Aristo Trie -- a Patricia Trie with Merkle hash labeled edges ============================================================= These data structures allows to overlay the *Patricia Trie* with *Merkel Trie* hashes. With a particular layout, the structure is called and *Aristo Trie* (Patricia = Roman Aristocrat, Patrician.) This description does assume familiarity with the abstract notion of a hexary *Merkle Patricia [Trie](https://en.wikipedia.org/wiki/Trie)*. Suffice it to say the state of a valid *Merkle Patricia Tree* is uniquely verified by its top level vertex. 1. Deleting entries in a compact *Merkle Patricia Tree* ------------------------------------------------------- The main feature of the *Aristo Trie* representation is that there are no double used nodes any sub-trie as it happens with the representation as a [compact Merkle Patricia Tree](http://archive.is/TinyK). For example, consider the following state data for the latter. leaf = (0xf,0x12345678) (1) branch = (a,a,a,,, ..) with a = hash(leaf) root = hash(branch) These two nodes, called *leaf* and *branch*, and the *root* hash are a state (aka key-value pairs) representation as a *compact Merkle Patricia Tree*. The actual state is 0x0f ==> 0x12345678 0x1f ==> 0x12345678 0x2f ==> 0x12345678 The elements from *(1)* can be organised in a key-value table with the *Merkle* hashes as lookup keys a -> leaf root -> branch This is a space efficient way of keeping data as there is no duplication of the sub-trees made up by the *Leaf* node with the same payload *0x12345678* and path snippet *0xf*. One can imagine how this property applies to more general sub-trees in a similar fashion. Now delete some key-value pair of the state, e.g. for the key *0x0f*. This amounts to removing the first of the three *a* hashes from the *branch* record. The new state of the *Merkle Patricia Tree* will look like leaf = (0xf,0x12345678) (2) branch1 = (,a,a,,, ..) root1 = hash(branch1) a -> leaf root1 -> branch1 A problem arises when all keys are deleted and there is no reference to the *leaf* data record, anymore. One should find out in general when it can be deleted, too. It might be unknown whether the previous states leading to here had only a single *Branch* record referencing to this *leaf* data record. Finding a stale data record can be achieved by a *mark and sweep* algorithm, but it becomes too clumsy to be useful on a large state (i.e. database). Reference counts come to mind but maintaining these is generally error prone when actors concurrently manipulate the state (i.e. database). 2. *Patricia Trie* example with *Merkle hash* labelled edges ------------------------------------------------------------ Continuing with the example from chapter 1, the *branch* node is extended by an additional set of structural identifiers *x, w, z*. It allows to handle the deletion of entries in a more benign way while keeping the *Merkle hashes* for validating sub-trees. A solution for the deletion problem is to represent the situation *(1)* as leaf-a = (0xf,0x12345678) copy of leaf from (1) (3) leaf-b = (0xf,0x12345678) copy of leaf from (1) leaf-c = (0xf,0x12345678) copy of leaf from (1) branch2 = ((x,y,z,,, ..)(a,b,c,,, ..)) root2 = (w,root) with root from (1) where a = hash(leaf-a) same as a from (1) b = hash(leaf-b) same as a from (1) c = hash(leaf-c) same as a from (1) w,x,y,z numbers, mutually different The records above are stored in a key-value database as w -> branch2 x -> leaf-a y -> leaf-b z -> leaf-c Then this structure encodes the key-value pairs as before 0x0f ==> 0x12345678 0x1f ==> 0x12345678 0x2f ==> 0x12345678 Deleting the data for key *0x0f* now results in the new state leaf-b = (0xf,0x12345678) (4) leaf-c = (0xf,0x12345678) branch3 = ((,y,z,,, ..)(,b,c,,, ..)) w -> branch3 y -> leaf-b z -> leaf-c Due to duplication of the *leaf* node in *(3)*, no reference count is needed in order to detect stale records cleanly when deleting key *0x0f*. Removing this key allows to remove hash *a* from *branch2* as well as also structural key *x* which will consequently be deleted from the lookup table. A minor observation is that manipulating a state entry, e.g. changing the payload associated with key *0x0f* to 0x0f ==> 0x987654321 the structural layout of the above trie will not change, that is the indexes *w, x, y, z* of the table that holds the data records as values. All that changes are values. leaf-d = (0xf,0x987654321) (5) leaf-b = (0xf,0x12345678) leaf-c = (0xf,0x12345678) branch3 = ((x,y,z,,, ..)(d,b,c,,, ..)) root3 = (w,hash(d,b,c,,, ..)) 3. Discussion of the examples *(1)* and *(3)* --------------------------------------------- Examples *(1)* and *(3)* differ in that the structural *Patricia Trie* information from *(1)* has been removed from the *Merkle hash* instances and implemented as separate table lookup IDs (called *vertexID*s later on.) The values of these lookup IDs are arbitrary as long as they are all different. In fact, the [Erigon](http://archive.is/6MJV7) project discusses a similar situation in **Separation of keys and the structure**, albeit aiming for a another scenario with the goal of using mostly flat data lookup structures. A graph for the example *(1)* would look like | root | +-------------+ | branch | +-------------+ | | | a a a | | | leaf while example *(2)* has (root) (6) | w | +-------------+ | branch2 | | (a) (b) (c) | +-------------+ / | \ x y z / | \ leaf-a leaf-b leaf-c The labels on the edges indicate the downward target of an edge while the round brackets enclose separated *Merkle hash* information. This last example (6) can be completely split into structural tree and Merkel hash mapping. structural trie hash map (7) --------------- -------- | (root) -> w w (a) -> x | (b) -> y +-------------+ (c) -> z | branch2 | +-------------+ / | \ x y z / | \ leaf-a leaf-b leaf-c 4. *Patricia Trie* node serialisation with *Merkle hash* labelled edges ----------------------------------------------------------------------- The data structure for the *Aristo Trie* forllows example *(7)* by keeping structural information separate from the Merkle hash labels. As for teminology, * an *Aristo Trie* is a pair *(structural trie, hash map)* where * the *structural trie* realises a haxary *Patricia Trie* containing the payload values in the leaf records * the *hash map* contains the hash information so that this trie operates as a *Merkle Patricia Tree*. In order to accommodate for the additional structural elements, a non RLP-based data layout is used for the *Branch*, *Extension*, and *Leaf* containers used in the key-value table that implements the *Patricia Trie*. It is now called *Aristo Trie* for this particular data layout. The structural keys *w, x, y, z* from the example *(3)* are called *vertexID* and implemented as 64 bit values, stored *Big Endian* in the serialisation. ### 4.1 Branch record serialisation 0 +--+--+--+--+--+--+--+--+--+ | | -- first vertexID 8 +--+--+--+--+--+--+--+--+--+ ... -- more vertexIDs +--+--+ | | -- access(16) bitmap +--+--+ || | -- marker(2) + unused(6) +--+ where marker(2) is the double bit array 00 For a given index *n* between *0..15*, if the bit at position *n* of the bit vector *access(16)* is reset to zero, then there is no *n*-th structural *vertexID*. Otherwise one calculates the n-th vertexID is at position Vn * 8 for Vn the number of non-zero bits in the range 0..(n-1) of access(16) Note that data are stored *Big Endian*, so the bits *0..7* of *access* are stored in the right byte of the serialised bitmap. ### 4.2 Extension record serialisation 0 +--+--+--+--+--+--+--+--+--+ | | -- vertexID 8 +--+--+--+--+--+--+--+--+--+ | | ... -- path segment +--+ || | -- marker(2) + pathSegmentLen(6) +--+ where marker(2) is the double bit array 10 The path segment of the *Extension* record is compact encoded. So it has at least one byte. The first byte *P0* has bit 5 reset, i.e. *P0 and 0x20* is zero (bit 4 is set if the right nibble is the first part of the path.) Note that the *pathSegmentLen(6)* is redunant as it is determined by the length of the extension record (as *recordLen - 9*.) ### 4.3 Leaf record serialisation 0 +-- .. ... -- payload (may be empty) +--+ | | ... -- path segment +--+ || | -- marker(2) + pathSegmentLen(6) +--+ where marker(2) is the double bit array 11 A *Leaf* record path segment is compact encoded. So it has at least one byte. The first byte *P0* has bit 5 set, i.e. *P0 and 0x20* is non-zero (bit 4 is also set if the right nibble is the first part of the path.) If present, the serialisation of the payload field can be either for account data, for RLP encoded or for unstructured data as defined below. ### 4.4 Leaf record payload serialisation for account data 0 +-- .. --+ | | -- nonce, 0 or 8 bytes +-- .. --+--+ | | -- balance, 0, 8, or 32 bytes +-- .. --+--+ | | -- storage ID, 0 or 8 bytes +-- .. --+--+ | | -- code hash, 0, 8 or 32 bytes +--+ .. --+--+ | | -- bitmask(2)-word array +--+ where each bitmask(2)-word array entry defines the length of the preceeding data fields: 00 -- field is missing 01 -- field lengthh is 8 bytes 10 -- field lengthh is 32 bytes Apparently, entries 0 and and 2 of the bitmask(2) word array cannot have the value 10 as they refer to the nonce and the storage ID data fields. So, joining the bitmask(2)-word array to a single byte, the maximum value of that byte is 0x99. ### 4.5 Leaf record payload serialisation for RLP encoded data 0 +--+ .. --+ | | | -- data, at least one byte +--+ .. --+ | | -- marker byte +--+ where the marker byte is 0xaa ### 4.6 Leaf record payload serialisation for unstructured data 0 +--+ .. --+ | | | -- data, at least one byte +--+ .. --+ | | -- marker byte +--+ where the marker byte is 0xff ### 4.7 Descriptor record serialisation 0 +-- .. ... -- recycled vertexIDs +--+--+--+--+--+--+--+--+ | | -- bottom of unused vertex IDs +--+--+--+--+--+--+--+--+ || | -- marker(2) + unused(6) +--+ where marker(2) is the double bit array 01 Currently, the descriptor record only contains data for producing unique vectorID values that can be used as structural keys. If this descriptor is missing, the value *(0x40000000,0x01)* is assumed. The last vertexID in the descriptor list has the property that that all values greater or equal than this value can be used as vertexID. The vertexIDs in the descriptor record must all be non-zero and record itself should be allocated in the structural table associated with the zero key. ### 4.7 Backend filter record serialisation 0 +--+--+--+--+--+ .. --+--+ .. --+ | | -- 32 bytes filter source hash 32 +--+--+--+--+--+ .. --+--+ .. --+ | | -- 32 bytes filter target hash 64 +--+--+--+--+--+ .. --+--+ .. --+ | | -- number of unused vertex IDs 68 +--+--+--+--+ | | -- number of structural triplets 72 +--+--+--+--+--+ .. --+ | | -- first unused vertex ID 80 +--+--+--+--+--+ .. --+ ... -- more unused vertex ID N1 +--+--+--+--+ || | -- flg(3) + vtxLen(29), 1st triplet +--+--+--+--+--+ .. --+ | | -- vertex ID of first triplet +--+--+--+--+--+ .. --+--+ .. --+ | | -- optional 32 bytes hash key +--+--+--+--+--+ .. --+--+ .. --+ ... -- optional vertex record N2 +--+--+--+--+ || | -- flg(3) + vtxLen(29), 2nd triplet +--+--+--+--+ ... where + minimum size of an empty filer is 72 bytes + the flg(3) represents the tuple (key-mode,vertex-mode) encoding the serialised storage states 0 -- encoded and present 1 -- not encoded, void => considered deleted 2 -- not encoded, to be ignored so, when encoded as flg(3) = key-mode * 3 + vertex-mode the the tuple (2,2) will never occur and flg(3) < 9 + the vtxLen(29) is the number of bytes of the optional vertex record which has maximum size 2^29-1 which is short of 512 MiB 5. *Patricia Trie* implementation notes --------------------------------------- ### 5.1 Database decriptor representation ^ +----------+ | | top | active delta layer, application cache | +----------+ | +----------+ ^ db | stack[n] | | desc | : | | optional passive delta layers, handled by obj | stack[1] | | transaction management (can be used to | | stack[0] | | successively recover the top layer) | +----------+ v | +----------+ | | roFilter | optional read-only backend filter | +----------+ | +----------+ | | backend | optional physical key-value backend database v +----------+ There is a three tier access to a key-value database entry as in top -> roFilter -> backend where only the *top* layer is obligatory. ### 5.2 Distributed access using the same backend There can be many descriptors for the same database. Due to delta layers and filters, each descriptor instance can work with a different state of the database. Although there is only one of the instances that can write the current state on the physical backend database, this priviledge can be shifted to any instance for the price of updating the *roFiters* for all other instances. #### Example: db1 db2 db3 -- db1, db2, .. database descriptors/handles | | | tx1 tx2 tx3 -- tx1, tx2, ..transaction/top layers | | | ø ø ø -- no backend filters yet \ | / \ | / PBE -- physical backend database After collapse/committing *tx1* and saving it to the physical backend database, the above architecture mutates to db1 db2 db3 | | | ø tx2 tx3 | | | ø ~tx1 ~tx1 -- filter reverting the effect of tx1 on PBE \ | / \ | / tx1+PBE -- tx1 merged into physical backend database When looked at descriptor API there are no changes when accessing data via *db1*, *db2*, or *db3*. In a different, more algebraic notation, the above tansformation is written as | tx1, ø | (8) | tx2, ø | PBE | tx3, ø | || \/ | ø, ø | (9) | tx2, ~tx1 | tx1+PBE | tx3, ~tx1 | The system can be further converted without changing the API by committing and saving *tx2* on the middle line of matrix (9) | ø, ø | (10) | ø, tx2+~tx1 | tx1+PBE | tx3, ~tx1 | || \/ | ø, ~(tx2+~tx1) | (11) | ø, ø | (tx2+~tx1)+tx1+PBE | tx3, ~tx1+~(tx2+~tx1) | The *+* notation just means the repeated application of filters in left-to-right order. The notation looks like algebraic group notation but this will not be analysed further as there is no need for a general theory for the current implementation. Suffice to say that the inverse *~tx* of *tx* is calculated against the current state of the physical backend database which makes it messy to formulate boundary conditions. Nevertheless, *(8)* can alse be transformed by committing and saving *tx2* (rather than *tx1*.) This gives | tx1, ~tx2 | (12) | ø, ø | tx2+PBE | tx3, ~tx2 | || \/ | ø, (tx1+~tx2) | (13) | ø, ø | tx2+PBE | tx3, ~tx2 | As *(11)* and *(13)* represent the same API, one has tx2+PBE =~ tx1+(tx2+~tx1)+PBE because of the middle rows (14) ~tx2 =~ ~tx1+~(tx2+~tx1) because of (14) (15) which looks like some distributive property in *(14)* and commutative property in *(15)* for this example (but it is not straight algebraically.) The *=~* operator above indicates that the representations are equivalent in the sense that they have the same effect on the backend database (looks a bit like residue classes.) It might be handy for testing/verifying an implementation using this example.