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2IL65 Algorithms Fall 2011 Lecture 1: Introduction 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. Data Structure a way to store and organize data to facilitate access and modifications. Algorithms research design and analysis of algorithms and data structures for computational problems. 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 Some administration first before we really get started … Organization Lecturer: Kevin Buchin, HG 7.32, k.a.buchin@tue.nl Web page: http://www.win.tue.nl/~kbuchin/ teaching/2IL65/ Book: T.H. Cormen, C.E. Leiserson, R.L. Rivest and C. Stein. Introduction to Algorithms (2nd or 3rd edition) mandatory 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 so we can get you caught up. Grading scheme 2IL65 1. 7 homework assignments, the best 5 of which each count for 10% of the final grade. 2. 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! Aim at at least 60% in the homework (45 pts on 5; 9 in average)!!! Homework Assignments Posted on web-page on Wednesdays before the lecture. Due Wednesday (one week later) before the lecture as paper copy. Late assignments will not be accepted. Only 5 out of 7 assignments count, hence there are no exceptions. Must be handed in by each student separately. Must be typeset in English. Any questions: Stop by my office (send email if you want to make sure that I have time) Homework Assignments 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. • You should formulate and write down your solution by yourself! • Handing in a rewording of the same solution is an obvious violation! Organization Components: 1. Lecture Wednesday 13:45 - 15:30 auditorium 16 2. “Big” tutorial Thursday 10:45 - 12:30 auditorium 12 During this tutorial you have the opportunity to work on this week's home work assignment. I will be present to answer questions. 3. “Small” tutorial Friday 13:45 - 15:15 auditorium 15 During these tutorials I will explain the solutions to the homework assignments of the previous week and answer any questions that arise. Check web-page for details Schedule The course again … 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 Why efficiency? Sorting 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 Describing algorithms A complete description of an algorithm consists of three parts: 1. the algorithm (expressed in whatever way is clearest and most concise, can be English and / or pseudocode) 2. a proof of the algorithm’s correctness 3. a derivation of the algorithm’s running time 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. Incremental algorithms Incremental algorithms process the input elements one-by-one and maintain the solution for the elements processed so far. Check book for more pseudocode conventions In pseudocode: IncAlg(A) // incremental algorithm which computes the solution of a problem with input A = {x1,…,xn} 1. initialize: compute the solution for {x1} 2. for j = 2 to n 3. do compute the solution for {x1,…,xj} using the (already computed) solution for {x1,…,xj-1} no “begin - end”, just indentation InsertionSort InsertionSort(A) // incremental algorithm that sorts array A[1..n] in non-decreasing order 1. initialize: sort A[1] 2. for j = 2 to A.length 3. do sort A[1..j] using the fact that A[1.. j-1] is already sorted InsertionSort InsertionSort(A) // incremental algorithm that sorts array A[1..n] in non-decreasing order 1. initialize: sort A[1] InsertionSort is an in place algorithm: 2. for j = 2 to A.length the numbers are rearranged within the 3. do key = A[j] array with only constant extra space. 4. i = j -1 5. while i > 0 and A[i] > key 6. do A[i+1] = A[i] 7. i ← i -1 8. A[i +1] = key 1 j n 1 3 14 17 28 6 … 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. 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. Correctness proof InsertionSort(A) Loop invariant 1. initialize: sort A[1] At the start of each iteration of the 2. for j = 2 to A.length “outer” for loop (indexed by j) the 3. do key = A[j] subarray A[1..j-1] consists of the 4. i = j -1 elements originally in A[1..j-1] but 5. while i > 0 and A[i] > key in sorted order. 6. do A[i+1] = A[i] 7. i = i -1 8. 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. Correctness proof InsertionSort(A) Loop invariant 1. initialize: sort A[1] At the start of each iteration of the 2. for j = 2 to A.length “outer” for loop (indexed by j) the 3. do key = A[j] subarray A[1..j-1] consists of the 4. i = j -1 elements originally in A[1..j-1] but 5. while i > 0 and A[i] > key in sorted order. 6. do A[i+1] = A[i] 7. i = i -1 8. 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. Correctness proof InsertionSort(A) Loop invariant 1. initialize: sort A[1] At the start of each iteration of the 2. for j = 2 to A.length “outer” for loop (indexed by j) the 3. do key = A[j] subarray A[1..j-1] consists of the 4. i = j -1 elements originally in A[1..j-1] but 5. while i > 0 and A[i] > key in sorted order. 6. do A[i+1] = A[i] 7. i = i -1 8. 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. Another sorting algorithm using a different paradigm … 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. Divide-and-conquer D&CAlg(A) // divide-and-conquer algorithm that computes the solution of a problem with input A = {x1,…,xn} 1. if # elements of A is small enough (for example 1) 2. then compute Sol (the solution for A) brute-force 3. else 4. split A in, for example, 2 non-empty subsets A1 and A2 5. Sol1 = D&CAlg(A1) 6. Sol2 = D&CAlg(A2) 7. compute Sol (the solution for A) from Sol1 and Sol2 8. return Sol MergeSort MergeSort(A) // divide-and-conquer algorithm that sorts array A[1..n] 1. if A.length == 1 2. then compute Sol (the solution for A) brute-force 3. else 4. split A in 2 non-empty subsets A1 and A2 5. Sol1 = MergeSort(A1) 6. Sol2 = MergeSort(A2) 7. compute Sol (the solution for A) from Sol1 en Sol2 MergeSort MergeSort(A) // divide-and-conquer algorithm that sorts array A[1..n] 1. if length[A] = 1 2. then skip 3. else 4. 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] 5. MergeSort(A1) 6. MergeSort(A2) 7. Merge(A, A1, A2) 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 MergeSort Merging A1 1 3 14 17 28 A2 4 7 8 21 35 A 1 3 4 7 8 14 17 21 28 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 MergeSort: correctness proof MergeSort(A) 1. if length[A] = 1 2. then skip 3. else 4. 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] 5. MergeSort(A1) 6. MergeSort(A2) Lemma 7. Merge(A, A1, A2) MergeSort sorts the array A[1..n] correctly. 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. ■ QuickSort another divide-and-conquer sorting algorithm… QuickSort QuickSort(A) // divide-and-conquer algorithm that sorts array A[1..n] 1. if length[A] ≤ 1 2. then skip 3. else 4. pivot = A[1] 5. move all A[i] with A[i] < pivot into auxiliary array A1 6. move all A[i] with A[i] > pivot into auxiliary array A2 7. move all A[i] with A[i] = pivot into auxiliary array A3 8. QuickSort(A1) 9. QuickSort(A2) 10. A = “A1 followed by A3 followed by A2” Analysis of algorithms some informal thoughts – for now … Analysis of algorithms Can we say something about the running time of an algorithm without implementing and testing it? InsertionSort(A) 1. initialize: sort A[1] 2. for j = 2 to A.length 3. do key = A[j] 4. i = j -1 5. while i > 0 and A[i] > key 6. do A[i+1] = A[i] 7. i = i -1 8. A[i +1] = key Analysis of algorithms Analyze the running time as a function of n (# of input elements) best case average case worst case 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). elementary operations add, subtract, multiply, divide, load, store, copy, conditional and unconditional branch, return … Analysis of algorithms: example n=10 n=100 n=1000 InsertionSort: 15 n2 + 7n – 2 1568 150698 1.5 x 107 MergeSort: 300 n lg n + 50 n 10466 204316 3.0 x 106 InsertionSort InsertionSort 6 x faster 1.35 x faster MergeSort 5 x faster n = 1,000,000 InsertionSort 1.5 x 1013 MergeSort 6 x 109 2500 x faster ! 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) Θ-notation Intuition: concentrate on the leading term, ignore constants 19 n3 + 17 n2 - 3n becomes Θ(n3) 2 n lg n + 5 n1.1 - 5 becomes Θ(n1.1) n - ¾ n √n becomes --- (precise definition next lecture …) Some rules and notation log n denotes log2 n We have for a, b, c > 0 : 1. logc (ab) = logc a + logc b 2. logc (ab) = b logc a 3. loga b = logc b / logc a 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 Some rules and notation log n denotes log2 n We have for a, b, c > 0 : 1. logc (ab) = logc a + logc b 2. logc (ab) = b logc a 3. loga b = logc b / logc a Announcements This week Lecture 2 on Friday no “small” tutorial Register on education.tue.nl!