This paper considers an optimal radial reconfiguration problem in multi-source distribution networks, where the goal is to find a radial configuration that minimizes quadratic distribution costs while ensuring all sink demands are met. This problem arises in critical infrastructure systems such as power distribution, water networks, and gas distribution, where radial configurations are essential for operational safety and efficiency. Optimal solution for this problem is known to be NP-hard. In this paper, we prove further that constructing a feasible radial distribution configuration is weakly NP-complete, making exact solution methods computationally intractable for large-scale networks. We propose FORWARD (Feasibility Oriented Random-Walk Inspired Algorithm for Radial Reconfiguration in Distribution Networks), a polynomial-time algorithm that leverages graph-theoretic decomposition and random walk principles to construct feasible radial configurations. Our approach introduces novel techniques including strategic graph partitioning at articulation points, dual graph condensation to address greedy shortsightedness, and capacity-aware edge swapping for infeasibility resolution. We provide rigorous theoretical analysis proving feasibility guarantees and establish a compositional framework enabling parallel processing while preserving optimality properties. Comprehensive numerical evaluation on networks ranging from IEEE standard test systems to 400-node small-world networks demonstrates that FORWARD consistently outperforms commercial MINLP solvers, achieving optimal or near-optimal solutions in seconds where traditional methods require hours or fail entirely. The algorithm's polynomial-time complexity and scalability make it particularly suitable for real-time distribution network management and as an effective initialization strategy for iterative optimization solvers.