How long are the paths in a transitive closure
{"Image DocumentName": "How long are the paths in a transitive closure?", "Image ImageDescription": "Let R be a relation on set A, and let A be a set with n elements. Rephrased: consider a graph G with n nodes and some number of edges. Lemma 1: If there is a path (of length at least 1) from a to b in R, then there is a path between a and b of length not exceeding n. Proof preparation: Suppose there is a path from a to b in R. Let the length of that path be m. Let the path be edges (x0, x1), (x1, x2), \u2026, (xm-1, xm) That\u2019s nodes x0, x1, x2, \u2026, xm-1, xm. If a node exists twice in our path, then it\u2019s not a shortest path. As we made no progress in our path between the two occurrences of the repeated node. Thus, each node may exist at most once in the path.", "Image XResolution": "37", "Image YResolution": "37", "Image ResolutionUnit": "Pixels/Centimeter", "Image Software": "SlidePlayer", "Image YCbCrPositioning": "Centered"}