Full fault dictionary storage based on labeled tree encoding

Parallel Algorithms for Dandelion-Like Codes

“…As an example consider the tree of 11 nodes, labeled from 0 to 10 in Fig.1a in which the path 1 0 is (1,4,6,8,3,0). We obtain f 0 = 1,…”

“…In our example, since f 0 = 1, f 1 = 8, and f 2 = 3, the algorithm set g [3] ← g [8] = 3, introducing a loop, and g [8] ← g [1] = 4, introducing a cycle. Fig.1d shows the resulting digraph: the associated codeword is W = [6,3,6,1,8,8,4,1,9].…”

“…As an example of decoding consider the codeword W = [6,3,6,1,8,8,4,1,9]. It is easy to see that, according with the reconstruction described above, we initially obtain the functional digraph represented in Fig.1d.…”

“…Thus we obtain f 0 = 1, f 1 = 8, f 2 = 3. Then we set g [1] ← g [8] = 4, g [8] ← g [3] = 3, and g [3] ← 0. Now G g is exactly the tree represented in Fig.1a. …”

“…For example, they are used in fault dictionary storage [1], distributed spanning tree maintenance [2], generation of random trees [3], Genetic Algorithms [4]. In this paper we restrict our attention to bijective string-based codes in which the length of the string must be equal to n − 2 [5] (n is the number of nodes of the encoded tree).…”