Monma-Paterson-Suri-Yao Algorithm

• The MPSY algorithm is a suite of geometric methods that partitions space into angular cones for efficient proximity queries and spanner construction.
• It achieves O(n log n) performance using techniques like radial sorting and Fortune-style sweepline algorithms for both Yao graph and MaxST computations.
• The method unifies classical proximity graphs by ensuring unique, provable spanner and spanning tree properties in normed planes even with non-distinct distances.

The Monma-Paterson-Suri-Yao (MPSY) algorithm encompasses a suite of geometric graph constructions and algorithms, most notably for generating Yao graphs as geometric spanners and for constructing maximum spanning trees (MaxST) in normed planes. Its fundamental approach, pioneered by Monma, Paterson, Suri, and Yao, partitions space around each point in a finite planar set into angular cones, using proximity queries within each cone to efficiently connect points. The theory and algorithmics of MPSY unify several classical proximity-graph notions—such as Yao graphs, minimum/maximum spanning trees, and furthest-neighbor structures—within a precise geometric and combinatorial framework.

1. Foundations and Core Definitions

The MPSY framework constructs graphs over a set $P$ of $n$ points in the plane ($\mathbb R^2$), generalized to arbitrary normed planes $(\mathbb R^2, \|\cdot\|)$. The central construction is as follows:

• For a fixed integer $k > 1$, partition the plane around each point $p \in P$ into $k$ disjoint closed cones of aperture $2\pi/k$. For each cone, identify the nearest neighbor to $p$ among all points in $P\setminus\{p\}$ lying in that cone. The directed edge $(p, q)$ is added to the graph for every such pair $(p, q)$.
• The undirected Yao graph $Y_k(P)$ is obtained by discarding the orientation.

Given all pairwise distances are distinct (or can be made so by a perturbation argument in strictly convex norms), these constructions admit unique spanner and spanning-tree properties with combinatorially robust uniqueness.

MPSY also captures the MaxST computation in normed planes. Here, the objective is to find a spanning tree $T$ on $P$ maximizing $L(T) = \sum_{\{u,v\}\in T}\|u-v\|$, where edge weights are given by the norm-induced metric and uniqueness follows from the assumption of distinct inter-point distances (Alonso et al., 20 Jan 2026).

2. Algorithmic Structure and Complexity

For Yao graphs, the basic construction algorithm per point $p \in P$ is:

1. For each cone $i = 0, ..., k-1$ at $p$, maintain the shortest distance to any $q \in P\setminus\{p\}$ lying in that cone.
2. For all pairs $(p, q)$, determine the cone index of $q$ relative to $p$ via polar coordinates, updating the best seen distance per cone.
3. Add edges $(p, q)$ for every cone where a nearest neighbor $q$ was found.

Efficient implementations use:

• Radial sorting of neighbors (per $p$) in $O(n\log n)$, followed by a linear pass to pick the closest per cone, yielding $O(n\log n + kn)$ total time for fixed $k$ (Bose et al., 2014).
• In practice, with small constant $k$, this approaches $O(n\log n)$.

Recent implementations optimize further by treating each direction (cone) separately using a Fortune-style sweepline algorithm for oriented Voronoi diagrams, yielding a practical $O(n \log n)$ construction for all $k$ cones (Funke et al., 2023). Algorithmic innovations include:

• Event queue partitioned into static (input points) and dynamic (intersections, deletions) segments.
• Beach-line data structures (e.g., leaf-linked AVL trees) for $O(\log n)$ per operation.
• Uniform grids for spatial bucketing in medium-size data, with typical performance far better than the worst-case $O(n^2)$. Empirical results confirm significant performance advantages over quadratic baseline methods (Funke et al., 2023).

For MaxST in normed planes:

1. Compute the furthest-neighbor graph FNG$(P)$: for each $p$, add a directed edge to its unique furthest neighbor $\mathrm{fn}(p)$.
2. Each component of FNG$(P)$ contains a spine (2-cycle) and decomposes into two clusters; clusters are organized via cyclic ordering around the convex hull of $P$.
3. Find maximal inter-cluster edges per adjacent pair in this ordering, omitting the minimum among them, then return the union of FNG$(P)$ and these selected edges as the MaxST (Alonso et al., 20 Jan 2026).
4. All steps are $O(n\log n)$ using geometric data structures for the underlying proximity queries.

3. Geometric and Combinatorial Properties

A crucial aspect is the spanner property: Yao graphs are $t$-spanners of the complete geometric graph for suitable $k$, with $t$ the stretch factor.

Key results:

• For $k \geq 7$, $t(k) = 1/(1 - 2\sin(\pi/k))$ (Bose et al., 2014).
• For odd $k \geq 5$, $t(k) = 1/(1 - 2\sin(3\pi/(4k)))$ yields improved bounds, e.g., $t(5) \leq 2+\sqrt{3}$, $t(6) \leq 5.8$, with tight lower bounds and refined geometric analyses for small $k$ (Barba et al., 2013).
• A Yao graph of $n$ points, parameter $k$, has $O(kn)$ edges; for fixed $k$, this is $O(n)$.

For MaxST, key combinatorial facts include:

• The furthest neighbor of any point lies on the convex hull $\partial\mathrm{conv}(P)$ (Lemma 2.1 in (Alonso et al., 20 Jan 2026)).
• The resulting tree structure is unique when all inter-point distances are distinct.
• All spines are 2-cycles in FNG$(P)$ whose endpoints lie alternately around the convex hull, forcing interleaving clusters that determine the admissible maximal edges to complete the MaxST.
• Key inequalities (such as the convex-quadrilateral and triangle-side bounds) ensure the cut properties and feasibility of the MPSY construction.

4. Handling Non-distinct Distances and General Norms

In strictly convex normed planes, input configurations with repeated distances are addressed via controlled perturbation:

• Each point is displaced by a vector of norm $<\epsilon/n$ to break any distance equalities, with total movement not exceeding $\epsilon$.
• Strict convexity ensures old equalities are destroyed, and no new ones are introduced.
• The property is established inductively by considering the rank of a point around the hull, guaranteeing eventual uniqueness of all inter-point distances (Alonso et al., 20 Jan 2026).
• After perturbation, the MPSY algorithm applies unchanged. Letting $\epsilon \to 0$ gives the exact solution for the unperturbed input.

For general (not strictly convex) norms, the essential steps require the construction of furthest-site Voronoi diagrams and separate correctness arguments.

5. Extensions, Applications, and Related Structures

MPSY constructions have deep implications in computational geometry and related fields:

• Spanner and Network Design: Yao graphs serve as sparse geometric spanners, crucial in wireless network topology control and approximation algorithms (Funke et al., 2023).
• Constrained and Visibility Graphs: The MPSY spanner analysis extends to constrained Yao graphs situated in static environments with obstacles; upper bounds on spanning ratios are preserved even under planarity constraints (Bose et al., 2014).
• Comparison to Θ-graphs: For $k=5$ and $k=6$, the separation between Yao and $\Theta$-graph spanning ratios is now rigorous, establishing circumstances where Yao graphs outperform $\Theta$-graphs of the same order (Barba et al., 2013).
• Dynamic and Higher-dimensional Extensions: While the planar case is algorithmically mature, higher-dimensional Yao graph construction and dynamic updates remain active research areas, with the MPSY approach forming a foundation for further developments (Funke et al., 2023).
• Practical Implementations: Optimized sweepline implementations based on MPSY are at least an order of magnitude faster than previous approaches, with empirical validation across real and synthetic datasets (Funke et al., 2023).

6. Summary Table of Main Results

Problem	MPSY Algorithm Time Complexity	Spanner Ratio $t(k)$	Uniqueness Condition
Yao-graph construction	$O(n\log n)$	$1/(1-2\sin(\pi/k))$ ($k\ge7$)	Cones, nearest neighbor
MaxST in normed plane	$O(n\log n)$	(Not a spanner)	All inter-point distances distinct
Yao-graph ($k=6$)	$O(n\log n)$	$\le5.8$	As above

This synthesis encapsulates the combinatorial, geometric, and algorithmic structure of the Monma-Paterson-Suri-Yao algorithm, its computational optimality, and the compositional principles spanning spanner theory and maximal-weight spanning tree construction in geometric and normed spaces (Alonso et al., 20 Jan 2026, Bose et al., 2014, Barba et al., 2013, O'Rourke, 2010, Funke et al., 2023).