Data structures as topological spaces (2002) [pdf]
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Data structures as topological spaces (2002) [pdf]
1–10 of 48 posts
Re: Data structures as topological spaces (2002) [pdf]
#2How is this fundamentally different than considering data structures as graphs?
Re: Data structures as topological spaces (2002) [pdf]
#3How is this fundamentally different than considering data structures as graphs?
Graphs are discrete, topologies are potentially continuous. Moreover, you can do different things with them such as create homeomorphisms to another topology much more easily than you can create bijections between graphs. In general, continuity lets you assume things that are impossible in discrete spaces. For example, many optimization problems are really easy in continuous spaces but really hard in discrete ones (linear programming for example)
Given the recent success with vectors as a general model for data (as witnessed by the continued success with deep neural networks), it's an interesting discussion to have.
Re: Data structures as topological spaces (2002) [pdf]
#4I wonder if this article is related to homotopy type theory at all, since they propose similar ideas.
Re: Data structures as topological spaces (2002) [pdf]
#5How is this fundamentally different than considering data structures as graphs?
Graphs are discrete, topologies are potentially continuous. Moreover, you can do different things with them such as create homeomorphisms to another topology much more easily than you can create bijections between graphs. In general, continuity lets you assume things that are impossible in discrete spaces. For example, many optimization problems are really easy in continuous spaces but really hard in discrete ones (l…
feels so true.
Re: Data structures as topological spaces (2002) [pdf]
#6How is this fundamentally different than considering data structures as graphs?
Graphs are discrete, topologies are potentially continuous. Moreover, you can do different things with them such as create homeomorphisms to another topology much more easily than you can create bijections between graphs. In general, continuity lets you assume things that are impossible in discrete spaces. For example, many optimization problems are really easy in continuous spaces but really hard in discrete ones (l…
Re: Data structures as topological spaces (2002) [pdf]
#7I'm left confused as to what the gluing in the rule replacement is. Must the boundary of a rule match on both sides? Also what examples there would be of what an example would of having topology that is not induced from a graph if it is at all possible.
Re: Data structures as topological spaces (2002) [pdf]
#8How is this fundamentally different than considering data structures as graphs?
Graphs are discrete, topologies are potentially continuous. Moreover, you can do different things with them such as create homeomorphisms to another topology much more easily than you can create bijections between graphs. In general, continuity lets you assume things that are impossible in discrete spaces. For example, many optimization problems are really easy in continuous spaces but really hard in discrete ones (l…
Discrete spaces can also be topological spaces, see discrete topology
Re: Data structures as topological spaces (2002) [pdf]
#9[flagged]
Re: Data structures as topological spaces (2002) [pdf]
#10Earlier quoted context omitted.
Graphs are discrete, topologies are potentially continuous. Moreover, you can do different things with them such as create homeomorphisms to another topology much more easily than you can create bijections between graphs. In general, continuity lets you assume things that are impossible in discrete spaces. For example, many optimization problems are really easy in continuous spaces but really hard in discrete ones (l…
Discrete spaces can also be topological spaces, see discrete topology
Did you not read the word “potentially”? Topological spaces are a more general case of spaces that contain the discrete case as a subset.