Consider a demand paging system with three frames, and the following page reference string: 1 2 3 4 5 4 1 6 4 5 1 3 2. The contents of the frames are as follows initially and after each reference (from left to right): The *-marked references cause page replacements. Which one or more of the following could be the page replacement policy/policies in use?

Let us trace the given reference string with 3 frames: 1 2 3 4 5 4 1 6 4 5 1 3 2 and verify each policy against the frame transitions given in the table:

1. Optimal (OPT) Replacement Policy:

   • At reference 4 (Frame: $\{1, 2, 3\}$): Future references are $5, 4, 1, 6, 4, 5, 1, 3, 2$. Among $\{1, 2, 3\}$, $1$ is accessed at index 7, $3$ at index 12, and $2$ at index 13. Since $2$ is needed furthest in the future, $2$ is replaced by $4$. Frame becomes $\{1, 4, 3\}$.

   • At reference 5 (Frame: $\{1, 4, 3\}$): Future references are $4, 1, 6, 4, 5, 1, 3, 2$. Among $\{1, 4, 3\}$, $4$ is accessed next, $1$ follows, and $3$ is accessed at index 12. Thus, $3$ is replaced by $5$. Frame becomes $\{1, 4, 5\}$.

   • References 4, 1: Hits. Frame remains $\{1, 4, 5\}$.

   • At reference 6 (Frame: $\{1, 4, 5\}$): Future references are $4, 5, 1, 3, 2$. Among $\{1, 4, 5\}$, $4$ and $5$ are accessed soon, while $1$ is accessed at index 11. Thus, $1$ is replaced by $6$. Frame becomes $\{6, 4, 5\}$.

   • References 4, 5: Hits. Frame remains $\{6, 4, 5\}$.

   • At reference 1 (Frame: $\{6, 4, 5\}$): Future references are $3, 2$. None of $6, 4, 5$ will be referenced again. Any of them can be replaced. Replacing $4$ gives $\{6, 1, 5\}$.

   • At reference 3 (Frame: $\{6, 1, 5\}$): Neither $6$ nor $5$ is used again. Replacing $5$ gives $\{6, 1, 3\}$.

   • At reference 2 (Frame: $\{6, 1, 3\}$): Neither $6$ nor $3$ is used again. Replacing $6$ gives $\{2, 1, 3\}$.

   All replacements perfectly match the table for Optimal Page Replacement. Hence, (D) is correct.

2. Least Recently Used (LRU):

   • At reference 4, the past accesses are $1, 2, 3$. The least recently used page is $1$. But in the table, $2$ is replaced. Hence, LRU cannot be in use.

3. Least Frequently Used (LFU):

   • At reference 6, past accesses are $1, 2, 3, 4, 5, 4, 1$. Frequency counts for pages in frames $\{1, 4, 5\}$ are: $\text{freq}(1) = 2$, $\text{freq}(4) = 2$, $\text{freq}(5) = 1$. LFU would replace page $5$. But the table replaces page $1$. Hence, LFU cannot be in use.

4. Most Frequently Used (MFU):

   • At reference 3 (step 12), past accesses are $1, 2, 3, 4, 5, 4, 1, 6, 4, 5, 1$. Frequency counts for pages in frames $\{6, 1, 5\}$ are: $\text{freq}(6) = 1$, $\text{freq}(1) = 3$, $\text{freq}(5) = 2$. MFU must replace page $1$ (highest frequency). However, the table replaces page $5$. Hence, MFU cannot be in use.

Therefore, only option (D) is correct.