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This overview explains how hybrid and partial-hybrid
strategies choose which graph regions to materialize.

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The chosen boundary changes work and representation, while
a reduced result may still need alignment for a dense request.

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The chosen boundary changes work and representation, while
a reduced result may still need alignment for a dense request.

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Consider a graph with two inner branches and an exclusive-or root.

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A full four-input vector has sixteen outputs.

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A materialization boundary marks where a computed value becomes stored data for later

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consumers, rather than remaining a pending operation.

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One strategy stores both branch vectors and then combines them.

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It makes each intermediate available for reuse, but also requires

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their construction and storage.

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If neither has another consumer, some of that storage may not provide a benefit.

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A hybrid strategy can retain a useful shared branch while evaluating

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another branch on demand.

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It must still produce every output required by the caller.

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Selecting internal boundaries is not permission to omit inconvenient

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assignments from a full-vector request.

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assignments from a full-vector request.

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Hybrid strategies can place a materialization boundary around a region.

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Partial-hybrid strategies preserve upper
structure while dispatching selected children.

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Follow the actual selected mode rather than
treating every hybrid label as the same algorithm.

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In S AND C AND D, OR S AND E, let S be A AND B.

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The left boundary encloses the whole eligible subtree.

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The right boundary encloses only the selected child.

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The parent and other child remain visible in the second diagram.

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At 11011 both paths must produce one.

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Compare all assignments before interpreting a structural difference.

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No-reinflation is an output contract too.

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A reduced packed result with live-variable metadata
is different from a fully broadcast ambient matrix.

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Current APIs expose admission and representation diagnostics;

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ask for dense output explicitly when that is the deliverable.

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Return to the four-input rule F equals A and B, exclusive-or C or D.

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The intervening five-input rule is a separate example; E is not part of this four-input rule.

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Fix C and D to zero.

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A and B remain free, so this is a restriction, not a complete assignment.

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After fixing C and D to zero, enumerate both possibilities for A and both for B.

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The first three assignments return zero and the last returns one.

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Omitting the last case would produce an incomplete result, even though only four outputs rather than sixteen are required.

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The restriction fixes C and D, while A and B remain free.

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Deliver all four remaining outputs: zero, zero, zero, one.

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This smaller task is complete because every assignment in its declared remaining universe is included; it is not sampling.

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Suppose the caller explicitly fixes C and D to zero and asks for the remaining function

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over A and B.

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The required universe now has four assignments.

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This is a smaller, declared task, not an incomplete attempt at

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the original sixteen-output task.

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The second branch becomes zero, so the final function is A and B.

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Its four outputs in zero-first order are zero zero zero one.

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All requested assignments are covered, even though the result is

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smaller than the unrestricted vector.

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Keep these two output contracts separate in the comparison.

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A selective materialization strategy is exact when it returns everything

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promised by its own contract.

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Whether it is faster still depends on the chosen boundaries, reuse, and delivery costs.

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Follow the actual selected mode and output representation.

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Charge alignment and broadcast whenever the caller requires the full layout.