Lecture 19: Topological sort Part 1: Problem and math basis. Topological sort. The problem. Why we are doing this!

Transcription

1 Chair of Software Engineering Einführung in die Programmierung Introduction to Programming Prof. r. Bertrand Meyer October 006 February 007 Lecture 9: Topological sort Part : Problem and math basis Intro. to Programming, lecture 9: Topological Sort I Intro. to Programming, lecture 9: Topological Sort I Topological sort by Caran d Ache From a given partial order, produce a compatible total order 3 Intro. to Programming, lecture 9: Topological Sort I The problem 4 Why we are doing this! From a given partial order, produce a compatible total order ¾Very Partial order: ordering constraints between elements of a set, e.g. ¾ Remove the dishes before discussing politics ¾ Walk to Üetliberg before lunch ¾ Take your medicine before lunch ¾ Finish lunch before removing dishes Total order: sequence including all elements of set Compatible: the sequence respects all ordering constraints ¾ Üetliberg, Medicine, Lunch, ishes, Politics : OK ¾ Medicine, Üetliberg, Lunch, ishes, Politics : OK ¾ Politics, Medicine, Lunch, ishes, Üetliberg : not OK Intro. to Programming, lecture 9: Topological Sort I common problem in many different areas efficient, non-trivial algorithm ¾Illustration of many algorithmic techniques ¾Illustration of data structures, complexity (big-oh notation), and other topics of last lecture ¾Illustration of software engineering techniques: from algorithm to component with useful API ¾Opportunity to learn or rediscover important mathematical concepts: binary relations (order relations in particular) and their properties ¾It s just beautiful! Today: problem and math basis Next time: detailed algorithm and component ¾Interesting, 5 Intro. to Programming, lecture 9: Topological Sort I 6

2 Reading assignment for next Monday Topological sort: example uses Touch of Class, chapter on topological sort: 7 From a dictionary, produce a list of definitions such that no word occurs prior to its definition Produce a complete schedule for carrying out a set of tasks, subject to ordering constraints (This is done for scheduling maintenance tasks for industrial tasks, often with thousands of constraints) Produce a version of a class with the features reordered so that no feature call appears before the feature s declaration Intro. to Programming, lecture 9: Topological Sort I 7 Intro. to Programming, lecture 9: Topological Sort I 8 Rectangles with overlap constraints Rectangles with overlap constraints Constraints: [B, A], [, A], [A, C], [B, ], [, C] Constraints: [B, A], [, A], [A, C], [B, ], [, C] Possible solution: B E A C B A C E B A C E Intro. to Programming, lecture 9: Topological Sort I 9 Intro. to Programming, lecture 9: Topological Sort I 0 The problem From a given partial order, produce a compatible total order Pictured as a graph Üetliberg Partial order: ordering constraints between elements of a set, e.g. Remove the dishes before discussing politics Walk to Üetliberg before lunch Take your medicine before lunch Finish lunch before removing dishes Total order: sequence including all elements of set Compatible: the sequence respects all ordering constraints Üetliberg, Medicine, Lunch, ishes, Politics : OK Medicine, Üetliberg, Lunch, ishes, Politics : OK Politics, Medicine, Lunch, ishes, Üetliberg : not OK Medicine Lunch ishes Politics Remove the dishes before discussing politics Walk to Üetliberg before lunch Take your medicine before lunch Finish lunch before removing dishes Intro. to Programming, lecture 9: Topological Sort I Intro. to Programming, lecture 9: Topological Sort I

3 (brother Sometimes there is no solution Overall structure () Introducing recursion requires that students know about stacks You must discuss abstract data types before introducing stacks Abstract data types rely on recursion The constraints introduce a cycle Given: A type G A set of elements of type G A set of constraints between these elements Required: An enumeration of the elements, in an order compatible with the constraints class ORERABLE [G] feature elements: LIST [G ] constraints: LIST [TUPLE [G, G ]] topsort : LIST [G ] is... ensure compatible (Result, constraints) end Intro. to Programming, lecture 9: Topological Sort I 3 Intro. to Programming, lecture 9: Topological Sort I 4 Some mathematical background... Binary relation on a set Any property that either holds or doesn t hold between two elements of a set On a set PERSON of persons, example relations are: mother : a mother b holds if and only if a is the mother of b father: child : sister: sibling : or sister) Notation: a r b to express that r holds of a and b. Note: relation names will appear in green Intro. to Programming, lecture 9: Topological Sort I 5 Intro. to Programming, lecture 9: Topological Sort I 6 Example: the before relation Some special relations on a set X Remove the dishes before discussing politics Walk to Üetliberg before lunch Take your medicine before lunch Finish lunch before removing dishes The set of interest: Tasks = {Politics, Lunch, Medicine, ishes, Üetliberg} universal [X ]: holds between any two elements of X id [X ]: holds between every element of X and itself empty [X ]: holds between no elements of X The constraining relation: ishes before Politics Üetliberg before Lunch Medicine before Lunch Lunch before ishes Intro. to Programming, lecture 9: Topological Sort I 7 Intro. to Programming, lecture 9: Topological Sort I 8 3

4 Relations: a more precise mathematical view We consider a relation r on a set P as: A set of pairs in P x P, containing all the pairs [x, y] such that x r y. Then x r y simply means that [x, y] r See examples on next slide Example: the before relation Remove dishes before discussing politics Walk to Üetliberg before lunch Take your medicine before lunch Finish lunch before removing dishes The set of interest: elements = {Politics, Lunch, Medicine, ishes, Üetliberg} The constraining relation: before = {[ishes, Politics], [Üetliberg, Lunch], [Medicine, Lunch], [Lunch, ishes]} Intro. to Programming, lecture 9: Topological Sort I 9 Intro. to Programming, lecture 9: Topological Sort I 0 Using ordinary set operators spouse = wife husband sibling = sister brother id [Person] sister sibling father ancestor universal [X ] = X x X (cartesian product) empty [X ] = Possible properties of a relation (On a set X. All definitions must hold for every a, b, c... X.) Total*: (a b) ((a r b) (b r a)) Reflexive: a r a Asymmetric: not ((a r b) (b r a)) Transitive: (a r b) (b r c) a r c *efinition of total is specific to this discussion (there is no standard definition). The other terms are standard. Intro. to Programming, lecture 9: Topological Sort I Intro. to Programming, lecture 9: Topological Sort I Examples (on a set of persons) Total order relation (strict) sibling sister family_head mother Reflexive, symmetric, transitive Symmetric, irreflexive Reflexive, antisymmetric (a family_head b means a is the head of b s family, with one head per family) Asymmetric, irreflexive Total: (a b) (a r b) (b r a) Reflexive: a r a Asymmetric: not ((a r b) (b r a)) Transitive: (a r b) (b r c) a r c Intro. to Programming, lecture 9: Topological Sort I 3 Relation is strict total order if: Total Irreflexive Transitive Asymmetric: not ((a r b) (b r a)) Transitive: (a r b) (b r c) a r c Example: less than < on natural numbers < 0 <, 0 < 3, 0 < 4,... < < 3 < 4,... < 3 < 4, Intro. to Programming, lecture 9: Topological Sort I 4 4

5 Theorem Total order relation (strict) A strict (total) order is asymmetric Relation is strict total order if: Total Irreflexive Transitive Asymmetric: not ((a r b) (b r a)) Transitive: (a r b) (b r c) a r c Theorem: A strict total order is asymmetric Intro. to Programming, lecture 9: Topological Sort I 5 Intro. to Programming, lecture 9: Topological Sort I 6 Total order relation (non-strict) Total order relation (strict) Relation is non-strict total order if: Total Reflexive Transitive Antisymmetric Transitive:(a r b) (b r c) a r c Relation is strict total order if: Total Irreflexive Transitive Transitive:(a r b) (b r c) a r c Example: less than or equal on natural numbers , 0, 0 3, 0 4,... 0, 0 3, 0 4,..., 3, 4,... 3, 4, Intro. to Programming, lecture 9: Topological Sort I 7 Intro. to Programming, lecture 9: Topological Sort I 8 Partial order relation (strict) Theorems Relation is strict partial order if: Total Irreflexive Transitive Transitive:(a r b) (b r c) a r c A strict (total) partial order is asymmetric y 0 [0, ] a [, ] b 3 [3, 0] c [4, ] d x Example: relation between points in a plane: p < q if both x p < x q y p < y q A total order is a partial order ( partial order really means possibly partial) Intro. to Programming, lecture 9: Topological Sort I 9 Intro. to Programming, lecture 9: Topological Sort I 30 5

6 y Example partial order Possible topological sorts y [0, ] a [, ] b [4, ] d p < q if both x p < x q y p < y q a b d 0 3 [3, 0] c 0 3 c Here the following hold: No link between a and c, b and c: a < b a < d c < d e.g. neither a < c nor c < a a < b a < d c < d Intro. to Programming, lecture 9: Topological Sort I 3 Intro. to Programming, lecture 9: Topological Sort I 3 Topological sort understood 0 y a b a < b a < d 3 c c < d d Here the relation < is: {[a, b], [a, d], [c, d]} One of the solutions is: a, b, c, d We are looking for a total order relation t such that < t Final statement of topological sort problem From a given partial order, produce a compatible total order where: A partial order p is compatible with a total order t if and only if p t Intro. to Programming, lecture 9: Topological Sort I 33 Intro. to Programming, lecture 9: Topological Sort I 34 From constraints to partial orders Powers and transitive closure of a relation Is a relation defined by a set of constraints, such as constraints = {[ishes, Politics], [Üetliberg, Lunch], [Medicine, Lunch], [Lunch, ishes]} always a partial order? r i + = r i ; r where ; is composition Ü L M P Üetliberg Medicine Lunch ishes Politics Transitive closure r + = r r... always transitive r r r 3 Intro. to Programming, lecture 9: Topological Sort I 35 Intro. to Programming, lecture 9: Topological Sort I 36 6

7 Reflexive transitive closure r 0 = id [X ] where X is the underlying set r i + = r i ; r where ; is composition Acyclic relation A relation r on a set X is acyclic if and only if: r + id [X ] = Ü M L P Ü M L P Transitive closure r + = r r... always transitive Reflexive transitive closure: r = r 0 r r... always transitive and reflexive r 0 r r r 3 Intro. to Programming, lecture 9: Topological Sort I 37 before + id [X] Intro. to Programming, lecture 9: Topological Sort I 38 Acyclic relations and partial orders Theorems: Any (strict) order relation is acyclic. A relation is acyclic if and only if its transitive closure is a (strict) order. From constraints to partial orders The partial order of interest is before + before = {[ishes, Politics], [Üetliberg, Lunch], [Medicine, Lunch], [Lunch, ishes]} (Also: if and only if its reflexive transitive closure is a nonstrict partial order) Üetliberg Medicine Lunch ishes Politics Intro. to Programming, lecture 9: Topological Sort I 39 Intro. to Programming, lecture 9: Topological Sort I 40 Back to software... The basic algorithm idea p q r t s v u w topsort Intro. to Programming, lecture 9: Topological Sort I 4 Intro. to Programming, lecture 9: Topological Sort I 4 7

8 What we have seen The topological sort problem and its applications Mathematical background: Relations as sets of pairs Properties of relations Order relations: partial/total, strict/nonstrict Transitive, reflexive-transitive closures The relation between acyclic and order relations The basic idea of topological sort Reading assignment for next Monday Touch of Class, chapter on topological sort: 7 Next: how to do it for Efficient operation (O (m+n) for m constraints & n items) Good software engineering: effective API Intro. to Programming, lecture 9: Topological Sort I 43 Intro. to Programming, lecture 9: Topological Sort I 44 End of lecture 9 Intro. to Programming, lecture 9: Topological Sort I 45 8