Recall that in a min-heap, the parent node is always smaller than or equal to its children. How can you leverage this property to efficiently locate a node's parent or children?

Consider how the decrease-key operation affects the heap property. What steps must you take to restore the heap property after decreasing a key, and how can you do this in logarithmic time?

Think about how to maintain a mapping between node values and their positions in the heap. How can this mapping help you quickly locate a node to perform decrease-key, and how can you update it efficiently during heap operations?