X + Y Sorting¶
The structured problem of ordering every pair from two numeric collections by its coordinate sum, while preserving the full Cartesian product and its duplicate sums.
Core Idea¶
X + Y sorting receives two numeric collections and orders the full Cartesian product by the additive key. With n inputs on each side, n² labeled pairs must be placed in nondecreasing order of x+y, including distinct pairs that happen to share a sum.
The research interest comes from structure: the n² keys are not independent because rows and columns share coordinates. A generic sort supplies the baseline, while specialized comparison schemes and lower bounds ask whether that dependence reduces comparisons or total time under a stated computational model.
Scope of Application¶
- Algorithm analysis. Separates output size, comparison count, and total running time.
- Computational geometry. Connects structured sums to arrangements and cell-counting bounds.
- Sparse polynomial work. Orders exponent sums arising from cross-products.
- Optimization applications. Organizes combinations whose costs are separable into two terms.
Clarity¶
State whether X and Y are sets or multisets, whether inputs are presorted, how ties are represented, and whether the output is explicit. Complexity claims must name the comparison, word-RAM, or bounded-integer model. Inclusion test: Given X and Y, enumerate or implicitly represent every cross-pair and return them in nondecreasing x+y order with duplicates and computational model specified. Exclusion test: Exclude finding one target sum, sorting only distinct sums, convolution alone, and sorting a precomputed unstructured list without preserving pair identity. Nearest boundary: Sumset sorting may collapse duplicate values; X + Y sorting ordinarily orders the full pair collection, so multiplicity and pair provenance remain. Exit condition: The task changes identity when pairs are omitted, the key is not addition, or output no longer represents the full ordered Cartesian product.
Manages Complexity¶
The abstraction isolates the gap between quadratic output and comparison overhead. It permits exploitation of monotone rows and columns without pretending the full result has subquadratic size.
Abstract Reasoning¶
- Normalize input and computational assumptions.
- Construct the conceptual n-by-n sum matrix.
- Preserve pair labels and multiplicities.
- Exploit row-column order where the algorithm permits.
- Report output cost, comparisons, and auxiliary operations separately.
Knowledge Transfer¶
The pattern transfers to separable keys f(x)+g(y) when monotonicity and output identity are preserved. Arbitrary pair scoring lacks the additive structure that motivates the problem.
Relationships to Other Abstractions¶
Current abstraction X + Y Sorting Domain-specific
Parents (1) — more general patterns this builds on
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X + Y Sorting is a kind of Computational problem Domain-specific
X+Y Sorting is a Computational Problem asking for all Cartesian-product pairs to be ordered by coordinate sum while preserving duplicate sums.
Hierarchy path (1) — routes to 1 parentless root
- X + Y Sorting → Computational problem → Function (Mapping)
Neighborhood in Abstraction Space¶
X + Y Sorting sits in a crowded region of the domain-specific corpus (36th percentile for distinctiveness): several abstractions share nearly its structure, so a description that fits it tends to fit its neighbors too.
Family — Matrices, Measures & Numeric Structures (30 abstractions)
Nearest neighbors
- Database Index — 0.89
- Distance Matrix — 0.88
- Ternary Operation — 0.88
- Maximum subarray problem — 0.88
- Partition problem — 0.87
Computed from structural-signature embeddings · 2026-10-08