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CS 3343: Analysis of Algorithms. Lecture 24: Graph searching, Topological sort. Review of MST and shortest path problem. Run Kruskal’s algorithm Run Prim’s algorithm Run Dijkstra’s algorithm. 2. f. c. 4. 7. 8. 8. a. d. g. 6. 7. 6. 1. 3. 5. 6. b. e. Graph Searching.

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CS 3343: Analysis of Algorithms

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## CS 3343: Analysis of Algorithms

Lecture 24: Graph searching, Topological sort

### Review of MST and shortest path problem

• Run Kruskal’s algorithm

• Run Prim’s algorithm

• Run Dijkstra’s algorithm

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### Graph Searching

• Given: a graph G = (V, E), directed or undirected

• Goal: methodically explore every vertex (and every edge)

• Ultimately: build a tree on the graph

• Pick a vertex as the root

• Find (“discover”) its children, then their children, etc.

• Note: might also build a forest if graph is not connected

• Here we only consider that the graph is connected

• “Explore” a graph, turning it into a tree

• Pick a source vertex to be the root

• Expand frontier of explored vertices across the breadth of the frontier

• Associate vertex “colors” to guide the algorithm

• White vertices have not been discovered

• All vertices start out white

• Grey vertices are discovered but not fully explored

• They may be adjacent to white vertices

• Black vertices are discovered and fully explored

• They are adjacent only to black and gray vertices

• Explore vertices by scanning adjacency list of grey vertices

BFS(G, s) {

initialize vertices;// mark all vertices as white

Q = {s};// Q is a queue; initialize to s

while (Q not empty) {

u = Dequeue(Q);

if (v.color == WHITE) {

v.color = GREY;

v.d = u.d + 1;

v.p = u;

Enqueue(Q, v);

}

u.color = BLACK;

}

}

What does v.d represent?

What does v.p represent?

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Touch every vertex: Θ(n)

u = every vertex, but only once (Why?)

v = every vertex that appears in some other vert’s adjacency list

Total: Θ(m)

### BFS: The Code Again

BFS(G, s) {

initialize vertices;

Q = {s};

while (Q not empty) {

u = Dequeue(Q);

if (v.color == WHITE) {

v.color = GREY;

v.d = u.d + 1;

v.p = u;

Enqueue(Q, v);

}

u.color = BLACK;

}

}

What will be the running time?

Total running time: Θ(n+m)

### Depth-First Search

• Depth-first search is another strategy for exploring a graph

• Explore “deeper” in the graph whenever possible

• Edges are explored out of the most recently discovered vertex v that still has unexplored edges

• When all of v’s edges have been explored, backtrack to the vertex from which v was discovered

### Depth-First Search

• Vertices initially colored white

• Then colored gray when discovered

• Then black when finished

DFS(G)

{

for each vertex u  G->V

{

u->color = WHITE;

}

time = 0;

for each vertex u  G->V

{

if (u->color == WHITE)

DFS_Visit(u);

}

}

DFS_Visit(u)

{

u->color = GREY;

time = time+1;

u->d = time;

{

if (v->color == WHITE)

DFS_Visit(v);

}

u->color = BLACK;

time = time+1;

u->f = time;

}

### Depth-First Search: The Code

DFS(G)

{

for each vertex u  G->V

{

u->color = WHITE;

}

time = 0;

for each vertex u  G->V

{

if (u->color == WHITE)

DFS_Visit(u);

}

}

DFS_Visit(u)

{

u->color = GREY;

time = time+1;

u->d = time;

{

if (v->color == WHITE)

DFS_Visit(v);

}

u->color = BLACK;

time = time+1;

u->f = time;

}

### Depth-First Search: The Code

What does u->d represent?

DFS(G)

{

for each vertex u  G->V

{

u->color = WHITE;

}

time = 0;

for each vertex u  G->V

{

if (u->color == WHITE)

DFS_Visit(u);

}

}

DFS_Visit(u)

{

u->color = GREY;

time = time+1;

u->d = time;

{

if (v->color == WHITE)

DFS_Visit(v);

}

u->color = BLACK;

time = time+1;

u->f = time;

}

### Depth-First Search: The Code

What does u->f represent?

DFS(G)

{

for each vertex u  G->V

{

u->color = WHITE;

}

time = 0;

for each vertex u  G->V

{

if (u->color == WHITE)

DFS_Visit(u);

}

}

DFS_Visit(u)

{

u->color = GREY;

time = time+1;

u->d = time;

{

if (v->color == WHITE)

DFS_Visit(v);

}

u->color = BLACK;

time = time+1;

u->f = time;

}

### Depth-First Search: The Code

Will all vertices eventually be colored black? (How about in BFS?)

DFS(G)

{

for each vertex u  G->V

{

u->color = WHITE;

}

time = 0;

for each vertex u  G->V

{

if (u->color == WHITE)

DFS_Visit(u);

}

}

DFS_Visit(u)

{

u->color = GREY;

time = time+1;

u->d = time;

{

if (v->color == WHITE)

DFS_Visit(v);

}

u->color = BLACK;

time = time+1;

u->f = time;

}

### Depth-First Search: The Code

What will be the running time?

DFS(G)

{

for each vertex u  G->V

{

u->color = WHITE;

}

time = 0;

for each vertex u  G->V

{

if (u->color == WHITE)

DFS_Visit(u);

}

}

DFS_Visit(u)

{

u->color = GREY;

time = time+1;

u->d = time;

{

if (v->color == WHITE)

DFS_Visit(v);

}

u->color = BLACK;

time = time+1;

u->f = time;

}

### Depth-First Search: The Code

How many times will DFS_Visit() be called?

DFS(G)

{

for each vertex u  G->V

{

u->color = WHITE;

}

time = 0;

for each vertex u  G->V

{

if (u->color == WHITE)

DFS_Visit(u);

}

}

DFS_Visit(u)

{

u->color = GREY;

time = time+1;

u->d = time;

{

if (v->color == WHITE)

DFS_Visit(v);

}

u->color = BLACK;

time = time+1;

u->f = time;

}

### Depth-First Search: The Code

How much time is needed within each DFS_Visit()?

DFS(G)

{

for each vertex u  G->V

{

u->color = WHITE;

}

time = 0;

for each vertex u  G->V

{

if (u->color == WHITE)

DFS_Visit(u);

}

}

DFS_Visit(u)

{

u->color = GREY;

time = time+1;

u->d = time;

{

if (v->color == WHITE)

DFS_Visit(v);

}

u->color = BLACK;

time = time+1;

u->f = time;

}

### Depth-First Search: The Code

So, running time of DFS = O(V+E)

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### DFS Example

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What is the structure of the grey vertices? What do they represent?

sourcevertex

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3 | 4

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sourcevertex

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sourcevertex

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sourcevertex

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sourcevertex

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### DFS and cycles in graph

• A graph G is acyclic iff a DFS of G yields no back edges

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### Directed Acyclic Graphs

• A directed acyclic graph or DAG is a directed graph with no directed cycles:

Cyclic

Acyclic

### Topological Sort

• Topological sort of a DAG:

• Linear ordering of all vertices in graph G such that vertex u comes before vertex v if edge (u, v)  G

• Real-world example: getting dressed

Underwear

Socks

Watch

Pants

Shoes

Shirt

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Tie

Jacket

Underwear

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### Topological Sort Algorithm

Topological-Sort()

{ // condition: the graph is a DAG

Run DFS

When a vertex is finished, output it

Vertices are output in reverse topological order

}

• Time: O(V+E)

• Correctness: Want to prove that(u,v)  G  uf > vf

### Correctness of Topological Sort

• Claim: (u,v)  G  uf > vf

• When (u,v) is explored, u is grey

• v = grey  (u,v) is back edge. Contradiction (Why?)

• v = white  v becomes descendent of u  vf < uf (since must finish v before backtracking and finishing u)

• v = black  v already finished  vf < uf

### Another Algorithm

• Store vertices in a priority min-queue, with thein-degree of the vertex as the key

• While queue is not empty

• Extract minimum vertex v, and give it next number

• Decrease keys of all adjacent vertices by 1

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### Another Algorithm

• Store vertices in a priority min-queue, with thein-degree of the vertex as the key

• While queue is not empty

• Extract minimum vertex v, and give it next number

• Decrease keys of all adjacent vertices by 1

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### Topological Sort Runtime

• Runtime:

• O(|V|) to build heap +O(|E|) DECREASE-KEY ops

• O(|V| + |E| log |V|) with a binary heap

• O(|V|2) with an array

Compare to DFS:

• O(|V|+|E|)