An instruction pipeline has five stages, namely, instruction fetch $$(IF),$$ instruction decode and register fetch $$(ID/RF)$$ instruction execution $$(EX),$$ memory access $$(MEM),$$ and register writeback $$(WB)$$ with stage latencies $$1$$ ns, $$2.2$$ $$ns,$$ $$2$$ $$ns,$$ $$1$$ $$ns,$$ and $$0.75$$ $$ns,$$ respectively ($$ns$$ stands for nanoseconds). To gain in terms of frequency, the designers have decided to split the $$ID/RF$$ stage into three stages $$(ID, RF1, RF2)$$ each of latency $$2.2/3$$ $$ns,$$ Also, the $$EX$$ stage is split into two stages $$(EX1, EX2)$$ each of latency $$1$$ ns. The new design has a total of eight pipeline stages. A program has $$20$$% branch instructions which execute in the $$EX$$ stage and produce the next instruction pointer at the end of the $$EX$$ stage in the old design and at the end of the $$EX2$$ stage in the new design. The IF stage stalls after fetching a branch instruction until the next instruction pointer is computed. All instructions other than the branch instruction have an average $$CPI$$ of one in both the designs. The execution times of this program on the old and the new design are $$P$$ and $$Q$$ nanoseconds, respectively. The value of $$P/Q$$ is _____________.

Consider the following processors ($$ns$$ stands for nanoseconds). Assume that the pipeline registers have zero latency.
$$P1:$$ Four-stage pipeline with stage latencies $$1$$ $$ns,$$ $$2$$ $$ns,$$ $$2$$ $$ns,$$ $$1$$ $$ns.$$
$$P2:$$ Four-stage pipeline with stage latencies $$1$$ $$ns,$$ 1$$.5$$ $$ns,$$ $$1.5$$ $$ns,$$ $$1.5$$ $$ns.$$
$$P3:$$ Five-stage pipeline with stage latencies $$0.5$$ $$ns,$$ $$1$$ $$ns,$$ $$1$$ $$ns,$$ $$0.6$$ $$ns,$$ $$1$$ $$ns.$$
$$P4:$$ Five-stage pipeline with stage latencies $$0.5$$ $$ns,$$ $$0.5$$ $$ns,$$ $$1$$ $$ns,$$ $$1$$ $$ns,$$ $$1.1$$ $$ns.$$

Which processor has the highest peak clock frequency?

Consider a hash table with 100 slots. Collisions are resolved using chaining. Assuming simple uniform hashing, what is the probability that the first 3 slots are unfilled after the first 3 insertions?

Suppose depth first search is executed on the graph below starting at some unknown vertex. Assume that a recursive call to visit a vertex is made only after first checking that the vertex has not been visited earlier. Then the maximum possible recursion depth (including the initial call) is _________.