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Greedy Algorithms ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]
Overview ,[object Object],[object Object],[object Object],[object Object],[object Object]
Greedy Strategy ,[object Object],[object Object],[object Object]
Activity-selection Problem ,[object Object],[object Object],[object Object],[object Object],[object Object],Example: Activities in each line are compatible.
Example: i  1 2 3 4 5 6 7 8 9 10 11 s i 1 3 0 5 3 5 6 8 8 2 12 f i 4 5 6 7 8 9 10 11 12 13 14 { a 3 ,  a 9 ,  a 11 }  consists of mutually compatible activities. But it is not a maximal set. { a 1 ,  a 4 ,  a 8 ,  a 11 }  is a largest subset of mutually compatible activities. Another largest subset is  { a 2 ,  a 4 ,  a 9 ,  a 11 } .
Optimal Substructure ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]
Recursive Solution ,[object Object],[object Object],[object Object],Recursive  Solution:               ij j k i ij S j k c k i c S j i c if } 1 ] , [ ] , [ max{ if 0 ] , [
Greedy-choice Property ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]
Recursive Algorithm ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],Initial Call:  Recursive-Activity-Selector (s, f, 0, n+1) Complexity:    (n) Straightforward to convert the algorithm to an iterative one. See the text.
Typical Steps ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]
Elements of Greedy Algorithms ,[object Object],[object Object],[object Object]
Minimum Spanning Trees ,[object Object],[object Object],[object Object],b c a d e f 5 11 0 3 1 7 -3 2 a b c f e d 5 3 -3 1 0 Acyclic  subset of edges( E ) that connects all vertices of  G . w ( T ) =  weight of  T :
Generic Algorithm ,[object Object],[object Object],[object Object],[object Object],A :=   ; while  A not complete tree  do find a safe edge (u, v); A := A    {(u, v)} od
Definitions ,[object Object],[object Object],b c a d e f 5 11 0 3 1 7 -3 2 a  light  edge crossing cut (could be more than one) cut that  respects  an edge set  A =  {(a, b), (b, c)} ,[object Object],[object Object],[object Object],[object Object],[object Object]
Theorem 23.1 Proof: Let  T  be a MST that includes  A . Case:  ( u ,  v ) in  T . We’re done. Case:  ( u ,  v ) not in  T .  We have the following:  u y x v edge in  A cut shows edges in T Theorem 23.1:  Let ( S ,  V  -  S ) be any cut that respects  A , and let ( u ,  v )  be a light edge crossing ( S ,  V  -  S ). Then, ( u , v) is safe for  A . ( x ,  y )  crosses cut. Let  T ´  = { T  - {( x ,  y )}}    {( u ,  v )} . Because ( u ,  v ) is light for cut, w ( u ,  v )     w ( x ,  y ). Thus,  w ( T ´) =  w ( T ) -  w ( x ,  y ) +  w ( u ,  v )     w ( T ). Hence,  T ´ is also a MST.  So,  ( u ,  v ) is safe for  A .
Corollary In general, A will consist of several connected components (CC). Corollary:  If ( u ,  v ) is a light edge connecting one CC in  G A   = ( V ,  A ) to another CC in  G A , then ( u ,  v ) is safe for  A .
Kruskal’s Algorithm ,[object Object],[object Object],[object Object],[object Object]
Prim’s Algorithm ,[object Object],[object Object],[object Object],[object Object]
Prim’s Algorithm ,[object Object],[object Object],implemented as a min-heap Max-heap as a binary tree. 11 14 13 18 17 19 20 18 24 26 1 2 3 4 5 6 7 8 9 10
[object Object],[object Object],[object Object],[object Object],Prim’s Algorithm
Prim’s Algorithm Q := V[G]; for  each u    Q  do key[u] :=   od ; key[r] := 0;  [r] := NIL; while  Q        do u := Extract - Min(Q); for  each v    Adj[u]  do if  v    Q    w(u, v) < key[v]  then  [v] := u; key[v] := w(u, v) fi od od Complexity: Using binary heaps: O(E lg V). Initialization – O(V). Building initial queue – O(V). V Extract-Min’s – O(V lgV). E Decrease-Key’s – O(E lg V). Using Fibonacci heaps: O(E + V lg V). (see book) Note:   A = {(v,   [v]) : v    v - {r} - Q}. decrease-key operation
Example of Prim’s Algorithm b/  c/  a/0 d/  e/  f/  5 11 0 3 1 7 -3 2 Q = a  b  c  d  e  f 0      Not in tree
Example of Prim’s Algorithm b/5 c/  a/0 d/11 e/  f/  5 11 0 3 1 7 -3 2 Q = b  d  c  e  f 5 11     
Example of Prim’s Algorithm Q = e  c  d  f 3  7 11   b/5 c/7 a/0 d/11 e/3 f/  5 11 0 3 1 7 -3 2
Example of Prim’s Algorithm Q = d  c  f 0  1  2 b/5 c/1 a/0 d/0 e/3 f/2 5 11 0 3 1 7 -3 2
Example of Prim’s Algorithm Q = c  f 1  2 b/5 c/1 a/0 d/11 e/3 f/2 5 11 0 3 1 7 -3 2
Example of Prim’s Algorithm Q = f -3 b/5 c/1 a/0 d/11 e/3 f/-3 5 11 0 3 1 7 -3 2
Example of Prim’s Algorithm Q =   b/5 c/1 a/0 d/11 e/3 f/-3 5 11 0 3 1 7 -3 2
Example of Prim’s Algorithm 0 b/5 c/1 a/0 d/0 e/3 f/-3 5 3 1 -3

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test pre

  • 1.
  • 2.
  • 3.
  • 4.
  • 5. Example: i 1 2 3 4 5 6 7 8 9 10 11 s i 1 3 0 5 3 5 6 8 8 2 12 f i 4 5 6 7 8 9 10 11 12 13 14 { a 3 , a 9 , a 11 } consists of mutually compatible activities. But it is not a maximal set. { a 1 , a 4 , a 8 , a 11 } is a largest subset of mutually compatible activities. Another largest subset is { a 2 , a 4 , a 9 , a 11 } .
  • 6.
  • 7.
  • 8.
  • 9.
  • 10.
  • 11.
  • 12.
  • 13.
  • 14.
  • 15. Theorem 23.1 Proof: Let T be a MST that includes A . Case: ( u , v ) in T . We’re done. Case: ( u , v ) not in T . We have the following: u y x v edge in A cut shows edges in T Theorem 23.1: Let ( S , V - S ) be any cut that respects A , and let ( u , v ) be a light edge crossing ( S , V - S ). Then, ( u , v) is safe for A . ( x , y ) crosses cut. Let T ´ = { T - {( x , y )}}  {( u , v )} . Because ( u , v ) is light for cut, w ( u , v )  w ( x , y ). Thus, w ( T ´) = w ( T ) - w ( x , y ) + w ( u , v )  w ( T ). Hence, T ´ is also a MST. So, ( u , v ) is safe for A .
  • 16. Corollary In general, A will consist of several connected components (CC). Corollary: If ( u , v ) is a light edge connecting one CC in G A = ( V , A ) to another CC in G A , then ( u , v ) is safe for A .
  • 17.
  • 18.
  • 19.
  • 20.
  • 21. Prim’s Algorithm Q := V[G]; for each u  Q do key[u] :=  od ; key[r] := 0;  [r] := NIL; while Q   do u := Extract - Min(Q); for each v  Adj[u] do if v  Q  w(u, v) < key[v] then  [v] := u; key[v] := w(u, v) fi od od Complexity: Using binary heaps: O(E lg V). Initialization – O(V). Building initial queue – O(V). V Extract-Min’s – O(V lgV). E Decrease-Key’s – O(E lg V). Using Fibonacci heaps: O(E + V lg V). (see book) Note: A = {(v,  [v]) : v  v - {r} - Q}. decrease-key operation
  • 22. Example of Prim’s Algorithm b/  c/  a/0 d/  e/  f/  5 11 0 3 1 7 -3 2 Q = a b c d e f 0   Not in tree
  • 23. Example of Prim’s Algorithm b/5 c/  a/0 d/11 e/  f/  5 11 0 3 1 7 -3 2 Q = b d c e f 5 11  
  • 24. Example of Prim’s Algorithm Q = e c d f 3 7 11  b/5 c/7 a/0 d/11 e/3 f/  5 11 0 3 1 7 -3 2
  • 25. Example of Prim’s Algorithm Q = d c f 0 1 2 b/5 c/1 a/0 d/0 e/3 f/2 5 11 0 3 1 7 -3 2
  • 26. Example of Prim’s Algorithm Q = c f 1 2 b/5 c/1 a/0 d/11 e/3 f/2 5 11 0 3 1 7 -3 2
  • 27. Example of Prim’s Algorithm Q = f -3 b/5 c/1 a/0 d/11 e/3 f/-3 5 11 0 3 1 7 -3 2
  • 28. Example of Prim’s Algorithm Q =  b/5 c/1 a/0 d/11 e/3 f/-3 5 11 0 3 1 7 -3 2
  • 29. Example of Prim’s Algorithm 0 b/5 c/1 a/0 d/0 e/3 f/-3 5 3 1 -3