
M = Main memory size in words
How do I calculate how many bits are needed for:
- Index
- Block offset
- Byte offset
- Tag
a) in Direct Mapped Cache
b) in Fully Associative Cache?
The address may be split up into the following parts:
[ tag | index | block or line offset | byte offset ]
Number of byte offset bits
0 for word-addressable memory, log2(bytes per word) for byte addressable memory
Number of block or line offset bits
log2(words per line)
Number of index bits
log2(CS), where CS is the number of cache sets.
For Fully Associative, CS is 1. Since log2(1) is 0, there are no
index bits.
For Direct Mapped, CS is equal to CL, the number of cache lines, so the number of index bits is log2(CS) === log2(CL).
For n-way Associative CS = CL ÷ n: log2(CL ÷ n) index bits.
How many cache lines you have got can be calculated by dividing the cache size by the block size = S/B (assuming they both do not include the size for tag and valid bits).
Number of tag bits
Length of address minus number of bits used for offset(s) and index. The Length of the the addresses can be calculated using the size of the main memory, as e.g. any byte needs to be addressed, if it's a byte addressable memory.
Source: http://babbage.cs.qc.edu/courses/cs343/cache_parameters.xhtml

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