Total-Bits Calculation

“How many total bits are required for a direct-mapped cache with … ?” is on every final. The data is only part of it — you also store a tag and a valid bit per line. Here's the reliable recipe.

1. The recipe

① Block size in bytes = (words/block) × 4

② Number of lines = data capacity ÷ block size

③ index = log₂(lines) · block offset = log₂(words/block) · byte offset = 2

④ tag = address bits − index − block offset − byte offset

⑤ bits/line = 1 (valid) + tag + (words/block × 32) · Total = lines × bits/line

\text{Total bits} = \text{lines}\times\big(\underbrace{1}_{\text{valid}} + \text{tag} + \underbrace{\text{words/block}\times 32}_{\text{data}}\big)

2. Worked — Autumn 2025 Q3(b)

Direct-mapped, 64 KB of data, 8-word (32-byte) blocks, 32-bit address. Find total bits.

Q3bbigTotal bits (tag + valid + data) for a 64 KB, 8-word-block direct-mapped cache, 32-bit address.

① Block size = 8 words × 4 = $32$ bytes $= 2^5$ .

② Lines = $\dfrac{64\text{ KB}}{32\text{ B}} = \dfrac{2^{16}}{2^{5}} = 2^{11} = 2048$ .

③ Offsets/index. byte offset = 2, block offset = $\log_2 8 = 3$ , index = $\log_2 2048 = 11$ .

④ Tag = $32 - 11 - 3 - 2 = 16$ bits.

⑤ Bits per line = valid 1 + tag 16 + data $(8\times32=256)$ = 273.

\text{Total} = 2048 \times 273 = \boxed{559{,}104\ \text{bits}} \;=\; 546\ \text{Kibits}

3. Index size from block counts — Q3(c)

A common variant: “a direct-mapped cache has m blocks and memory has n blocks — find the index size.” The index just selects one of the $m$ cache lines:

\text{index bits} = \log_2 m \qquad\qquad \text{tag bits} = \log_2 n - \log_2 m = \log_2\!\frac{n}{m}

e.g. $m = 2^{10}=1024$ cache blocks ⇒ index = 10 bits, regardless of $n$ .

4. Live calculator

Data capacity (KB) Words / block Address bits

Block size = 32 B · Lines = 2,048 (index 11 bits) · block offset 3 · byte offset 2
Tag = 32 − 11 − 3 − 2 = 16 bits
Bits/line = 1 + 16 + 256 = 273
Total = 2,048 × 273 = 559,104 bits(546 Kibits)

Don't forget the +1 valid bit per line and the tag — a very common mistake is to report only the data bits.

Section checklist

lines = capacity ÷ block size; index = log₂(lines).
tag = address − index − block offset − byte offset.
total = lines × (1 + tag + data); data = words/block × 32.