$(18$ points$)$ Segmentation and Paging

Suppose that a small computer system uses $8-$bit virtual and physical addresses, and each address represents one byte. Let the size of the computer's physical memory be $256$ bytes. Answering the following

questions:

$($a$)(5$ points$)$ First, suppose the Base & Bounds approach is adopted in memory management. Assume that one process has been loaded into the memory. The value of the base register for heap segment is $12$ and the bound of the heap segment $($representing the size of the heap$)$ is $18.$ Given the following offsets of the virtual address of $6,20,14,30,$ which offsets would probably cause the segmentation fault? For those offsets which would not cause the segmentation fault, what are the corresponding physical addresses?

$($b$)$ Suppose we use a two$-$level paging in memory management, and the page directory has $4$ entries, the page table $($PT$)$ has $8$ entries. The two$-$level page table is shown in shown below. $($Black area represents an empty space in the page directory or page table$)$

$($i$)(3$ points$)$ How many bits are used for the offset? What is the size of one page?

$($ii$)(4$ points$)$ Assume that one entry in a page directory and page table needs one byte. If we use the two$-$level page table above, how many bytes of memory space would be saved compare with single$-$level page table $($using the same page size$)?$

$($iii$)(6$ points$)$ Given the following virtual addresses: $95,29,182,$ which of them would be mapped? If the virtual address is mapped, what is its corresponding physical address?