This paper proposes earliest and latest path algorithms based on binary weight allocation, assigning weights of 2(i-1) and 2(m-i) to the i-th arc in a network. While traditional shortest path algorithms optimize only distance, our approach leverages Binary-Addition-Tree ordering to efficiently identify lexicographically smallest and largest paths that establish connectivity. These paths partition the solution space into three regions: guaranteed disconnection, transitional connectivity, and guaranteed no simple paths. Our weight allocation enables implicit encoding of multiple objectives directly in binary representations, maintaining the O((|V|+|E|)log|V|) complexity of Dijkstra's algorithm while allowing simultaneous optimization of competing factors like reliability and cost. Experimental validation demonstrates significant computational time reduction compared to traditional multi-objective methods. Applications span telecommunications, transportation networks, and supply chain management, providing efficient tools for network planning and reliability analysis under multiple constraints.