Partial fromvtow v Orders u w implieswalkfromutow AlbertRMeyer - PowerPoint PPT Presentation

�������� Walks in digraph G Mathematics for Computer Science MIT 6.042J/18.062J walk from u to v and Partial from v to w v Orders u w implies walk from u to w Albert R Meyer March 22, 2013 Albert R Meyer March 22, 2013 po’s.1 po’s.2 Walks in digraph G Walks in digraph G walk from u to v and transitive relation R: from v to w, implies u R v AND v R w u R v AND v R w R v AND v R w walk from u to w: IMPLIES u R w ES u R w u R w IMPLI G + v AN D v G + w u G + u G + v AND v G + w v AND v G + w G + is transitive u G + w IMPLIES u G + IMPLIES u G + w w Albert R Meyer March 22, 2013 Albert R Meyer March 22, 2013 po’s.3 po’s.4 ��

�������� Paths in DAG D transitivity Theorem: pos length path from R is a transitive iff u to v implies R = G + for some no path from v to u digraph G u D + v IMPLIES NOT (v D + u) Albert R Meyer March 22, 2013 Albert R Meyer March 22, 2013 po’s.5 po’s.6 Paths in DAG D strict partial orders asymmetric relation R: y transitive & u R v IMPLIES NOT (v R u) u R v IMPLIES NOT (v R u) R v IMPLIES N OT (v R u) asymmetric D + is asymmetric Albert R Meyer March 22, 2013 Albert R Meyer March 22, 2013 po’s.7 po’s.8 ��

�������� strict partial orders strict partial orders examples: Theorem: • ⊂ on sets R is a SPO iff • “indirect prerequisite” on R = D + for some MIT subjects • less than, < , on real DAG D numbers Albert R Meyer March 22, 2013 Albert R Meyer March 22, 2013 po’s.9 po’s.10 linear orders linear orders basic example: Given any two elements, < or ≤ on the Reals: one will be “bigger than” if x ≠ y, then either the other one. x < y OR y < x Albert R Meyer March 22, 2013 Albert R Meyer March 22, 2013 po’s.11 po’s.12 ��

�������� linear orders linear orders R is linear: The whole partial order is a chain � no incomparable elements if x ≠ y, then either x R y OR y R x OR Albert R Meyer March 22, 2013 Albert R Meyer March 22, 2013 po’s.13 po’s.14 weak partial orders linear orders same as a strict partial A topological sort turns order R, except that a partial order into a a R a always holds linear order � …in a way examples: that is consistent ≤ is weak p.o. on R with the partial order ⊆ is weak p.o. on sets Albert R Meyer March 22, 2013 Albert R Meyer March 22, 2013 po’s.15 po’s.16 ��

�������� reflexivity antisymmetry relation R on set A binary relation R is is reflexive iff antisymmetric iff a R a for all a ∈ ∈ A it is asymmetric except for a R a case. G * is reflexive Albert R Meyer March 22, 2013 Albert R Meyer March 22, 2013 po’s.17 po’s.18 A / Antisymmetry antisymmetry antisymmetric relation R: y y minor difference: u R v IMPLIES NOT (v R u) u R v IMPLIES NOT (v R u) R v IMPLI ES NOT (v R u) whether aRa is allowed for u ≠ v for u ≠ v u ≠ v D * is antisymmetric for sometimes never DAG D Albert R Meyer March 22, 2013 Albert R Meyer March 22, 2013 po’s.19 po’s.20 ��

�������� weak partial orders weak partial orders Theorem: transitive, R is a WPO iff antisymmetric & R = D * for some reflexive DAG D Albert R Meyer March 22, 2013 Albert R Meyer March 22, 2013 po’s.21 po’s.22 ��

MIT OpenCourseWare http://ocw.mit.edu 6.042J / 18.062J Mathematics for Computer Science Spring 20 15 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.