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Consider a program that consists of $8$ pages (from $0$ to $7$) and we have $4$ page frames in the physical memory for the pages. The page reference string is :

$1\; 2\; 3\; 2\; 5\; 6\; 3\; 4\; 6\; 3\; 7\; 3\; 1\; 5\; 3\; 6\; 3\; 4\; 2\; 4\; 3\; 4\; 5\; 1$

The number of page faults in LRU and optimal page replacement algorithms are respectively (without including initial page faults to fill available page frames with pages) :

1. $9$ and $6$
2. $10$ and $7$
3. $9$ and $7$
4. $10$ and $6$
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+1 vote

LRU Page Replacement:-

 1 2 3 2 5 6 3 4 6 3 7 3 1 5 3 6 3 4 2 4 3 4 5 1
 5 5 5 5 5 5 7 7 7 7 7 6 6 6 6 6 6 6 5 5 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 2 2 2 2 2 2 4 4 4 4 4 1 1 1 1 1 4 4 4 4 4 4 4 1 1 1 1 1 6 6 6 6 6 6 6 6 5 5 5 5 5 2 2 2 2 2 1 F F F F F F F F F F

Number of Page Fault = 10

Optimal Page Replacement :-

 1 2 3 2 5 6 3 4 6 3 7 3 1 5 3 6 3 4 2 4 3 4 5 1
 5 5 5 4 4 4 7 7 7 5 5 5 5 5 5 5 5 5 5 1 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 2 2 2 2 6 6 6 6 6 6 6 6 6 6 6 6 4 4 4 4 4 4 4 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 F F F F F F F

Number of Page Fault = 7

Hence,Option(B)10 and 7 is the correct choice.

edited
+1 vote

In LRU : it will replace the page which is least recently used

1     2     3   2.   5  6 (r) 3   4(r)  6   3   7(r)  3 1(r)  5r 3   6(r)  3  4r     2r     4  3      5R 1R

 5 5 5 5 5 5 7 7 7 7 7 6 6 6 6 6 6. 6 5 1 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3. 3 3 3 2 2 2 2 2 2 4 4 4 4 4 1 1 1 1 1 4 4 4 4. 4 4 4 1 1 1 1 1 6 6 6 6 6 6 6 6 5 5 5 5 5 2 2 2.  2 2 2

Therefore total replacement at LRU I s 10

optimal page replacement: Replace the page which is not used for longest duration of time in future

no of page fault occur : 7

b option

LRU

no of page faults(without including initial page faults to fill available page frames with pages) is 10

Optimal Page replacement

no of page faults(without including initial page faults to fill available page frames with pages) is 7

Hence option is B