NAME

Net::OnlineCode::Bones - Graph decoding internals

DESCRIPTION

This page gives an overview of how the decoding algorithm for Online Codes work.

The decoding algorithm can be described in one of two ways:

  • in terms of solving a set of algebraic equations; and

  • in terms of resolving a graph.

The first of these explains what the algorithm does, while the second describes how it does it.

Solving a system of algebraic equations

Recall that the Online Codes algorithm works with:

  • message blocks, which are portions of the original file

  • auxiliary blocks, which are the XOR sum of one or more message blocks.

  • check blocks, which are the XOR sum of one or more message and/or auxiliary blocks.

On the encoder side, the algorithm generates auxiliary blocks by using a pseudo-random number generator (PRNG). These blocks are stored locally by the encoder, but are never transmitted. However, by sending the seed value for the PRNG to the decoder, the decoder knows how the auxiliary blocks were constructed, even though it does not know the values of them. In other words, give the PRNG seed value, the decoder can construct a set of equations, one for each auxiliary block:

 aux   = msg   XOR msg   XOR ...
    1       x         y         ...
 aux   = ...
    2
   :

Initially, all the values in these equations are unknown on the decoder side.

As for check blocks, the encoder picks a random seed value for its PRNG and uses this to generate a list of message and/or check blocks to XOR together to calculate the check block's value. It sends both the seed used and the final XOR value to the decoder. As with auxiliary blocks, the decoder can use the PRNG with the transmitted seed value to construct an equation for a received check block:

 chk  = msg_or_aux  XOR msg_or_aux   XOR ...
    1             x               y

Unlike the equations constructed for auxiliary blocks, however, the value of the check block is also sent to the decoder, so each equation includes a single known value on the left-hand side of the equation.

Before the first check block is received, the decoder has a set of equations involving unknown values. As check blocks are received, eventually one of them will be composed of just a single message or auxiliary block. In algebraic terms, we have:

 chk    = msg_or_aux
    x               y

Since there is just a single unknown value in the equation, we can reverse the order of it and use the new form of the equation

 msg_or_aux    = chk
           y        x

Since we have a single unknown value on the left side and only known values on the right side, this new rule solves the value on the left. Now wherever this message/aux block appears in another equation, we can substitute the right side of the equation. This removes one unknown value from the set of equations each time this step is taken.

Decoding progresses in this way by finding an equation with only a single unknown value, solving that unknown value then substituting the result into any other equation that mentions this value. This proceeds until there are no unknowns left. At that point the entire file has been "solved".

Solution in terms of a graph

The method of solving all of the equations above can be re-expressed in terms of a graph. Nodes in the graph represent blocks, while the edges capture the relation between blocks on the left side of an equation and those on the right. So for example, a check block C (on the left hand side of an equation) is composed of an auxiliary node A and a message node M is represented by:

  • three nodes M, A and C

  • an edge between M and C

  • an edge between A and C

There is also an additional structure imposed on the nodes in the graph so that edges can be unambiguously identified as belonging to a particular equation. Technically, the graph is a bipartite graph. It keeps each of the block types grouped with other blocks of that type and orders the groups like so:

 message blocks < auxiliary blocks < check blocks

Graphically, the example rule above could be illustrated as follows:

  message           auxiliary           check
 
     M <--------------------------------- C
                                        /
                        A <------------/
 

This diagram could equally have been written with the check blocks on the left and the message blocks on the right, or turned 90 degrees. It's merely a matter of convention, similar to the two ways of writing out the equation as either:

 C <- M xor A

or

 M xor A -> C

For the remainder of the document, I'll go with the convention of saying that auxiliary blocks are to the right of the message blocks and check blocks are to the right of both of them. (My code uses a different convention again and talks about check nodes being higher than auxiliary and message nodes).

Besides information about edges, the graph also stores a status bit for each node to indicate whether that node is known (solved) or unknown. Check nodes are always taken to be solved since the encoder sends the value of that block, whereas message and auxiliary nodes are all initially unknown/unsolved.

In the explanation of the algebraic interpretation, I talked about finding an equation that had just a single unknown and rearranging it so that the single unknown value moves to one side and all the other knowns move to the other side. There is an analoguous operation on the graph, and this is named the "propagation rule".

The propagation rule involves finding a known node which has exactly one unsolved neighbour on the left. In the above example, if we are considering whether to propagate from node C (which is known) both M and A are unknown, so the rule does not match. If, on the other hand, one of M or A are known, the rule does match.

When the propagation rule matches, the solution for the newly-solved node on the left becomes the XOR of the node on the right plus all the other nodes emanating from that (right) node. When a node is solved in this way, all edges from the node on the right are removed from the graph.

In my code, the propagation rule is handled in a routine called resolve().

Cascades

When matched, the propagation rule solves one extra node somewhere to the left of the starting node. In the algebraic interpretation, I talked about substituting a newly-solved variable into all other equations where the variable appeared. There is an analogous procedure in the graph-based implementation, which is implemenented in the cascade() routine.

For the sake of discussion, let's assume that the message block M was solved by the propagation rule and that it had the solution:

 M <- A xor C

To simulate substituting M into all other equations where it appears, we need to work backwards (from left to right) from node M and see if any of those nodes now match the propagation rule. Since there will be one rightward edge in the graph from that node for each equation the left node appears in, this effectively reaches all equations that could could become solvable.

In the case where the left node which has become solved is an auxiliary block, the cascade() routine also queues up the auxiliary node itself for checking the propagation rule.

Special handling for auxiliary nodes

Although in theory the propagation rule could be applied to unsolved auxiliary nodes, in practice this has proved troublesome, so I have not implemented it. Instead I have implemented a special "auxiliary rule" that gives comparable results.

Recall that the propagation rule works with a single known node on the right and a single unknown node on the left. It is also possible to devise a rule where there is a message node on the left and an unsolved aux rule on the right. If the auxiliary node has only one unsolved left neighbour (ie the message node) and that message node becomes solved, then the auxiliary block can be solved too.

Initially, each auxiliary block will be composed of some number of message blocks:

 aux   = msg   xor  msg   xor ...
    x       i          j

When the last unsolved message block on the right becomes solved then this equation has no more unknowns apart from the aux block itself. Therefore, it can be marked as solved (with the above solution) and we can cascade up from that aux block to see if it solves any more equations.

Optimising by tracking counts of unsolved left nodes

When aux or check nodes are created, the number of unknown/unsolved edges that they are comprised of is calculated. Whenever a node becomes solved, each of the nodes that include that node in its list of edges has their unsolved count decremented.

This improves performance by avoiding having to scan the node's full list of leftward edges when it is considered for resolving.

"Bones" ("Bundles of Node Elements")

In a previous version of the program, edges in the graph were stored by keeping track of the left (bottom) end of the edge in a hash, while the right (top) end was stored in a list. I also had a separate array for storing the solutions of each node. The "Bones" structure essentially combines the top part of the edge and the solution into a single fixed-size array. This was done to improve performance by eliminating lots of list copying as the graph was processed.

The Bones structure is a fixed-sized array with three elements:

  • count of unsolved left (down) edges

  • node ids of unsolved left (down) edges

  • list of known node ids

Bones can also be viewed as encapsulating an algebraic equation of the form:

 [unknown nodes] <- [known nodes]

At the start of decoding, each auxiliary node has a Bone object created for it:

 [aux node, message nodes] <- []

That is, the aux node and all its constituent message nodes are all marked as unknown/unsolved (there are no knowns in the equation).

The "Bone" is attached to the auxiliary node and reciprocal links (the other end of the edges) are created in each of the component message nodes. All the nodes in the left hand side except the aux node itself are considered to be the top parts of edges.

When a check node is created, its Bone is of the form:

 [unsolved msg/aux nodes] <- [ check node, solved msg/aux nodes ]

The check node is placed on the right along with other known nodes because the decoder knows the value of all received check nodes. Reciprocal links are only created for unsolved nodes.

As can be seen, Bones have aspects of algebraic equations, but they also encapsulate edge structure.

As nodes become solved by the propagation or auxiliary rule, elements are shifted in the array to take this form:

 [newly-solved node] <- [list of nodes to XOR to get value]

This is exactly the form of a solution to a node, so the Bone is stored in the solution array. The right-hand side is also scanned and any reciprocal links are deleted, as is the top part of the edge.

In summary, a Bone always represents an equation. At the start of the decoding process it also encapsulates edge structure, but at the end it becomes a solution for either a message block or an auxiliary block.

From version 0.04 of the Net::OnlineCode modules onwards, the resolver returns a Bone object for each solved node. It will always be an array of the form mentioned above, and encoded as follows:

 [
    1,           # the number of "unknowns"
    msg_or_aux,  # the node that was just solved
    list of component nodes 
 ]