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Kuhn munkres algorithm

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Abdullah Nasser

The Kuhn-Munkres algorithm is used to find the maximum weighted matching in a weighted bipartite graph. It works by finding an initial feasible labeling and matching. It then searches for augmenting paths in the equality graph to improve the matching, or improves the labeling if no augmenting path exists to obtain a larger equality graph. The algorithm repeats this process until finding a perfect matching. An example applies the algorithm to find the optimal matching for a weighted bipartite graph.

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The Kuhn-Munkres Algorithm step by step
The Kuhn-Munkres algorithm  The Kuhn-Munkres Algorithm: It’s algorithm to find the maximum weight perfect matching in a weighted complete bipartite graph  Kuhn-Munkres algorithm consider assignment problem in terms of bipartite graph
The Kuhn-Munkres algorithm  Key idea  find a good feasible labeling that remains enough edges in equality graph.  to ensure perfect matching can be done  Algorithm describe  Start with any feasible labeling L and some matching M in GL  While M is not perfect matching, repeat: Find an augmenting path in GL to increase the size of M or if no path exists, improve L to L’ such that GL ⊂ GL’
Before the algorithm Y1 1 Y2 6 Y3 8 1 6 X1 Y1 0 1 6 4 X2 Original Graph Y2 0 8 1 6 X3 X1 6 Y3 0 Y1 0 Y2 0 6 X3 4 8 4 4 X2 8 Initial feasible labeling Y3 0 X1 6 X2 8 X3 4 Equality Graph + Matching l ( x) l ( y ) w( xy) 1
Before the algorithm  Updating feasible labeling: Or how we compute: l  min l ( x) l ( y) w( xy) x S y T ^ l (v ) l (v ) l , if v S l (v ) l , if v T l (v) , otherwise If we have the following labeling L. Assume S={X1,X2} T={y2} Y1 0 1 Y2 0 6 8 1 6 X1 6 Y3 0 X2 8 4 X3 4 Compute l only for x1,x2 With all Y except y2 x S &y T Find minimum of: x1, y1 = l(x1)+l(y1)-w(x1,y2) = 6+0–1= 5 x2, y3 = l(x2)+l(y3)-w(x1,y2) = 8+0–6= 2 Y1 0 1 0 Y2 2 6 8 1 6 6 X1 4 Y3 0 8 X2 6 4 X3 4 Subtract 2 from x1,x2 Add 2 to y2
Algorithm Generate initial labeling L and matching M in GL  Step 1. If X is M-saturateda (perfect matching), stop Otherwise, let u be an M-unsaturated vertex. Set S = {u} and T = .  Step 2. If N G ( S ) T , go to step 3. Otherwise, Compute l l ^ l (v ) min l ( x) l ( y) w( xy) x S y T l (v ) l , if v S l (v ) l , if v T l (v) , otherwise Replace L by L’ and G by Gl’
Algorithm y N Gl ( S ) T yz M M M E (P)
Example :  Apply Kuhn- Munkers algorithm to find an optimal matching: Y1 1 Y2 6 Y3 8 1 6 X1 4 X2 X3 Original Graph
Example : Y1 1 X1 min l ( x) l ( y) w( xy) l l (v ) 6 Y3 8 1 6 N Gl ( S ) T ^ Y2 x S y T l (v ) l , if v S l (v ) l , if v T l (v) , otherwise y N Gl ( S ) T yz M M M E (P) 4 X2 Original Graph X3
Example : Y1 0 1 min l ( x) l ( y) w( xy) l l (v ) 6 x S y T l (v ) l , if v S l (v ) l , if v T l (v) , otherwise y N Gl ( S ) T yz M M M E (P) Y3 0 8 1 6 N Gl ( S ) T ^ Y2 0 4 X1 6 X2 8 Original Graph X3 4
Example : Y1 0 1 min l ( x) l ( y) w( xy) l l (v ) 6 x S y T 1 l l Y2 X3 4 Y3 , if v S l (v ) 4 X1 6 X2 8 Original Graph Y1 l (v ) Y3 0 8 6 N Gl ( S ) T ^ Y2 0 , if v T l (v) , otherwise y N Gl ( S ) T X1 X2 X3 Equality Graph + Matching yz M M M E (P)
Example : Y1 0 1 min l ( x) l ( y) w( xy) l l (v ) 6 x S y T 1 l l Y2 X3 4 Y3 , if v S l (v ) 4 X1 6 X2 8 Original Graph Y1 l (v ) Y3 0 8 6 N Gl ( S ) T ^ Y2 0 , if v T l (v) , otherwise y N Gl ( S ) T X1 X2 X3 Equality Graph + Matching yz M M M E (P)
Example : Y1 0 1 Y2 0 6 8 1 6 N Gl ( S ) T min l ( x) l ( y) w( xy) l ^ l (v ) x S y T l l Y2 X3 4 Y3 , if v S l (v ) 4 X1 6 X2 8 Original Graph Y1 l (v ) Y3 0 , if v T l (v) , otherwise y N Gl ( S ) T X1 X2 X3 Equality Graph + Matching yz M M M E (P)
Example : Y1 0 1 Y2 0 6 8 1 6 N Gl ( S ) T min l ( x) l ( y) w( xy) l ^ l (v ) x S y T l l Y2 X3 4 Y3 , if v S l (v ) 4 X1 6 X2 8 Original Graph Y1 l (v ) Y3 0 , if v T l (v) , otherwise y N Gl ( S ) T X1 X2 X3 Equality Graph + Matching yz M M M E (P)
Example : Y1 0 1 Y2 0 6 8 1 6 N Gl ( S ) T min l ( x) l ( y) w( xy) l ^ l (v ) x S y T l l X3 4 Y2 Y3 , if v S l (v ) 4 X1 6 X2 8 Original Graph Y1 l (v ) Y3 0 , if v T l (v) , otherwise y N Gl ( S ) T X1 X2 X3 Equality Graph + Matching S = {X1} yz M T = {} M M E (P) N Gl (S ) {y2}
Example : Y1 0 1 Y2 0 6 8 1 6 N Gl ( S ) T min l ( x) l ( y) w( xy) l ^ l (v ) x S y T l l X3 4 Y2 Y3 , if v S l (v ) 4 X1 6 X2 8 Original Graph Y1 l (v ) Y3 0 , if v T l (v) , otherwise y N Gl ( S ) T X1 X2 X3 Equality Graph + Matching S = {X1} yz M T = {} M M E (P) N Gl (S ) {y2}
Example : Y1 0 1 Y2 0 6 8 1 6 N Gl ( S ) T min l ( x) l ( y) w( xy) l ^ l (v ) x S y T l l X3 4 Y2 Y3 , if v S l (v ) 4 X1 6 X2 8 Original Graph Y1 l (v ) Y3 0 , if v T l (v) , otherwise y N Gl ( S ) T X1 X2 X3 Equality Graph + Matching S = {X1} yz M T = {} M M E (P) N Gl (S ) {y2}
Example : Y1 0 1 Y2 0 6 8 1 6 N Gl ( S ) T min l ( x) l ( y) w( xy) l ^ l (v ) x S y T l l X3 4 Y2 Y3 , if v S l (v ) 4 X1 6 X2 8 Original Graph Y1 l (v ) Y3 0 , if v T l (v) , otherwise y N Gl ( S ) T X1 X2 X3 Equality Graph + Matching S = {X1} yz M T = {} M M E (P) N Gl (S ) {y2}
Example : Y1 0 1 Y2 0 6 8 1 6 N Gl ( S ) T min l ( x) l ( y) w( xy) l ^ l (v ) x S y T l l X3 4 Y2 Y3 , if v S l (v ) 4 X1 6 X2 8 Original Graph Y1 l (v ) Y3 0 , if v T l (v) , otherwise y N Gl ( S ) T X1 X2 X3 Equality Graph + Matching S = {X1} yz M T = {} M M E (P) N Gl (S ) {y2}
Example : Y1 0 1 min l ( x) l ( y) w( xy) l l (v ) 6 x S y T 1 l l X3 4 Y2 Y3 , if v S l (v ) 4 X1 6 X2 8 Original Graph Y1 l (v ) Y3 0 8 6 N Gl ( S ) T ^ Y2 0 , if v T l (v) , otherwise y N Gl ( S ) T X1 X2 X3 Equality Graph + Matching S = {X1} yz M T = {} M M E (P) N Gl (S ) {y2}
Example : Y1 0 1 min l ( x) l ( y) w( xy) l l (v ) 6 x S y T 1 l l X3 4 Y2 Y3 , if v S l (v ) 4 X1 6 X2 8 Original Graph Y1 l (v ) Y3 0 8 6 N Gl ( S ) T ^ Y2 0 , if v T l (v) , otherwise y N Gl ( S ) T X1 X2 X3 Equality Graph + Matching yz M Go to 2 S = {X1, X2} T = {y2} M M E (P) N Gl (S ) {y2}
Example : Y1 0 1 min l ( x) l ( y) w( xy) l l (v ) 6 x S y T 1 l l X3 4 Y2 Y3 , if v S l (v ) 4 X1 6 X2 8 Original Graph Y1 l (v ) Y3 0 8 6 N Gl ( S ) T ^ Y2 0 , if v T l (v) , otherwise y N Gl ( S ) T X1 X2 X3 Equality Graph + Matching S = {X1, X2} yz M T = {y2} M M E (P) N Gl (S ) {y2}
Example : Y1 0 1 min l ( x) l ( y) w( xy) =2 x S l y T l (v ) 6 1 l l X3 4 Y2 Y3 , if v S l (v ) 4 X1 6 X2 8 Original Graph Y1 l (v ) Y3 0 8 6 N Gl ( S ) T ^ Y2 0 , if v T l (v) , otherwise y N Gl ( S ) T X1 X2 X3 Equality Graph + Matching S = {X1, X2} yz M T = {y2} M M E (P) N Gl (S ) {y2}
Example : Y1 0 1 min l ( x) l ( y) w( xy) l l (v ) 6 x S y T 1 l l X3 4 Y2 Y3 , if v S l (v ) 4 X1 4 X2 6 Original Graph Y1 l (v ) Y3 0 8 6 N Gl ( S ) T ^ Y2 2 , if v T l (v) , otherwise y N Gl ( S ) T X1 X2 X3 Equality Graph + Matching S = {X1, X2} yz M T = {y2} M M E (P) N Gl (S ) {y2}
Example : Y1 0 1 min l ( x) l ( y) w( xy) l l (v ) 6 x S y T 1 l l X3 4 Y2 Y3 , if v S l (v ) 4 X1 4 X2 6 Original Graph Y1 l (v ) Y3 0 8 6 N Gl ( S ) T ^ Y2 2 , if v T l (v) , otherwise y N Gl ( S ) T X1 X2 X3 Equality Graph + Matching S = {X1, X2} yz M T = {y2} M M E (P) N Gl (S ) {y2}
Example : Y1 0 1 min l ( x) l ( y) w( xy) l l (v ) 6 x S y T 1 l l Y2 X3 4 Y3 , if v S l (v ) 4 X1 4 X2 6 Original Graph Y1 l (v ) Y3 0 8 6 N Gl ( S ) T ^ Y2 2 , if v T l (v) , otherwise y N Gl ( S ) T X1 X2 X3 Equality Graph + Matching S = {X1, X2} yz M T = {y2} M M E (P) N Gl (S ) {y2,y3}
Example : Y1 0 1 min l ( x) l ( y) w( xy) l l (v ) 6 x S y T 1 l l Y2 X3 4 Y3 , if v S l (v ) 4 X1 4 X2 6 Original Graph Y1 l (v ) Y3 0 8 6 N Gl ( S ) T ^ Y2 2 , if v T l (v) , otherwise y N Gl ( S ) T X1 X2 X3 Equality Graph + Matching S = {X1, X2} yz M T = {y2} M M E (P) N Gl (S ) {y2,y3}
Example : Y1 0 1 min l ( x) l ( y) w( xy) l l (v ) 6 x S y T 1 l l Y2 X3 4 Y3 , if v S l (v ) 4 X1 4 X2 6 Original Graph Y1 l (v ) Y3 0 8 6 N Gl ( S ) T ^ Y2 2 , if v T l (v) , otherwise y N Gl ( S ) T X1 X2 X3 Equality Graph + Matching S = {X1, X2} yz M T = {y2} M M E (P) N Gl (S ) {y2,y3}
Example : Y1 0 1 min l ( x) l ( y) w( xy) l l (v ) 6 x S y T 1 l l Y2 X3 4 Y3 , if v S l (v ) 4 X1 4 X2 6 Original Graph Y1 l (v ) Y3 0 8 6 N Gl ( S ) T ^ Y2 2 , if v T l (v) , otherwise y N Gl ( S ) T X1 X2 X3 Equality Graph + Matching S = {X1, X2} yz M T = {y2} M M E (P) N Gl (S ) {y2,y3}
Example : Y1 0 1 min l ( x) l ( y) w( xy) l l (v ) 6 x S y T 1 l l Y2 X3 4 Y3 , if v S l (v ) 4 X1 4 X2 6 Original Graph Y1 l (v ) Y3 0 8 6 N Gl ( S ) T ^ Y2 2 , if v T l (v) , otherwise y N Gl ( S ) T X1 X2 X3 Equality Graph + Matching S = {X1, X2} yz M T = {y2} M M E (P) N Gl (S ) {y2,y3}
Example : Y1 0 1 min l ( x) l ( y) w( xy) l l (v ) 6 x S y T 1 l l Y2 X3 4 Y3 , if v S l (v ) 4 X1 4 X2 6 Original Graph Y1 l (v ) Y3 0 8 6 N Gl ( S ) T ^ Y2 2 , if v T l (v) , otherwise y N Gl ( S ) T X1 X2 X3 Equality Graph + Matching S = {X1, X2} yz M T = {y2} M M E (P) Go to 1 N Gl (S ) {y2,y3}
Example : Y1 0 stop 1 min l ( x) l ( y) w( xy) l l (v ) 6 x S y T 1 l l Y2 X3 4 Y3 , if v S l (v ) 4 X1 4 X2 6 Original Graph Y1 l (v ) Y3 0 8 6 N Gl ( S ) T ^ Y2 2 , if v T l (v) , otherwise y N Gl ( S ) T X1 X2 X3 Equality Graph + Matching S = {X1, X2} yz M T = {y2} M M E (P) Go to 1 N Gl (S ) {y2,y3}
Thanks

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Kuhn munkres algorithm

  • 2. The Kuhn-Munkres algorithm  The Kuhn-Munkres Algorithm: It’s algorithm to find the maximum weight perfect matching in a weighted complete bipartite graph  Kuhn-Munkres algorithm consider assignment problem in terms of bipartite graph
  • 3. The Kuhn-Munkres algorithm  Key idea  find a good feasible labeling that remains enough edges in equality graph.  to ensure perfect matching can be done  Algorithm describe  Start with any feasible labeling L and some matching M in GL  While M is not perfect matching, repeat: Find an augmenting path in GL to increase the size of M or if no path exists, improve L to L’ such that GL ⊂ GL’
  • 4. Before the algorithm Y1 1 Y2 6 Y3 8 1 6 X1 Y1 0 1 6 4 X2 Original Graph Y2 0 8 1 6 X3 X1 6 Y3 0 Y1 0 Y2 0 6 X3 4 8 4 4 X2 8 Initial feasible labeling Y3 0 X1 6 X2 8 X3 4 Equality Graph + Matching l ( x) l ( y ) w( xy) 1
  • 5. Before the algorithm  Updating feasible labeling: Or how we compute: l  min l ( x) l ( y) w( xy) x S y T ^ l (v ) l (v ) l , if v S l (v ) l , if v T l (v) , otherwise If we have the following labeling L. Assume S={X1,X2} T={y2} Y1 0 1 Y2 0 6 8 1 6 X1 6 Y3 0 X2 8 4 X3 4 Compute l only for x1,x2 With all Y except y2 x S &y T Find minimum of: x1, y1 = l(x1)+l(y1)-w(x1,y2) = 6+0–1= 5 x2, y3 = l(x2)+l(y3)-w(x1,y2) = 8+0–6= 2 Y1 0 1 0 Y2 2 6 8 1 6 6 X1 4 Y3 0 8 X2 6 4 X3 4 Subtract 2 from x1,x2 Add 2 to y2
  • 6. Algorithm Generate initial labeling L and matching M in GL  Step 1. If X is M-saturateda (perfect matching), stop Otherwise, let u be an M-unsaturated vertex. Set S = {u} and T = .  Step 2. If N G ( S ) T , go to step 3. Otherwise, Compute l l ^ l (v ) min l ( x) l ( y) w( xy) x S y T l (v ) l , if v S l (v ) l , if v T l (v) , otherwise Replace L by L’ and G by Gl’
  • 7. Algorithm y N Gl ( S ) T yz M M M E (P)
  • 8. Example :  Apply Kuhn- Munkers algorithm to find an optimal matching: Y1 1 Y2 6 Y3 8 1 6 X1 4 X2 X3 Original Graph
  • 9. Example : Y1 1 X1 min l ( x) l ( y) w( xy) l l (v ) 6 Y3 8 1 6 N Gl ( S ) T ^ Y2 x S y T l (v ) l , if v S l (v ) l , if v T l (v) , otherwise y N Gl ( S ) T yz M M M E (P) 4 X2 Original Graph X3
  • 10. Example : Y1 0 1 min l ( x) l ( y) w( xy) l l (v ) 6 x S y T l (v ) l , if v S l (v ) l , if v T l (v) , otherwise y N Gl ( S ) T yz M M M E (P) Y3 0 8 1 6 N Gl ( S ) T ^ Y2 0 4 X1 6 X2 8 Original Graph X3 4
  • 11. Example : Y1 0 1 min l ( x) l ( y) w( xy) l l (v ) 6 x S y T 1 l l Y2 X3 4 Y3 , if v S l (v ) 4 X1 6 X2 8 Original Graph Y1 l (v ) Y3 0 8 6 N Gl ( S ) T ^ Y2 0 , if v T l (v) , otherwise y N Gl ( S ) T X1 X2 X3 Equality Graph + Matching yz M M M E (P)
  • 12. Example : Y1 0 1 min l ( x) l ( y) w( xy) l l (v ) 6 x S y T 1 l l Y2 X3 4 Y3 , if v S l (v ) 4 X1 6 X2 8 Original Graph Y1 l (v ) Y3 0 8 6 N Gl ( S ) T ^ Y2 0 , if v T l (v) , otherwise y N Gl ( S ) T X1 X2 X3 Equality Graph + Matching yz M M M E (P)
  • 13. Example : Y1 0 1 Y2 0 6 8 1 6 N Gl ( S ) T min l ( x) l ( y) w( xy) l ^ l (v ) x S y T l l Y2 X3 4 Y3 , if v S l (v ) 4 X1 6 X2 8 Original Graph Y1 l (v ) Y3 0 , if v T l (v) , otherwise y N Gl ( S ) T X1 X2 X3 Equality Graph + Matching yz M M M E (P)
  • 14. Example : Y1 0 1 Y2 0 6 8 1 6 N Gl ( S ) T min l ( x) l ( y) w( xy) l ^ l (v ) x S y T l l Y2 X3 4 Y3 , if v S l (v ) 4 X1 6 X2 8 Original Graph Y1 l (v ) Y3 0 , if v T l (v) , otherwise y N Gl ( S ) T X1 X2 X3 Equality Graph + Matching yz M M M E (P)
  • 15. Example : Y1 0 1 Y2 0 6 8 1 6 N Gl ( S ) T min l ( x) l ( y) w( xy) l ^ l (v ) x S y T l l X3 4 Y2 Y3 , if v S l (v ) 4 X1 6 X2 8 Original Graph Y1 l (v ) Y3 0 , if v T l (v) , otherwise y N Gl ( S ) T X1 X2 X3 Equality Graph + Matching S = {X1} yz M T = {} M M E (P) N Gl (S ) {y2}
  • 16. Example : Y1 0 1 Y2 0 6 8 1 6 N Gl ( S ) T min l ( x) l ( y) w( xy) l ^ l (v ) x S y T l l X3 4 Y2 Y3 , if v S l (v ) 4 X1 6 X2 8 Original Graph Y1 l (v ) Y3 0 , if v T l (v) , otherwise y N Gl ( S ) T X1 X2 X3 Equality Graph + Matching S = {X1} yz M T = {} M M E (P) N Gl (S ) {y2}
  • 17. Example : Y1 0 1 Y2 0 6 8 1 6 N Gl ( S ) T min l ( x) l ( y) w( xy) l ^ l (v ) x S y T l l X3 4 Y2 Y3 , if v S l (v ) 4 X1 6 X2 8 Original Graph Y1 l (v ) Y3 0 , if v T l (v) , otherwise y N Gl ( S ) T X1 X2 X3 Equality Graph + Matching S = {X1} yz M T = {} M M E (P) N Gl (S ) {y2}
  • 18. Example : Y1 0 1 Y2 0 6 8 1 6 N Gl ( S ) T min l ( x) l ( y) w( xy) l ^ l (v ) x S y T l l X3 4 Y2 Y3 , if v S l (v ) 4 X1 6 X2 8 Original Graph Y1 l (v ) Y3 0 , if v T l (v) , otherwise y N Gl ( S ) T X1 X2 X3 Equality Graph + Matching S = {X1} yz M T = {} M M E (P) N Gl (S ) {y2}
  • 19. Example : Y1 0 1 Y2 0 6 8 1 6 N Gl ( S ) T min l ( x) l ( y) w( xy) l ^ l (v ) x S y T l l X3 4 Y2 Y3 , if v S l (v ) 4 X1 6 X2 8 Original Graph Y1 l (v ) Y3 0 , if v T l (v) , otherwise y N Gl ( S ) T X1 X2 X3 Equality Graph + Matching S = {X1} yz M T = {} M M E (P) N Gl (S ) {y2}
  • 20. Example : Y1 0 1 min l ( x) l ( y) w( xy) l l (v ) 6 x S y T 1 l l X3 4 Y2 Y3 , if v S l (v ) 4 X1 6 X2 8 Original Graph Y1 l (v ) Y3 0 8 6 N Gl ( S ) T ^ Y2 0 , if v T l (v) , otherwise y N Gl ( S ) T X1 X2 X3 Equality Graph + Matching S = {X1} yz M T = {} M M E (P) N Gl (S ) {y2}
  • 21. Example : Y1 0 1 min l ( x) l ( y) w( xy) l l (v ) 6 x S y T 1 l l X3 4 Y2 Y3 , if v S l (v ) 4 X1 6 X2 8 Original Graph Y1 l (v ) Y3 0 8 6 N Gl ( S ) T ^ Y2 0 , if v T l (v) , otherwise y N Gl ( S ) T X1 X2 X3 Equality Graph + Matching yz M Go to 2 S = {X1, X2} T = {y2} M M E (P) N Gl (S ) {y2}
  • 22. Example : Y1 0 1 min l ( x) l ( y) w( xy) l l (v ) 6 x S y T 1 l l X3 4 Y2 Y3 , if v S l (v ) 4 X1 6 X2 8 Original Graph Y1 l (v ) Y3 0 8 6 N Gl ( S ) T ^ Y2 0 , if v T l (v) , otherwise y N Gl ( S ) T X1 X2 X3 Equality Graph + Matching S = {X1, X2} yz M T = {y2} M M E (P) N Gl (S ) {y2}
  • 23. Example : Y1 0 1 min l ( x) l ( y) w( xy) =2 x S l y T l (v ) 6 1 l l X3 4 Y2 Y3 , if v S l (v ) 4 X1 6 X2 8 Original Graph Y1 l (v ) Y3 0 8 6 N Gl ( S ) T ^ Y2 0 , if v T l (v) , otherwise y N Gl ( S ) T X1 X2 X3 Equality Graph + Matching S = {X1, X2} yz M T = {y2} M M E (P) N Gl (S ) {y2}
  • 24. Example : Y1 0 1 min l ( x) l ( y) w( xy) l l (v ) 6 x S y T 1 l l X3 4 Y2 Y3 , if v S l (v ) 4 X1 4 X2 6 Original Graph Y1 l (v ) Y3 0 8 6 N Gl ( S ) T ^ Y2 2 , if v T l (v) , otherwise y N Gl ( S ) T X1 X2 X3 Equality Graph + Matching S = {X1, X2} yz M T = {y2} M M E (P) N Gl (S ) {y2}
  • 25. Example : Y1 0 1 min l ( x) l ( y) w( xy) l l (v ) 6 x S y T 1 l l X3 4 Y2 Y3 , if v S l (v ) 4 X1 4 X2 6 Original Graph Y1 l (v ) Y3 0 8 6 N Gl ( S ) T ^ Y2 2 , if v T l (v) , otherwise y N Gl ( S ) T X1 X2 X3 Equality Graph + Matching S = {X1, X2} yz M T = {y2} M M E (P) N Gl (S ) {y2}
  • 26. Example : Y1 0 1 min l ( x) l ( y) w( xy) l l (v ) 6 x S y T 1 l l Y2 X3 4 Y3 , if v S l (v ) 4 X1 4 X2 6 Original Graph Y1 l (v ) Y3 0 8 6 N Gl ( S ) T ^ Y2 2 , if v T l (v) , otherwise y N Gl ( S ) T X1 X2 X3 Equality Graph + Matching S = {X1, X2} yz M T = {y2} M M E (P) N Gl (S ) {y2,y3}
  • 27. Example : Y1 0 1 min l ( x) l ( y) w( xy) l l (v ) 6 x S y T 1 l l Y2 X3 4 Y3 , if v S l (v ) 4 X1 4 X2 6 Original Graph Y1 l (v ) Y3 0 8 6 N Gl ( S ) T ^ Y2 2 , if v T l (v) , otherwise y N Gl ( S ) T X1 X2 X3 Equality Graph + Matching S = {X1, X2} yz M T = {y2} M M E (P) N Gl (S ) {y2,y3}
  • 28. Example : Y1 0 1 min l ( x) l ( y) w( xy) l l (v ) 6 x S y T 1 l l Y2 X3 4 Y3 , if v S l (v ) 4 X1 4 X2 6 Original Graph Y1 l (v ) Y3 0 8 6 N Gl ( S ) T ^ Y2 2 , if v T l (v) , otherwise y N Gl ( S ) T X1 X2 X3 Equality Graph + Matching S = {X1, X2} yz M T = {y2} M M E (P) N Gl (S ) {y2,y3}
  • 29. Example : Y1 0 1 min l ( x) l ( y) w( xy) l l (v ) 6 x S y T 1 l l Y2 X3 4 Y3 , if v S l (v ) 4 X1 4 X2 6 Original Graph Y1 l (v ) Y3 0 8 6 N Gl ( S ) T ^ Y2 2 , if v T l (v) , otherwise y N Gl ( S ) T X1 X2 X3 Equality Graph + Matching S = {X1, X2} yz M T = {y2} M M E (P) N Gl (S ) {y2,y3}
  • 30. Example : Y1 0 1 min l ( x) l ( y) w( xy) l l (v ) 6 x S y T 1 l l Y2 X3 4 Y3 , if v S l (v ) 4 X1 4 X2 6 Original Graph Y1 l (v ) Y3 0 8 6 N Gl ( S ) T ^ Y2 2 , if v T l (v) , otherwise y N Gl ( S ) T X1 X2 X3 Equality Graph + Matching S = {X1, X2} yz M T = {y2} M M E (P) N Gl (S ) {y2,y3}
  • 31. Example : Y1 0 1 min l ( x) l ( y) w( xy) l l (v ) 6 x S y T 1 l l Y2 X3 4 Y3 , if v S l (v ) 4 X1 4 X2 6 Original Graph Y1 l (v ) Y3 0 8 6 N Gl ( S ) T ^ Y2 2 , if v T l (v) , otherwise y N Gl ( S ) T X1 X2 X3 Equality Graph + Matching S = {X1, X2} yz M T = {y2} M M E (P) Go to 1 N Gl (S ) {y2,y3}
  • 32. Example : Y1 0 stop 1 min l ( x) l ( y) w( xy) l l (v ) 6 x S y T 1 l l Y2 X3 4 Y3 , if v S l (v ) 4 X1 4 X2 6 Original Graph Y1 l (v ) Y3 0 8 6 N Gl ( S ) T ^ Y2 2 , if v T l (v) , otherwise y N Gl ( S ) T X1 X2 X3 Equality Graph + Matching S = {X1, X2} yz M T = {y2} M M E (P) Go to 1 N Gl (S ) {y2,y3}