An Algorithm for: Explaining Algorithms

1. An Algorithm for:Explaining Algorithms Tomasz Müldner September 12

2. Vision = what is where by looking Visualization = the power or process of forming a mental image of vision of something not actually present to the sight You have 10s to find this image September 12

3. Dijkstra feared… “…permanent mental damage for most students exposed to program visualization software …” September 12

4. Contents • Preface • Introduction to Algorithm Visualization, AV • Examples of AV • Algorithm Explanation, AE • Examples of AE • Conclusions & Future Work September 12

5. Preface • Under Construction • Early version • Invitation to collaborate September 12

6. Al-Khorezmi -> Algorithm The ninth century: • the chief mathematician in the academy of sciences in Baghdad September 12

7. Introduction to AV AV uses multimedia: • Graphics • Animation • Auralization to show abstractions of data September 12

8. Examples of AV • Multiple Sorting • Duke • More • AIA September 12

9. Typical Approach in AV • take the description of the algorithm • graphically represent data in the code using bars, points, etc. • use animation to represent the flow of control • show the animated algorithm and hope that the learner will now understand the algorithm September 12

10. Problems with AV • Graphical language versus text • Low level of abstraction (code stepping) • Emphasis on meta-tools • Students perform best if they are asked to develop visualizations • no attempt to visualize or even suggest essential properties, such as invariants • Very few attempts to visualize recursive algorithms September 12

11. Introduction to AE • systematic procedure to explain algorithms: an algorithm for explaining algorithms • Based on findings from Cognitive Psychology, Constructivism Theory, Software Engineering • visual representation is used to help reason about the textual representation • Use multiple abstraction levels to focus on selected issues • Designed by experts September 12

12. Goals of AE • Understanding of both, what the algorithm is doing and how it works • Ability to justify the algorithm correctness (why the algorithm works) • Ability to code the algorithm in any programming language • Understanding of time complexity of the algorithm September 12

13. Requirements for AE • The algorithm is presented at several levels of abstraction • Each level of abstraction is represented by the abstract data model and pseudocode • The design supports active learning • The design helps to understand time complexity September 12

14. Levels of Abstraction public static void selection(List aList) { for (int i = 0; i < aList.size(); ++i) swap(smallest(i, aList) , i, aList); } Primitive operations can be: • Explained at different abstraction level • inlined September 12

15. AE Catalogue Entries • Multi-leveled Abstract Algorithm Model • Example of an abstract implementation of the Abstract Algorithm Model • Tools that can be used to help to predict the algorithm complexity • Questions for students September 12

16. MAK • Uses multimedia: • Graphics • Animation • Auralization to show abstractions of data • Interacts with the student; e.g. by providing post-tests • Uses a student model for adaptive behavior September 12

17. MAK Selection Sort Insertion Sort Quick Sort September 12

18. Selection Sort: Abstract Data Model • Sequences of elements of type T, denoted by Seq<T> with a linear order defined in one of three ways: • type T supports the function int compare(const T x) • type T supports the “<” relation • there is a global function int comparator(const T x, const T y) September 12

19. Selection Sort: Top level of Abstraction • Abstract Data Model • Type T also supports the function • swap(T el1, T el2) • The following operations on Seq<T> are available: • a sequence t can be divided into prefix and suffix • the prefix can be incremented (which will decrement the suffix) • first(suffix) • T smallest(seq<T> t, Comparator comp) September 12

20. Selection Sort: Top level of Abstraction Pseudocode void selection(Seq<T> t, Comparator comp) { for(prefix = NULL; prefix != t; increment prefix by one element) swap( smallest(suffix, comp), first(suffix) ); } September 12

21. Visualization void selection(Seq<T> t, Comparator comp) { for(prefix = NULL; prefix != t; increment prefix by one element) swap( smallest(suffix, comp), first(suffix) ); } List two invariants September 12

22. void selection(Seq<T> t, Comparator comp) { for(prefix = NULL; prefix != t; increment prefix by one element) swap( smallest(suffix, comp), first(suffix) ); } List two invariants September 12

23. Selection Sort: Low level of Abstraction Pseudocode T smallest(Seq<T> t, Comparator comp) { smallest = first element of t; for(traverse t forward) if(smallest & current are out of order) smallest = current; } September 12

24. Visualization T smallest(Seq<T> t, Comparator comp) { smallest = first element of t; for(traverse t forward) if(smallest & current out of order) smallest = current; } September 12

25. Abstract Implementation The Abstract Iterator Implementation Model assumes • There is an Iterator type, where iterations are performed over a half-closed interval [a, b) • Iterator type supports the following operations: • two iterators can be compared for equality and inequality • there are operations to provide various kinds of traversals; for example forward and backward • an iterator can be dereferenced to access the object it points to September 12

26. Abstract Implementation The Abstract Iterator Implementation Model assumes (Cont.): • The domain Seq<T> supports Seq<T>::Iterator • The following two operations are defined on sequences: • Iterator t.begin() • Iterator t.end() September 12

27. Selection Sort: Abstract Implementation Pseudocode void selection(Seq<T> t, Comparator comp) { Seq<T>::Iterator eop; // end of prefix for(eop = t.begin(); eop != t.end(); ++eop) swap(smallest(eop, t.end(), comp), eop); } T smallest(Seq<T> t, Comparator comp) { Iterator small = t.begin(); Iterator current; for(current = t.begin(); current != t.end(); ++current) if(value of current < value of small) small = current; return value of small; } September 12

28. void selection(T *x, T* const end, int comparator(const T, const T)) { T* eop; for(eop = x; eop != end; ++eop) swap( smallest(eop, end, comparator), eop ); } T *smallest(T * const first, T * const last, int comparator(const T, const T) ) { T *small = first; T *current; for(current = first; current != last; ++ current) if(comparator(* current, *small) < 0) small = current; return s; } void selection(Seq<T> t, Comparator comp) { Seq<T>::Iterator eop; // end of prefix for(eop = t.begin(); eop != t.end(); ++eop) swap(smallest(eop, t.end(), comp), eop); } T smallest(Seq<T> t, Comparator comp) { Iterator small = t.begin(); Iterator current; for(current = t.begin(); current != t.end(); ++current) if(value of current < value of small) small = current; return value of small; } C implementation September 12

29. typedef struct { int a; int b; } T; #define SIZE(x) (sizeof(x)/sizeof(T)) T x[ ] = { {1, 2}, {3, 7}, {2, 4}, {11, 22} }; #define S SIZE(x) int comparator1(const T x, const T y) { if(x.a == y.a) return 0; if(x.a < y.a) return -1; return 1; } int comparator2(const T x, const T y) { if(x.a == y.a) if(x.b = y.b) return 0; else if(x.b < y.b) return -1; else return 1; if(x.a < y.a) return -1; return 1; } int main() { printf("original sequence\n"); show(x, S); selection(x, x+S, comparator1); printf("sequence after first sort\n"); show(x, S); selection(x, x+S, comparator2); printf("sequence after second sort\n"); show(x, S); } September 12

30. template <typename Iterator, typename Predicate> void selection(Iterator first, Iterator last, Predicate compare) { Iterator eop; for(eop = first; eop != last; ++eop) swap(*min_element(eop, last, compare), *eop); } void selection(Seq<T> t, Comparator comp) { Seq<T>::Iterator eop; // end of prefix for(eop = t.begin(); eop != t.end(); ++eop) swap(smallest(eop, t.end(), comp), eop); } T smallest(Seq<T> t, Comparator comp) { Iterator small = t.begin(); Iterator current; for(current = t.begin(); current != t.end(); ++current) if(value of current < value of small) small = current; return value of small; } C++ implementation September 12

31. public static void selection(List aList, Comparator aComparator) { for (int i = 0; i < aList.size(); i++) swap(smallest(i, aList, aComparator) , i, aList); } private static int smallest(int from, List aList, Comparator aComp) { int minPos = from; int count = from; for (ListIterator i = aList.listIterator(from); i.hasNext(); ++count) if (aComp.compare(i.next(), aList.get(minPos)) < 0) minPos = count; return minPos; } void selection(Seq<T> t, Comparator comp) { Seq<T>::Iterator eop; // end of prefix for(eop = t.begin(); eop != t.end(); ++eop) swap(smallest(eop, t.end(), comp), eop); } T smallest(Seq<T> t, Comparator comp) { Iterator small = t.begin(); Iterator current; for(current = t.begin(); current != t.end(); ++current) if(value of current < value of small) small = current; return value of small; } Java implementation September 12

32. Algorithm Complexity Three kinds of tools: • to experiment with various data sizes and plot a function that approximates the time spent on execution with this data. • a visualization that helps to carry out time analysis of the algorithm • questions regarding the time complexity September 12

33. Post Test • What is the number of comparisons and swaps performed when selection sort is executed for: • sorted sequence • sequence sorted in reverse • What is the time complexity of the function isSorted(t), which checks if t is a sorted sequence? • Hand-execute the algorithm for a sample set of input data of size 4. • Hand-execute the next step of the algorithm for the specified state • What’s the last step of the algorithm? • There are two invariants of this algorithm; which one is essential for the correctness of swap(smallest(), eop), and why. • “do it yourself “ September 12

34. Quick Sort Pseudocode void quick(Seq<T> t, Comparator comp) { if( size(t) <= 1) return; pivot = choosePivot(t); divide(pivot, t, t1, t2, t3, comp); quick(t1, comp); quick(t3, comp); concatenate(t, t1, t2, t3); } September 12

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38. Future Work • evaluation (eye movement?) • student model • different visualizations • more complex algorithms • Algorithmic design patterns • generic approach • MAK September 12