Master Theorem, Introduction to Graphs

Transcription

1 Master Theorem, Introduction to Graphs CSE21 Winter 2017, Day 10 (B00), Day 6-7 (A00) February 1,

2 Divide & Conquer: General Strategy Divide the problem of size n into a subproblems of size n/b. Recursively solve each subproblem. Conquer the problem of size n by combining solutions of subproblems. Rosen 5.4 has more examples

3 Divide & Conquer: General Strategy Divide the problem of size n into a subproblems of size n/b. Recursively solve each subproblem. Conquer the problem of size n by combining solutions of subproblems. T(n) = time to solve problem of size n T(1)=constant g(n) = time to do the conquer step to solve problem of size n Recurrence for T(n)? A. T(n) = a*t(n/b) + a*g(n) C. T(n) = a*t(n/b) + g(n) B. T(n) = T(a*n/b) + g(n) D. T(n) = g(n)*a*t(n/b)

4 Solving Divide and Conquer Recurrence Relations Divide and conquer algorithms have recurrences of the form T(n) = a*t(n/b) + g(n). If g(n) is a polynomial, there is a nice theorem called the Master Theorem that allows us to quickly estimate the time complexity of many divide and conquer algorithms. T(n) = time to solve problem of size n T(1)=constant g(n) = time to do the conquer step to solve problem of size n

5 Master Theorem

6 Master Theorem a log b n Size 1

7 Master Theorem for Mergesort If T MS (n) is runtime of MergeSort on list of size n, T MS (0) = c 0 T MS (1) = c' T MS (n) = 2T MS (n/2) + cn where c 0, c, c' are some constants a=2, b=2, d=1 so a=b d O(n 1 log n)

8 Master Theorem for Mergesort If T MS (n) is runtime of MergeSort on list of size n, T MS (0) = c 0 T MS (1) = c' T MS (n) = 2T MS (n/2) + cn where c 0, c, c' are some constants Not much work! a=2, b=2, d=1 so a=b d O(n 1 log n)

9 Master Theorem for Binary Search Do one comparison to decide which half to search in. Then repeat on a list of half the size. T(0) = c 0 T(1) = c' T(n) = T(n/2) + c where c 0, c, c' are some constants a=1, b=2, d=0 so a=b d O(n 0 log n) = O(log n)

10 Divide & Conquer Wins Big Mergesort Dividing into two subproblems each with half the size is a big win over other sorting algorithms. Binary Search Dividing into one subproblem with half the size is a big win over linear search. Will this work in other contexts?

11 Multiplication: WHAT Given two n-digit (or bit) integers a = a n-1 a 1 a 0 and b = b n-1 b 1 b 0 return the decimal (or binary) representation of their product. Rosen p x

12 Multiplication: HOW Given two n-digit (or bit) integers a = a n-1 a 1 a 0 and b = b n-1 b 1 b 0 return the decimal (or binary) representation of their product. Rosen p x Compute partial products (using single digit multiplications), shift, then add. How many operations?

13 Multiplication: HOW Given two n-digit (or bit) integers a = a n-1 a 1 a 0 and b = b n-1 b 1 b 0 return the decimal (or binary) representation of their product. Rosen p x Compute partial products (using single digit multiplications), shift, then add. How many operations? O(n 2 )

14 Multiplication: HOW Divide and conquer? Divide n-digit numbers into two n/2-digit numbers. If a = and b = , we can write To multiply: a = (1234) * (5678) b = (2468) * (1357) ((1234) * (5678))((2468) * (1357))= (1234)(2468) * (1234)(1357) * (2468)(5678) * (1357)(5678)

15 One 8-digit multiplication Multiplication: WHEN ( )( )=((1234) * (5678))((2468) * (1357))= (1234)(2468) * (1234)(1357) * (2468)(5678) * (1357)(5678) Four 4-digit multiplications (plus some shifts, sums)

16 One 8-digit multiplication Multiplication: WHEN ( )( )=((1234) * (5678))((2468) * (1357))= (1234)(2468) * (1234)(1357) * (2468)(5678) * (1357)(5678) Four 4-digit multiplications (plus some shifts, sums) A. a = 4, b=4, d=0 C. a=4, b=2, d=1 B. a = 4, b=4, d=1 D. a=4, b=2, d=0

17 Multiplication: WHEN a=4, b=2, d=1 so a>b d O(n log_2(4) ) = O(n 2 )

18 Multiplication: WHEN a=4, b=2, d=1 so a>b d O(n log_2(4) ) = O(n 2 ) A. This is good news! B. This is bad news! C. I m not sure.

19 Enter Anatoly Karatsuba... Rosen p. 528 Insight: replace one (of the 4) multiplications by (linear time) subtraction

20 Karatsuba Multiplication: HOW Rosen p. 528 ( )( )=((1234) * (5678))((2468) * (1357))= (1234)(2468) * (1234)(1357) * (2468)(5678) * (1357)(5678) (1234)(2468) * ( ) + [(1234) - (5678)][ (1357)-(2468) ] * (1357)(5678) * ( ) Insight: replace one (of the 4) multiplications by (linear time) subtraction

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22 Karatsuba Multiplication: WHEN Instead of T(n) = 4 T(n/2) + cn T K (n) = 3 T K (n/2) + pn with c a constant, now we have with p a constant. Rosen p. 528 a=3, b=2, d=1 so a>b d O(n log_2(3) ) = O(n )

23 Karatsuba Multiplication: WHEN Rosen p. 528 n 1.58 is better than n 2 Progress since then This is good news! 1963: Toom and Cook develop series of algorithms that are time O(n 1+ ). 2007: Furer uses number theory to achieve the best known time for multiplication. 2016: Still open whether there is a linear time algorithm for multiplication.

24 What is a graph? A (directed) graph G is A nonempty set of vertices V, also called nodes and A set of edges E, each pointing from one (tail) vertex to another (head) vertex. (A directed edge is denoted with an arrow ) head tail

25 Variants of graphs Undirected graph: no arrows on edges. Rosen p. 644 If there s an edge between v and w then there's an edge between w and v. (More precisely: an edge e vw connects the unordered pair of vertices {v,w}.) Multigraph: undirected graph that may have multiple edges between a pair of nodes. Such edges are sometimes called parallel edges. Simple graph: undirected graph with no self-loops (edge from v to v) and no parallel edges. Mixed graph: directed graph that may have multiple edges between a pair of nodes as well as self-loops.

26 Graphs are everywhere

27 Graphs are everywhere The internet graph

28 Map coloring Graphs are everywhere

29 Path planning for robots Graphs are everywhere

30 Graphs are everywhere

31 Are these the same graph? A. Yes: the set of vertices is the same. B. Yes: we can rearrange the vertices so that the pictures look the same. C. No: the pictures are different. D. No: the left graph has a crossing and the right one doesn't. E. None of the above.

32 Representing directed graphs Diagrams with vertices and edges How many vertices? For each ordered pair of vertices (v,w) how many edges go from v to w?

33 Representing directed graphs Diagrams with vertices and edges How many vertices? n For each ordered pair of vertices (v,w) how many edges go from v to w? How many ordered pairs of vertices are there? A. n B. 2n C. n 2 D. n(n-1)/2 E. 2 n

34 Representing directed graphs Diagrams with vertices and edges How many vertices? n For each ordered pair of vertices (v,w) how many edges go from v to w? Need to store n 2 ints

35 Representing directed graphs Adjacency matrix n x n matrix: entry in row i and column j is the number of edges from vertex i to vertex j Rosen p. 669

36 Representing directed graphs Adjacency matrix n x n matrix: entry in row i and column j is the number of edges from vertex i to vertex j What can you say about the adjacency matrix of a loopless graph? A. It has all zeros. B. All the elements below the diagonal are 1. C. All the elements are even. D. All the elements on the diagonal are 0. E. None of the above. Rosen p. 669

37 Representing directed graphs Adjacency matrix n x n matrix: entry in row i and column j is the number of edges from vertex i to vertex j What can you say about the adjacency matrix of a graph with no parallel edges? A. It has no zeros. B. It is symmetric. C. All the entries above the diagonal are 0. D. All entries are either 0 or 1. E. None of the above. Rosen p. 669

38 Representing directed graphs Adjacency matrix n x n matrix: entry in row i and column j is the number of edges from vertex i to vertex j What can you say about the adjacency matrix of an undirected graph? A. It has no zeros. B. It is symmetric. C. All the entries above the diagonal are 0. D. All entries are either 0 or 1. E. None of the above. Rosen p. 669

39 Representing undirected graphs Simple undirected graph: * Only need to store the adjacency matrix above diagonal. What's the maximum number of edges a simple undirected graph with n vertices can have? A. n 2 B. n 2 /2 C. n(n-1)/2 D. n(n+1)/2 E. n

40 Efficiency? When is an adjacency matrix an inefficient way to store a graph? When there is a high density of edges compared to number of vertices??? When there is a low density of edges compared to number of vertices???

41 Representing directed graphs Adjacency list (list of lists): for each vertex v, associate list of all neighbors of v. Rosen p. 668

42 Neighbors The neighbors of a vertex v are all the vertices w for which there is an edge whose endpoints are v,w. If two vertices are neighbors then they are called adjacent to one another. Rosen p. 651

43 Degree The degree of a vertex in an undirected graph is the total number of edges incident with it, except that a loop contributes twice. What's the maximum degree of a vertex in this graph? A. 0. B. 1 C. 2 D. 3 E. None of the above. Rosen p. 652

44 What's the degree of vertex 5? A. 5 B. 3 C. 2 D. 1 E. None of the above. Degree

45 What's the degree of vertex 0? A. 5 B. 3 C. 2 D. 1 E. None of the above. Degree

46 Handshakes If there are n people in a room, and each shakes hands with d people, how many handshakes take place? A. n B. d C. nd D. (nd)/2 E. None of the above.

47 Handshakes If there are n people in a room, and each shakes hands with d people, how many handshakes take place? A. n B. d C. nd D. (nd)/2 E. None of the above. Don't double-count each handshake!

48 Handshakes "in" graphs If a simple graph has n vertices and each vertex has degree d, how many edges are there? 2 E = n*d

49 Handshakes in graphs If any graph has n vertices, then 2 E = sum of degrees of all vertices

50 Handshakes "in" graphs If any graph has n vertices, then 2 E = sum of degrees of all vertices What can we conclude? A. Every degree in the graph is even. B. The number of edges is even. C. The number of vertices with odd degree is even. D. The number self loops is even. E. None of the above.

51 Puzzles

52 Tartaglia's Pouring Problem Large cup: contains 8 ounces, can hold more. Medium cup: is empty, has 5 ounce capacity. Small cup: is empty, has 3 ounce capacity You can pour from one cup to another until the first is empty or the second is full. Can we divide the coffee in half? How? A. Yes B. No

53 Tartaglia's Pouring Problem Large cup: contains 8 ounces, can hold more. Medium cup: is empty, has 5 ounce capacity. Small cup: is empty, has 3 ounce capacity You can pour from one cup to another until the first is empty or the second is full. Can we divide the coffee in half? How? Hint: configurations (l,m,s) code # ounces in each cup A. Yes B. No

54 Tartaglia's Pouring Problem

55 Tartaglia's Pouring Problem Rephrasing the problem: Looking for path from (8,0,0) to (4,4,0)

56 Path Sequence (v 0, e 1, v 1, e 2, v 2,, e k, v k ) describes a route through the graph from to start vertex v 0 end vertex v k

57 Tartaglia's Pouring Problem Rephrasing the problem: (1) Is there a path from (8,0,0) to (4,4,0)? (2) If so, what's the best path? "Best" means "shortest length"

58 What's the shortest length of a path from (8,0,0) to (4,4,0)? A. 7 B. 8 C. 14 D. 16 E. None of the above. Tartaglia's Pouring Problem

59 Algorithmic questions related to paths Reachability: // decision Does there exist a path from vertex v to vertex w? Path: // construction Find a path from vertex v to vertex w. Distance: // optimization What s the length of the shortest path from vertex v to vertex w?

60 Seating Chart to be posted on Website Good luck! Exam Announcements Exam on Monday Covers through Day 9 (no graphs) Bring student ID. One handwritten note sheet (8.5 x 11, both sides). Look up seat assignment. No calculators. No blue books. Review Session covering the Practice Midterm Saturday 1-3pm Sunday 12-2pm (selected by Piazza vote)