EMI 2 – Matching problem Javier Belenguer Faguás Catalin Costin Stanciu Maria Cabezuelo...
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Transcript of EMI 2 – Matching problem Javier Belenguer Faguás Catalin Costin Stanciu Maria Cabezuelo...
![Page 1: EMI 2 – Matching problem Javier Belenguer Faguás Catalin Costin Stanciu Maria Cabezuelo Sepúlveda Jaime Barrachina Verdia.](https://reader035.fdocuments.in/reader035/viewer/2022062421/56649ce55503460f949b298e/html5/thumbnails/1.jpg)
EMI 2 – Matching problemEMI 2 – Matching problem
Javier Belenguer FaguásCatalin Costin Stanciu
Maria Cabezuelo SepúlvedaJaime Barrachina Verdia
![Page 2: EMI 2 – Matching problem Javier Belenguer Faguás Catalin Costin Stanciu Maria Cabezuelo Sepúlveda Jaime Barrachina Verdia.](https://reader035.fdocuments.in/reader035/viewer/2022062421/56649ce55503460f949b298e/html5/thumbnails/2.jpg)
The problem:
Let G = {{a,b,c,d,e},{e,f,g,h,i,j}} be complete
(this means that for every x from X and y from Y there exists a (x,y) that pertains to E)
H is a weighted graph whose weights are given by:
f g h i j
a 3 5 5 4 1 5
b 2 2 0 2 2 2
c 2 4 4 1 0 4
d 0 1 1 0 0 1
e 1 2 1 3 3 3
0 0 0 0 0
![Page 3: EMI 2 – Matching problem Javier Belenguer Faguás Catalin Costin Stanciu Maria Cabezuelo Sepúlveda Jaime Barrachina Verdia.](https://reader035.fdocuments.in/reader035/viewer/2022062421/56649ce55503460f949b298e/html5/thumbnails/3.jpg)
The graph:
![Page 4: EMI 2 – Matching problem Javier Belenguer Faguás Catalin Costin Stanciu Maria Cabezuelo Sepúlveda Jaime Barrachina Verdia.](https://reader035.fdocuments.in/reader035/viewer/2022062421/56649ce55503460f949b298e/html5/thumbnails/4.jpg)
Step 1 → u = {a}; V(T) = {a}; E(T) = 0; V(T,E) = 0;Step 2 → Ad(a) = {g,h};
g → M-insaturated → go to Step 3Step 3 → (a – g); V(T) = E(T) = 0; go to Step 1;
![Page 5: EMI 2 – Matching problem Javier Belenguer Faguás Catalin Costin Stanciu Maria Cabezuelo Sepúlveda Jaime Barrachina Verdia.](https://reader035.fdocuments.in/reader035/viewer/2022062421/56649ce55503460f949b298e/html5/thumbnails/5.jpg)
Step 1 → u = {b}; V(T) = {b}; E(T) = 0; V(T,E) = 0;Step 2 → Ad(b) = {f,g,i,j};
f → M-insaturated → go to Step 3Step 3 → (b-f); V(T) = E(T) = 0; go to Step 1;
![Page 6: EMI 2 – Matching problem Javier Belenguer Faguás Catalin Costin Stanciu Maria Cabezuelo Sepúlveda Jaime Barrachina Verdia.](https://reader035.fdocuments.in/reader035/viewer/2022062421/56649ce55503460f949b298e/html5/thumbnails/6.jpg)
Step 1 → u = {c}; V(T) = {c}; E(T) = 0; V(T,E) = 0;Step 2 → Ad(c) = {g,h};
g → M-saturated ^ g pertains to V(T)→ go to Step 4;Step 4 → (g,a) edge; V(T)={a,c,g}; E(T)={cg,ga};
V(T,E)={g}; Go to Step 2;h → M-insaturated → Step 3 → (h,c) → go to Step 1
![Page 7: EMI 2 – Matching problem Javier Belenguer Faguás Catalin Costin Stanciu Maria Cabezuelo Sepúlveda Jaime Barrachina Verdia.](https://reader035.fdocuments.in/reader035/viewer/2022062421/56649ce55503460f949b298e/html5/thumbnails/7.jpg)
Step 1 → u = {d}; V(T) = {d}; E(T) = {}; V(T,E) = {d};Step 2 → Ad(d) = {g,h};
g → M-saturated ^g does not pertain to V(T); → go to 4Step 4 → (g,a) edge → V(T) = {d,g,a}; E(T) = {dg,ga};
V(T,E) = {d,a};h → M-saturated ^h does not pertain to V(T): → go to 4Step 4 → (h,c) edge → V(T) = {a,c,d,g,h};
V(T,E) = {d,a,c}; E(T) = {dh,hc,dg,ga}; go to Step 2
![Page 8: EMI 2 – Matching problem Javier Belenguer Faguás Catalin Costin Stanciu Maria Cabezuelo Sepúlveda Jaime Barrachina Verdia.](https://reader035.fdocuments.in/reader035/viewer/2022062421/56649ce55503460f949b298e/html5/thumbnails/8.jpg)
We choose a; Ad(a) = {g,h};g → M-saturated ^h does not pertain to V(T)
→ go to Step 5;Step 5 → No perfect matching.
![Page 9: EMI 2 – Matching problem Javier Belenguer Faguás Catalin Costin Stanciu Maria Cabezuelo Sepúlveda Jaime Barrachina Verdia.](https://reader035.fdocuments.in/reader035/viewer/2022062421/56649ce55503460f949b298e/html5/thumbnails/9.jpg)
Now we recalculate the labels:
f g h i j L1 L2
a 3 5 5 4 1 5(e) 4
b 2 2 0 2 2 2 2
c 2 4 4 1 0 4(e) 3
d 0 1 1 0 0 1(e) 0
e 1 2 1 3 3 3 3
L1 0 0(o) 0(o) 0 0
L2 0 1 1 0 0
alpha = min(L(x) + L(y) – p(x,y)) = 1 Odd → L(v) + alphaEven → L(v) - alpha
![Page 10: EMI 2 – Matching problem Javier Belenguer Faguás Catalin Costin Stanciu Maria Cabezuelo Sepúlveda Jaime Barrachina Verdia.](https://reader035.fdocuments.in/reader035/viewer/2022062421/56649ce55503460f949b298e/html5/thumbnails/10.jpg)
Step 1 → u = {e}; V(T) = {d,g,h,a,c,e}; V(T,E) = {d,c,a,e};Step 2 → Ad(e) = {i,j};
j → M-insaturated → go to Step 3Step 3 → (e-j) →there's an M-augmenting path → Go to 1
![Page 11: EMI 2 – Matching problem Javier Belenguer Faguás Catalin Costin Stanciu Maria Cabezuelo Sepúlveda Jaime Barrachina Verdia.](https://reader035.fdocuments.in/reader035/viewer/2022062421/56649ce55503460f949b298e/html5/thumbnails/11.jpg)
Step 1 → u = {d}; V(T) = {d}; V(T,E) = {};Step 2 → Ad(d) = {g,h};
g → M-saturated ^ doesn't belong to V(T) → go to Step 4Step 4 → (g,a) pertains to M → V(T)={g,a,d}
V(T,E)={a,d} ; E(T)={dg,ga}; return to step 2h→ M-saturated ^ doesn't belong to V(T) → go to Step 4Step 4 → (h,c) pertains to M → V(T)={g,a,d,c,h}
V(T,E)={a,d,c} ; E(T)={dg,ga,dh,hc}; return to step 2
![Page 12: EMI 2 – Matching problem Javier Belenguer Faguás Catalin Costin Stanciu Maria Cabezuelo Sepúlveda Jaime Barrachina Verdia.](https://reader035.fdocuments.in/reader035/viewer/2022062421/56649ce55503460f949b298e/html5/thumbnails/12.jpg)
Step 2 →v= a; Ad(a) = {g,h,i}; i → M-insaturated → go to Step 3Step 3 →There's a path d-g-a-i ^ (ga) belongs to MDelete the edge (ga) from M ; Add (ai) to M;Delete V(T) , E(T), V(T,E); go to Step 1;
![Page 13: EMI 2 – Matching problem Javier Belenguer Faguás Catalin Costin Stanciu Maria Cabezuelo Sepúlveda Jaime Barrachina Verdia.](https://reader035.fdocuments.in/reader035/viewer/2022062421/56649ce55503460f949b298e/html5/thumbnails/13.jpg)
Step 1 → u=d; V(T)={d}; V(T,E)={d}; E(T)=0;Step 2 → v=d; Ad(d)={g,h};
g → M-insaturated → go to Step 3;Step 3 → (dg) is an M-augmenting path;Add (dg) to M; Delete V(T), V(T,E), E(T);Go to step 1;
![Page 14: EMI 2 – Matching problem Javier Belenguer Faguás Catalin Costin Stanciu Maria Cabezuelo Sepúlveda Jaime Barrachina Verdia.](https://reader035.fdocuments.in/reader035/viewer/2022062421/56649ce55503460f949b298e/html5/thumbnails/14.jpg)
Step 2 →v= a; Ad(a) = {g,h,i}; i → M-insaturated → go to Step 3Step 3 →There's a path d-g-a-i ^ (ga) belongs to MDelete the edge (ga) from M ; Add (ai) to M;Delete V(T) , E(T), V(T,E); go to Step 1;
![Page 15: EMI 2 – Matching problem Javier Belenguer Faguás Catalin Costin Stanciu Maria Cabezuelo Sepúlveda Jaime Barrachina Verdia.](https://reader035.fdocuments.in/reader035/viewer/2022062421/56649ce55503460f949b298e/html5/thumbnails/15.jpg)
Step 1 → All x are saturated so M is a perfect matching