Continuous Edit Distance (CED)

• Continuous Edit Distance (CED) is a metric that generalizes the classical edit distance to continuous, elastic, and geometric data using real-valued cost operations.
• It quantifies dissimilarity by integrating continuous edit operations like insertion, deletion, and substitution, and applies dynamic programming for efficient computation.
• CED enables robust alignment and averaging in applications such as speech analysis, topological data analysis, and geometric curve matching through specialized strategies.

Continuous Edit Distance (CED) defines a metrized notion of dissimilarity that extends the classical edit (Levenshtein) distance from discrete structures to settings involving continuous, elastic, or geometric data. Prominent instantiations quantize the cost of similarity transformation between objects such as time-varying persistence diagrams, real-exponent strings, or polygonal curves, using a continuous generalization of edit operations. Several recent frameworks—including exp-edit distance on exponent-strings, CED for time-varying persistence diagrams, and the continuous Fréchet-edit distance—illustrate the core principles, algorithmic strategies, and structural properties of CED-type metrics (Baek, 23 Aug 2024, Tchitchek et al., 15 Dec 2025, Fox et al., 19 Mar 2024).

1. Formal Definitions and Underlying Structures

CED generalizes classical edit distance by enabling non-integer “amounts” of symbol insertion, deletion, and substitution within mathematical objects that carry continuous or elastic structure.

Exponent-strings: A $\mathbb{R}^+$-exponent-string over a finite alphabet $\Sigma$ is a finite sequence $$p = \langle(\sigma_1, s_1), \dots, (\sigma_n, s_n)\rangle$$ where $\sigma_i \in \Sigma$, $s_i \in \mathbb{R}^+$, and adjacent runs of the same symbol are contracted. Concatenation merges adjacent runs of the same symbol, summing their exponents (Baek, 23 Aug 2024).

Time-Varying Persistence Diagrams (TVPDs): Objects indexed by time, with each timepoint equipped with a persistence diagram. For two TVPDs $P = (P_i)_{i=1}^{N_P}$ and $Q = (Q_j)_{j=1}^{N_Q}$, the metric acts on piecewise-constant representations over intervals of length $\Delta$ (Tchitchek et al., 15 Dec 2025).

Polygonal Curves and Fréchet-Edit Distance: Discrete or continuous polygonal curves $\pi$, $\sigma$ in $\mathbb{R}^d$, with edits enabling vertex insertion and deletion. In the continuous variant, all possible edited versions of $\sigma$ are encoded in a combinatorial “DAG complex” product (Fox et al., 19 Mar 2024).

2. Edit Operations and Cost Functions

CED frameworks define real-valued costs for infinitesimal edit operations:

Model	Edit Operations	Edit Cost Function
Exponent-strings	$\lambda \to a^q$, $a^q \to \lambda$, $a^q \to b^q$	$w(\lambda\to b^q) = q\,w(\lambda\to b)$; $w(a^q\to \lambda) = q\,w(a\to\lambda)$; $w(a^q\to b^q)=q\,w(a\to b)$
TVPDs (CED for diagrams)	Block alignment, gaps (insert/delete)	Integration over local block costs: substitution, insertion, deletion with scalar weights $\alpha$, $\beta$
Fréchet-edit distance	Vertex insertion, deletion	Unit cost per edit (discrete); geometric propagation for continuous settings

Costs are extended to continuous values by linearization (for exponent-strings) or integration (for TVPD block alignment). Discrete steps become special cases when exponents or block durations are integral.

3. Algorithmic Computation

Exponent-Strings (Exp-Edit Distance)

If all exponents are rational, multiply all exponents by the least common multiple $C$ of denominators, reducing the computation to ordinary run-length encoded (RLE) string edit distance on integer runs. The final distance is scaled by $1/C$. The edit distance can thus be computed in $O(|w_1||w_2|\log|w_1 w_2|)$ time for input strings $w_1, w_2$ (Baek, 23 Aug 2024).

TVPDs (CED Metric on Diagrams)

Local blockwise costs $D^\alpha_\Delta(P_i,Q_j)$ are precomputed via integration of Wasserstein distances; dynamic programming fills an $N_P \times N_Q$ cost table in $O(N_P N_Q)$, with $O(N_P N_Q p^3)$ for local costs ($p$ diagram size). CED-geodesic steps and barycenter computation rely on dynamic programming and piecewise-constant approximations (Tchitchek et al., 15 Dec 2025).

Continuous Fréchet-Edit Distance

Continuous variants model edited curves as walks in product DAG complexes, with “free-space” propagation yielding a polynomial bound. Deletion-only, insertion-only, and mixed-edit cases are handled by layered construction and enumeration of canonical subcurves, with computational complexity ranging from $O(m n^3)$ (deletion-only, $m,n$ curve lengths) up to $O((m+n)^3 n m^3)$ (both edits). Weak variants (permitting backtracking walks) are NP-hard even in low-dimensional settings (Fox et al., 19 Mar 2024).

4. Metric and Algebraic Properties

All major instantiations of CED satisfy the following properties:

• Non-negativity: $\mathrm{dist}(p,q)\geq 0$
• Identity of indiscernibles: $\mathrm{dist}(p,p)=0$
• Symmetry: For symmetric base weights and operations.
• Triangle inequality: Concatenation or composition of optimal sequences yields metricity.
• Prefix-invariance: For exponent-strings, $$\mathrm{dist}(p\cdot u, p\cdot v) = \mathrm{dist}(u,v)$$.

The CED on TVPDs is a true metric, supports explicit geodesics, and admits well-defined barycenters under the CED-Fréchet energy (Tchitchek et al., 15 Dec 2025). Exponent-strings with unit costs furnish a metric on $\Sigma_{\mathbb{R}^+}^*$ (Baek, 23 Aug 2024).

5. Applications and Empirical Properties

Exponent-Strings and Speech Analysis

Exp-edit distance quantifies mismatches between phonetic transcriptions with real-valued durations, supporting tasks such as segmental comparison, dialectometry, paraphasia detection, and evaluation in prosodic TTS systems (Baek, 23 Aug 2024). The formalism bridges discrete and continuous analysis in linguistics.

TVPDs and Topological Data Analysis

CED underpins robust, interpretable alignment, averaging, and clustering of time-varying persistence diagrams. The metric demonstrates robustness to both temporal and spatial noise, recovers temporal shifts in dynamical-system datasets, and achieves or surpasses the performance of standard elastic dissimilarities (L2, Fréchet, TWED, DTW) in clustering applications (Tchitchek et al., 15 Dec 2025).

Geometric Curve Matching

Continuous Fréchet-edit distance captures dissimilarity in the presence of outliers, noise, or structural differences by permitting localized edits. Deletion-only, insertion-only, and full edit variants can be applied in trajectory simplification and geometric pattern matching (Fox et al., 19 Mar 2024).

6. Hardness and Algorithmic Complexity

Whereas the discrete strong variants admit polynomial time algorithms, all weak Fréchet-edit variants (removing monotonicity constraints) are NP-hard, even for deletion-only or insertion-only cases in one (discrete) or two (continuous) dimensions (Fox et al., 19 Mar 2024). For metric CED on TVPDs or exponent-strings, efficient solutions exist for moderate input size, with complexity dominated by the structure of the underlying data objects.

Variant	Complexity (Best)	Hardness
Discrete Fréchet-edit	$O(m^2 + m n)$	P
Continuous Fréchet-edit	$O((m+n)^3 n m^3)$	P
Weak variants (cont./discr.)	—	NP-hard
Exp-edit (rational exponents)	$O(|w_1||w_2|\log|w_1w_2|)$	P
TVPD CED (CED metric/dyn. prog.)	$O(N_P N_Q p^3)$	P

7. Significance and Ongoing Developments

CED unifies a range of continuous, elastic, and geometric edit distances, providing a metrically principled, efficiently computable similarity measure for applications where objects accommodate continuous or run-length structure. The robust alignment and averaging capabilities of CED metrics have proven empirically advantageous in fields such as speech analysis and topological data science. The formulation admits generalization across data types—from strings with real-valued exponents to multivariate diagrams and curves—while preserving both algorithmic tractability and interpretability. Open challenges include further mitigation of worst-case complexity for high-dimensional or weak variants, and extension to new application domains where conventional discrete edit metrics are insufficient (Baek, 23 Aug 2024, Tchitchek et al., 15 Dec 2025, Fox et al., 19 Mar 2024).