abhi shelat

Transcription

1 L abhi shelat

2 Huffman

3 image: wikimedia

4 Alice m Bob

5 m Alice m Bob

6 MOSCOW President Vladimir V. Putin s typically theatrical order to withdraw the bulk of Russian forces from Syria, a process that the Defense Ministry said it began on Tuesday, seemingly caught Washington, Damascus and everybody in between off guard just the way the Russian leader likes it. By all accounts, Mr. Putin delights at creating surprises, reinforcing Russia s newfound image as a sovereign, global heavyweight and keeping him at the center of

7 m MOSCOW President Vladimir V. Putin s typically theatrical order to withdraw the bulk of Russian forces from Syria, a process that the Defense Ministry said it began on Tuesday, seemingly caught Washington, Damascus and everybody in between off guard just the way the Russian leader likes it. By all accounts, Mr. Putin delights at creating surprises, reinforcing Russia s newfound image as a sovereign, global heavyweight and keeping him at the center of

8 e: 25 i: 200 o: 170 u: 87 p: 78 g: 47 b: 40 f:

9 e: i: o: u: p: g: b: f:

10 def: cost of an encoding B(T,{f c })= X c2c f c `c e: i: o: u: p: g: b: f:

11 character frequency e: 2480 i: a: o: r: n: t: s: 1928 l: c: 1007 u: p: m: d: h: y: g: b: 4051 f: v: 2010 k: w: 1825 z: 849 x: 6926 q: 729 j: e i a o r n t s l c u p m d h y g b f v k w z x q j

12 morse code image

13 Morse code

14 def: prefix-free code

15 def: prefix-free code

16 def: prefix code e: 25 0 i: o: u: p: g: b: f:

17 decoding a prefix code e: 25 0 i: o: u: p: g: b: f:

18 code to binary tree e: 25 0 i: o: u: p: g: b: f:

19 prefix code binary tree

20 use tree to encode e: i: o: u: p: e i o u p

21 given the Goal

22 Goal given the character frequencies {f c } c C produce a prefix code T with smallest cost min T B(T,{f c})

23 Property Lemma:Optimal tree is full. y x a b

24 Divide & conquer?

25 e: 2 i: 25 o: 20 u: 18 p: 5 counter-example

26 e i o u p g b f

27 e i o u p 47 g 40 b f

28 e i o u p 40 b f 47 g

29 e i o u p 40 b f 47 g

30 e i o u p 40 b f 47 g

31 25 e 200 i 170 o g 87 u 78 p 40 b 24 f

32 e 200 i 170 o u 78 p g b f

33 e 200 i 170 o u 78 p g 40 b 24 f

34 i 170 o 25 e u 78 p g 40 b 24 f

35 200 i o e e: i: o: u: p: g: b: f: u 78 p g 40 b 24 f

36 200 i o e e: i: o: u: p: g: b: f: u 78 p g 40 b 24 f

37 objective

38 lemma: exchange argument

39 Exchange argument LEMMA: Let x, y C be characters with smallest frequencies f x,f y. There exists an optimal prefix code T for C in which x, y are siblings. That is, the codes for x, y have the same length and only di er in the last bit. The optimal solution for consists of computing an optimal solution for T y x a b

40 LEMMA: Let x, y C be characters with smallest frequencies f x,f y. There exists an optimal prefix code T for C in which x, y are siblings. That is, the codes for x, y have the same length and only di er in the last bit. The optimal solution for consists of computing an optimal solution for T T y b x a a b x y

41 T T y y x f a f b a a b f x f a x b

42 T T y y x f x f a a a b x b B(T )= f c c + f x x + f a a B(T )= f c c + f x x + f a a c c B(T ) B(T ) 0

43 exchange argument T T y b a a x b x y B(T ) B(T ) 0

44 T T T y y b x a a a b x b x y

45 T T T y y b x a a a b x b x y B(T ) B(T ) 0 B(T ) B(T ) 0 Tis also optimal

46 exchange argument LEMMA: Let x, y C be characters with smallest frequencies f x,f y. There exists an optimal prefix code T for C in which x, y are siblings. That is, the codes for x, y have the same length and only di er in the last bit. The optimal solution for consists of computing an optimal solution for T T y b x a a b x y

47 optimal sub-structure f c f x 40 f y 24 f c problem of size n-1 64 f z

48 optimal sub-structure f c f x 40 f y 24 f c problem of size n-1 64 f z Lemma: The optimal solution for T consists of computing an optimal solution for T and replacing the left z with a node having children x, y.

49 T z z

50 T T z x y B(T ) B(T )

51 T T z x y B(T ) B(T ) B(T )=B(T ) f x f y

52 Suppose T is not

53 Suppose T is not U B(U) <B(T ) x y

54 Suppose T is not U B(U) <B(T ) x y U z

55 Suppose T is not U B(U) <B(T ) x U y B(U )=B(U) f x f y < B(t) - fx - fy z But this implies that B(T ) was not optim

56 therefore T z x y

57 summary of argument

58 MST

59 connecting houses image: thefranciscofamily.org,

60 connecting houses image: thefranciscofamily.org,

61 connecting houses image: thefranciscofamily.org,

62 connecting houses image: thefranciscofamily.org,

63 connecting houses image: thefranciscofamily.org,

64 connecting houses image: thefranciscofamily.org,

65 connecting houses image: thefranciscofamily.org,

66 connecting houses image: thefranciscofamily.org,

67 b 8 d 8 g 10 2 a 9 e 5 9 i c 1 f 6 h

68 graphs clrs [ch 22]

69

70 representation G =(V,E) adjacency list space: time list neighbors: time check an edge:

71 representation G =(V,E) adjacency matrix space: time list neighbors: time check an edge:

72 definition: path a sequence of nodes with the property that simple path: cycle:

73 definition:tree connected graph: a tree is

74 what we want: b 8 d 8 g 10 2 a 9 e 5 9 i c 1 f 6 h

75 minimum spanning tree looking for a set of edges that (a) connects all vertices (b) has the least cost min (u,v) T w(u, v) T E

76 facts looking for a set of edges that (a) connects all vertices (b) has the least cost min (u,v) T w(u, v) T E how many edges does solution have? does solution have a cycle?

77 strategy start with an empty set of edges A repeat for v-1 times: add lightest edge that does not create a cycle

78 example b 8 d 8 g a 9 e 5 9 i c 1 f 6 h

79 kruskal b 8 d 8 g a 9 e 5 9 i c 1 f 6 h

80 kruskal b 8 d 8 g a 9 e 5 9 i c 1 f 6 h

81 kruskal b 8 d 8 g a 9 e 5 9 i c 1 f 6 h

82 kruskal b 8 d 8 g a 9 e 5 9 i c 1 f 6 h

83 kruskal b 8 d 8 g a 9 e 5 9 i c 1 f 6 h

84 kruskal b 8 d 8 g a 9 e 5 9 i c 1 f 6 h

85 kruskal b 8 d 8 g a 9 e 5 9 i c 1 f 6 h

86 kruskal b 8 d 8 g a 9 e 5 9 i c 1 f 6 h

87 kruskal e d f c b a i g h 8 7 e d f c b a i g h 8 7 e d f c b a i g h 8 7 e d f c b a i g h 8 7 e d f c b a i g h 8 7 e d f c b a i g h 8 7 e d f c b a i g h 8 7 e d f c b a i g h 8 7

88 why does this work? 1 T 2 repeat V 1 times: add to T the lightest edge e E that does not create a cycle