Weighted finite-state automata (WSFAs) arecommonly used in NLP. Failure transitions area useful extension for compactly representingbackoffs or interpolation in n-gram modelsand CRFs, which are special cases of WFSAs.Unfortunately, applying standard algorithmsfor computing the pathsum requires expand-ing these compact failure transitions. As aresult, na ̈ıve computation of the pathsum inacyclic WFSAs with failure transitions runs inO(|Q|2|Σ|) (O(|Q||Σ|) for deterministic WF-SAs) while the equivalent algorithm in normalWFSAs runs in O(|E|), where E representsthe set of transitions, Q the set of states, andΣ the alphabet. In this work, we present moreefficient algorithms for computing the pathsumin sparse acyclic WFSAs, i.e., WFSAs with av-erage out symbol fraction s ≪ 1. In those,backward runs in O(s|Q||Σ|). We proposean algorithm for semiring-weighted automatawhich runs in O(|E| + s|Σ||Q||Tmax| log |Σ|),where |Tmax| is the size of the largest con-nected component of failure transitions. Ad-ditionally, we propose faster algorithms fortwo specific cases. For ring-weighted WF-SAs we propose an algorithm with complex-ity O(|E| + s|Σ||Q||πmax|), where |πmax| de-notes the longest path length of failure transi-tions stemming from q and Σ(q) the set of sym-bols on the outgoing transitions from q. Forsemiring-weighted WFSAs whose failure tran-sition topology satisfies a condition exemplifiedby CRFs, we propose an algorithm with com-plexity O(|E| + s|Σ||Q| log |Σ|).