CS 323 Lecture 1. Design and Analysis of Algorithms. Hoeteck Wee

Transcription

1 { CS 323 Lecture 1 } Design and Analysis of Algorithms Hoeteck Wee hoeteck@cs.qc.cuny.edu

2 Algorithmic ideas are pervasive APPLICATIONS. Economics, auctions and game theory Biology, genes and genomes Combinatorial optimization Internet search and massive data sets THIS COURSE. techniques for the design and analysis of algorithms also, implementation of these algorithms in python prerequisites: CS 220, CS 313, basic programming skills Hoeteck Wee CS 323 Aug 30, / 17

3 Administration BIRDS ACCOUNT. login via ssh to owl.cs.qc.edu username/password = first two letters of last name + first two letters of first name + last 4 digits of student ID instructor has read access to your home directory. HOMEWORK. written and programming assignments. no late submissions for homework drop two lowest homework scores programming assignments submitted via birds account. Homework 1 out by this Wed (Sep 1) and due Sep 12 (Sun) MIDTERMS. dates are tentatively Oct 6, Nov 3, Dec 1. no rescheduling. Hoeteck Wee CS 323 Aug 30, / 17

4 Homework 0 TASKS. 1. change your password. 2. create a directory submit in your home directory. 3. create a file submit/my .txt containing your address. 4. play with python. DEADLINE. sep 8 (wed) at pm Hoeteck Wee CS 323 Aug 30, / 17

5 Python overview BASICS. interpreted language (i.e. no need to compile) get help in interpreter using help(<object>) or dir() PROPERTIES. dynamically, implicitly typed (i.e. you don t have to declare variables) case sensitive object-oriented (i.e. everything is an object) % python >>> a = [1, 2, 3] >>> a[0] 1 >>> a[1:] [2, 3] >>> i, j = 1, 2 Hoeteck Wee CS 323 Aug 30, / 17

6 Python overview a = [2,5,8] for i in range(3): # range(3) returns [0,1,2] if a[i] > 2: print a[i] SYNTAX. no mandatory statement termination characters (vs ; in java) blocks are specified by indentation (vs { } in java) FLOW CONTROL STATEMENTS. if, for and while Hoeteck Wee CS 323 Aug 30, / 17

7 A SAMPLE OF ALGORITHMS

8 Computing the maximum PROBLEM. given a list a of n numbers, compute the maximum. ALGORITHM. max = a[0] for i in range(1, n): if a[i] > max: max = a[i] return max RUNNING TIME. linear time O(n), i.e. running time is at most a constant factor times input size Hoeteck Wee CS 323 Aug 30, / 17

9 Merging sorted lists PROBLEM. combine two sorted lists a, b into a single sorted list. ALGORITHM. i, j, out = 0, 0, [] while both lists are non-empty: if a[i] <= b[j]: append a[i] to out and increment i else: append b[j] to out and increment j append remainder of nonempty list to out RUNNING TIME. linear time O(n) Hoeteck Wee CS 323 Aug 30, / 17

10 Sorting a list PROBLEM. sort a list a of n numbers. ALGORITHM. divide-and-conquer def mergesort(a): if len(a) <= 1: return a else: mid = len(a)/2 return merge(mergesort(a[:mid]), mergesort(a[mid:])) RUNNING TIME. O(n log n) Hoeteck Wee CS 323 Aug 30, / 17

11 Closest pair of points PROBLEM. given a list of n points in the plane (x 1, y 1),..., (x n, y n), find the pair that is closest. ALGORITHM. try all pairs of points min = (x 1 x 2) 2 + (y 1 y 2) 2 for i in range(1, n+1): for j in range(i+1, n+1): d = (x i x j) 2 + (y i y j) 2 if d < min: min = d return min RUNNING TIME. quadratic time O(n 2 ) Hoeteck Wee CS 323 Aug 30, / 17

12 Large contiguous sums PROBLEM. Given an array a of n integers positive and negative, find the largest sum of contiguous integers. EXAMPLE. a = [ 1, 3, 2, 7, 4, 2, 2, 3, 1], output: = 7. ALGORITHM. Compute sums of all sequences of contiguous integers maxsum = 0 for i in range(n): for j in range(i+1, n): sum = 0 # compute a[i]+...+a[j-1] for k in range(i,j): sum += a[k] if sum > maxsum: maxsum = sum return maxsum RUNNING TIME. O(n 3 ) Hoeteck Wee CS 323 Aug 30, / 17

13 Binary search PROBLEM. given a sorted list a of n numbers, determine if item is in a. ALGORITHM. binary search def bsearch(a, item): if len(a) == 0: return False else: mid = len(a)/2 if a[mid]==item: return True else: if item < a[mid]: return bsearch(a[:mid],item) else: return bsearch(a[mid+1:],item) RUNNING TIME. O(log n), due to repeated halving. Hoeteck Wee CS 323 Aug 30, / 17

14 Integer square root PROBLEM. given an n-bit integer A, compute the integer square root of A. ALGORITHM. use binary search to find the square root def isqrt(a): lower, upper = 1, A while upper-lower>1: mid = (lower+upper)/2 sq = mid*mid if sq > A: upper = mid else: lower = mid if (lower*lower == n): return lower else: return -1 RUNNING TIME. O(n 3 ) Hoeteck Wee CS 323 Aug 30, / 17

15 Integer square root input input Hoeteck Wee CS 323 Aug 30, / 17

16 THE END