An augmented metric space (X, d_X, f_X) is a metric space (X, d_X) equipped with a function f_X: X \to R. It arises commonly in practice, e.g, a point cloud X in R^d where each point x\in X has a density function value f_X(x) associated to it. Such an augmented metric space  naturally gives rise to a 2-parameter filtration  . However, the resulting 2-parameter persistence module_{\bullet}(K) } could still be of wild representation type, and may not have simple indecomposables. In this paper, motivated by the elder-rule for the zeroth homology of a 1-parameter filtration, we propose a barcode-like summary, called the elder-rule-staircode, as a way to encode the zeroth homology of the 2-parameter filtration induced by a finite augmented metric space . Specifically, given a finite (X, d_X, f_X), its elder-rule-staircode consists of n = |X| number of staircase-like blocks in the plane. We show that the fibered barcode , the fibered merge tree,_0(\mathcal{K}) is interval decomposable, then the barcode of \mathrm{H}_0(\mathcal{K}) is equal to the elder-rule-staircode. Furthermore, regardless of the interval decomposability, the fibered barcode, the dimension function (a.k.a. the Hilbert function), } and the graded Betti numbers associated to the zeroth homology of the 2-parameter filtration induced by (X, d_X, f_X_0(\mathcal{K}} ) can all be efficiently computed once the elder-rule-staircode is given . Furthermore, for certain special cases, this staircode corresponds exactly to the set of indecomposables of the zeroth homology of the 2-parameter filtration . Finally, we develop and implement an efficient algorithm to compute the elder-rule-staircode in O(n^2\log n) time, which can be improved to O(n^2\alpha(n)) if X is from a fixed dimensional Euclidean space \mathbb{R}^d, where \alpha(n) is the inverse Ackermann function.
An augmented metric space is a metric space (X, d_X) equipped with a function f_X: X \to R. This type of data arises commonly in practice, e.g, a point cloud X in R^d where each point x\in X has a density function value f_X(x) associated to it. An augmented metric space (X, d_X, f_X) naturally gives rise to a 2-parameter filtration \mathcal{K . However, the resulting 2-parameter persistent homology \mathrm{H_{\bullet}(K) } could still be of wild representation type, and may not have simple indecomposables. In this paper, motivated by the elder-rule for the zeroth homology of 1-parameter filtration, we propose a barcode-like summary, called the elder-rule-staircode, as a way to encode \mathrm{H . Specifically, if n = |X|, the elder-rule-staircode consists of n number of staircase-like blocks in the plane. We show that if \mathrm{H_0(\mathcal{K}) is interval decomposable, then the barcode of \mathrm{H}_0(\mathcal{K}) is equal to the elder-rule-staircode. Furthermore, regardless of the interval decomposability, the fibered barcode, the dimension function (a.k.a. the Hilbert function), } and the graded Betti numbers of \mathrm{H_0(\mathcal{K}} ) can all be efficiently computed once the elder-rule-staircode is given . Finally, we develop and implement an efficient algorithm to compute the elder-rule-staircode in O(n^2\log n) time, which can be improved to O(n^2\alpha(n)) if X is from a fixed dimensional Euclidean space \mathbb{R}^d, where \alpha(n) is the inverse Ackermann function.
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