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DAST, Spring 2006. © L. Joskowicz 1 Data Structures – LECTURE 1 Introduction Motivation: algorithms and abstract data types Easy problems, hard problems Examples: sorting Course contents
DAST, Spring 2006. © L. Joskowicz 2 Programs and algorithms Why do we need algorithms? to solve problems with a computing device What is the difference between an algorithm and a program? a program is an implementation of an algorithm to be run on a specific computer and operating system. an algorithm is more abstract – it does not deal with machine specific details – think of it as a method to solve a problem. The course emphasis is on algorithms.
DAST, Spring 2006. © L. Joskowicz 3 Data structures A data structure is a method of storing data for the purpose of efficient computation variables, arrays, linked lists, binary trees How data is stored is key for how a problem will be solved. Assumptions about the data determine what data structure and algorithm will be used sorting integers vs. words Data structures and algorithm development go together! You cannot have one without the other!
DAST, Spring 2006. © L. Joskowicz 4 Abstract Data Types –ADT An abstract data type is a collection of formal specifications of data-storing entities with a well designed set of operations. The set of operations defined with the ADT specification are the operations it “supports” What is the difference between a data structure (or a class of objects) and an ADT? The data structure or class is an implementation of the ADT to be run on a specific computer and operating system. Think of it as an abstract JAVA class. The course emphasis is on ADTs.
DAST, Spring 2006. © L. Joskowicz 5 Focus of the course In this course, we will study algorithms and ADTs for solving the most common computational problems: searching, sorting, indexing,... We will learn how to rigorously analyze an algorithms in terms of space and time complexity is A 1 always better than A 2 ? We will learn how to adapt know algorithms and develop new ones.
DAST, Spring 2006. © L. Joskowicz 6 Algorithms and problem solving Say you have a computational problem to solve Is there an algorithm that solves it? not always! Example: the halting problem. Is there an efficient algorithm that solves it? not always! Example: packing problem. Is my algorithm the best possible algorithm? not necessarily! Example: sorting in O(n 2 ) What is the best algorithm we can develop? sorting takes Ω(nlogn) time and Ω(n) space.
DAST, Spring 2006. © L. Joskowicz 7 Easy problems, hard problems Over the past 50 years (and especially the last 30 years), many algorithms for a wide variety of computational tasks have been developed A classification of hard and easy problems has also been developed, together with formal tools to prove what is their complexity and how they are related to each other. Equivalence classes of complexity – Ω(n) – linear; Ω(nlogn); – Ω(n 2 ) – quadratic; Ω(n k ) – polynomial; – Ω(2 n ) – exponential; Ω( 2 2 n ) doubly exponential – unsolvable!
DAST, Spring 2006. © L. Joskowicz 8 Easy problem: shortest path planning Find the shortest path (minimum number of changes and stops) between two stations in the Paris metro O(n) in the number of segments
DAST, Spring 2006. © L. Joskowicz 9 Bin packing: a hard problem! Given a board and a set of parts, pack them without overlap so that they occupy the smallest rectangle. 7 parts, 30 squares
DAST, Spring 2006. © L. Joskowicz 10 Bin packing: possible solutions 40 squares 36 squares
DAST, Spring 2006. © L. Joskowicz 11 Bin packing: optimal solution Algorithm 1. generate all legal combinations 2. record area covered 3. keep the best one 30 squares Number of legal configurations: combinatorial There is no better solution in the worst case!!
DAST, Spring 2006. © L. Joskowicz 12 What kind of efficiency? Given an algorithm A, we can ask the following questions on its time and space complexity. Best case: what is the complexity for the most favorable kind of input? Worst case: what is the complexity for the least favorable kind of input? Average case: what is the complexity for the average kind of input? Upper and lower bounds: what is the best we can do for this problem? Trade-off between time and space.
DAST, Spring 2006. © L. Joskowicz 13 Efficiency: sorting Given an array of n integers A[i], sort them in increasing order. Two algorithms (among many others) to do this: BubbleSort: compare two adjacent numbers, and exchange them if A[i-1] < A[i]. Repeat n times. MergeSort: recursively split the array in half, sort each part, and then merge them together.
DAST, Spring 2006. © L. Joskowicz 14 846155101835229747 Bubble sort (1) One pass start 846155101835 229747 2 nd iteration 84 61 55 10 18 35229747 1 st iteration 976155101835 22 97 84 n th iteration … …
DAST, Spring 2006. © L. Joskowicz 15 Bubble sort (2) 84 61 55101835 229747 2 nd pass 84 22 55101835 229761 3 rd pass 8461 47101822 35 97 55 n th pass … … n passes 1 st pass 976155101835 22 97 84
DAST, Spring 2006. © L. Joskowicz 16 Merge sort (1) 54261326 62316245 45623162 54261326 Split phase 0 1 2 3 level
DAST, Spring 2006. © L. Joskowicz 17 Merge sort (2) 66543221 54261326 54623162 63216542 Merge phase 0 1 2 3 level
DAST, Spring 2006. © L. Joskowicz 18 Comparison SPACET Best I M Worst E Average Bubble Sort nn one pass n 2 n passes n 2 /2 n/2 passes Merge Sort n log n MergeSort: Number of levels: 2 l = n l = log 2 n Time for merge: n
DAST, Spring 2006. © L. Joskowicz 19 Other types of algorithms and analyses Up to now, you have studied exact, deterministic algorithms. There are other types as well: Randomized algorithms: makes random choices during execution: pick a random element from an array instead of the first one minimize the chances of always picking a bad one! Probabilistic analysis for randomized algorithms Approximation algorithms: instead of finding an optimal solution, find one close to it bin packing.
DAST, Spring 2006. © L. Joskowicz 20 Course topics (1) Techniques for formal analysis of asymptotic algorithm complexity with recurrence equations Techniques for solving recurrence equations: substitution, recursion-tree, master method. Proving upper and lower bounds Sorting, in-depth: merge sort, quick sort, counting sort, radix sort, bucket sort.
DAST, Spring 2006. © L. Joskowicz 21 Course topics (2) Common ADTs and their algorithms: heaps, priority queues, binary trees, AVL trees, B-trees Hash tables and hash functions Graph algorithms: Breadth-First Search, Depth- First Search, Shortest path algorithms, Minimum Spanning Trees, Strongly Connected Components. Union-Find of sets (time permitting).
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