Path planning is a key element in navigation of a mobile robot. Several algorithms such as a gradient method have been successfully implemented so for. Although the gradient method can provide the global optimal path, it computes the navigation function over the whole environment at all times, which result in high computational cost. This paper proposes a high-speed path planning scheme, called a gradient method with topological information, in which the search space for computation of a navigation function can be remarkably reduced by exploiting the characteristics of the topological information reflecting the topology of the navigation path. The computing time of the gradient method with topological information can therefore be significantly decreased without losing the global optimality. This reduced path update period allows the mobile robot to find a collision-free path even in the dynamic environment.