Total Orders. Lecture 41 Section 8.5. Robb T. Koether. Hampden-Sydney College. Mon, Apr 8, 2013
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1 Total Orders Lecture 41 Section 8.5 Robb T. Koether Hampden-Sydney College Mon, Apr 8, 2013 Robb T. Koether (Hampden-Sydney College) Total Orders Mon, Apr 8, / 30
2 1 Total Orders 2 Topological Sorting 3 PERT and CPM 4 Assignment Robb T. Koether (Hampden-Sydney College) Total Orders Mon, Apr 8, / 30
3 Outline 1 Total Orders 2 Topological Sorting 3 PERT and CPM 4 Assignment Robb T. Koether (Hampden-Sydney College) Total Orders Mon, Apr 8, / 30
4 Comparable Elements Definition (Comparable elements) Let R be a relation on a set A and let a, b A. Then a and b are comparable if (a, b) R or (b, a) R. In the following examples, which elements are comparable? The relation on {1, 2, 3, 4, 5}. The relation divides on {1, 2, 3, 4, 6, 8, 12, 24}. The relation on P({a, b, c}). Robb T. Koether (Hampden-Sydney College) Total Orders Mon, Apr 8, / 30
5 Total Orders Definition (Comparable elements) Let R be a relation on a set A. Then R is a total order on A if R is a partial order on A and for all a, b A, a and b are comparable. Which of the previous examples are total orders? Robb T. Koether (Hampden-Sydney College) Total Orders Mon, Apr 8, / 30
6 Sorting When designing a C++ class, we define a function lessthan(). We use lessthan() to define the operator <. We use the operator < to define the operators <=, >, and >=. We use these operators to sort an array in a sort() function. What properties must lessthan() have for the sort() function to work? Robb T. Koether (Hampden-Sydney College) Total Orders Mon, Apr 8, / 30
7 Sorting For sort() to work properly, the operator <= (and the operator >=) must be a total order relation. In the Point class of two-dimensional points (x, y), define (x, y) < (z, w) to mean that x < z and y < w. Apply a bubble sort algorithm to the list {(1, 0), (0, 2), (0, 0), (0, 2), (1, 1), (1, 0)}. What happens if we replace and with or? What would be an acceptable choice for lessthan() in the Point class? Robb T. Koether (Hampden-Sydney College) Total Orders Mon, Apr 8, / 30
8 Chains Definition Let A be a set with a partial order relation. A set of elements in A is called a chain if every pair of elements in the set are comparable. The length of the chain is one less than the number of elements in the chain. It is clear that is a total order on the chain. Robb T. Koether (Hampden-Sydney College) Total Orders Mon, Apr 8, / 30
9 Example Let be the relation divides on {1, 2, 3, 4, 6, 8, 12, 24}. Give examples of three distinct chains. Robb T. Koether (Hampden-Sydney College) Total Orders Mon, Apr 8, / 30
10 Maximal and Minimal Elements Definition Let A be a set with partial order. An element a A is a maximal element of A if for all b A, either b a or a and b are not comparable. An element a A is a minimal element of A if for all b A, either a b or a and b are not comparable. An element a A is a greatest element of A if for all b A, b a. An element a A is a least element of A if for all b A, a b. A set may have many maximal elements, but only one greatest element. Similarly for minimal and least. Robb T. Koether (Hampden-Sydney College) Total Orders Mon, Apr 8, / 30
11 Examples Consider the relation on the set {{a}, {b}, {c}, {a, b}, {a, c}, {b, c}}. This set has three maximal elements and three minimal elements, but no greatest or least element. Consider the relation on the set P({a, b, c}). This set has a greatest element {a, b, c} and a least element. Robb T. Koether (Hampden-Sydney College) Total Orders Mon, Apr 8, / 30
12 Outline 1 Total Orders 2 Topological Sorting 3 PERT and CPM 4 Assignment Robb T. Koether (Hampden-Sydney College) Total Orders Mon, Apr 8, / 30
13 The Painter s Algorithm In Computer Graphics, when drawing objects on the screen, a problem is to determine which objects obscure which objects, from the point of view of the viewer (which may change). An early algorithm was the Painter s Algorithm: Draw the objects, beginning with the most distant and working back to the nearest. In other words, sort the objects by their distance from the viewer. Robb T. Koether (Hampden-Sydney College) Total Orders Mon, Apr 8, / 30
14 The Painter s Algorithm Eye This presented a serious difficulty. The objects are not points, so how do we determine their distance from the viewer? Robb T. Koether (Hampden-Sydney College) Total Orders Mon, Apr 8, / 30
15 The Painter s Algorithm We could define an order relation on the objects: a b means that b obscures part of a or a = b. However, this does not satisfy antisymmetry or transitivity. Give counterexamples to antisymmetry and transitivity. Thus, a sort() function will not work. What if we defined a b to mean that b completely obscures a or a = b? Robb T. Koether (Hampden-Sydney College) Total Orders Mon, Apr 8, / 30
16 Topological Sorting Definition Let and be two partial orders on a set A. Then is compatible with if, for all a, b A, a b implies a b. The relation is a topological sorting for if is a total order that is compatible with. Robb T. Koether (Hampden-Sydney College) Total Orders Mon, Apr 8, / 30
17 Examples Let be the divides relation on {1, 2, 3, 4, 6, 8, 12, 24}. One possible topological sorting for is, because a b means a b and if a b, then a b. Let be the relation on P({a, b, c}). One possible topological sorting for is defined by {a} {b} {c} {a, b} {a, c} {b, c} {a, b, c}. This is the lexicographical order. It is also an order determined by cardinality. Robb T. Koether (Hampden-Sydney College) Total Orders Mon, Apr 8, / 30
18 Topological Sorting Algorithm Given a partial order, the following algorithm will produce a topological sorting for. Let be a partial order on a nonempty finite set A. Choose any minimal element m A. A A {m}. While A, Choose any minimal element m in A. Define m m. A A {m } m m Robb T. Koether (Hampden-Sydney College) Total Orders Mon, Apr 8, / 30
19 Example Let A = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} and let be the divides relation. Robb T. Koether (Hampden-Sydney College) Total Orders Mon, Apr 8, / 30
20 Example is the least element Robb T. Koether (Hampden-Sydney College) Total Orders Mon, Apr 8, / 30
21 Example Remove 1 Robb T. Koether (Hampden-Sydney College) Total Orders Mon, Apr 8, / 30
22 Example is a minimal element Robb T. Koether (Hampden-Sydney College) Total Orders Mon, Apr 8, / 30
23 Example Remove 7 Robb T. Koether (Hampden-Sydney College) Total Orders Mon, Apr 8, / 30
24 Example is a minimal element Robb T. Koether (Hampden-Sydney College) Total Orders Mon, Apr 8, / 30
25 Example Remove 2 Robb T. Koether (Hampden-Sydney College) Total Orders Mon, Apr 8, / 30
26 Example is a minimal element Robb T. Koether (Hampden-Sydney College) Total Orders Mon, Apr 8, / 30
27 Example Remove 5 Robb T. Koether (Hampden-Sydney College) Total Orders Mon, Apr 8, / 30
28 Example is a minimal element Robb T. Koether (Hampden-Sydney College) Total Orders Mon, Apr 8, / 30
29 Example Remove 3 Robb T. Koether (Hampden-Sydney College) Total Orders Mon, Apr 8, / 30
30 Example is a minimal element Robb T. Koether (Hampden-Sydney College) Total Orders Mon, Apr 8, / 30
31 Example Remove 4 Robb T. Koether (Hampden-Sydney College) Total Orders Mon, Apr 8, / 30
32 Example Thus, one possible topological sorting is Robb T. Koether (Hampden-Sydney College) Total Orders Mon, Apr 8, / 30
33 Outline 1 Total Orders 2 Topological Sorting 3 PERT and CPM 4 Assignment Robb T. Koether (Hampden-Sydney College) Total Orders Mon, Apr 8, / 30
34 PERT and CPM PERT = Program Evaluation and Review Technique. CPM = Critical Path Method, also called Critical Path Analysis. A project is broken down into its component tasks. Some component tasks cannot be begun until other are finished. Some component tasks take longer than others. Robb T. Koether (Hampden-Sydney College) Total Orders Mon, Apr 8, / 30
35 PERT and CPM In what sequence should the subtasks be done in order to minimize the total time for the project? Define the relation x y to mean either x = y or x must be completed before y can be begun. Show that is a partial order relation. Robb T. Koether (Hampden-Sydney College) Total Orders Mon, Apr 8, / 30
36 Example For example, suppose that the tasks are Write the Employee class, Test the Employee class, Write the HourlyEmployee class, Test the HourlyEmployee class, Write the application program, and Test the application program. Robb T. Koether (Hampden-Sydney College) Total Orders Mon, Apr 8, / 30
37 Example Write Employee Class Test Employee Class Write Hourly Employee Class Test Hourly Employee Class Test the application program Write the application program Robb T. Koether (Hampden-Sydney College) Total Orders Mon, Apr 8, / 30
38 Example Consider a more complex project with the following components and requirements. Immediately Time Needed to Task Preceding Task Perform Task 1 7 hours hours hours hours 5 2, 3 3 hours hours 7 2, 3 1 hours 8 4, 5 2 hours 9 6, 7, 8 5 hours Robb T. Koether (Hampden-Sydney College) Total Orders Mon, Apr 8, / 30
39 Example Task 4 Task 2 Task 6 Task 1 Task 5 Task 9 Task 3 Task 7 Task 8 How long will it take to complete the project? Robb T. Koether (Hampden-Sydney College) Total Orders Mon, Apr 8, / 30
40 Example 6 Hours 6 Hours 1 Hour 7 Hours 3 Hours 5 Hours 3 Hours 1 Hour 2 Hours How long will it take to complete the project? Robb T. Koether (Hampden-Sydney College) Total Orders Mon, Apr 8, / 30
41 Outline 1 Total Orders 2 Topological Sorting 3 PERT and CPM 4 Assignment Robb T. Koether (Hampden-Sydney College) Total Orders Mon, Apr 8, / 30
42 Assignment Assignment Read Sections 8.5, pages Exercises 21, 23, 27, 28, 45, 47, 50, 51, page 513. Robb T. Koether (Hampden-Sydney College) Total Orders Mon, Apr 8, / 30
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