arXiv subjects

Md. Rawha Siddiqi Riad

Publications and source records attributed to Md. Rawha Siddiqi Riad.

3 recordsLinked to original sources

arXiv2026

Exact and Asymptotic Analysis of Greedy Online Facility Assignment on Cycle Metrics

We identify the free-facility gaps of greedy online assignment on a cycle with the discrete additive coalescent and use this connection to determine the exact expected assignment cost. The cycle has circumference $n$, with one unit-capacity facility at each integer point and $n$ independent uniform arrivals. Each request is assigned immediately to a nearest free facility. The conditional one-step cost is the sum of squared gap lengths divided by $4n$. A uniform cyclic-order argument identifies the unordered gap process with additive coalescence. The classical Cayley-tree representation then reduces the accumulated cost to a two-vertex distance functional, yielding \[ G_n=\frac n4\sum_{d=0}^{n-1}\prod_{j=1}^{d}\left(1-\frac jn\right) \sim\frac14\sqrt{\frac\pi2}\,n^{3/2}. \] The formula requires a linear number of scalar recurrence steps to evaluate. We also obtain an asymptotic upper bound of $π\sqrt3/2$ on the ratio of expected greedy cost to expected offline optimum under this arrival model. An exact rational Bellman calculation shows that greedy is optimal for three facilities but exceeds the optimal online expected cost by $1/512$ for four facilities. Independent exact recurrences and finite tree enumeration provide reproducible checks of the analysis.

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arXiv2026

Competitive Analysis of Online Facility Assignment Algorithms on Discrete Grid Graphs: Performance Bounds and Remediation Strategies

We study the \emph{Online Facility Assignment} (OFA) problem on a discrete $r\times c$ grid graph under the standard model of Ahmed, Rahman, and Kobourov: a fixed set of facilities is given, each with limited capacity, and an online sequence of unit-demand requests must be irrevocably assigned upon arrival to an available facility, incurring Manhattan ($L_1$) distance cost. We investigate how the discrete geometry of grids interacts with capacity depletion by analyzing two natural baselines and one capacity-aware heuristic. First, we give explicit adversarial sequences on grid instances showing that purely local rules can be forced into large competitive ratios: (i) a capacity-sensitive weighted-Voronoi heuristic (\textsc{CS-Voronoi}) can suffer cascading \emph{region-collapse} effects when nearby capacity is exhausted; and (ii) nearest-available \textsc{Greedy} (with randomized tie-breaking) can be driven into repeated long reassignments via an \emph{oscillation} construction. These results formalize geometric failure modes that are specific to discrete $L_1$ metrics with hard capacities. Motivated by these lower bounds, we then discuss a semi-online extension in which the algorithm may delay assignment for up to $τ$ time steps and solve each batch optimally via a min-cost flow computation. We present this batching framework as a remediation strategy and delineate the parameters that govern its performance, while leaving sharp competitive guarantees for this semi-online variant as an open direction.

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arXiv2026

Expected Cost of Greedy Online Facility Assignment on Regular Polygons (v3)

We study a greedy online facility assignment process on a regular $n$-gon, where unit-capacity facilities occupy the vertices and customers arrive sequentially at uniformly random locations on polygon edges. Each arrival is irrevocably assigned to the nearest currently free facility under the shortest edge-walk metric, with uniform tie-breaking among equidistant choices. Our main theoretical result is an exact value-function characterization: for every occupancy state $S\subseteq V$, the expected remaining cost $V(S)$ satisfies a finite-horizon integral recurrence obtained by conditioning on the random arrival edge and position. To make this recurrence computationally effective, we exploit dihedral symmetry of the regular polygon and show that $V(S)$ is invariant under rotations and reflections, enabling canonicalization and symmetry-reduced dynamic programming. For small $n$, we evaluate the recurrence accurately using deterministic numerical integration over piecewise-linear distance regions,; for larger $n$, we estimate the expected total cost via direct Monte Carlo simulation of the online process and report $95\%$ confidence intervals. Our computations validate the recurrence (including a closed-form check for the square, $n=4$) and indicate that the total expected cost increases with $n$, while the per-customer expected travel distance grows gradually as remaining free vertices become farther on average. \keywords{Online algorithms \and Facility assignment \and Expected cost \and Regular polygons \and Symmetry reduction \and Monte Carlo}

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