Review: Big O Notation Let T(n) be a function that defines the - - PowerPoint PPT Presentation

Review: Big O Notation

Let T(n) be a function that defines the worst-case running time of an algorithm. T(n) is O(f(n)) if
 T(n) ≤ c ∙ f(n), where c ≥ 0 for all n ≥ n0 Example:
 Let T(n) = 3n + 2
 T(n) is O(n) because
 T(n) ≤ 4n for all n ≥ 2 O(n) is the asymptotic upper bound of T(n).

1

Some Asymptotic Orderings

Logarithms:
 logan is O(nd), for all bases a and all degrees d
 ➔ All logarithms grow slower than all 
 polynomials

25 50 75 100 1 2 3 4 5 6 7 8 9 10

y = x2 y = log2 x

2

Some Asymptotic Orderings

Exponential functions:
 nd is O(rn) when r > 1
 ➔ Polynomials grow no more quickly than
 exponential functions.

y = x2 y = 2x

300 600 900 1200 1 2 3 4 5 6 7 8 9 10

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Ordering of Common O() Functions

O(1) O(log n) O(n) O(n log n) O(nd) O(rn)

4

Linear Time - O(n)

Linear time. Running time is at most a constant factor times the size of the input. Computing the maximum. Compute maximum of n numbers a1, …, an.

5-1

Linear Time - O(n)

Linear time. Running time is at most a constant factor times the size of the input. Computing the maximum. Compute maximum of n numbers a1, …, an. max = a1 for i = 2 to n { if (ai > max) max = ai }

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Cost?

• Merge. Combine two sorted lists A = a1,a2,

…,an with B = b1,b2,…,bn into sorted whole.

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Logarithmic time - O(log n)

Logarithmic time. Do a constant amount of work to discard a constant fraction of the input (often 1/2 ) Binary search: Search a sorted collection

7-1

Logarithmic time - O(log n)

Logarithmic time. Do a constant amount of work to discard a constant fraction of the input (often 1/2 ) Binary search: Search a sorted collection

low = 0; high = a.length - 1; while( low <= high ) { middle = ( low + high ) / 2; if( key == a[ middle ] ) return middle; else if( key < a[ middle ] ) high = middle - 1; else low = middle + 1; } return -1;

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Linearithmic time - O(n log n)

Linearithmic time. Divide-and-conquer Repeatedly divide a problem in half (gives the log n part) Solve the problem in constant time Combine the results of the divided problems in linear time Mergesort: Repeatedly divide collection in half Sort when collections have size 2 Merge resulting lists

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Mergesort

13 17 6 3 9 2 16 1 13 17 6 3 9 2 16 1 13 17 6 3 9 2 16 1

Divide

13 17 3 6 2 9 1 16

Sort

3 6 13 17 1 2 9 16 1 2 3 6 9 13 16 17

Conquer

9

Quadratic Time - O(n2)

Quadratic time. Examine all pairs of input Typically involves nested loops Closest pair of points in a plane. Given a list of n points in the plane (x1, y1), …, (xn, yn), find the pair that is closest.

10-1 Slides04 O() and Graphs.key - February 4, 2019

Quadratic Time - O(n2)

Quadratic time. Examine all pairs of input Typically involves nested loops Closest pair of points in a plane. Given a list of n points in the plane (x1, y1), …, (xn, yn), find the pair that is closest.

min = (x1 - x2)2 + (y1 - y2)2 for i = 1 to n { for j = i+1 to n { d = (xi - xj)2 + (yi - yj)2 if (d < min) min = d } }

10-2

Beyond Polynomial Time

O(N!). Consider all permutations and pick the best. Traveling salesperson. Given n cities with the distances between each pair of cities. Find the shortest path that visits each city exactly once.

11-1

Beyond Polynomial Time

O(N!). Consider all permutations and pick the best. Traveling salesperson. Given n cities with the distances between each pair of cities. Find the shortest path that visits each city exactly once. min = sum of distances visiting cities in order c1, c2, ... cn shortest_path = c1, c2, ... cn for every other permutation p { cost = sum of distances of that permutation if (cost < min) { min = cost shortest_path = p } }

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Running Times as Functions of Input Size

12

Asymptotic Analysis of Gale-Shapley

Initialize each college and student to be free. while (some college is free and hasn't accepted every student) { Choose such a college c s = 1st student on c’s list that c has not yet accepted if (s is free) assign c and s to each other else if (s prefers c to her current college c’) assign s and c to each other, and c' to be free else s rejects c }

13-1

Asymptotic Analysis of Gale-Shapley

Initialize each college and student to be free. while (some college is free and hasn't accepted every student) { Choose such a college c s = 1st student on c’s list that c has not yet accepted if (s is free) assign c and s to each other else if (s prefers c to her current college c’) assign s and c to each other, and c' to be free else s rejects c } Cost 2n 1 1 1 1 1 1 1

13-2 Slides04 O() and Graphs.key - February 4, 2019

Asymptotic Analysis of Gale-Shapley

Initialize each college and student to be free. while (some college is free and hasn't accepted every student) { Choose such a college c s = 1st student on c’s list that c has not yet accepted if (s is free) assign c and s to each other else if (s prefers c to her current college c’) assign s and c to each other, and c' to be free else s rejects c } Cost 2n 1 1 1 1 1 1 1 1

13-3

Asymptotic Analysis of Gale-Shapley

Initialize each college and student to be free. while (some college is free and hasn't accepted every student) { Choose such a college c s = 1st student on c’s list that c has not yet accepted if (s is free) assign c and s to each other else if (s prefers c to her current college c’) assign s and c to each other, and c' to be free else s rejects c }

14-1

Asymptotic Analysis of Gale-Shapley

Initialize each college and student to be free. while (some college is free and hasn't accepted every student) { Choose such a college c s = 1st student on c’s list that c has not yet accepted if (s is free) assign c and s to each other else if (s prefers c to her current college c’) assign s and c to each other, and c' to be free else s rejects c } Cost Reps 2n 1 1 ≤ n2 1 ≤ n2 1 ≤ n2 1 ≤ n2 1 ≤ n2 1 ≤ n2 1 ≤ n2 1 ≤ n2

14-2 Slides04 O() and Graphs.key - February 4, 2019

Asymptotic Analysis of Gale-Shapley

Initialize each college and student to be free. while (some college is free and hasn't accepted every student) { Choose such a college c s = 1st student on c’s list that c has not yet accepted if (s is free) assign c and s to each other else if (s prefers c to her current college c’) assign s and c to each other, and c' to be free else s rejects c } Cost Reps 2n 1 1 ≤ n2 1 ≤ n2 1 ≤ n2 1 ≤ n2 1 ≤ n2 1 ≤ n2 1 ≤ n2 1 ≤ n2 1

14-3

Undirected Graph

Undirected graph. G = (V , E) V = nodes. E = edges between pairs of nodes. Captures pairwise relationship between

• bjects.

Graph size parameters: n = |V|, m = |E|. V = {1, 2, 3, 4, 5} E = {(1,2), (1,4), (1,5), (2,3), (2,4), (3,5)} n=5 m=6

1 3 5 4 2

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Food “chain”

http:// www.twingroves.district96.k 12.il.us/Wetlands/ Salamander/SalGraphics/ salfoodweb.gif

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Flight Carrier Routes

http:/ /www.continental.com/web/ en-US/content/travel/routes/co-us_200702.pdf

17

ISP Graph

http:/ /www.cybergeography.org/atlas/williamscommunications_large.gif

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Internet Backbone (2006)

• pentransit-2.ar2.NYC1.gblx.net

• zit.com

• rb.wojomedia.com

• ther

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Webpage Graph

View of www.mtholyoke.edu generated by 
 http:/ /www.aharef.info/static/htmlgraph/

20 My Facebook Friends

Created with Social Graph http:/ /apps.facebook.com/socgraph/

21

9-11 Terrorist Social Network

Valdis Krebs, http:// www.firstmonday.org/ issues/issue7_4/krebs

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Common Graph Questions

Given a maze, how do you get from the start to the finish. (Is there a path from node A to node B?) Google Maps: What is the shortest path from A to B? If I broadcast a message on a network, will it reach all the computers I think it should? (Is there a path between every pair of nodes in a graph?)

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Path

A path in an undirected graph G = (V , E) is a sequence P of nodes v1, v2, …, vk-1, vk with the property that each consecutive pair vi, vi+1 is joined by an edge in E.

1 3 5 4 2

1-4-2 is a path. 1-3-4 is NOT a path.

1 3 5 4 2

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Cycle

A cycle is a path v1, v2, …, vk-1, vk in which v1 = vk, k > 2, and the first k-1 nodes are all distinct.

1 3 5 4 2

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Connectivity

An undirected graph is connected if for every pair of nodes u and v, there is a path between u and v.

1 3 5 4 2

is a connected graph.

1 3 5 4 2

is NOT a connected graph.

26

Trees

A tree is an undirected graph that is connected and does not contain a cycle.

1 3 5 4 2

is NOT a tree

1 3 5 4 2

is a tree

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Trees

1 3 5 4 2 1 3 5 4 2 1 3 5 4 2

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Trees

1 3 5 4 2 1 3 5 4 2 1 3 5 4 2

http:/ /www.offbeattravel.com/MoCA.html

(Upside-down)

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