telecommunication operators

Abstract: In the CONGEST model, exact single-source shortest paths (SSSP) with nonnegative weights and exact maximum $st$-flow in directed planar networks have both been solved in $\tilde O(D^2)$ rounds, where $D$ is the hop diameter, while the lower bound is $Ω(D)$. Given the approximate distance labels of a recent preprint (arXiv:2606.31192), which we use as a black box, we solve exact directed SSSP in $\tilde O(D)$ rounds. It rests on a lemma that answers queries from a virtual source with the labels, and on a corrected discretization of the reduction of exact to approximate distances (arXiv:2210.16351). All further bounds of the form $\tilde O(D)$ and $D\cdot n^{o(1)}$ use these labels. Through the reduction of arXiv:2303.00811, SSSP with negative weights, or the detection of a negative cycle, takes $D\cdot n^{o(1)}$ rounds with high probability. We weight the face-disjoint graph of arXiv:2411.11718 so that its distances are dual distances, which turns distance problems in the dual into problems in a planar network of hop diameter $O(D)$. This gives exact maximum $st$-flow in directed planar networks in $\min\{\tilde O(D^2),D\cdot n^{o(1)}\}$ rounds with high probability, and in $\tilde O(D)$ rounds with $s,t$ on a common face. In undirected planar networks, the value of a minimum $st$-cut and a minimum cut take $\tilde O(D)$ rounds. For this we repair the auxiliary graphs of arXiv:2608.09500 at $s$ and $t$, and we show that its recursion, as we read it, can miss the minimum cut, also with exact shortest paths: on a network with $11$ vertices it reports $37$ for a minimum cut of $15$. Two more shortest-path computations per level make it exact. The virtual-source lemma, the weighting of the face-disjoint graph and the corrected discretization are machine-checked in Lean 4 with Mathlib; all other arguments are proved on paper only. Numerical checks are not used in any proof.

Abstract: In integrated sensing and communication (ISAC) systems involving multiple operators, unmanned aerial vehicle (UAV) tracking can benefit from sensing at multiple base stations, while raw sensing records, channel information, and transmit decisions remain local to each operator. The resulting problem is to obtain a reliable UAV prediction from different track estimates and use it to guide transmission without centralizing their data or control variables. To resolve this, we propose a two-timescale framework comprising slow-timescale motion-uncertainty calibration and online track fusion with predictive beamforming. Local radar innovations calibrate the uncertainty assigned by each tracker to unmodeled UAV motion, with only the resulting parameters shared across operators. During online tracking, covariance intersection (CI) combines local estimates without requiring knowledge of correlations between their estimation errors. The fused prediction then guides each operator's transmission design subject to communication QoS, power, and tracking constraints. We further establish exact rank one recovery for the communication covariances and finite interval bounds on local tracking uncertainty. Simulations show improved consistency of tracking uncertainty under substantial motion model mismatch. CI avoids overconfidence under correlated errors, while using the current fused prediction reduces RMSE by about 31% relative to transmission designed for communication alone, with less than 2% additional power. Ablation further shows that the power difference between current and outdated predictions is driven mainly by predictive covariance rather than predictive mean.