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This metric induces the discrete topology, which isn't very useful. Regarding its metric structure, you could apply the contraction mapping theorem but I can't
by dls2016 7y ago
This metric induces the discrete topology, which isn't very useful. Regarding its metric structure, you could apply the contraction mapping theorem but I can't think of any interesting contractions off the top of my head.
I'm just spit-balling here... but when a natural metric induces the discrete topology then it's probably more useful to look at connectivity graphs induced by the metric. For instance the graph on all strings with two strings connected by an edge if their edit distance equals 1. Restrict to words in a dictionary and you have a basic spelling suggestion algorithm.
Edit: the analyst in me really wants a translation-invariant abelian group structure ... then we can do Fourier analysis...
- olooney 7y ago> Restrict to words in a dictionary and you have a basic spelling suggestion algorithm. Yep. https://norvig.com/spell-correct.html https://norvig.com/spell-correct.html
- mrkeen 7y agoThe upside of it being metric is that you can optimise lookups by storing the strings in a BK-tree.
- jointpdf 7y agoI’m kind of tacking this on here, but does anyone have a good recommendation for notes or a book on applied topology for someone with a decent but not extensive background in applied math? In particular, my goal is to more deeply understand manifold learning techniques like MDS and UMAP.