As integrated circuit technology continues to advance, clock tree synthesis has become increasingly significant in the design of ultra-large-scale integrated circuits. Traditional clock tree synthesis methods often face challenges such as insufficient computational resources and buffer fan-out limitations when dealing with ultra-large-scale clock trees. To address this issue, this paper proposes an improved clock tree synthesis algorithm called incomplete balanced KSR (IB-KSR). Building upon the KSR algorithm, this proposed algorithm efficiently reduces the consumption of computational resources and constrains the fan-out of each buffer by incorporating incomplete minimum spanning tree (IMST) technology and a clustering strategy grounded in Balanced Split. In experiments, the IB-KSR algorithm was compared with the GSR algorithm. The results indicated that IB-KSR reduced the global skew of the clock tree by 43.4% and decreased the average latency by 34.3%. Furthermore, during program execution, IB-KSR maintained low computational resource consumption.

Rigid-body chain following on piecewise analytic paths is a fundamental subroutine in motion planning and multibody simulation. The problem is nontrivial when only the leader trajectory of the first node is available: enforcing fixed inter-node distances reduces to circle–curve intersection, which is generally multi-valued and becomes particularly challenging near non-smooth junctions. We present a Dichotomy Geometric Path Search (DGPS) framework that converts each constraint into a one-dimensional root-finding task and resolves the branch selection through no-backtracking ordering: at every time step, the admissible solution for the current node is the nearest feasible root in the past relative to its immediately preceding node. DGPS combines backward bracketing with bisection, achieving robust convergence. Compared with the inverse Jacobian method, which maps end-effector velocities to joint velocities via explicit inversion, the proposed approach avoids Jacobian inversion and globally coupled nonlinear solves. We further characterize the local structure of the zero set and establish monotonicity/uniqueness conditions that justify stable root selection across piecewise junctions. Extensive tests on representative piecewise trajectories (line–arc–line, polylines with corners, piecewise sinusoids, and time reparameterization) show that DGPS enforces distance constraints to near machine precision, produces interpretable speed/acceleration transients around non-smooth events, and exhibits computational costs consistent with iteration difficulty. The results support DGPS as a general, efficient solver requiring only the prescribed leader trajectory.