Shortest-Distance Problems and Algorithms
Shortest-distance problems arise in a variety of applications.
- General Shortest-Distance Problems: Traditional single-source or
all-pair shortest-distance problems were introduced in the
tropical semiring ((min,+)-semiring). These problems can be
generalized to the case of an arbitrary semiring. There exists a
simple and generic single-source shortest-distance algorithm that
works with any k-closed semiring and that is correct
regardless of the queue discipline chosen for its
implementation. Classical algorithms such as those of Bellman-Ford,
Dijkstra, or Lawler, are all specific instances of that generic
algorithm. The classical all-pairs shortest-distance algorithm of
Floyd-Warshall can also be straight-forwardedly generalized to the
case of closed semirings, non-necessarily idempotent.
- n-Best-Strings Problems: The problem of determining the
n shortest paths of a weighted directed graph is a
well-studied problem and admits a large number of variants. The
related problems that arise in text and speech processing applications
are different. The weighted graphs considered in such applications are
weighted automata and it is often desirable to determine not the
n best paths, but the n best distinct
strings of the automaton with the lowest costs.
Frameworks and Algorithms for Shortest-Distance Problems.
Journal of Automata, Languages and Combinatorics,
Mehryar Mohri and Michael
Efficient Algorithm for the N-Best-Strings Problem.
In Proceedings of the International Conference on Spoken Language
Processing 2002 (ICSLP '02).
Denver, Colorado, September 2002.
General Algebraic Frameworks and Algorithms for Shortest-Distance
Technical Memorandum 981210-10TM, AT&T Labs - Research, 62 pages, 1998.