“It ain’t no good if it ain’t snappy enough.” (Efficient Computations) COS 116, Spring 2012 Adam FinkelsteinToday’s focus: efficiency in computation “If it is worth doing, it is worth doing well, and fast.” Recall: our model of “computation”: pseudocodeQuestion: How do we measure the “speed” of an algorithm? Ideally, should be independent of: machine technology“Running time” of an algorithm Definition: the number of “elementary operations” performed by the algorithm Elementary operations: +, -, *, /, assignment, evaluation of conditionals (discussed also in pseudocode handout) “Speed” of computer: number of elementary operations it can perform per second (Simplified definition) Do not consider this in “running time” of algorithm; technology-dependent.Example: Find Min n items, stored in array A Variables are i, best best ← 1 Do for i = 2 to n { if (A[ i ] < A[best]) then { best ← i } }Example: Find Min n items, stored in array A Variables are i, best best ← 1 Do for i = 2 to n { if (A[ i ] < A[best]) then { best ← i } } How many operations executed before the loop? A: 0 B: 1 C: 2 D: 3Example: Find Min n items, stored in array A Variables are i, best best ← 1 Do for i = 2 to n { if (A[ i ] < A[best]) then { best ← i } } How many operations per iteration of the loop? A: 0 B: 1 C: 2 D: 3Example: Find Min n items, stored in array A Variables are i, best best ← 1 Do for i = 2 to n { if (A[ i ] < A[best]) then { best ← i } } How many times does the loop run? A: n B: n+1 C: n-1 D: 2n “iterations”Example: Find Min n items, stored in array A Variables are i, best best ← 1 Do for i = 2 to n { if (A[ i ] < A[best]) then { best ← i } } Uses at most 2(n – 1) + 1 operations Initialization Number of iterations 1 assignment & 1 comparison = 2 operations per loop iteration } (roughly = 2n)Efficiency of Selection Sort Do for i = 1 to n – 1 { Find cheapest bottle among those numbered i to n Swap that bottle and the i’th bottle. } For the i’th round, takes at most 2(n – i ) + 3 To figure out running time, need to figure out how to sum (n – i) for i = 1 to n – 1 …and then double the result. About 2(n – i) steps 3 stepsGauss’s trick : Sum of (n – i) for i = 1 to n – 1 S = 1 + 2 + … + (n – 2) + (n – 1) + S = (n – 1) + (n – 2) + … + 2 + 1 2S = n + n + … + n + n 2S = n(n – 1) So total time for selection sort is ≤ n(n – 1) + 3n n – 1 timesDiscussion Time “20 Questions”: I have a number between 1 and a million in mind. Guess it by asking me yes/no questions, and keep the number of questions small. Question 1: “Is the number bigger than half a million?” No Question 2: “Is the number bigger than a quarter million?” Strategy: Each question halves the range of possible answers. NoPseudocode: Guessing number from1 to n Lower ← 1 Upper ← n Found ← 0 Do while (Found=0) { Guess ←Round( (Lower + Upper)/2 ) If (Guess = True Number) { Found ← 1 Print(Guess) } If (Guess < True Number) { Lower ← Guess } else { Upper← Guess } } Binary Search How many times does the loop run??Brief detour: Logarithms (CS view) log2 n = K means 2K-1 < n ≤ 2K In words: K is the number of times you need to divide n by 2 in order to get a number ≤ 1 John Napier 16 1024 1048576 8388608 log2 n 4 10 20 23 nRunning times encountered in this lecture n= 8 n= 1024 n= 1048576 n=8388608 log2 n 3 10 20 23 n 8 1024 1048576 8388608 n2 64 1048576 1099511627776 70368744177664 Efficiency really makes a difference!“There are only 10 types of people in the world – those who know binary and those who don’t.” Next….Binary search and binary representation of numbers Say we know 0 ≤ number < 2K Is 2K / 2 ≤ number < 2K? No Yes Is 2K / 4 ≤ number < 2K / 2? No Yes Is 2K × 3/8 ≤ number < 2K / 2? No Yes … … 0 2KBinary representations (cont’d) In general, each number can be uniquely identified by a sequence of yes/no answers to these questions. Correspond to paths down this “tree”: Is 2K / 2 ≤ number < 2K? No Yes Is 2K / 4 ≤ number < 2K / 2? No Yes Is 2K / 8 ≤ number < 2K / 4? No Yes … … Is 2K × 3/8 ≤ number < 2K / 2? No Yes … … …Binary representation of n (the more standard definition) n = 2k bk + 2k-1 bk-1 + … + 2 b2 + b1 where the b’s are either 0 or 1) The binary representation of n is: ⎣n⎦2 = bk bk – 1 … b2 b1Efficiency of Effort: A lens on the world QWERTY keyboard “UPS Truck Driver’s Problem” (a.k.a. Traveling Salesman Problem or TSP) CAPTCHA’s Quantum computing [Jim Loy]Can n particles do 2n “operations” in a single step? Or is Quantum Mechanics not quite correct? SIAM J. Computing 26(5) 1997 Computational efficiency has a bearing on physical
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