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201 changes: 201 additions & 0 deletions challenges/medium/86_paged_attention/challenge.html
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<p>
Implement decode-phase attention over a <strong>paged KV cache</strong>. In LLM serving systems (e.g., vLLM),
the key and value tensors for each sequence are stored in fixed-size memory <em>blocks</em> (pages) that
may be scattered non-contiguously across a shared GPU memory pool. A <code>block_table</code> maps each
sequence's logical block indices to physical block indices in the cache pool. Given a single query vector
per sequence (one new token being generated), compute the attention output by gathering the relevant
K/V blocks via the block table and computing scaled dot-product attention over the full context.
</p>

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<text x="190" y="92" fill="#888" font-size="9" text-anchor="middle">values = physical block indices in pool &#x2193;</text>

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<text x="16" y="118" fill="#666" font-size="10">K_cache / V_cache pool (GPU memory)</text>

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<text x="310" y="208" text-anchor="middle" fill="#ccc" font-size="11" font-weight="bold">Decode Attention (per sequence s, per head h)</text>

<text x="36" y="232" fill="#ffcc66" font-size="10">1.</text>
<text x="56" y="232" fill="#aaa" font-size="10">Gather K, V: token t is at pool[ block_table[s, t/B] ], offset t%B</text>

<text x="36" y="254" fill="#ffcc66" font-size="10">2.</text>
<text x="56" y="254" fill="#aaa" font-size="10">scores[t] = Q[s,h] &#xb7; K[s,h,t] / &#x221a;head_dim for t = 0 .. context_lens[s]-1</text>

<text x="36" y="276" fill="#ffcc66" font-size="10">3.</text>
<text x="56" y="276" fill="#aaa" font-size="10">output[s,h] = &#x2211;_t softmax(scores)[t] &#xb7; V[s,h,t]</text>
</svg>

<h2>Implementation Requirements</h2>
<p>
Implement the function <code>solve(Q, K_cache, V_cache, block_table, context_lens, output, batch_size, num_heads, head_dim, block_size, max_blocks_per_seq)</code>
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don't really have to say this in the implementation requirements. @claude change this to match the format of other challenge's implementation requirements

that computes paged decode-phase attention:
</p>
<ul>
<li><code>Q</code>: query tensor of shape <code>(batch_size, num_heads, head_dim)</code>, dtype <code>float32</code> — one query per sequence</li>
<li><code>K_cache</code>: paged key cache of shape <code>(num_blocks, block_size, num_heads, head_dim)</code>, dtype <code>float32</code></li>
<li><code>V_cache</code>: paged value cache of shape <code>(num_blocks, block_size, num_heads, head_dim)</code>, dtype <code>float32</code></li>
<li><code>block_table</code>: physical block indices of shape <code>(batch_size, max_blocks_per_seq)</code>, dtype <code>int32</code></li>
<li><code>context_lens</code>: number of valid KV tokens per sequence, shape <code>(batch_size,)</code>, dtype <code>int32</code></li>
<li><code>output</code>: result of shape <code>(batch_size, num_heads, head_dim)</code>, dtype <code>float32</code></li>
</ul>
<p>
For each sequence \(s\) and each attention head \(h\), compute:
</p>
<ol>
<li>
Gather the \(\text{context_lens}[s]\) key and value vectors from the paged cache using
\(\text{block_table}[s]\). Token at logical position \(t\) lives in physical block
\(\text{block_table}[s,\;\lfloor t / B \rfloor]\) at offset \(t \bmod B\) within that block,
where \(B = \text{block_size}\).
</li>
<li>
Compute scaled dot-product attention:
\[\text{scores}[t] = \frac{Q[s, h] \cdot K[s, h, t]}{\sqrt{\text{head_dim}}}\]
</li>
<li>
Apply softmax over all \(\text{context_lens}[s]\) positions to get attention weights.
</li>
<li>
Compute: \(\displaystyle \text{output}[s, h] = \sum_{t} \text{softmax}(\text{scores})[t] \cdot V[s, h, t]\)
</li>
</ol>
<p>
Do not use external libraries beyond the framework you select. Keep the function signature unchanged.
Write results directly into <code>output</code>.
</p>

<h2>Example</h2>
<p>
Input: <code>batch_size</code> = 1, <code>num_heads</code> = 1, <code>head_dim</code> = 4,
<code>block_size</code> = 2, <code>context_lens</code> = [2], <code>block_table</code> = [[0]]
</p>
<p>
\(Q[0, 0] = \begin{bmatrix} 1.0 & 1.0 & 0.0 & 0.0 \end{bmatrix}\)
</p>
<p>
Keys gathered from block 0 (2 tokens):
\[
K_0 = \begin{bmatrix} 1.0 & 0.0 & 0.0 & 0.0 \end{bmatrix}, \quad
K_1 = \begin{bmatrix} 0.0 & 1.0 & 0.0 & 0.0 \end{bmatrix}
\]
Values gathered from block 0:
\[
V_0 = \begin{bmatrix} 2.0 & 0.0 & 0.0 & 0.0 \end{bmatrix}, \quad
V_1 = \begin{bmatrix} 0.0 & 4.0 & 0.0 & 0.0 \end{bmatrix}
\]
</p>
<p>
Scores (before softmax):
\[
s_0 = \frac{Q \cdot K_0}{\sqrt{4}} = \frac{1}{2} = 0.5, \quad
s_1 = \frac{Q \cdot K_1}{\sqrt{4}} = \frac{1}{2} = 0.5
\]
Attention weights: \(\text{softmax}([0.5, 0.5]) = [0.5, 0.5]\)
\[
\text{output}[0, 0] = 0.5 \cdot V_0 + 0.5 \cdot V_1 =
\begin{bmatrix} 1.0 & 2.0 & 0.0 & 0.0 \end{bmatrix}
\]
</p>

<h2>Constraints</h2>
<ul>
<li>1 &le; <code>batch_size</code> &le; 32</li>
<li>1 &le; <code>num_heads</code> &le; 64</li>
<li>1 &le; <code>head_dim</code> &le; 256; <code>head_dim</code> is a multiple of 8</li>
<li>1 &le; <code>block_size</code> &le; 64; <code>block_size</code> is a power of 2</li>
<li>1 &le; <code>context_lens[s]</code> &le; 8,192 for all sequences <code>s</code></li>
<li>All input tensors are on the GPU and in <code>float32</code> (except <code>block_table</code> and <code>context_lens</code> which are <code>int32</code>)</li>
<li><code>block_table[s, i]</code> is a valid index into the first dimension of <code>K_cache</code> for all <code>i &lt; ceil(context_lens[s] / block_size)</code></li>
<li>Performance is measured with <code>batch_size</code> = 8, <code>num_heads</code> = 32, <code>head_dim</code> = 128, <code>block_size</code> = 16, <code>context_lens[s]</code> = 2,048 for all sequences</li>
</ul>
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