Algorithmic skill for optimally partitioning LLM context or task DAGs across K agents. Covers two modes: (A) online/reactive spawning under context pressure, and (B) static pre-planned DAG execution with graph min-cut optimization. Encodes algorithms for sequence-stable clustering, dynamic K selection, token-budget-aware partitioning, and handoff protocol design. Use when designing multi-agent systems, deciding when and how many agents to spawn, or optimizing a WinDAGs-style DAG for token efficiency.
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Algorithmic skill for optimally partitioning LLM context or task DAGs across K agents. Covers two modes: (A) online/reactive spawning under context pressure, and (B) static pre-planned DAG execution with graph min-cut optimization. Encodes algorithms for sequence-stable clustering, dynamic K selection, token-budget-aware partitioning, and handoff protocol design. Use when designing multi-agent systems, deciding when and how many agents to spawn, or optimizing a WinDAGs-style DAG for token efficiency.
{"kind":"deliverable","produces":[{"kind":"design-doc","description":"Partition plan assigning context and task-DAG segments across K agents with budgets and handoff protocol"},{"kind":"code","description":"Partitioning implementation (online reactive spawning or static min-cut) for the target orchestrator"}]}
Agent Context Partitioner
Purpose
When a single LLM context grows unwieldy, or when you are pre-planning a complex task DAG, you need to decide:
How many agents K to use — and when K should grow
Which context goes to which agent — without blindly copying everything everywhere
What to communicate between agents — the minimum necessary, not everything
How to avoid order-dependent mistakes — the greedy epsilon-similarity trap
This skill provides the algorithmic vocabulary and pseudocode for all four decisions.
Core Mental Model: Agents as Context Protectors
Agents are not a hivemind that shares a global context. They are context protectors: each agent carries deep specialized knowledge and is over-qualified for its task. The cost of spawning a new agent is not just the system prompt — it is the semantic coherence loss from splitting context that belongs together, plus the communication overhead of routing results across agents.
A single session accumulates context. At time t_i, context pressure triggers the question: spawn a new agent, and if so, what does it inherit?
Key decisions at each time step:
K(t_i): how many agents right now?
Partition π(t_i): which context chunks go to which agent?
Handoff H(a→b): what is the minimal context summary to send?
Mode B: Static DAG Execution
Given a pre-planned task DAG (WinDAGs-style), assign nodes to agents before execution begins. Sequential dependent nodes can share one agent (saving context-copy overhead); parallel independent nodes go to separate agents. This is a graph partitioning problem.
The Sequence-Dependence Problem (Lessons from Newsle)
The Epsilon-Similarity Triangle Trap
Greedy online clustering is order-dependent. Given three context chunks:
a = database schema docs
b = ORM configuration
c = migration scripts
Where sim(a,b) > ε, sim(b,c) > ε, but sim(a,c) < ε.
Arrival order [c, a, b]: c and a are assigned to different agents (sim < ε), then b must join one of them. Whichever wins, the wrong agent owns two of the three tightly related chunks.
Arrival order [a, b, c]: a and b cluster together (sim > ε), then c joins. Now one agent owns all three. Correct.
The same data yields different (often wrong) partitions depending on arrival order. This is the fundamental failure mode of greedy online context assignment.
The Fix: Consensus Before Commitment
Never make irrevocable assignment decisions from a single ordering. Three practical implementations:
Algorithm 1: Evidence Accumulation Clustering (EAC) — Best for Offline/Batch
Origin: Fred & Jain 2002, "Data Clustering Using Evidence Accumulation"
Run N random centroid initializations, build a co-association matrix, cluster on it.
defeac_context_partition(chunks, K, n_trials=50):
"""
Order-independent context partitioning via consensus clustering.
chunks: list of ContextChunk(id, embedding, token_count, dep_set)
"""
n = len(chunks)
embeddings = np.array([c.embedding for c in chunks])
sim_matrix = cosine_similarity(embeddings)
dep_graph = build_dependency_graph(chunks)
co_assoc = np.zeros((n, n))
for _ inrange(n_trials):
# Randomize which centroid gets offered the chunk first
initial_centroids = random.sample(range(n), K)
labels = greedy_assign_with_centroids(sim_matrix, initial_centroids,
dep_graph) # preserves causal orderfor i inrange(n):
for j inrange(n):
if labels[i] == labels[j]:
co_assoc[i][j] += 1
co_assoc /= n_trials
dist_matrix = 1 - co_assoc
# Single-linkage on co-association distances
final_labels = agglomerative_clustering(dist_matrix, K, linkage='single')
return enforce_causal_closure(final_labels, dep_graph)
Complexity: O(n_trials × n × K). For n=1000 chunks, n_trials=50: fast enough for pre-planned DAGs.
Key insight: the randomization is of centroid selection order, not chunk order. Chunk causal order is a hard constraint maintained in greedy_assign_with_centroids.
Algorithm 2: Persistent Homology (Vietoris-Rips) — Best for Unknown ε
When you don't know the right epsilon, persistent homology finds topologically stable clusters across all scales.
defpersistent_homology_partition(chunks, K):
"""
Use 0-th persistent homology to find epsilon-independent clusters.
Requires: ripser or gudhi library.
"""
embeddings = np.array([c.embedding for c in chunks])
dist_matrix = 1 - cosine_similarity(embeddings)
# H_0 persistence diagram: (birth, death) pairs for connected components
diagrams = ripser(dist_matrix, metric='precomputed', maxdim=0)
h0 = diagrams['dgms'][0]
# The K most persistent components = stable clusters# Avoid the K-1 highest-persistence merges to keep K clusters
finite_pairs = h0[h0[:, 1] < np.inf]
persistences = finite_pairs[:, 1] - finite_pairs[:, 0]
sorted_deaths = np.sort(finite_pairs[:, 1])[::-1]
if K <= len(sorted_deaths):
# Cut between the (K-1)th and Kth most persistent merges
epsilon_threshold = (sorted_deaths[K-2] + sorted_deaths[K-1]) / 2else:
epsilon_threshold = sorted_deaths[-1] / 2
labels = connected_components_at_threshold(dist_matrix, epsilon_threshold)
return enforce_causal_closure(labels, build_dependency_graph(chunks))
Key insight: low-persistence merges are "accidents of epsilon" — they only become connected at extreme epsilon values. High-persistence components are the true semantic clusters. The VR filtration gives you a stability certificate for every cluster assignment.
When to use: when the task domain is unknown and you cannot calibrate epsilon empirically.
Algorithm 3: RANSAC-Bootstrap — Best for Real-Time
defbootstrap_stable_partition(chunks, K, n_bootstrap=100, sample_frac=0.8,
confidence_threshold=0.8):
"""
Real-time stable partitioning via bootstrap sampling.
Only commits chunks with >confidence_threshold consensus.
"""
n = len(chunks)
vote_matrix = np.zeros((n, n))
co_occurrence = np.zeros((n, n))
for _ inrange(n_bootstrap):
sample_idx = sorted(random.sample(range(n), int(n * sample_frac)))
sample_embeds = np.array([chunks[i].embedding for i in sample_idx])
labels = kmeans(sample_embeds, K)
for a, i inenumerate(sample_idx):
for b, j inenumerate(sample_idx):
co_occurrence[i][j] += 1if labels[a] == labels[b]:
vote_matrix[i][j] += 1
vote_matrix /= np.maximum(co_occurrence, 1) # normalize# Committed chunks: those with >threshold consistent vote# Deferred chunks: wait for more context before assignment
final_labels = spectral_clustering(vote_matrix, K)
confidence = np.array([
max(sum(vote_matrix[i][j] for j inrange(n)
if final_labels[j] == final_labels[i]) / n
for i inrange(n))
])
deferred = np.where(confidence < confidence_threshold)[0]
# Deferred chunks get the label of their most confident neighborfor i in deferred:
neighbors = np.argsort(vote_matrix[i])[::-1][:5]
committed_neighbors = [j for j in neighbors if j notin deferred]
if committed_neighbors:
final_labels[i] = final_labels[committed_neighbors[0]]
return enforce_causal_closure(final_labels, build_dependency_graph(chunks))
Key insight: chunks that are truly between clusters will have inconsistent assignment across bootstrap samples (noisy vote_matrix rows). Don't commit them until they gather more votes. The confidence_threshold is your "conviction gate."
Algorithm 4: Streaming BIRCH with Causal Closure — Best for O(n) Online
For high-throughput streaming where you cannot afford EAC or RANSAC overhead:
classCausalBIRCH:
"""
BIRCH (Zhang et al. 1996) adapted for causally-ordered context chunks.
Each CF-node represents a cluster with causal closure enforced.
"""def__init__(self, K_max, token_budget, radius_threshold=0.3):
self.clusters = [] # list of CFNodeself.K_max = K_max
self.token_budget = token_budget
self.T = radius_threshold
defabsorb(self, chunk):
best_cluster, best_dist = None, float('inf')
for cluster inself.clusters:
# Can we absorb without exceeding budget or radius?if cluster.tokens + chunk.token_count > self.token_budget:
continue
new_radius = cluster.prospective_radius(chunk.embedding)
if new_radius <= self.T and new_radius < best_dist:
best_cluster = cluster
best_dist = new_radius
if best_cluster isNone:
iflen(self.clusters) < self.K_max:
# Spawn new cluster
new_cluster = CFNode()
new_cluster.add(chunk)
# Enforce causal closure: pull in all deps of this chunkfor dep in chunk.dep_set:
dep_cluster = self.find_cluster_of(dep)
if dep_cluster and dep_cluster != new_cluster:
# Chunk depends on something in another cluster → merge or copyif dep_cluster.tokens + new_cluster.tokens <= self.token_budget:
new_cluster.merge(dep_cluster)
self.clusters.remove(dep_cluster)
else:
# Mark as cross-agent dependency (communication edge)
new_cluster.add_remote_dep(dep, dep_cluster.agent_id)
self.clusters.append(new_cluster)
return new_cluster
else:
# At K_max: absorb into closest cluster regardless of radius
best_cluster = min(self.clusters,
key=lambda c: c.prospective_radius(chunk.embedding))
best_cluster.add(chunk)
return best_cluster
Complexity: O(K) per chunk. The causal closure check is the critical addition over vanilla BIRCH.
Algorithm 5: DAG-METIS + FM Refinement (Mode B)
For pre-planned DAGs, use multilevel graph partitioning:
defpartition_task_dag(dag, K, token_budget_per_agent):
"""
Phase 1: Augment edge weights with communication + sync cost
Phase 2: METIS multilevel partitioning
Phase 3: Schedulability enforcement (no cycles in agent-level DAG)
Phase 4: EAC stabilization across 5 coarsening strategies
Phase 5: Fiduccia-Mattheyses (FM) refinement
Phase 6: Token budget enforcement with agent splitting
"""# Phase 1: Edge weights = communication tokens + sync penaltyfor u, v in dag.edges():
critical_path_fraction = (critical_path_length(dag, start=v) /
dag.total_critical_path())
dag.set_edge_weight(u, v,
estimate_comm_tokens(u, v) +
COMM_LATENCY * critical_path_fraction)
# Phase 2: Undirected METIS with node weights = estimated token cost
weighted = dag.to_weighted_undirected(edge_weight='max')
partition = metis_partition(weighted, K,
node_weights={v: dag.token_cost(v) for v in dag},
balance=1.1)
# Phase 3: Enforce no cycles in agent-level dependency graph
agent_dag = build_agent_dag(dag, partition)
while has_cycle(agent_dag):
cycle = find_cycle(agent_dag)
partition = break_cycle_by_merging(dag, partition, cycle)
agent_dag = build_agent_dag(dag, partition)
# Phase 4: EAC over 5 METIS coarsening strategies
strategies = ['heavy_edge', 'random', 'sorted_heavy_edge', 'fc', 'greedy']
nodes = list(dag.nodes())
n = len(nodes)
co_assoc = np.zeros((n, n))
for s in strategies:
p = metis_partition(weighted, K, coarsening=s,
node_weights={v: dag.token_cost(v) for v in dag})
for i, ni inenumerate(nodes):
for j, nj inenumerate(nodes):
if p[ni] == p[nj]:
co_assoc[i][j] += 1
co_assoc /= len(strategies)
partition = spectral_from_coassoc(co_assoc, K, nodes)
partition = enforce_causal_closure(partition, dag)
# Phase 5: FM refinement (up to 20 passes)for _ inrange(20):
partition, improved = fm_refinement_pass(dag, partition, K)
ifnot improved:
break# Phase 6: Token budget enforcementfor agent_id inrange(K):
agent_tokens = sum(dag.token_cost(v)
for v, a in partition.items() if a == agent_id)
while agent_tokens > token_budget_per_agent:
moved = move_cheapest_boundary_node(dag, partition, agent_id)
ifnot moved:
K += 1
partition = split_largest_agent(dag, partition)
agent_tokens = sum(dag.token_cost(v)
for v, a in partition.items() if a == agent_id)
return partition, build_execution_schedule(dag, partition)
FM Refinement (Fiduccia-Mattheyses)
deffm_refinement_pass(dag, partition, K):
"""O(|E|) per pass. Find node moves that reduce communication cost."""
gains = {v: compute_node_gain(dag, partition, v) for v in dag.nodes()}
locked = set()
moves = []
whileTrue:
best = max(
((v, t, gains[(v,t)]) for v in dag.nodes() if v notin locked
for t inrange(K) if t != partition[v]),
key=lambda x: x[2], default=None
)
if best isNone:
break
node, target, gain = best
old = partition[node]
partition[node] = target
locked.add(node)
moves.append((node, old, target, gain))
# Update gains for neighborsfor neighbor in dag.neighbors(node):
if neighbor notin locked:
gains.update({(neighbor, t): compute_node_gain(dag, partition, neighbor)
for t inrange(K)})
# Find best prefix of moves (maximize cumulative gain)
best_cut, best_i, cumulative = 0, 0, 0for i, (node, old, new, gain) inenumerate(moves):
cumulative += gain
if cumulative > best_cut:
best_cut, best_i = cumulative, i + 1# Roll back moves after best prefixfor node, old, new, gain in moves[best_i:]:
partition[node] = old
return partition, best_cut > 0
Algorithm 6: Dynamic K Selection
Gap Statistic (Tibshirani 2001) — Offline
defgap_statistic_k(embeddings, K_max=10, n_reference=20):
"""Find optimal K by comparing observed dispersion to null distribution."""
gaps = []
stderrs = []
for K inrange(1, K_max + 1):
labels = kmeans(embeddings, K)
W_K = within_cluster_dispersion(embeddings, labels)
bounds = (embeddings.min(axis=0), embeddings.max(axis=0))
ref_log_W = []
for _ inrange(n_reference):
ref = np.random.uniform(bounds[0], bounds[1], size=embeddings.shape)
ref_labels = kmeans(ref, K)
ref_log_W.append(np.log(within_cluster_dispersion(ref, ref_labels)))
gap = np.mean(ref_log_W) - np.log(W_K)
stderr = np.std(ref_log_W) * np.sqrt(1 + 1 / n_reference)
gaps.append(gap)
stderrs.append(stderr)
# Tibshirani criterion: smallest K where gap(K) >= gap(K+1) - stderr(K+1)for K inrange(len(gaps) - 1):
if gaps[K] >= gaps[K + 1] - stderrs[K + 1]:
return K + 1return K_max
BIC for Streaming GMM — Online
defstreaming_bic_spawn_check(new_embedding, gmm_k, gmm_params, n_samples):
"""
Returns (should_spawn: bool, new_K: int).
Split if BIC improves by more than BIC_THRESHOLD.
"""
d = len(new_embedding)
p = 2 * d # params per GMM component (diagonal covariance)
log_lik_K = compute_log_likelihood(gmm_params, new_embedding)
bic_K = -2 * log_lik_K + gmm_k * p * np.log(n_samples)
# Try splitting the highest-variance component
candidate = split_highest_variance_component(gmm_params)
run_em_steps(candidate, n_steps=10) # partial convergence, not full EM
log_lik_Kplus1 = compute_log_likelihood(candidate, new_embedding)
bic_Kplus1 = -2 * log_lik_Kplus1 + (gmm_k + 1) * p * np.log(n_samples)
BIC_THRESHOLD = 10# prevents hair-trigger splittingif bic_Kplus1 < bic_K - BIC_THRESHOLD:
returnTrue, gmm_k + 1, candidate
returnFalse, gmm_k, gmm_params
MDL Spawn Decision — Principled
MDL(K) = K × d × log(n) [model description length]
+ Σ_i [-log P(c_i | cluster)] [data given model]
+ comm_cost(K) [token cost of cross-agent communication]
Spawn when: MDL(K+1) + comm_cost(K+1) < MDL(K)
The MDL formulation naturally penalizes:
Creating new agents (larger model description)
Copying context to multiple agents (larger comm_cost)
Poor cluster fit (larger data description length)
Causal Closure: The Hard Constraint
Unlike news articles, LLM context chunks have causal ordering and dependency graphs. This is a hard constraint that all partitioning algorithms must enforce.
defenforce_causal_closure(labels, dep_graph):
"""
If chunk j is assigned to agent A but chunk i (which j depends on)
is assigned to agent B, then either:
(a) merge A and B if token budgets allow, or
(b) mark i as a "shared" chunk (copied to both agents)
"""
changed = Truewhile changed:
changed = Falsefor chunk_id in dep_graph.nodes():
agent = labels[chunk_id]
for dep_id in dep_graph.predecessors(chunk_id):
dep_agent = labels[dep_id]
if dep_agent != agent:
# Violation: chunk depends on content in another agent# Resolution: merge agents or flag as cross-agent depif can_merge(agent, dep_agent):
labels = merge_agents(labels, agent, dep_agent)
changed = Trueelse:
# Copy dep_id to agent's context (communication cost)
mark_as_shared(dep_id, agent)
return labels
Handoff Protocol: Minimal Context Transfer
When a chunk must cross agent boundaries, send the minimum necessary:
defcompute_handoff(from_agent, to_agent, dep_graph):
"""
Compute the minimal context summary for cross-agent dependency.
Returns: (summaries: list[CompressedChunk], full_chunks: list[Chunk])
"""# Find what to_agent actually needs from from_agent
needed_ids = transitive_deps(
to_agent.pending_task_ids,
exclude=set(to_agent.context_chunk_ids)
)
needed_ids = {cid for cid in needed_ids
if chunk_owner(cid) == from_agent}
# Separate boundary chunks (near the partition cut) from interior
boundary = {cid for cid in needed_ids
ifmax(embedding_sim(cid, c) for c in to_agent.context_chunk_ids) > 0.5}
interior = needed_ids - boundary
return Handoff(
# Interior chunks: full copy (to_agent needs them verbatim)
full_chunks=[chunks[cid] for cid in interior],
# Boundary chunks: compressed summary (to_agent can infer the rest)
summaries=[compress(chunks[cid]) for cid in boundary],
# Dependency map so to_agent can request more if needed
dep_map={cid: dep_graph.predecessors(cid) for cid in needed_ids}
)
Key principle: compress chunks that are near the partition boundary (high semantic overlap with to_agent's existing context) — the receiving agent can fill in gaps via its own inference. Send full copies only for chunks with no semantic overlap with the receiver.
Communication Complexity Lower Bound (Yao 1979)
You cannot eliminate all communication if agents' contexts have mutual information.
For agents A (context C_A) and B (context C_B) jointly answering query q:
Communication ≥ Ω(I(C_A; C_B | q))
where I is mutual information. This means:
Find partitions that minimize MI across the cut, not just cosine distance
Cosine similarity is a proxy for MI (valid for normalized Gaussian-ish embeddings)
Practically: weight edges by max(0, cosine_similarity(u, v) - 0.5) when building the graph for partitioning, rather than raw cosine similarity. This calibrates toward information-theoretic optimality.
Production Systems Gap Analysis
System
K Selection
Sequence Stability
Token Budget
Communication Model
LangGraph
Manual
None
None
Unbounded
AutoGen
Fixed upfront
None
None
Message board (all-to-all)
CrewAI
Role-based
None
None
Task output passing
TEGP (this skill)
BIC/Gap/MDL
EAC/Homology/Bootstrap
Explicit constraint
MI-weighted min-cut
The core failures shared by all production systems:
No sequence-dependence mitigation (order-dependent greedy routing)
No formal communication cost model (cross-agent reference is treated as free)
K is fixed by the human, not computed from the task structure
No partition refinement after initial assignment
Practical Heuristics (Implementable Today)
Chunk at semantic boundaries — paragraph breaks, function boundaries, section headers — not fixed token counts. NLTK's TextTilingTokenizer or LLM-based segmentation works.
Embed every chunk immediately — use text-embedding-3-small (cheap, fast). Store (chunk_id, embedding, token_count, dep_set).
Maintain the causal dep graph — as you chunk, record which chunks reference which earlier chunks (via explicit reference or sequential proximity).
Use BIRCH + causal closure for online mode — O(n) per chunk, handles streaming naturally. Set radius threshold by calibrating on 10 sample chunks (the 75th percentile pairwise distance is a good starting T).
Run 20 EAC trials in background — every 10 new chunks, run 5 trials and update co_assoc. After 20 background trials, apply any corrections with >50% co_assoc agreement.
Spawn at 20% budget remaining, not 0% — you need budget for the handoff protocol itself (system prompt for new agent + context copy).
Handoff = summary + dep list, not full context copy — compress boundary chunks (those semantically overlapping the receiving agent's existing context), send full copy only for chunks with zero overlap.
For static DAGs: METIS then 20 FM passes — never use round-robin or role-keyword routing. Even a random initial partition + FM refinement is 2-3x better than greedy role assignment.
Calibrate ε empirically — don't use persistent homology unless you truly don't know ε. Run EAC on 50 sample chunks from your domain, find the natural persistence gap in H_0.
BIC threshold of 10 — prevents hair-trigger agent spawning. Only spawn when BIC(K+1) < BIC(K) - 10.
When to Apply Each Algorithm
Scenario
Algorithm
Why
Offline pre-planning, known K
EAC + spectral
Best stability, order-independent
Unknown K, unknown ε
Persistent homology
Scale-free, stability-certified
High-throughput online
BIRCH + causal closure
O(n), streaming-native
Real-time with budget
RANSAC-bootstrap
Confidence-gated, adaptive
Static task DAG
METIS + FM
Batch-optimal, graph-aware
Dynamic context growth
Streaming Leiden
Local updates, community structure
K selection, offline
Gap statistic
Gold standard, well-studied
K selection, online
Streaming BIC
Incremental, scalable
Key Papers
EAC: Fred & Jain, "Data Clustering Using Evidence Accumulation", ICPR 2002
