Greedy Point Insertion Algorithms

• Greedy point insertion algorithms are incremental procedures that construct sparse structures by sequentially adding elements that satisfy feasibility constraints.
• They are applied in diverse fields such as geometric spanners, binary search trees, metric sampling, kernel methods, and sparse optimization.
• Modern implementations combine greedy selection with corrective updates and approximations to yield efficient, certifiable solutions across various problem domains.

Searching arXiv for recent and foundational papers on greedy point insertion algorithms across geometry, BSTs, kernel methods, metric sampling, and measure optimization. Search query: "greedy point insertion algorithms arXiv greedy permutations greedy spanner GreedyASS measure spaces" Greedy point insertion algorithms are incremental construction procedures in which a structured object is built by repeatedly adding a new point, atom, vertex, or edge selected by a greedy rule. Across the literature, the phrase covers several distinct but closely related paradigms: greedy edge insertion in geometric spanners and triangulations, minimal point addition in the geometric model of binary search trees, farthest-point traversal in metric spaces, stabilized sample selection for kernel approximation, greedy marginal-gain maximization for determinantal point process MAP inference, constructive growth of independent sets in graphs, and sparse optimization over spaces of Radon measures by insertion of Dirac deltas (Alewijnse et al., 2014, Fox, 2011, Chubet et al., 2024, Wenzel et al., 2019, Han et al., 2017, Brændeland, 2015, Hnatiuk et al., 5 Aug 2025). In all of these settings, greediness is combined with a feasibility constraint and, in the more advanced variants, with corrective updates that reoptimize the current representation after insertion.

1. General schema and problem classes

The common schema is an iterative transition from a current partial structure to an enlarged one. The current state may be a Euclidean graph, an arborally satisfied point set, a prefix of a greedy permutation, a finite dictionary of kernel centers, a subset in a submodular maximization problem, a family of independent vertex sets, or a sparse measure of the form

$u = \sum_{j=1}^N \lambda^j \delta_{x^j}.$

At each step, one defines a feasible candidate set, evaluates candidates by a local or global score, inserts the best candidate according to that score, and optionally performs a correction step on coefficients, positions, or local combinatorial structure (Chubet et al., 2024, Hnatiuk et al., 5 Aug 2025).

The greedy score varies by domain. In geometric $t$-spanners, the score is implicit in the distance order of candidate pairs and in the test $\delta_G(u,v) > t|uv|$; in metric sampling, it is the distance to the already inserted set; in kernel approximation it may be the residual, the power function, or a stabilized restriction thereof; in DPP MAP inference it is the marginal gain

$$\Delta(i\mid X) = \log\det(L_{X\cup\{i\}}) - \log\det(L_X);$$

in graph-independent-set growth it is a heuristic on non-neighbors; and in sparse measure optimization it is the maximization of the dual certificate magnitude $|p_u(x)|$, where $p_u = -K_* \nabla F(Ku)$ (Wenzel et al., 2019, Han et al., 2017, Brændeland, 2015, Hnatiuk et al., 5 Aug 2025).

A second unifying feature is that modern greedy point insertion methods rarely rely on insertion alone. They typically combine insertion with a corrective phase: local Dijkstra searches and WSPD filtering in greedy spanners, coefficient or sliding updates in sparse inverse problems, finite-dimensional reoptimization in kernel and measure-space settings, or local subtree/path rearrangements in the BST model. This suggests that, in the contemporary literature, “greedy point insertion” usually denotes an insertion-centered outer logic rather than a purely myopic one-step algorithm.

2. Euclidean geometric graphs: greedy spanners and triangulations

In planar geometric graph theory, the canonical example is the greedy $t$-spanner. Given a point set $P \subset \mathbb{R}^2$, a Euclidean graph $G=(P,E)$ is a $t$-spanner if every pair $t$0 has a $t$1-path, that is,

$t$2

The standard greedy spanner is exactly a greedy edge insertion algorithm on the complete geometric graph: all pairs are processed in ascending order of Euclidean length, and an edge $t$3 is added iff the current graph has no $t$4-path between $t$5 and $t$6. The straightforward implementation examines $t$7 pairs and performs shortest-path checks, yielding $t$8 time, while all earlier greedy constructions were at least quadratic in $t$9 (Alewijnse et al., 2014).

The main distribution-sensitive improvement is based on the observation that many point sets have many short greedy edges that are locally decidable and few or no long ones. To formalize this, the paper introduces bridging pairs and the notions of $\delta_G(u,v) > t|uv|$0-bridged and locally-$\delta_G(u,v) > t|uv|$1-bridged point sets. The key lemma states that if $\delta_G(u,v) > t|uv|$2 is $\delta_G(u,v) > t|uv|$3-bridged, then a Euclidean graph $\delta_G(u,v) > t|uv|$4 is a $\delta_G(u,v) > t|uv|$5-spanner iff all pairs with $\delta_G(u,v) > t|uv|$6 have $\delta_G(u,v) > t|uv|$7-paths. On uniformly random points in a $\delta_G(u,v) > t|uv|$8 square, there exists a constant $\delta_G(u,v) > t|uv|$9 such that, with high probability, $$\Delta(i\mid X) = \log\det(L_{X\cup\{i\}}) - \log\det(L_X);$$0 is locally-$$\Delta(i\mid X) = \log\det(L_{X\cup\{i\}}) - \log\det(L_X);$$1-bridged. Consequently, the greedy spanner has no edges longer than $$\Delta(i\mid X) = \log\det(L_{X\cup\{i\}}) - \log\det(L_X);$$2, and the greedy construction becomes effectively local on those inputs (Alewijnse et al., 2014).

The algorithmic realization of this local-to-global phenomenon combines three mechanisms. First, all greedy edges of length at most $$\Delta(i\mid X) = \log\det(L_{X\cup\{i\}}) - \log\det(L_X);$$3 are computed by grid bucketing and local Dijkstra searches in

$$\Delta(i\mid X) = \log\det(L_{X\cup\{i\}}) - \log\det(L_X);$$4

expected time. Second, long-edge candidates are filtered through a well-separated pair decomposition together with the path-hyperbola

$$\Delta(i\mid X) = \log\det(L_{X\cup\{i\}}) - \log\det(L_X);$$5

which certifies distant $$\Delta(i\mid X) = \log\det(L_{X\cup\{i\}}) - \log\det(L_X);$$6-paths from local information. Third, only a residual set of “hard” WSPD pairs is passed to a full global greedy check. The resulting algorithm computes the greedy $$\Delta(i\mid X) = \log\det(L_{X\cup\{i\}}) - \log\det(L_X);$$7-spanner in

$$\Delta(i\mid X) = \log\det(L_{X\cup\{i\}}) - \log\det(L_X);$$8

expected time on uniformly distributed points and has worst-case time

$$\Delta(i\mid X) = \log\det(L_{X\cup\{i\}}) - \log\det(L_X);$$9

on arbitrary inputs (Alewijnse et al., 2014).

A related greedy insertion paradigm appears in greedy triangulations, where edges are processed in non-decreasing order of length and inserted whenever they do not cross earlier edges. According to the available summary, given the Delaunay triangulation $|p_u(x)|$0, one can find the greedy triangulation $|p_u(x)|$1 in linear time, and, conversely, given $|p_u(x)|$2, it is possible to compute $|p_u(x)|$3 in linear expected time. The same summary states that a hierarchical clustering algorithm originally derived from $|p_u(x)|$4 remains almost correct for any triangulation of bounded dilation, and that the properties of the greedy triangulation suffice to guarantee a linear-time bound (Mulzer et al., 2013). This places greedy edge insertion in triangulations within the broader bounded-dilation and geometric-spanner framework.

3. Geometric BSTs: arboral satisfaction and online point addition

In the geometric model of binary search trees, greedy point insertion is literal point addition in the plane. A successful access sequence $|p_u(x)|$5 is represented by the point set

$|p_u(x)|$6

with key on the $|p_u(x)|$7-axis and time on the $|p_u(x)|$8-axis. A point set is arborally satisfied if every pair of points is either horizontally or vertically aligned or defines an axis-aligned rectangle containing a third point. Arborally satisfied supersets are exactly the geometric images of valid BST executions. GreedyASS, the geometric counterpart of GreedyFuture, processes the access points online and, at time $|p_u(x)|$9, adds the unique minimal set of points on the line $p_u = -K_* \nabla F(Ku)$0 needed to make all rectangles involving $p_u = -K_* \nabla F(Ku)$1 satisfied (Fox, 2011).

This insertion rule is both minimal and immediate. If the rectangle between the new access point $p_u = -K_* \nabla F(Ku)$2 and an earlier point $p_u = -K_* \nabla F(Ku)$3 is unsatisfied, then the only admissible repair at time $p_u = -K_* \nabla F(Ku)$4 is to insert $p_u = -K_* \nabla F(Ku)$5. Performing this over all unsatisfied rectangles yields the unique minimal set $p_u = -K_* \nabla F(Ku)$6. An equivalent operational view maintains the last access time $p_u = -K_* \nabla F(Ku)$7 of each key and sweeps to the right and left from $p_u = -K_* \nabla F(Ku)$8, adding a point whenever the running maximum of last-access times increases. The analysis then introduces inclusive neighborhoods $p_u = -K_* \nabla F(Ku)$9, sizes $t$0, ranks

$t$1

and the potential $t$2, leading to the access lemma

$t$3

for the amortized cost. From this, GreedyFuture inherits balance, static optimality, static finger, working-set, and sequential-access bounds (Fox, 2011).

A second geometric BST line studies the Greedy algorithm through its touch matrix. At time $t$4, given the current access $t$5, Greedy touches exactly the columns in the stair $t$6, namely those columns whose rectangles to their last-touch points would otherwise be unsatisfied. This is again a greedy point insertion rule: all newly exposed unsatisfied rectangles incident to the current access are fixed immediately on the current row. The paper develops hidden-element arguments and input-revealing gadgets showing that the execution log partially reveals the combinatorial structure of the input sequence. For permutation accesses $t$7, where $t$8 is the pattern-avoidance parameter, the touch matrix satisfies forbidden-submatrix constraints that imply

$t$9

For preorder traversals, this yields an almost linear bound, and with linear-cost preprocessing Greedy becomes linear on preorder sequences (Chalermsook et al., 2015).

These BST results are significant because they show that maximally greedy local repairs need not preclude strong global behavior. At the same time, they also mark a boundary: neither the access-lemma results for GreedyFuture nor the forbidden-submatrix analysis for Greedy resolves dynamic optimality, which remains open (Fox, 2011, Chalermsook et al., 2015).

4. Metric sampling, greedy permutations, and greedy trees

In finite metric spaces, greedy point insertion usually means farthest-point traversal. For a finite metric space $P \subset \mathbb{R}^2$0, a greedy permutation $P \subset \mathbb{R}^2$1 satisfies

$P \subset \mathbb{R}^2$2

and an ordering is $P \subset \mathbb{R}^2$3-greedy if

$P \subset \mathbb{R}^2$4

The first $P \subset \mathbb{R}^2$5 points of an exact greedy permutation give a $P \subset \mathbb{R}^2$6-approximation to $P \subset \mathbb{R}^2$7-center clustering, and the same insertion order underlies metric sampling, approximation of metric spaces, and sparse filtrations in topological data analysis (Chubet et al., 2024).

The paper introduces the greedy tree as a hereditary ball tree that records an approximate nearest predecessor for each inserted point. If $P \subset \mathbb{R}^2$8 is inserted from the Voronoi cell of $P \subset \mathbb{R}^2$9, then $G=(P,E)$0 and the insertion distance is

$G=(P,E)$1

An $G=(P,E)$2-greedy tree encodes geometric decay of insertion distances, $G=(P,E)$3-approximate predecessors, and a heap-ordered traversal whose output is a $G=(P,E)$4-approximate greedy permutation. This separates the construction of a multiscale hierarchical object from the extraction of an ordering, a distinction that the paper exploits algorithmically (Chubet et al., 2024).

The principal algorithmic contribution is a simplification of earlier $G=(P,E)$5 algorithms in doubling metrics. Clarkson’s finite Voronoi diagram machinery is adapted so that cells store greedy-tree nodes rather than raw points; strict heap ordering is relaxed to a bucket queue with a backburner; and the crucial primitive becomes linear-time merging of two approximate greedy trees. Two $G=(P,E)$6-greedy trees can be merged into a $G=(P,E)$7-greedy tree in $G=(P,E)$8 time. A divide-and-conquer recurrence then gives a deterministic $G=(P,E)$9 construction of a constant-factor approximate greedy tree in linear space, and a refinement procedure extracts a $t$0-approximate greedy permutation in the same asymptotic time (Chubet et al., 2024).

This line makes explicit that greedy point insertion need not mean operating directly on points. The insertion policy can be encoded in a tree, the tree can be merged before the order is fully known, and the ordering can then be refined from the hierarchy. A plausible implication is that, in doubling metrics, the greedy tree is the central structural object and the permutation is a derived view.

5. Greedy selection in approximation and probabilistic subset design

In kernel interpolation, greedy point insertion chooses sample points for building nested approximation spaces

$t$1

Classical rules maximize the residual, the power function $t$2, or the normalized residual $t$3. The stabilized framework introduces the admissible set

$t$4

and then performs greedy selection only within $t$5. When $t$6, the restriction disappears; when $t$7, every $t$8-stabilized rule degenerates to pure $t$9-greedy. The key point is that the power-function restriction enforces spacing and hence stability, while leaving a scale of strategies between purely function-driven and purely worst-case-driven sampling (Wenzel et al., 2019).

Under Sobolev-kernel assumptions, the paper proves for any $t$00-stabilized greedy algorithm the upper bound

$t$01

which matches the optimal rate up to constants. It also proves optimal-order fill-distance decay, lower bounds on separation distance, a Lebesgue-constant estimate

$t$02

and optimal approximation rates even outside the native space. Here greedy point insertion is not merely a combinatorial heuristic; it is a mechanism for constructing asymptotically quasi-uniform, stable sampling sets with provable convergence and conditioning properties (Wenzel et al., 2019).

A different approximation-theoretic use of greedy insertion arises in DPP MAP inference. For a DPP kernel $t$03, the objective

$t$04

is submodular, and greedy subset construction adds the item with maximal marginal gain $t$05. The computational bottleneck is repeated evaluation of log-determinants or Schur complements. The paper accelerates this by combining first-order matrix-log-determinant expansions, candidate partitioning, and high-order scalar-log approximations with stochastic trace estimators. For the single-item greedy algorithm, this reduces dense complexity from $t$06 to $t$07; for the approximate-greedy output $t$08, assuming $t$09, the value satisfies

$t$10

where $t$11 is the maximal marginal-gain approximation error (Han et al., 2017).

These two literatures emphasize complementary aspects of greedy insertion. Kernel methods use greediness to regulate geometry, stability, and approximation error in continuous domains, whereas DPP MAP uses greediness to navigate a submodular combinatorial objective with heavy linear-algebraic cost. In both cases, exact maximization is computationally expensive, and the modern algorithmic trend is to replace it by structured approximations without losing the essential theoretical guarantees.

6. Discrete graph growth and sparse optimization over measures

In combinatorial graph optimization, a particularly direct constructive example is the family $t$12 for maximum independent sets. The algorithm starts from all independent sets of cardinality $t$13, maintains a family $t$14 of equally sized independent sets, and for each $t$15 computes the set of non-neighbors that can be added while preserving independence. It then evaluates those candidates by a heuristic and inserts the best one, producing a new family of larger independent sets. Heuristic $t$16 chooses the candidate with the most non-neighbors of its own; heuristic $t$17 chooses the one whose non-neighbors induce the most stable graph, with stability defined by

$t$18

The process stops when no current set can be enlarged, at which point all returned sets are maximal independent sets (Brændeland, 2015).

The reported behavior is strongly dependent on the heuristic and on the seed size $t$19. On random graphs with $t$20, the failure rate of $t$21 rises to about $t$22 at $t$23, $t$24 remains around $t$25, and $t$26 is reported to maintain a $t$27 success rate throughout the tested range. Even when the greedy method fails, the observed gap $t$28 is typically at most $t$29, with only very rare cases of $t$30 in the broader experiments. This is a purely discrete greedy point insertion pattern: the inserted “point” is a feasible vertex, and parallel growth from many seeds acts as diversification rather than as correction (Brændeland, 2015).

At the opposite end of the spectrum, greedy point insertion over spaces of Radon measures treats sparse measures as finite sums of Dirac deltas and solves

$t$31

Classical point insertion selects a new support point by maximizing $t$32, where $t$33, then reoptimizes coefficients or both coefficients and positions. Exact coefficient-only updates yield asymptotic linear convergence, and exact sliding updates can yield finite-step convergence under the structural assumptions on the dual certificate and the sparse minimizer (Hnatiuk et al., 5 Aug 2025).

The lazified framework replaces exact subproblem solutions by certified inexact ones. In lazy generalized conditional gradient, a direction $t$34 is accepted if it satisfies

$t$35

and exact maximization is performed only when no such lazy direction exists. This preserves monotonic descent and gives global $t$36 convergence. The more elaborate LPDAP combines DropStep, inexact coefficient optimization controlled by a finite-dimensional dual gap, local support improvement, and lazy GCG, recovering asymptotic linear convergence. Finally, NLGCG couples lazy insertion with clustering and Newton steps in the finite-dimensional weight-position variables; after finite identification of the correct number of atoms, it achieves

$t$37

that is, asymptotic quadratic convergence (Hnatiuk et al., 5 Aug 2025). In this setting, greediness is no longer the entire algorithmic story, but it remains the globalization mechanism that keeps sparse inverse problems both constructive and convergent.

7. Cross-cutting principles and unresolved questions

Across these domains, several structural motifs recur. First, insertion is always constrained: noncrossing in triangulations, $t$38-path sufficiency in spanners, arboral satisfaction in BST geometry, admissibility sets in kernel selection, submodular marginal gains in DPPs, non-neighbor feasibility in independent-set growth, and sparsity-regularized descent in measure spaces. Second, the best-performing methods exploit locality or hierarchy. Bridgedness makes long spanner edges unnecessary on random inputs, greedy trees encode multiscale metric structure, GreedyASS acts only on the current time line, and measure-space algorithms use local support improvement and local merging (Alewijnse et al., 2014, Chubet et al., 2024, Fox, 2011, Hnatiuk et al., 5 Aug 2025).

A third theme is that exact greediness is often too expensive, and the technically strongest papers replace it by controlled approximation. The kernel literature does so through $t$39-restriction, the DPP literature through approximate log-determinant evaluation, and the measure-space literature through lazy directions and inexact coefficient subsolvers (Wenzel et al., 2019, Han et al., 2017, Hnatiuk et al., 5 Aug 2025). This suggests that the decisive property is not exact maximization per se, but rather the availability of computable certificates that lower-bound the usefulness of an insertion or upper-bound the loss from inexactness.

Several central questions remain open. The BST literature does not prove dynamic optimality of GreedyFuture or Greedy, though it establishes the first non-trivial upper bounds for GreedyFuture and strong pattern-avoidance bounds for Greedy (Fox, 2011, Chalermsook et al., 2015). In geometric spanners, the distribution-sensitive algorithm is near-linear in expectation on uniform random inputs but remains near-quadratic in the worst case (Alewijnse et al., 2014). In kernel approximation, the choice of $t$40 exposes a persistent trade-off between local error reduction and numerical stability (Wenzel et al., 2019). In measure optimization, the new lazy and Newton-based variants show that inexactness can coexist with linear or quadratic convergence, suggesting that future work may continue to shift greedy point insertion away from exact global solves toward adaptive, certificate-driven local computation (Hnatiuk et al., 5 Aug 2025).

Taken together, these works show that greedy point insertion algorithms are not a single method but a research program. Their common principle is constructive sparsification by sequential augmentation; their differences lie in what counts as a point, how feasibility is encoded, which quantity is maximized, and whether insertion is followed by local or global correction. The resulting theory ranges from Euclidean spanners and triangulations to BST geometry, doubling-metric hierarchies, kernel and DPP approximation, graph optimization, and sparse inverse problems over measure spaces.