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A challenge towards Quality of Service (QoS) is to provide uninterrupted network services in the event of link failure. Path protection is a promising solution by employing a link-disjoint secondary backup path to protect the primary path to ensure the continuity of network services. Existing path protection schemes are classified into dedicated-path protection (e.g., 1+1 protection) and shared-path protection (e.g., 1:N protection, M:N protection). However, these schemes either consume substantial network resource, or cannot respond to link failure quickly enough for real-time services. This paper focuses on an end-to-end partial bandwidth protection scheme denoted as α+1 protection. In α+1 protection, only critical or real-time information of the primary path is duplicated and transferred through the secondary path. The ratio of the protection bandwidth (of secondary path) to the full bandwidth (of primary path) is defined as ±. The benefit of α+1 protection is twofold. Firstly, since only a portion of the primary bandwidth is protected, it reduces the redundancy of data transmission, and hence improves the efficiency of resource utilization. Secondly, ± is a flexible parameter that can be adjusted according to the requirements of different services or networks, as long as minimum protection for uninterrupted services is guaranteed. However, finding the optimal pair of primary-secondary paths for α+1 protection remains challenging. In this paper, we study the optimality conditions of the pair of paths with respect to any given value of ±, and devise a new optimum paths-finding algorithm for α+1 protection.