1. Spring 2015 Lecture 1: Introduction
2IL50 Data Structures
Spring Lecture 1: Introduction

2. Algorithms
Algorithm a well-defined computational procedure that takes some value, or a set of values, as input and produces some value, or a set of values, as output.
Algorithm sequence of computational steps that transform the input into the output.
Algorithms research design and analysis of algorithms and data structures for computational problems.

3. Data structures
Data Structure a way to store and organize data to facilitate access and modifications.
Abstract data type describes functionality (which operations are supported).
Implementation a way to realize the desired functionality
how is the data stored (array, linked list, …)
which algorithms implement the operations

4. The course
Design and analysis of efficient algorithms for some basic computational problems.
Basic algorithm design techniques and paradigms
Algorithms analysis: O-notation, recursions, …
Basic data structures
Basic graph algorithms

5. Some administration first
before we really get started …

6. Organization
Lecturer: Prof. Dr. Bettina Speckmann, MF 4.105,
I’m here every day but Friday …
Web page:
Book: T.H. Cormen, C.E. Leiserson, R.L. Rivest and C. Stein. Introduction to Algorithms (3rd edition) mandatory

7. Prerequisites
Being able to work with basic programming constructs such as linked lists, arrays, loops …
Being able to apply standard proving techniques such as proof by induction, proof by contradiction ...
Being familiar with sums and logarithms such as discussed in Chapter 3 and Appendix A of the textbook.
If you think you might lack any of this knowledge, please come and talk to me immediately!

8. Grading scheme 2IL50
6 homework assignments, the best 5 of which each count for 10% of the final grade.
A written exam (closed book) which counts for the remaining 50% of the final grade.
If you reach less than 50% of the possible points on the homework assignments, then you are not allowed to participate in the final exam nor in the second chance exam. You will fail the course and your next chance will be next year. Your grade will be the minimum of 5 and the grade you achieved. If you reach less than 50% of the points on the final exam, then you will fail the course, regardless of the points you collected with the homework assignments. However, you are allowed to participate in the second chance exam. The grade of the second chance exam replaces the grade for the first exam, that is, your homework assignments always count for 50% of your grade.
Do the homework assignments!

9. Homework Assignments Posted on web-page on Monday before lecture.
Due Sundays at 23:59 as .pdf in the electronic mailbox of your instructor.
Late assignments will not be accepted. Only 5 out of 6 assignments count, hence there are no exceptions.
Must be typeset – use Latex! See example file on web-page.
Name scheme: Ai-LastName.pdf. If your name is Anton van Gelderland and you submit the 1st assignment, then your file must be named A1-vanGelderland.pdf.
Use the tag [2IL50] in the subject line of your .
Any questions: Stop by my office whenever you want (except Fridays!) (send if you want to make sure that I have time)

10. Academic Dishonesty
Academic Dishonesty All class work has to be done independently. You are of course allowed to discuss the material presented in class, homework assignments, or general solution strategies with me or your classmates, but you have to formulate and write up your solutions by yourself. You must not copy from the internet, your friends, or other textbooks. Problem solving is an important component of this course so it is really in your best interest to try and solve all problems by yourself. If you represent other people's work as your own then that constitutes fraud and will be dealt with accordingly.

11. Organization Components: Lectures Monday 5+6 AUD 6 Wednesday 3+4 AUD 3
Tutorials Wednesday see web-page for rooms and instructors! The instructors will explain the solutions to the homework assign-ments of the previous week and answer any questions that arise.
Office hours Thursday 17:00-18:00 around MF 4.104b
Reading and writing mathematical proofs

12. How to read and write mathematical proofs
Lecturer: Dr. Kevin Verbeek, MF 4.106,
Web page: Part of Data Structures:
Book: Daniel Solow How to Read and Do Proofs (5th edition) not mandatory

14. Signing up for groups
Group registration is open from today at 18:00 until tomorrow at 18:00.

15. Some statistics …

16. Sorting
let’s get started …

17. The sorting problem Input: a sequence of n numbers ‹a1, a2, …, an›
Output: a permutation of the input such that ‹ai1 ≤ … ≤ ain›
The input is typically stored in arrays
Numbers ≈ Keys
Additional information (satellite data) may be stored with keys
We will study several solutions ≈ algorithms for this problem

18. Describing algorithms
A complete description of an algorithm consists of three parts:
the algorithm
(expressed in whatever way is clearest and most concise, can be English and / or pseudocode)
a proof of the algorithm’s correctness
a derivation of the algorithm’s running time

19. InsertionSort Like sorting a hand of playing cards:
start with empty left hand, cards on table
remove cards one by one, insert into correct position
to find position, compare to cards in hand from right to left
cards in hand are always sorted
InsertionSort is
a good algorithm to sort a small number of elements
an incremental algorithm
Incremental algorithms process the input elements one-by-one and maintain the solution for the elements processed so far.

20. Incremental algorithms
Incremental algorithms process the input elements one-by-one and maintain the solution for the elements processed so far.
In pseudocode:
IncAlg(A)
// incremental algorithm which computes the solution of a problem with input A = {x1,…,xn}
initialize: compute the solution for {x1}
for j = 2 to n
do compute the solution for {x1,…,xj} using the (already computed) solution for {x1,…,xj-1}
Check book for more pseudocode conventions
no “begin - end”, just indentation

21. InsertionSort InsertionSort(A)
// incremental algorithm that sorts array A[1..n] in non-decreasing order
initialize: sort A[1]
for j = 2 to A.length
do sort A[1..j] using the fact that A[1.. j-1] is already sorted

22. InsertionSort InsertionSort(A)
// incremental algorithm that sorts array A[1..n] in non-decreasing order
initialize: sort A[1]
for j = 2 to A.length
do key = A[j]
i = j -1
while i > 0 and A[i] > key
do A[i+1] = A[i]
i = i -1
A[i +1] = key
InsertionSort is an in place algorithm: the numbers are rearranged within the array with only constant extra space. …
j n

23. Correctness proof
Use a loop invariant to understand why an algorithm gives the correct answer.
Loop invariant (for InsertionSort) At the start of each iteration of the “outer” for loop (indexed by j) the subarray A[1..j-1] consists of the elements originally in A[1..j-1] but in sorted order.

24. Correctness proof
To proof correctness with a loop invariant we need to show three things:
Initialization Invariant is true prior to the first iteration of the loop.
Maintenance If the invariant is true before an iteration of the loop, it remains true before the next iteration.
Termination When the loop terminates, the invariant (usually along with the reason that the loop terminated) gives us a useful property that helps show that the algorithm is correct.

25. Correctness proof
InsertionSort(A)
initialize: sort A[1]
for j = 2 to A.length
do key = A[j]
i = j -1
while i > 0 and A[i] > key
do A[i+1] = A[i]
i = i -1
A[i +1] = key
Initialization Just before the first iteration, j = 2 ➨ A[1..j-1] = A[1], which is the element originally in A[1], and it is trivially sorted.
Loop invariant At the start of each iteration of the “outer” for loop (indexed by j) the subarray A[1..j-1] consists of the elements originally in A[1..j-1] but in sorted order.

26. Correctness proof
InsertionSort(A)
initialize: sort A[1]
for j = 2 to A.length
do key = A[j]
i = j -1
while i > 0 and A[i] > key
do A[i+1] = A[i]
i = i -1
A[i +1] = key
Maintenance Strictly speaking need to prove loop invariant for “inner” while loop. Instead, note that body of while loop moves A[j-1], A[j-2], A[j-3], and so on, by one position to the right until proper position of key is found (which has value of A[j]) ➨ invariant maintained.
Loop invariant At the start of each iteration of the “outer” for loop (indexed by j) the subarray A[1..j-1] consists of the elements originally in A[1..j-1] but in sorted order.

27. Correctness proof
InsertionSort(A)
initialize: sort A[1]
for j = 2 to A.length
do key = A[j]
i = j -1
while i > 0 and A[i] > key
do A[i+1] = A[i]
i = i -1
A[i +1] = key
Termination The outer for loop ends when j > n; this is when j = n+1 ➨ j-1 = n. Plug n for j-1 in the loop invariant ➨ the subarray A[1..n] consists of the elements originally in A[1..n] in sorted order.
Loop invariant At the start of each iteration of the “outer” for loop (indexed by j) the subarray A[1..j-1] consists of the elements originally in A[1..j-1] but in sorted order.

28. Another sorting algorithm
using a different paradigm …

29. MergeSort A divide-and-conquer sorting algorithm.
Divide-and-conquer break the problem into two or more subproblems, solve the subproblems recursively, and then combine these solutions to create a solution to the original problem.

30. Divide-and-conquer D&CAlg(A)
// divide-and-conquer algorithm that computes the solution of a problem with input A = {x1,…,xn}
if # elements of A is small enough (for example 1)
then compute Sol (the solution for A) brute-force
else
split A in, for example, 2 non-empty subsets A1 and A2
Sol1 = D&CAlg(A1)
Sol2 = D&CAlg(A2)
compute Sol (the solution for A) from Sol1 and Sol2
return Sol

31. MergeSort MergeSort(A)
// divide-and-conquer algorithm that sorts array A[1..n]
if A.length == 1
then compute Sol (the solution for A) brute-force
else
split A in 2 non-empty subsets A1 and A2
Sol1 = MergeSort(A1)
Sol2 = MergeSort(A2)
compute Sol (the solution for A) from Sol1 en Sol2

32. MergeSort MergeSort(A)
// divide-and-conquer algorithm that sorts array A[1..n]
if A.length == 1
then skip
else
n = A.length ; n1 = n/2 ; n2 = n/2 ;
copy A[1.. n1] to auxiliary array A1[1.. n1]
copy A[n1+1..n] to auxiliary array A2[1.. n2]
MergeSort(A1)
MergeSort(A2)
Merge(A, A1, A2)

33. MergeSort 3 14 1 28 17 8 21 7 4 35 1 3 4 7 8 14 17 21 28 35 3 14 1 28 17 8 21 7 4 35 1 3 14 17 28 4 7 8 21 35 3 14 1 28 17 3 14 1 17 28 3 14

34. MergeSort
Merging 1 3 14 17 28 A1 4 7 8 21 35 A2 A 1 3 4 7 8 14 17 21 28 35

35. MergeSort: correctness proof
Induction on n (# of input elements)
proof that the base case (n small) is solved correctly
proof that if all subproblems are solved correctly, then the complete problem is solved correctly

36. MergeSort: correctness proof
MergeSort(A)
if length[A] = 1
then skip
else
n = A.length ; n1 = n/2 ; n2 = n/2 ;
copy A[1.. n1] to auxiliary array A1[1.. n1]
copy A[n1+1..n] to auxiliary array A2[1.. n2]
MergeSort(A1)
MergeSort(A2)
Merge(A, A1, A2)
Proof (by induction on n)
Base case: n = 1, trivial ✔
Inductive step: assume n > 1. Note that n1 < n and n2 < n.
Inductive hypothesis ➨ arrays A1 and A2 are sorted correctly
Remains to show: Merge(A, A1, A2) correctly constructs a sorted array A out of the sorted arrays A1 and A2 … etc ■
Lemma MergeSort sorts the array A[1..n] correctly.

37. another divide-and-conquer sorting algorithm…
QuickSort
another divide-and-conquer sorting algorithm…

38. QuickSort QuickSort(A)
// divide-and-conquer algorithm that sorts array A[1..n]
if length[A] ≤ 1
then skip
else
pivot = A[1]
move all A[i] with A[i] < pivot into auxiliary array A1
move all A[i] with A[i] > pivot into auxiliary array A2
move all A[i] with A[i] = pivot into auxiliary array A3
QuickSort(A1)
QuickSort(A2)
A = “A1 followed by A3 followed by A2”

39. Analysis of algorithms
some informal thoughts – for now …

40. Analysis of algorithms
Can we say something about the running time of an algorithm without implementing and testing it?
InsertionSort(A)
initialize: sort A[1]
for j = 2 to A.length
do key = A[j]
i = j -1
while i > 0 and A[i] > key
do A[i+1] = A[i]
i = i -1
A[i +1] = key

41. Analysis of algorithms
Analyze the running time as a function of n (# of input elements)
best case
average case
worst case
elementary operations add, subtract, multiply, divide, load, store, copy, conditional and unconditional branch, return …
An algorithm has worst case running time T(n) if for any input of size n the maximal number of elementary operations executed is T(n).

42. Analysis of algorithms: example
n= n= n=1000
x 107
x 106
InsertionSort
6 x faster
1.35 x faster
MergeSort
5 x faster
InsertionSort: n2 + 7n – 2
MergeSort: n lg n + 50 n
n = 1,000, InsertionSort x 1013
MergeSort x x faster !

43. Analysis of algorithms
It is extremely important to have efficient algorithms for large inputs
The rate of growth (or order of growth) of the running time is far more important than constants
InsertionSort: Θ(n2)
MergeSort: Θ(n log n)

44. Θ-notation
Intuition: concentrate on the leading term, ignore constants
19 n n2 - 3n becomes Θ(n3)
2 n lg n n becomes
n - ¾ n √n becomes
(precise definition next lecture…)
Θ(n1.1)
---

45. Some rules and notation
log n denotes log2 n
We have for a, b, c > 0 :
logc (ab) =
logc a + logc b
b logc a
loga b =
logc b / logc a

46. Find the leading term lg35n vs. √n ?
logarithmic functions grow slower than polynomial functions
lga n grows slower than nb for all constants a > 0 and b > 0
n100 vs. 2 n ?
polynomial functions grow slower than exponential functions
na grows slower than bn for all constants a > 0 and b > 1

47. Announcements Today 7+8 Reading and writing mathematical proofs
Emergency Latex office hour in MF 11
Today 18:00
Group registration opens on OASE
Wednesday 1+2
tutorials, discuss solutions to homework A1 from (see web-page)