A core Course on Modeling

1. A core Course on Modeling Introductionto Modeling 0LAB0 0LBB0 0LCB0 0LDB0 c.w.a.m.v.overveld@tue.nl v.a.j.borghuis@tue.nl P.15

2. Criteria for modeling: scalability Scalability: towhichextentcansomecharacteristicdimensions of the problemincrease, where the model stillfunctions? Scale: n (=number of elements, states, records, time steps, …) Performance: • O(1) – solving effort (SE) does not depend on n • O(log n) – SE hardlydepends on n • O(n) – SE proportionalto n • O(n log n) – SE slightlyworsethanproportional • O(np) – SE polynomial • O(2n), O(n!), … – SE worsethanpolynomial

3. Criteria for modeling: scalability Example: tofindsomething in a database usingindexingtakes no more time ifther are more stored records. Comparethistousingan index in a thickbook: no needto browse over the pages. • O(1) – solving effort (SE) independent of n • O(log n) – SE hardlydepends on n • O(n) – SE proportionalto n • O(n log n) – SE slightlyworsethanproportional • O(np) – SE polynomial • O(2n), O(n!), … – SE worsethanpolynomial O(1) n

4. Criteria for modeling: scalability Example: tofindsomething in anordered, non-indexed set takes placebyrepeatedlyhalving (cf. guessing a numberbetween 0 and 1024: this takes no more than 10 higher-lowerguesses). WithN trials you search a set of size 2N, so a set of size M is searched in log M trials. • O(1) – solving effort (SE) independent of n • O(log n) – SE hardlydepends on n • O(n) – SE proportionalto n • O(n log n) – SE slightlyworsethanproportional • O(np) – SE polynomial • O(2n), O(n!), … – SE worsethanpolynomial O(log n) n

5. Criteria for modeling: scalability Manyproblemsrequirethatsomething (with a fixedamount of effort) needs do bedoneforeach item in a set. In such cases, the total effort growsproportially with the size of the set. • O(1) – solving effort (SE) independent of n • O(log n) – SE hardlydepends on n • O(n) – SE proportionalto n • O(n log n) – SE slightlyworsethanproportional • O(np) – SE polynomial • O(2n), O(n!), … – SE worsethanpolynomial O(n) n

6. Criteria for modeling: scalability Sorted sets allowthingstobedone (e.g., searching) muchfaster. Soifyoufrequentlyneedto search some set, itmaypay off tosortitonceand keep itsorted. Sorting a set of n elementscanbedone in O(n log n) steps. • O(1) – solving effort (SE) independent of n • O(log n) – SE hardlydepends on n • O(n) – SE proportionalto n • O(n log n) – SE slightlyworsethanproportional • O(np) – SE polynomial • O(2n), O(n!), … – SE worsethanpolynomial O(n logn) n

7. Criteria for modeling: scalability The dot product of twovectors takes O(n) in the dimension n. Multiplying a vector with a matrix O(n2). Multiplyingtwo matrices O(n3). Solving a linear set of equations O(n3). Many of these operations allowalternative approaches to win some efficiency (http://en.wikipedia.org/wiki/Computational_complexity_of_mathematical_operations). • O(1) – solving effort (SE) independent of n • O(log n) – SE hardlydepends on n • O(n) – SE proportionalto n • O(n log n) – SE slightlyworsethanproportional • O(np) – SE polynomial • O(2n), O(n!), … – SE worsethanpolynomial O(np) n

8. Criteria for modeling: scalability • O(1) – solving effort (SE) independenten of n • O(log n) – SE hardlydepends on n • O(n) – SE proportionalto n • O(n log n) – SE slightlyworsethanproportional • O(np) – SE polynomial • O(2n), O(n!), … – SE worsethanpolynomial Given n points; askedfor the shortest route visitingthemall (traveling salesman problem, TSP). Manyproblemscanbetransformedtothis type. O(1) n