Counting Unit Circular Arc Intersections

Given a set of $n$ circular arcs of the same radius in the plane, we consider the problem of computing the number of intersections among the arcs. The problem was studied before and the previously best algorithm solves the problem in $O(n^{4/3+ε})$ time [Agarwal, Pellegrini, and Sharir, SIAM J. Comput., 1993], for any constant $ε>0$. No progress has been made on the problem for more than 30 years. We present a new algorithm of $O(n^{4/3}\log^{16/3}n)$ time and improve it to $O(n^{1+ε}+K^{1/3}n^{2/3}(\frac{n^2}{n+K})^ε\log^{16/3}n)$ time for small $K$, where $K$ is the number of intersections of all arcs.

💡 Research Summary

The paper addresses the classic computational geometry problem of counting all intersection points among a set S of n circular arcs that share the same radius (assumed to be 1 for simplicity). Historically, the best known deterministic algorithm for this problem was due to Agarwal, Pellegrini, and Sharir (1993), running in O(n^{4/3+ε}) time for any arbitrarily small constant ε>0. A matching lower bound of Ω(n^{4/3}) exists in the so‑called partition‑algorithm model, originally proved for line‑segment intersections and applicable to unit‑radius arcs as well. Despite this, no improvement had been reported for more than three decades.

The authors present two major improvements. First, they design a deterministic algorithm that runs in O(n^{4/3}·log^{16/3} n) time, shaving off the extra n^{ε} factor and replacing it with a polylogarithmic term. Second, they develop a K‑sensitive algorithm whose running time depends on the actual number K of intersection points. The bound is
O(n^{1+ε} + K^{1/3} n^{2/3} · (n^{2}/(n+K))^{ε} · log^{16/3} n).
When K=Θ(n^{2}) (the worst case) this collapses to the first bound; when K=O(n^{2−δ}) for any constant δ>0 the running time becomes o(n^{4/3}), a genuine improvement over the previous worst‑case bound.

The technical core relies on hierarchical (1/r)-cuttings adapted to unit circular arcs. For a parameter r≤n, the authors construct a cutting of size O(r^{2}) consisting of pseudo‑trapezoids (each cell has two vertical sides and top/bottom sides that are portions of unit circles). This cutting can be built in O(n r) time, and its size and construction time can be expressed in terms of K as well (Theorem 1). The cutting partitions the plane so that each cell contains only O(n/r) arcs, and each arc intersects only a constant number of cells.

Next, the plane is covered by a grid C of axis‑parallel squares of side length 1/√2. For each square C, a constant‑size neighbourhood N(C) is precomputed; any arc intersecting C must have its centre in one of the squares of N(C). Lemma 1 guarantees that each arc touches only O(1) squares, and the total number of centre points over all neighbourhoods is linear in n. Consequently, the global counting problem reduces to counting intersections inside each square C, considering only arcs whose centres lie in N(C).

Inside a fixed square C, the authors further split arcs into short arcs (at least one endpoint lies inside the square) and long arcs (both endpoints lie on the square’s boundary). Intersections are classified into four types:

1. long–long,
2. short–short,
3. long–short,
4. short–long.

Type 2 (short–short) is handled by the classic Agarwal‑Pellegrini‑Sharir algorithm in O(m^{4/3+ε}) time, where m is the number of short arcs in the cell. Types 3 and 4 are symmetric; the paper focuses on long–short intersections. These are further divided into (3.1) pairs intersecting exactly once and (3.2) pairs intersecting twice. For (3.1) the authors introduce two geometric observations:

Observation 1 uses the lune formed by the two unit disks centered at the endpoints of a short arc. A long arc intersects the short arc exactly once iff its centre lies inside the lune but outside the other disk. This reduces counting to a 2‑D range‑search problem over centre points.

Observation 2 exploits the fact that a long arc’s underlying circle intersects the square’s boundary at two points; the angular interval of possible intersection with a short arc can be expressed as a range on the circle’s parameter space.

By building appropriate range‑search data structures (e.g., 2‑D Fenwick trees or segment trees) the authors achieve an O(m^{2}/log m + n log^{2} m) time bound for (3.1). For (3.2) (double intersections) a similar approach is applied twice, incurring only a constant factor overhead. Type 1 (long–long) intersections are handled by noting that each long arc contributes at most two boundary points; after enumerating all such boundary points, the problem reduces to counting intersecting intervals on a line, which can be done in linear time per cell.

Putting everything together, each pseudo‑trapezoid cell τ of the cutting contains n_τ = O(n/r) arcs. The total work over all cells is
O( Σ_τ ( n_τ^{4/3}·log^{16/3} n ) ) = O( n^{4/3}·log^{16/3} n ) when r = n^{1/3}. By choosing r adaptively based on K, the authors derive the K‑sensitive bound mentioned earlier.

The paper also shows that the same framework solves the bichromatic intersection problem (counting intersections between red and blue arcs) with identical asymptotic complexity, because the algorithm never relies on the colour of an arc beyond the classification into two sets.

In the related work discussion, the authors compare their results with algorithms for arcs of varying radii (O(n^{3/2+ε})), for circles (Katz‑Sharir O(n^{4/3} log n) and Agarwal‑Aronov‑Sharir‑Suri expected O(n^{4/3} log^{2/3} n)), and for line segments (several O(n^{4/3}) and O(n^{1.695}) results). They emphasize that the presence of up to two intersections per pair of arcs makes the problem fundamentally harder than the segment case, preventing a direct transfer of the recent O(n^{4/3}) algorithm for segments (Chan‑Zheng).

In conclusion, the paper delivers the first improvement over the three‑decade‑old bound for unit‑radius arc intersection counting, narrowing the gap between known upper and lower bounds to a polylogarithmic factor. The combination of hierarchical cuttings, pseudo‑trapezoid cells, and careful geometric observations about unit arcs constitutes a versatile toolkit that may be applicable to other problems involving unit‑radius objects, such as proximity queries, motion planning, and geometric graph constructions. Future directions include randomized or parallel variants that could further reduce the logarithmic overhead, and extensions to other constant‑curvature curves.