Improved Lower Bounds for Graph Edit Distance

The problem of deriving lower and upper bounds for the edit distance between undirected, labeled graphs has recently received increasing attention. However, only one algorithm has been proposed that allegedly computes not only an upper but also a lower bound for non-uniform edit costs and incorporates information about both node and edge labels. In this paper, we demonstrate that this algorithm is incorrect. We present a corrected version <inline-formula> <tex-math notation="LaTeX">$\mathsf {B\scriptstyle{RANCH}}$</tex-math><alternatives> <inline-graphic xlink:href="blumenthal-ieq1-2772243.gif"/></alternatives></inline-formula> that runs in <inline-formula> <tex-math notation="LaTeX">$\mathcal{O}(n^2\Delta ^3+n^3)$</tex-math><alternatives> <inline-graphic xlink:href="blumenthal-ieq2-2772243.gif"/></alternatives></inline-formula> time, where <inline-formula> <tex-math notation="LaTeX">$\Delta$</tex-math><alternatives><inline-graphic xlink:href="blumenthal-ieq3-2772243.gif"/> </alternatives></inline-formula> is the maximum of the maximum degrees of input graphs <inline-formula> <tex-math notation="LaTeX">$G$</tex-math><alternatives><inline-graphic xlink:href="blumenthal-ieq4-2772243.gif"/> </alternatives></inline-formula> and <inline-formula><tex-math notation="LaTeX">$H$</tex-math><alternatives> <inline-graphic xlink:href="blumenthal-ieq5-2772243.gif"/></alternatives></inline-formula>. We also develop a speed-up <inline-formula><tex-math notation="LaTeX">$\mathsf {B\scriptstyle{RANCH}}\mathsf{F\scriptstyle{AST}}$</tex-math> <alternatives><inline-graphic xlink:href="blumenthal-ieq6-2772243.gif"/></alternatives></inline-formula> that runs in <inline-formula><tex-math notation="LaTeX">$\mathcal{O}(n^2\Delta ^2+n^3)$</tex-math><alternatives> <inline-graphic xlink:href="blumenthal-ieq7-2772243.gif"/></alternatives></inline-formula> time and computes an only slightly less accurate lower bound. The lower bounds produced by <inline-formula><tex-math notation="LaTeX">$\mathsf {B\scriptstyle{RANCH}}$</tex-math><alternatives><inline-graphic xlink:href="blumenthal-ieq8-2772243.gif"/> </alternatives></inline-formula> and <inline-formula><tex-math notation="LaTeX">$\mathsf {B\scriptstyle{RANCH}}\mathsf{F\scriptstyle{AST}}$</tex-math><alternatives> <inline-graphic xlink:href="blumenthal-ieq9-2772243.gif"/></alternatives></inline-formula> are shown to be pseudo-metrics on a collection of graphs. Finally, we suggest an anytime algorithm <inline-formula> <tex-math notation="LaTeX">$\mathsf {B\scriptstyle{RANCH}}\mathsf{T\scriptstyle{IGHT}}$</tex-math><alternatives> <inline-graphic xlink:href="blumenthal-ieq10-2772243.gif"/></alternatives></inline-formula> that iteratively improves <inline-formula><tex-math notation="LaTeX">$\mathsf {B\scriptstyle{RANCH}}$</tex-math><alternatives> <inline-graphic xlink:href="blumenthal-ieq11-2772243.gif"/></alternatives></inline-formula>’s lower bound. <inline-formula><tex-math notation="LaTeX">$\mathsf {B\scriptstyle{RANCH}}\mathsf{T\scriptstyle{IGHT}}$</tex-math> <alternatives><inline-graphic xlink:href="blumenthal-ieq12-2772243.gif"/></alternatives></inline-formula> runs in <inline-formula><tex-math notation="LaTeX">$\mathcal{O}(n^3\Delta ^2+I(n^2\Delta ^3+n^3))$</tex-math><alternatives> <inline-graphic xlink:href="blumenthal-ieq13-2772243.gif"/></alternatives></inline-formula> time, where the number of iterations <inline-formula><tex-math notation="LaTeX">$I$</tex-math><alternatives> <inline-graphic xlink:href="blumenthal-ieq14-2772243.gif"/></alternatives></inline-formula> is controlled by the user. A detailed experimental evaluation shows that all suggested algorithms are Pareto optimal, that they are very effective when used as filters for edit distance range queries, and that they perform excellently when used within classification frameworks.

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