overview overview who practicalities introduction data structures and algorithms lecture 1 sorting insertion sort pseudo code merge sort

Transcription

1 overview data structures and algorithms lecture 1 overview who lectures: Femke van Raamsdonk f.van.raamsdonk at vu.nl T446 exercise classes: Paul Ursulean Petar Vukmirovic

2 when and where tests in week 36 42: lectures: Mondays in KC159 Thursdays in HG10A00 two groups for the exercise classes: Tuesdays in M655 (group 1) and in M639 (group 2) Fridays in different rooms written exam (closed book) in week 8 of the course there is a resit in January mid-term exam in week 4 of the course recommended but not obligatory if mid-term exam better than exam, then the mid-term mark contributes for 30% to the exam-mark practical work recommended but not obligatory for bonus more news end of the week material Introduction to Algorithms by Cormen, Leiserson, Rivest, Stein overview

3 data structures and algorithms: context what this course is about some problems cannot be solved some problems cannot be solved efficiently some problems can be solved efficiently for some problems we do not know whether they can be (efficiently) solved if P NP then the NP-complete problems cannot be efficiently solved we will study basic data structures and analysis and design of algorithms prerequisite: elementary programming but this is not a programming course prerequisite: elementary (discrete) mathematics and graph theory but this is not a pure theory course we study: algorithmic design, data structures, efficiency of algorithms example algorithm: baking a cake example algorithm: Euclid s gcd software / data structures : tools input : ingredients program / algorithm : recipee hardware: oven compute the greatest common divisor of two non-negative numbers a b: if b = 0 then return a if b 0, then compute the gcd of b and (a mod b) the second line contains a recursive call output: cake

4 algorithm complexity an algorithm is a list of instructions, the essence of a program what are important aspects? correctness does the algorithm meet the requirements? termination does the algorithm eventually produce an output? efficiency or complexity how much time and memory space does it use? algorithms that do the same may differ in performance time complexity: how much time does the algorithm use? time as function of the input space complexity: how much space does the algorithm use? space as function of the input we care about time complexity hence we care about steps or elementary operations example: sort a finite sequence of length n of numbers assumption: our computer performs 10 9 operations per second : uses say 2 n 2 steps : uses say 50 n log n steps a sequence of length n = 10 7 takes for : seconds (55 hours) for : about 12 seconds assumption: our computer performs 10 9 operations per second number of operations time s s s s min h47min yr

5 overview : specification input: a finite sequence of elements output: an ordered or sorted permutation of the input-sequence what do we sort? sorted: definitions elements are usually integers or natural numbers an element may occur more than once the input-sequence is assumed to be an array (an array starts at index 1) an ordering is a binary relation such that n n reflexivity if m n and n p then m p transitivity if m n and n m then m = n anti-symmetry an ordering is total if every pair of elements can be compared a sequence a 1 a 2... a n is ordered if it is non-decreasing that is, a 1 a 2... a n usually we consider natural numbers or integers with

6 overview : idea and example the sequence consists of a sorted part followed by a non-sorted part initially: the sorted part consists only of the first element loop: while the non-sorted part is non-empty insert the first element of the non-sorted part in the correct position of the sorted part [5, 3, 4, 7, 1] [3, 5, 4, 7, 1] [3, 4, 5, 7, 1] [3, 4, 5, 7, 1] [1, 3, 4, 5, 7] : pseudo-code : correctness Algorithm insertionsort(a, n): for j := 2 to n do key := A[j] i := j 1 while i 1 and A[i] > key do A[i + 1] := A[i] i := i 1 A[i + 1] := key loop invariant I : at the start of the for-loop, the subarray A[1... j 1] is a sorted permutation of the sub-array A[1... j 1] of the input-array init: I is true initially, that is: for j = 2 loop: I remains true during the loop (!) end: after termination, that is: for j = n + 1, I gives correctness

7 : worst-case time complexity overview running time depends on the input: best case if input is sorted, worst case if input is inverse sorted running time depends on the size of the input: we want to express running time as function T of size of the input which running time? best case is useless; average case is interesting; worst case gives guarantee worst-case time complexity is quadratic in the size of the input (more details in the next lecture) pseudo-code pseudo-code: globally input we write algorithms in pseudo-code pseudo-code resembles a programming language but is independent of specific syntax output using return block structure via indentation declare and call procedures declare and use data structures recursive calls objects with attributes, for example A.length

8 pseudo-code: calculating pseudo-code: control booleans: true, false calculating with booleans: integers and, or (short-circuiting) declare and use variables assignment declare and update arrays and array elements calculating with integers: addition, subtraction, multiplication, modulo if then, while do, for do, repeat, elementary tests on integers: greater than, less than overview : idea how do we sort? loop: while the sequence contains more than 1 element split the sequence into two parts of (almost) equal length sort the two subsequences using mergesort (recursion) merge the two sorted sublists how do we merge? loop: as long as both sequences are non-empty compare the first elements of both sequences remove the smallest and move it to the end of the output-sequence last: if one of the sequences is empty, add the other to the tail of the output-sequence

9 : example mergesort: some pseudo-code each node represents a recursive call Algorithm mergesort(a, p, r): if p < r then q := (p + r)/2 mergesort(a, p, q) mergesort(a, q + 1, r) Merge(A, p, q, r) where is the work done? for pseudo-code Merge see the book Merge: example question input: A = [1, 3, 5, 7, 2, 4, 6, 8] p = 1 q = 4 give the recursion tree of applying mergesort to the sequence r = 8 A[1... 4] is sorted and A[5... 8] is sorted

10 : time complexity mergesort: inventor mergesort was invented by John von Neumann ( ) in 1945 is more efficient (in time) than its worst-case time complexity is on Θ(n log n) (more details in the next lecture) material for this lecture: Book Chapters 1 and 2 addtional reading: book on algorithms by David Harel additional reading: art of computer programming by Donald Knuth