Data structures

Figure 2.16:

This transducer modifications are very powerful on the one hand. On the other hand, it introduces also some complexity that might not be required for many appli- cations. However, the technique shows to be helpful for applications discussed in the thesis. It is possible to use a global set of variables, e.g. a list of proper names which should be searched on a web-page. Further, the concept can be used to handle context free languages when a stack memory is introduced and used as global variable for both λIandλF. However, there are also some disadvantages. First, the transducer construc-

tion may need some more attention because the deterministic-assumption is difficult to predict forλ declarations. Second, the transducer minimization, determinization and composition operators do not evaluate anyλ declaration and it assumes that all λ declarations are non overlapping andε-free.

2.8

Data structures

A transducer is treated as directed graph for defining a suitable data structure. The structure has a significant impact on both, the memory consumption and the com- putational time it takes to apply operators. In speech recognition, the transducer is optimized for efficient decoding on the one hand. On the other hand, the size should not exceed certain limitations, e.g. constant storage, random access and cache sizes. In the following, are common data structures described: adjacency-, incidence matrix and list. The spectral graph theory uses both, adjacency and incidence matrix to close the gap between linear algebra and graph theory. This was introduced, e.g. by Brouwer et al. [46]. A data structure can be optimized for certain operators such as composi- tion, decoding, etc. Let V = {v1, .., vn} be the set of states and let E = {e1, .., em} be the

set of transitions for a graph G = (V, E ). Note, a transducer is defined by augmenting the states and transitions with some more information, e.g. the marker of a state is initial, final, weighted or the transitions input, output symbol and weight.

An adjacency matrix for G is defined as follows:

ai j =

¨

wi j if(i , j ) ∈ E

0 else.

for w_{i j} the label between state i and j . It takes a_{O (n}2_{) memory given n states. An ex-} ample is shown in Figure 2.17. A undirected graph has a symmetric matrix. Hence, the

   a b c d a 0 46 0 0 b 0 0 1 32 c 0 5 0 0 d 0 0 0 0   

Figure 2.17:

matrix has a complete set of real eigenvalues and an orthogonal eigenvector basis. In spectral graph theory, these eigenvalues are the spectrum of the graph, e.g. introduced by Brouwer et al. [46]. A graph with no self-loops has zeros on the main diagonal. In addition, the matrix product An for a not weighted graph gives the number of walks between two states of length n . Adjacency matrices are used for various applications, e.g., ranking of web-pages according to hyper-links. In contrast to the adjacency, the incidence matrix stores the transitions for each state. An example is shown in Figure 2.18. It takes_{O (n · m) memory where m is the number of transitions in G . This data}

   a , b b , c c , b b , d a 46 0 0 o b 0 1 0 32 c 0 0 5 0 d 0 0 0 0   

Figure 2.18:

structure is more suitable for graphs with far more states than transitions. Formally, the matrix is defined as following:

bi j =

¨

wi j if vi∈ ej

2.8. DATA STRUCTURES 29

Figure 2.19:

An decoding efficient data structure for graphs is list based. A common represen- tation uses adjacent lists. It is a list of stages with each stage linked to a transition, e.g. let x and y be two stages so that E _{⊂ V × V and (x , y ) ∈ E . For each x ∈ E , a list A(x )} of all x -following states is defined as follows:

A(x ) = {y ∈ V : (x , y ) ∈ E }.

The intention is a stage-oriented processing for which each stage can be hashed, ef- ficiently. An example is given in Figure 2.19, where the set of outgoing transitions is represented as linked list. The links are denoted by arrow and terminals are marked by a ground symbol. Usually, the stages are organized in an array whereas the outgo- ing transitions for each stage are linked lists. In this thesis, an array is also used for transitions. A stage is linked to the set of all its outgoing transitions. This reduces the memory requirement significantly.

The concept of adjacent lists was taken up, e.g. by Goodrich et al.[47] to represent a graph with an incidence list. In contrast to previous structure, it is a list of transitions where each transition points to a source and destination stages The list of transitions can be organized as an array, similar to stages in adjacent lists. Also linked list are often used. Figure 2.20 on next page shows an example and is formalized as:

I(x ) = {(x , y ) ∈ E : x ∈ V }

Both list representations are suited to store weighted finite-state transducers. Each transition stores an output and input symbol together with a weight. In addition, a state needs to be marked as initial or final state. Usually a second array is used to quickly access these states. The data structure is suited for embedded usage because it enables an efficient block decoding by evaluating all outgoing transitions at once.

Figure 2.20:

A garbage collector is not required for not mutable transducers. For other transduc- ers, a concept needs to be implemented that enables to change the transducer without rebuilding it from the very beginning. In practice, the mutable transducer is used dur- ing decoding. It is built during decoding and represents the current search sub space. Typically, stages or transitions need to be deleted or the target of a transition changes. In practice, it is also common to use smart pointers. The concept is used in Section 4 where it is used to build the word history tree during decoding. An example implemen- tation for weighted grammars and transducers for language processing is provided by Allauzen et al. [48], [49].