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Table 2

Summary of characteristics of N-best algorithms. For the time complexities, the alphabet is taken to be constant and the input is represented as a wta. The two right-most columns indicate whether the algorithms compute best contexts as a preprocessing step, and what optimization methods are used to find new candidate objects.

Algorithm	Objects	Time complexity	Best contexts	Search-space expansion
Eppstein (1998)	Paths	$O (n log n + N n + m)$	No	Adding sidetracks to implicit heap representations of paths
Mohri and Riley (2002)	Strings	No formal analysis provided	Yes	On-the-fly determinization
Huang and Chiang (2005)	Runs	$O (m log n + s_{max} N log N)$ ^⁵	No	Increment
Best Trees v.1	Trees	$O (N^{2} n (n^{2} + m log N))$	Yes	Eppstein’s algorithm
Best Trees	Trees	$O (N m (log m + log N))$	Yes	Increment
Best Trees	Runs	$O (N (log m + log N))$ ^⁶	Yes	Increment

Algorithm	Objects	Time complexity	Best contexts	Search-space expansion
Eppstein (1998)	Paths	$O (n log n + N n + m)$	No	Adding sidetracks to implicit heap representations of paths
Mohri and Riley (2002)	Strings	No formal analysis provided	Yes	On-the-fly determinization
Huang and Chiang (2005)	Runs	$O (m log n + s_{max} N log N)$ ^⁵	No	Increment
Best Trees v.1	Trees	$O (N^{2} n (n^{2} + m log N))$	Yes	Eppstein’s algorithm
Best Trees	Trees	$O (N m (log m + log N))$	Yes	Increment
Best Trees	Runs	$O (N (log m + log N))$ ^⁶	Yes	Increment

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