Corner Block List
The idea
The corner block list describes a mosaic floorplan: a dissection of the die into exactly n rooms, one block per room, with no empty space and every internal junction a T rather than a crossing. Its reason to exist is that mosaic floorplans include non-slicing ones, layouts no sequence of straight cuts can produce, which slicing trees cannot represent at all.
The encoding comes from a demolition. Repeatedly delete the block in the top-right corner. Each corner block is attached in one of two ways, and each deletion records three things:
| list | records |
|---|---|
| S | the block names, in the order they are inserted (reverse of deletion) |
| L | each block's orientation: H if it spans the top, V if it spans the right |
| T | how many T-junctions the block's inner edge covers |
Replaying the three lists in order rebuilds the floorplan exactly, and both the demolition and the rebuild run in linear time. A T-count above zero is precisely where non-slicing structure appears: a block whose edge straddles more than one neighbor.
One structural difference from the O-tree and B-tree is worth naming. Those decode a compacted packing whose geometry depends on the block sizes. A corner block list fixes the room structure* independent of sizes: the blocks are poured into the rooms afterward. That makes it natural for soft blocks and for keeping a topology fixed while dimensions change.
The recipe
Step 1 (find the corner). Take the block in the top-right corner of the current floorplan.
Step 2 (record). Append its name to S, its orientation (H for top, V for right) to L, and the number of T-junctions its inner edge covers to T.
Step 3 (delete). Remove it, letting the covered neighbors expand into the freed space, and repeat from Step 1 until one block remains.
Step 4 (decode). To rebuild, start from the base block and insert the rest in order: an H block enters along the top covering its T top-boundary junctions, a V block enters along the right covering its T right-boundary junctions.
Watch it work: the pinwheel, encoded and rebuilt
The five-room pinwheel is the canonical non-slicing floorplan. The walkthrough names its corner block, lays out the three lists, then decodes them insertion by insertion, showing the T-junction each block covers as it enters.
Pros, cons, and the trade-off
Strengths
- Non-slicing native: it represents mosaic floorplans, including the non-slicing ones slicing trees cannot express.
- Size-independent topology: it fixes the room structure, not a packing, so blocks can be resized without changing the representation.
- Linear encode and decode: both the corner-block demolition and the insertion rebuild are linear time.
- Clean combinatorics: mosaic floorplans are counted by the Baxter numbers, and corner-block-style encodings are how those bijections are proved.
Weaknesses
- Empty space needs dummies: a mosaic room cannot be vacant, so whitespace must be modeled as explicit dummy blocks.
- Redundancy: several corner block lists can map to the same floorplan unless extra rules are imposed, which its successors tightened.
- Two-part decode: fixing rooms first and sizing after is an extra step compared to the packing-native trees.
The trade-off
The corner block list trades the packing trees' directness for reach and flexibility. It captures non-slicing mosaic floorplans and keeps the topology independent of block sizes, at the cost of modeling empty space with dummy blocks and tolerating some encoding redundancy. Where the O-tree and B*-tree hand you a finished packing, the corner block list hands you a room structure to be sized, which is the right currency for soft blocks and combinatorial enumeration.
Is it actually used today?
The corner block list is chiefly a research and theory representation, and the ancestor of a family of mosaic encodings.
- Mosaic and soft-block floorplanning: it is a reference representation for non-slicing floorplans where topology should survive resizing.
- Enumeration theory: it appears in the combinatorial literature on counting floorplans, tied to the Baxter numbers.
- A family, refined by successors: twin binary sequences and later mosaic encodings tightened its dummy-block and redundancy weaknesses, so its ideas persist through descendants more than through direct use.
- Note on production tools: commercial floorplanners do not publish their internal representations, so any specific use of a corner block list there is not verifiable. The evidence above is the academic record.