Sungsoo Na

Two-piece self-rearrangements of triangles and of polygons in general position

A self-rearrangement cuts a polygon into polygons and moves them by isometries to tile it again; it is nontrivial if the set of positioned pieces changes. We classify the triangles that admit a nontrivial two-piece self-rearrangement, with arbitrary polygonal pieces and a cut of...

When does a random disk first overlap an earlier one? An expansion of the geometric birthday problem

The expected index of the first disk that overlaps an earlier one, when disks of diameter σ are placed independently and uniformly with centres in a disk of radius R, has the formal expansion √(π/(2q)) + 2/3 + √3/(4π) + X√q + c2q in the pair-overlap probability q, whose constant...

The height of symmetric digital search trees: two-point concentration and the law of the height

For each integer b ≥ 2, let H_n be the height of the symmetric b-ary digital search tree on n independent uniform keys, and e_D(n) the expected number of depth-D nodes. We prove Pr (H_n < D) = exp (−e_D(n)) + o_b(1) uniformly in integers D > log_b n. Thus H_n concentrates on two...

The longest induced path in the n×n grid

A snake in a graph is an induced path. Let a(n) be the largest number of cells of a snake in the n × n grid. We prove that a(n) = (2n2 − 4n + c(n))/3, where c(n) is an explicit constant depending only on n modulo 6, for every n ≥ 13 except n = 15; with exhaustive values for n ≤ 1...

The height of symmetric digital search trees: two-point concentration and the law of the height

For each integer b ≥ 2, let H_n be the height of the symmetric b-ary digital search tree on n independent uniform keys, and e_D(n) the expected number of depth-D nodes. We prove Pr (H_n < D) = exp (−e_D(n)) + o_b(1) uniformly in integers D > log_b n. Thus H_n concentrates on two...

Two Algebraic Thresholds for Relaxations of Stable Metric TSP

Bilu–Linial stability asks whether an optimal solution remains uniquely optimal after every independent multiplicative increase of the input costs by a prescribed factor. For symmetric metric TSP, every 1.8-stable instance is solvable in polynomial time, and both the subtour rela...

When does a random disk first overlap an earlier one? An expansion of the geometric birthday problem

The expected index of the first disk that overlaps an earlier one, when disks of diameter σ are placed independently and uniformly with centres in a disk of radius R, has the formal expansion √(π/(2q)) + 2/3 + √3/(4π) + X√q + c2q in the pair-overlap probability q, whose constant...

Two-piece self-rearrangements of triangles and of polygons in general position

A self-rearrangement cuts a polygon into polygons and moves them by isometries to tile it again; it is nontrivial if the set of positioned pieces changes. We classify the triangles that admit a nontrivial two-piece self-rearrangement, with arbitrary polygonal pieces and a cut of...

The longest induced path in the n×n grid

A snake in a graph is an induced path. Let a(n) be the largest number of cells of a snake in the n × n grid. We prove that a(n) = (2n2 − 4n + c(n))/3, where c(n) is an explicit constant depending only on n modulo 6, for every n ≥ 13 except n = 15; with exhaustive values for n ≤ 1...

Determining a Regular n-Gon from Points on Distinct Sides

A convex regular n-gon has four Euclidean degrees of freedom. Four generic distinguishable points in the plane lie on four distinct supporting side-lines of exactly (n − 1)(n − 2)(n − 3) geometric regular n-gons, one per injective assignment of the points to sides modulo a common...

Two-piece self-rearrangements of convex polygons: decision theorems

A nontrivial two-piece self-rearrangement of a polygon cuts it into two polygons and reassembles them by isometries into the same polygon with a different positioned partition. For convex polygons with exact real-algebraic vertex coordinates and two arbitrary polygonal pieces, we...

A random point of the circumdisk of a random triangle in a parallelogram

For every parallelogram we evaluate in closed form the probability that a uniform random point of the circumdisk of a random triangle lies in the triangle, for three independent uniform vertices in the parallelogram. Its dilogarithm part has eleven functions with coefficients rat...

The minimum order of a non-Hamiltonian inscribable simplicial polyhedron

We prove, without assuming the auxiliary conjectures in Dillencourt's 1996 study, that the minimum order of a non-Hamiltonian inscribable simplicial polyhedron is twenty. Complete enumeration gives the same numbers of eighteen- and nineteen-vertex 1-supertough candidates as his c...

An integer-visiting process with real multipliers: coverage and rates

A positive noninteger start has its integer part and zero visited; its fractional part is repeatedly multiplied by a real β > 1, visiting and subtracting each unvisited integer part. For every β > 1 and almost every start (or every start and almost every β), this process visits e...

Random simplices with small circumradius in a convex window: shape law and window effect

For d + 1 independent uniform points in a convex body P ⊂ Rd with circumradius R ≤ R⋆, we expand the conditional mean of every bounded shape functional to first order in R⋆, with explicit remainder: the limit is the typical Poisson-Delaunay mean, the first-order term a mean-width...

The height of symmetric digital search trees: two-point concentration and the law of the height

For each integer b ≥ 2, let H_n be the height of the symmetric b-ary digital search tree on n independent uniform keys, and e_D(n) the expected number of depth-D nodes. We prove Pr (H_n < D) = exp (−e_D(n)) + o_b(1) uniformly in integers D > log_b n. Thus H_n concentrates on two...

A random point of the circumdisk of a random triangle in a parallelogram

For every parallelogram we evaluate in closed form the probability that a uniform random point of the circumdisk of a random triangle lies in the triangle, for three independent uniform vertices in the parallelogram. Its dilogarithm part has eleven functions with coefficients rat...

Two Algebraic Thresholds for Relaxations of Stable Metric TSP

Bilu–Linial stability asks whether an optimal solution remains uniquely optimal after every independent multiplicative increase of the input costs by a prescribed factor. For symmetric metric TSP, every 1.8-stable instance is solvable in polynomial time, and both the subtour rela...

Determining a Regular n-Gon from Points on Distinct Sides

A convex regular n-gon has four Euclidean degrees of freedom. Four generic distinguishable points in the plane lie on four distinct supporting side-lines of exactly (n − 1)(n − 2)(n − 3) geometric regular n-gons, one per injective assignment of the points to sides modulo a common...

Coalition wins in multiplayer k-in-a-row

In cyclic k-in-a-row on Z2 with p ≥ 3 players, a player has a winning strategy against all the others exactly when it moves first and either k = 1, or k = 2 and p ≤ 8. Against every other player, the target, the others force a win of one of them. When 2p > k they win in round k,...