Unidad 2: Problemas de camino m´ınimo: Representaci´ on
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Representaci´ on Recorrido de grafos Cam on Camin ino o m´ m´ın ınimo imo
Unidad 2: Proble Problemas mas de camino m´ınimo Representaci´ on on Matriz de adyacencia Matriz de incidencia Listas de vecinos Recorrido de grafos Estructuras de datos BFS DFS Cami Ca mino no m´ın ınim imoo Dijkstra & Moore Bellman & Ford Dantzig & Floyd
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Representaci´ on Recorrido de grafos Cam on Camin ino o m´ m´ın ınimo imo
Matriz de adyacencia Matr Matriz iz de incidencia incidencia Listas de de vecinos vecinos
Representaci´on on de grafos
Matriz de adyacencia Matriz de incidencia Listas de vecinos vecinos (para (para cada v´ ertice, ertice, se indica la lista de sus vecinos). Cantidad Cantid ad de v´ertices ertic es y lista de aristas (se asumen asume n los v´ertices ertic es numerados de 1 a n).
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Representaci´ on Recorrido de grafos Cam on Camin ino o m´ m´ın ınimo imo
Matriz de adyacencia Matr Matriz iz de incidencia incidencia Listas de de vecinos vecinos
Matriz de adyacencia
Dado un grafo G cuyos c uyos v´erti ertice cess est´ est´an an nume numera rado doss de 1 a n, definimos la matriz de adyacencia de G como M ∈ {0, 1}n×n donde ertice erticess i y j son adyacentes y 0 en otro caso. M (i , j ) = 1 si los v´ Ej: 1
2
3
5
0 1 0 0
1 0 0 1
0 0 0 0
0 1 0 0
0 0 1 1
0 0 1 1
0 1 0 1
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Representaci´ on Recorrido de grafos Cam on Camin ino o m´ m´ın ınimo imo
Matriz de adyacencia Matr Matriz iz de incidencia incidencia Listas de de vecinos vecinos
Propiedades de la matriz de adyacencia
k Si AG es la matriz de adyacencia de G , entonces AG [i , j ] es la cantidad de caminos (no necesariamente simples) distintos de longitud k entre i y j . 2 En particular, AG [i , i ] = d G G (i ). k Y adem´ adem´as, as, un graf grafoo G es bipartito si y s´olo olo si AG [i , i ] = 0 para todo k impar, o sea, si la diagonal de la matriz n/2 2 j +1 j +1 tiene la diagonal diagonal nula. A j =0 j =0 G
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Representaci´ on Recorrido de grafos Cam on Camin ino o m´ m´ın ınimo imo
Matriz de adyacencia Matr Matriz iz de incidencia incidencia Listas de de vecinos vecinos
Matriz de incidencia
Numerando las aristas de G de 1 a m, definimos la matriz de incidencia de G como M ∈ {0, 1}m×n donde M (i , j ) = 1 si el v´ertice j es uno de los extremos de la arista i y 0 en otro caso. Ej: 1
2
3
5
1 0 0 0 0
1 0 1 0 1 0 0 1 0 1
0 1 0 0 0
0 0 0 1 0
0 0 0 0 1
0 0 1 0 0
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Representaci´ on Recorrido de grafos Cam on Camin ino o m´ m´ın ınimo imo
Matriz de adyacencia Matr Matriz iz de incidencia incidencia Listas de de vecinos vecinos
Listas de vecinos
Ej: 2
1
3
5 4
6
L1 L2 L3 L4 L5 L6
:2 : 1 → 4 → : 5 → 6 : 2 → 5 → : 3 → 4 → : 3 → 4 →
7 6 → 7 6 5 → 7
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Representaci´ on Recorrido de grafos Cam on Camin ino o m´ m´ın ınimo imo
Lista de aristas
Ej: 1
2
3
Matriz de adyacencia Matr Matriz iz de incidencia incidencia Listas de de vecinos vecinos
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Representaci´ on Recorrido de grafos Cam on Camin ino o m´ m´ın ınimo imo
Estructura Estru cturass de datos BFS DFS
Algoritmos para recorrido de grafos Descripci´ on del problema a grandes rasgos: on Tengo un grafo descripto de cierta forma y quiero recorrer sus
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Representaci´ on Recorrido de grafos Cam on Camin ino o m´ m´ın ınimo imo
Estructura Estru cturass de datos BFS DFS
Estructuras de datos
Una pila Una pila (stack ) es una estructura de datos donde el ultimo u ´ltimo en entrar es el primero en salir.
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Representaci´ on Recorrido de grafos Cam on Camin ino o m´ m´ın ınimo imo
Estructura Estru cturass de datos BFS DFS
Estructuras de datos
Una cola Una cola (queue ) es una estructura de datos donde el primero en llegar es el primero en salir.
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Representaci´ on Recorrido de grafos Cam on Camin ino o m´ m´ın ınimo imo
Estructura Estru cturass de datos BFS DFS
Recorrido BFS BFS: Breadth-First Search, o sea, recorrido a lo ancho. A partir de BFS: un v´ ertice, ertice, recorre recorre sus vecinos, vecinos, luego los vecinos de sus vecinos, vecinos, y as´ as´ı sucesivament sucesi vamente.... e.... si al finalizar finaliz ar quedan queda n v´ertices ertic es sin visitar, se repite a partir de un v´ ertice ertice no visitado.
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Representaci´ on Recorrido de grafos Cam on Camin ino o m´ m´ın ınimo imo
Recorrido BFS: pseudoc´ odigo odigo
Estructura Estru cturass de datos BFS DFS
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Representaci´ on Recorrido de grafos Cam on Camin ino o m´ m´ın ınimo imo
Recorrido BFS: propiedades
Estructura Estru cturass de datos BFS DFS
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Representaci´ on Recorrido de grafos Cam on Camin ino o m´ m´ın ınimo imo
Recorrido BFS: propiedades
Estructura Estru cturass de datos BFS DFS
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Representaci´ on Recorrido de grafos Cam on Camin ino o m´ m´ın ınimo imo
Estructura Estru cturass de datos BFS DFS
Recorrido BFS: reconocimiento de grafos bipartitos
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Representaci´ on Recorrido de grafos Cam on Camin ino o m´ m´ın ınimo imo
Recorrido DFS
Estructura Estru cturass de datos BFS DFS
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Representaci´ on Recorrido de grafos Cam on Camin ino o m´ m´ın ınimo imo
Dijkstra Dijks tra & Moor Moore e Bellman & Ford Ford Dantzig & Floyd Floyd
Teorema Sea G = (V , E ), s ∈ V , sin ciclos negativos. Para cada j ∈ V , sea d j la longitud de un camino de s a j . Los n´ umeros umeros d j representan las distancias de s a j si y s´olo olo si d s s = 0 y d j ≤ d i i + (ij ) ∀ij ∈ E . Demo: ⇒) Es claro que d s s = 0. Supongamos que d j > d i i + (ij ) para alg´un un ij ∈ E . Como d i i es la longitud de un camino P entre entre s e i , entonces (P + ij ) = d i i + (ij ) < d j , por lo tanto d j no es la distancia entre s y j . ⇐) Sea P = s = i 1 i 2 . . . i k k = j un camino de s a j . d j = d i ik k d i ik k 1 −
≤ ≤
d i ik k d i ik k
1
−
2
−
+ (i k k −1 i k k ) + (i k k −2 i k k −1 )
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Representaci´ on Recorrido de grafos Cam on Camin ino o m´ m´ın ınimo imo
Dijkstra Dijks tra & Moor Moore e Bellman & Ford Ford Dantzig & Floyd Floyd
Algoritm Alg oritmo o gen´erico erico de corre correcci´ cci´ on de etiquetas on
vectores d y P de longitud n; d (s ) = 0; P (s ) = 0; d ( j ) = ∞ para 1 ≤ j ≤ n, j = s ;
Mientras exista ij ∈ E con d ( j ) > d (i ) + (ij ) d ( j ) = d (i ) + (ij ); P ( j ) = i ;
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Representaci´ on Recorrido de grafos Cam on Camin ino o m´ m´ın ınimo imo
Dijkstra Dijks tra & Moor Moore e Bellman & Ford Ford Dantzig & Floyd Floyd
Propiedad Todo ij en el grafo de predecesores tiene (ij ) + d i i − d j ≤ 0. Demo: Cuando se agrega una arista ij , d j = d i i + (ij ) ⇒ Demo: Cuando (ij ) + d i i − d j = 0. En las siguientes iteraciones, si d i i disminuye, entonces (ij ) + d i i − d j ≤ 0. si d j disminuye, se elimina la arista ij del grafo.
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Representaci´ on Recorrido de grafos Cam on Camin ino o m´ m´ın ınimo imo
Dijkstra Dijks tra & Moor Moore e Bellman & Ford Ford Dantzig & Floyd Floyd
Propiedad Si no hay ciclos negativos, el grafo de predecesores tiene un ´unico camino desde s hacia ha cia to todo do otro otr o v´ertice ert ice k alcanzable desde s , y este camino tiene longitud a lo sumo d k k . Demo: Por inducci´on Demo: Por on en la cantidad de iteraciones del algoritmo, hay un unico u ´nico camino desde s hacia ha cia to todo do otro otr o v´ertice ert ice k alcanzable desde s (se usa que no hay ciclos negativos para probar que a s nunca nunca se le asigna un padre 0 ≥ ij ∈P ((ij ) + d i i − d j ) distinto de 0). Sea P el camino de s a k . = ij ∈P (ij ) + d s s − d k k = ij ∈P (ij ) − d k k ⇒ ij ∈P ≤ d k k .
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Representaci´ on Recorrido de grafos Cam on Camin ino o m´ m´ın ınimo imo
Dijkstra Dijks tra & Moor Moore e Bellman & Ford Ford Dantzig & Floyd Floyd
Algoritmo de Bellman & Ford vectores d k , k = 0, . . . , n, de longitud n; = 1; k = 1; d 0 (1) = 0; d 0 (i ) = ∞ ∀i T 1 = falso; T 2 = falso; Mientras ¬(T 1 ∨ T 2 ) d k (1) = 0; Para i desde 2 hasta n d k (i ) = m´ın{d k −1 (i ), m´ın ji ∈E (d k −1 ( j ) + ( ji ))}; Si d k (i ) = d k −1 (i ) para todo i T 1 = verdadero;
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Representaci´ on Recorrido de grafos Cam on Camin ino o m´ m´ın ınimo imo
Dijkstra Dijks tra & Moor Moore e Bellman & Ford Ford Dantzig & Floyd Floyd
Algoritmo de Bellman & Ford
El invariante del algoritmo es d k (i ) = camino m´ınimo de 1 a i
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Representaci´ on Recorrido de grafos Cam on Camin ino o m´ m´ın ınimo imo
Dijkstra Dijks tra & Moor Moore e Bellman & Ford Ford Dantzig & Floyd Floyd
Algoritmo de Dantzig Los algoritmos de Dantzig y Floyd calculan los caminos m´ınimos entre todos tod os los pares de v´ertices. ertice s. Sea L[i , j ] la distancia de i a j , (i , j ) la longitud de la arista ij o ∞ si no existe. El algoritmo de Dantzig en cada paso k considera el subgrafo el subgrafo inducidoo por los v´ertices inducid ertice s 1, . . . , k .
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Representaci´ on Recorrido de grafos Cam on Camin ino o m´ m´ın ınimo imo
Dijkstra Dijks tra & Moor Moore e Bellman & Ford Ford Dantzig & Floyd Floyd
Algoritmo de Floyd El algoritmo de Floyd en cada paso k calcula el camino el camino m´ınimo de i a j co con n v´ertices ertic es inter intermedio medioss en el conjun conjunto to {1, . . . , k }.
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