Ramer–Douglas–Peucker algorithm: Difference between revisions

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== Complexity ==

The running time of this algorithm when run on a polyline consisting of {{math|''n'' – 1}} segments and {{mvar|n}} vertices is given by the recurrence {{math|''T''(''n'') {{=}} ''T''(''i'' + 1) + ''T''(''n'' − ''i'') + [[Big O notation|''O''(''n'')]]}} where {{math|''i'' {{=}} 1, 2,..., ''n'' − 2}} is the value of <code>index</code> in the pseudocode. In the worst case, {{math|''i'' {{=}} 1}} or {{math|''i'' {{=}} ''n'' − 2}} at each recursive invocation and this algorithm has a running time of {{math|[[Big theta|''Θ''(''n''²)]]}}. In the best case {{math|''i'' {{=}} {{sfrac|''n''|2}}}} (or {{math|''i'' {{=}} {{sfrac|''n'' ± 1|2}}}}) at each recursive invocation in which case the running time has the well-known solution (via the [[master theorem (analysis of algorithms)|master theorem for divide-and-conquer recurrences]]) of {{math|''O''(''n'' log ''n'')}}.

Using (fully or semi-) [[dynamic convex hull]] data structures, the simplification performed by the algorithm can be accomplished in {{math|''O''(''n'' log ''n'')}} time.<ref>{{cite techreport |last1 = Hershberger |first1 = John |first2 = Jack |last2 = Snoeyink |title = Speeding Up the Douglas-Peucker Line-Simplification Algorithm |date = 1992 | url = http://www.bowdoin.edu/~ltoma/teaching/cs350/spring06/Lecture-Handouts/hershberger92speeding.pdf}}</ref>