Consider the virtual page reference string $$$1,2,3,2,4,1,3,2,4,1$$$
On a demand paged virtual memory system running on a computer system that has main memory size of $$3$$ page frames which are initially empty. Let $$LRU,$$ $$FIFO$$ and $$OPTIMAL$$ denote the number of page faults under the corresponding page replacement policy. Then

A processor uses 36 bit physical addresses and 32 bit virtual addresses, with a page frame size of 4 Kbytes. Each page table entry is of size 4 bytes. A three level page table is used for virtual to physical address translation, where the virtual address is used as follows:
$$ * \,\,\,\,\,\,$$ Bits 30-31 are used to index into the first level page table
$$ * \,\,\,\,\,\,$$ Bits 21-29 are used to index into the second level page table
$$ * \,\,\,\,\,\,$$ Bits 12-20 are used to index into the third level page table, and
$$ * \,\,\,\,\,\,$$ Bits 0-11 are used as offset within the page

The number of bits required for addressing the next level page table (or page frame) in the page table entry of the first, second and third level page tables are respectively

A process has been allocated $$3$$ page frames. Assume that none of the pages of the process are available in the memory initially. The process makes the following sequence of page references (reference string): $$$1, 2, 1, 3, 7, 4, 5, 6, 3, 1$$$

If optimal page replacement policy is used, how many page faults occur for the above reference string?

A process has been allocated $$3$$ page frames. Assume that none of the pages of the process are available in the memory initially. The process makes the following sequence of page references (reference string): $$$1, 2, 1, 3, 7, 4, 5, 6, 3, 1$$$

Least Recently Used (LRU) page replacement policy is a practical approximation to optimal page replacement. For the above reference string, how many more page faults occur with LRU than with the optimal page replacement policy?