Two Algorithms for Finding $k$ Shortest Paths of a Weighted Pushdown Automaton

Weighted pushdown automata (WPDAs) have recently been adopted in some applications such as machine translation [Iglesias et al., 2011] as a more compact alternative to weighted finite-state automata (WFSAs) for representing a weighted set of strings. Allauzen and Riley [2012] introduce a set of basic algorithms for construction and inference of WPDAs, and the corresponding implementation as an extension of the open source finite-state transducer toolkit OpenFst.1 Although a shortest-path algorithm for WPDAs with bounded stack is described in Allauzen and Riley [2012], it does not give a k-shortest-path algorithm, which finds the k shortest accepting paths of the given automaton. Other than just the single shortest path, k shortest paths are useful for many purposes such as reranking the output in parsing [Collins and Koo, 2005] or tuning feature weights in machine translation [Chiang et al., 2009]. One existing work-around is to first expand the WPDA into an equivalent WFSA and then find the k shortest paths of the WFSA using the k-shortest-path algorithm for WFSAs (the expansion approach). Since the WPDA expansion has an exponential time and space complexity with respect to the size of the automaton, one usually has to prune the WPDA before expansion (the pruned expansion approach), i.e. remove those transitions and states that are not on any accepting path with a weight at most a given threshold greater than the shortest distance. However, setting an adequate threshold that neither prunes nor keeps too many states or transitions a priori is almost impossible in practice. In this paper, we introduce two efficient algorithms for finding the k shortest paths of a WPDA, both derived from the same weighted deductive logic description of the execution of a WPDA using different search strategies.