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tree_traversal.py

def DFS_dist_from_node(query_node, parents): | |

"""Return dictionary containing distances of parent GO nodes from the query""" | |

result = {} | |

stack = [] | |

stack.append( (query_node, 0) ) | |

while len(stack) > 0: | |

print("stack=", stack) | |

node, dist = stack.pop() | |

result[node] = dist | |

if node in parents: | |

for parent in parents[node]: | |

# Get the first member of each tuple, see | |

# http://stackoverflow.com/questions/12142133/how-to-get-first-element-in-a-list-of-tuples | |

stack_members = [x[0] for x in stack] | |

if parent not in stack_members: | |

stack.append( (parent, dist+1) ) | |

return result | |

def BFS_dist_from_node(query_node, parents): | |

"""Return dictionary containing minimum distances of parent GO nodes from the query""" | |

result = {} | |

queue = [] | |

queue.append( (query_node, 0) ) | |

while queue: | |

print("queue=", queue) | |

node, dist = queue.pop(0) | |

result[node] = dist | |

if node in parents: # If the node *has* parents | |

for parent in parents[node]: | |

# Get the first member of each tuple, see | |

# http://stackoverflow.com/questions/12142133/how-to-get-first-element-in-a-list-of-tuples | |

queue_members = [x[0] for x in queue] | |

if parent not in result and parent not in queue_members: # Don’t visit a second time | |

queue.append( (parent, dist+1) ) | |

return result | |

if __name__ == "__main__": | |

parents = dict() | |

parents = ‘N1‘: [‘N2‘, ‘N3‘, ‘N4‘], ‘N3‘: [‘N6‘, ‘N7‘], ‘N4‘: [‘N3‘], ‘N5‘: [‘N4‘, ‘N8‘], ‘N6‘: [‘N13‘], | |

‘N8‘: [‘N9‘], ‘N9‘: [‘N11‘], ‘N10‘: [‘N7‘, ‘N9‘], ‘N11‘: [‘N14‘], ‘N12‘: [‘N5‘] | |

print("Depth-first search:") | |

dist = DFS_dist_from_node(‘N1‘, parents) | |

print(dist) | |

print("Breadth-first search:") | |

dist = BFS_dist_from_node(‘N1‘, parents) | |

print(dist) | |

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# 7.9. Implementing Breadth First Search ¶

With the graph constructed we can now turn our attention to the

algorithm we will use to find the shortest solution to the word ladder

problem. The graph algorithm we are going to use is called the “breadth

first search” algorithm. **Breadth first search** (**BFS**) is one of

the easiest algorithms for searching a graph. It also serves as a

prototype for several other important graph algorithms that we will

study later.

Given a graph \(G\) and a starting vertex \(s\), a breadth

first search proceeds by exploring edges in the graph to find all the

vertices in \(G\) for which there is a path from \(s\). The

remarkable thing about a breadth first search is that it finds *all* the

vertices that are a distance \(k\) from \(s\) before it

finds *any* vertices that are a distance \(k+1\). One good way to

visualize what the breadth first search algorithm does is to imagine

that it is building a tree, one level of the tree at a time. A breadth

first search adds all children of the starting vertex before it begins

to discover any of the grandchildren.

To keep track of its progress, BFS colors each of the vertices white,

gray, or black. All the vertices are initialized to white when they are

constructed. A white vertex is an undiscovered vertex. When a vertex is

initially discovered it is colored gray, and when BFS has completely

explored a vertex it is colored black. This means that once a vertex is

colored black, it has no white vertices adjacent to it. A gray node, on

the other hand, may have some white vertices adjacent to it, indicating

that there are still additional vertices to explore.

The breadth first search algorithm shown in Listing 2 below uses the

adjacency list graph representation we developed earlier. In addition it uses a `Queue`

,

a crucial point as we will see, to decide which vertex to explore next.

In addition the BFS algorithm uses an extended version of the `Vertex`

class. This new vertex class adds three new instance variables:

distance, predecessor, and color. Each of these instance variables also

has the appropriate getter and setter methods. The code for this

expanded Vertex class is included in the `pythonds`

package, but we

will not show it to you here as there is nothing new to learn by seeing

the additional instance variables.

BFS begins at the starting vertex `s`

and colors `start`

gray to

show that it is currently being explored. Two other values, the distance

and the predecessor, are initialized to 0 and `None`

respectively for

the starting vertex. Finally, `start`

is placed on a `Queue`

. The

next step is to begin to systematically explore vertices at the front of

the queue. We explore each new node at the front of the queue by

iterating over its adjacency list. As each node on the adjacency list is

examined its color is checked. If it is white, the vertex is unexplored,

and four things happen:

- The new, unexplored vertex
`nbr`

, is colored gray. - The predecessor of
`nbr`

is set to the current node`currentVert`

- The distance to
`nbr`

is set to the distance to`currentVert + 1`

`nbr`

is added to the end of a queue. Adding`nbr`

to the end of

the queue effectively schedules this node for further exploration,

but not until all the other vertices on the adjacency list of`currentVert`

have been explored.

**Listing 2**

from pythonds.graphs import Graph, Vertexfrom pythonds.basic import Queuedef bfs(g,start): start.setDistance(0) start.setPred(None) vertQueue = Queue() vertQueue.enqueue(start) while (vertQueue.size() > 0): currentVert = vertQueue.dequeue() for nbr in currentVert.getConnections(): if (nbr.getColor() == 'white'): nbr.setColor('gray') nbr.setDistance(currentVert.getDistance() + 1) nbr.setPred(currentVert) vertQueue.enqueue(nbr) currentVert.setColor('black')

Let’s look at how the `bfs`

function would construct the breadth first

tree corresponding to the graph in Figure 1 . Starting

from fool we take all nodes that are adjacent to fool and add them to

the tree. The adjacent nodes include pool, foil, foul, and cool. Each of

these nodes are added to the queue of new nodes to expand.

Figure 3 shows the state of the in-progress tree along with the

queue after this step.

Figure 3: The First Step in the Breadth First Search

In the next step `bfs`

removes the next node (pool) from the front of

the queue and repeats the process for all of its adjacent nodes.

However, when `bfs`

examines the node cool, it finds that the color of

cool has already been changed to gray. This indicates that there is a

shorter path to cool and that cool is already on the queue for further

expansion. The only new node added to the queue while examining pool is

poll. The new state of the tree and queue is shown in Figure 4 .

Figure 4: The Second Step in the Breadth First Search

The next vertex on the queue is foil. The only new node that foil can

add to the tree is fail. As `bfs`

continues to process the queue,

neither of the next two nodes add anything new to the queue or the tree.

Figure 5 shows the tree and the queue after expanding all the

vertices on the second level of the tree.

Figure 5: Breadth First Search Tree After Completing One Level

FIgure 6: Final Breadth First Search Tree

You should continue to work through the algorithm on your own so that

you are comfortable with how it works. Figure 6 shows the

final breadth first search tree after all the vertices in

Figure 3 have been expanded. The amazing thing about the

breadth first search solution is that we have not only solved the

FOOL–SAGE problem we started out with, but we have solved many other

problems along the way. We can start at any vertex in the breadth first

search tree and follow the predecessor arrows back to the root to find

the shortest word ladder from any word back to fool. The function below ( Listing 3 ) shows how to follow the predecessor links to

print out the word ladder.

**Listing 3**

def traverse(y): x = y while (x.getPred()): print(x.getId()) x = x.getPred() print(x.getId())traverse(g.getVertex('sage'))

Next Section – 7.10. Breadth First Search Analysis

search

## Simplest programming tutorials for beginners

# Breadth first search

Traversal means visiting all the nodes of a graph. Breadth first traversal or Breadth first Search is a recursive algorithm for searching all the vertices of a graph or tree data structure. In this article, you will learn with the help of examples the BFS algorithm, BFS pseudocode and the code of the breadth first search algorithm with implementation in C++, C, Java and Python programs.

## BFS algorithm

A standard DFS implementation puts each vertex of the graph into one of two categories:

- Visited
- Not Visited

The purpose of the algorithm is to mark each vertex as visited while avoiding cycles.

The algorithm works as follows:

- Start by putting any one of the graph’s vertices at the back of a queue.
- Take the front item of the queue and add it to the visited list.
- Create a list of that vertex’s adjacent nodes. Add the ones which aren’t in the visited list to the back of the queue.
- Keep repeating steps 2 and 3 until the queue is empty.

The graph might have two different disconnected parts so to make sure that we cover every vertex, we can also run the BFS algorithm on every node

## BFS example

Let’s see how the Breadth First Search algorithm works with an example. We use an undirected graph with 5 vertices.

We start from vertex 0, the BFS algorithm starts by putting it in the Visited list and putting all its adjacent vertices in the stack.

Next, we visit the element at the front of queue i.e. 1 and go to its adjacent nodes. Since 0 has already been visited, we visit 2 instead.

Vertex 2 has an unvisited adjacent vertex in 4, so we add that to the back of the queue and visit 3, which is at the front of the queue.

Only 4 remains in the queue since the only adjacent node of 3 i.e. 0 is already visited. We visit it.

Since the queue is empty, we have completed the Depth First Traversal of the graph.

## BFS pseudocode

create a queue Q mark v as visited and put v into Q while Q is non-empty remove the head u of Q mark and enqueue all (unvisited) neighbours of u

## BFS code

The code for the Breadth First Search Algorithm with an example is shown below. The code has been simplified so that we can focus on the algorithm rather than other details.

### BFS in C

```
#include <stdio.h>
#include <stdlib.h>
#define SIZE 40
struct queue int items[SIZE]; int front; int rear;
;
struct queue* createQueue();
void enqueue(struct queue* q, int);
int dequeue(struct queue* q);
void display(struct queue* q);
int isEmpty(struct queue* q);
void printQueue(struct queue* q);
struct node int vertex; struct node* next;
;
struct node* createNode(int);
struct Graph int numVertices; struct node** adjLists; int* visited;
;
struct Graph* createGraph(int vertices);
void addEdge(struct Graph* graph, int src, int dest);
void printGraph(struct Graph* graph);
void bfs(struct Graph* graph, int startVertex);
int main() struct Graph* graph = createGraph(6); addEdge(graph, 0, 1); addEdge(graph, 0, 2); addEdge(graph, 1, 2); addEdge(graph, 1, 4); addEdge(graph, 1, 3); addEdge(graph, 2, 4); addEdge(graph, 3, 4); bfs(graph, 0); return 0;
void bfs(struct Graph* graph, int startVertex) struct queue* q = createQueue(); graph->visited[startVertex] = 1; enqueue(q, startVertex); while(!isEmpty(q)) printQueue(q); int currentVertex = dequeue(q); printf("Visited %d\n", currentVertex); struct node* temp = graph->adjLists[currentVertex]; while(temp) int adjVertex = temp->vertex; if(graph->visited[adjVertex] == 0) graph->visited[adjVertex] = 1; enqueue(q, adjVertex); temp = temp->next;
struct node* createNode(int v) struct node* newNode = malloc(sizeof(struct node)); newNode->vertex = v; newNode->next = NULL; return newNode;
struct Graph* createGraph(int vertices) struct Graph* graph = malloc(sizeof(struct Graph)); graph->numVertices = vertices; graph->adjLists = malloc(vertices * sizeof(struct node*)); graph->visited = malloc(vertices * sizeof(int)); int i; for (i = 0; i < vertices; i++) graph->adjLists[i] = NULL; graph->visited[i] = 0; return graph;
void addEdge(struct Graph* graph, int src, int dest) // Add edge from src to dest struct node* newNode = createNode(dest); newNode->next = graph->adjLists[src]; graph->adjLists[src] = newNode; // Add edge from dest to src newNode = createNode(src); newNode->next = graph->adjLists[dest]; graph->adjLists[dest] = newNode;
struct queue* createQueue() struct queue* q = malloc(sizeof(struct queue)); q->front = -1; q->rear = -1; return q;
int isEmpty(struct queue* q) if(q->rear == -1) return 1; else return 0;
void enqueue(struct queue* q, int value) if(q->rear == SIZE-1) printf("\nQueue is Full!!"); else if(q->front == -1) q->front = 0; q->rear++; q->items[q->rear] = value;
int dequeue(struct queue* q) int item; if(isEmpty(q)) printf("Queue is empty"); item = -1; else item = q->items[q->front]; q->front++; if(q->front > q->rear) printf("Resetting queue"); q->front = q->rear = -1; return item;
void printQueue(struct queue *q) int i = q->front; if(isEmpty(q)) printf("Queue is empty"); else printf("\nQueue contains \n"); for(i = q->front; i < q->rear + 1; i++) printf("%d ", q->items[i]);
```

## Breadth First Search in C++

```
#include <iostream>
#include <list>
using namespace std;
class Graph int numVertices; list
``` *adjLists; bool* visited;
public: Graph(int vertices); void addEdge(int src, int dest); void BFS(int startVertex);
;
Graph::Graph(int vertices) numVertices = vertices; adjLists = new list[vertices];
void Graph::addEdge(int src, int dest) adjLists[src].push_back(dest); adjLists[dest].push_back(src);
void Graph::BFS(int startVertex) visited = new bool[numVertices]; for(int i = 0; i < numVertices; i++) visited[i] = false; list queue; visited[startVertex] = true; queue.push_back(startVertex); list::iterator i; while(!queue.empty()) int currVertex = queue.front(); cout << "Visited " << currVertex << " "; queue.pop_front(); for(i = adjLists[currVertex].begin(); i != adjLists[currVertex].end(); ++i) int adjVertex = *i; if(!visited[adjVertex]) visited[adjVertex] = true; queue.push_back(adjVertex);
int main() Graph g(4); g.addEdge(0, 1); g.addEdge(0, 2); g.addEdge(1, 2); g.addEdge(2, 0); g.addEdge(2, 3); g.addEdge(3, 3); g.BFS(2); return 0;

### BFS Java code

```
import java.io.*;
import java.util.*;
class Graph private int numVertices; private LinkedList<Integer> adjLists[]; private boolean visited[]; Graph(int v) numVertices = v; visited = new boolean[numVertices]; adjLists = new LinkedList[numVertices]; for (int i=0; i
``` i = adjLists[currVertex].listIterator(); while (i.hasNext()) int adjVertex = i.next(); if (!visited[adjVertex]) visited[adjVertex] = true; queue.add(adjVertex); public static void main(String args[]) Graph g = new Graph(4); g.addEdge(0, 1); g.addEdge(0, 2); g.addEdge(1, 2); g.addEdge(2, 0); g.addEdge(2, 3); g.addEdge(3, 3); System.out.println("Following is Breadth First Traversal "+ "(starting from vertex 2)"); g.BFS(2);
}

## BFS in Python

```
import collections
def bfs(graph, root): visited, queue = set(), collections.deque([root]) visited.add(root) while queue: vertex = queue.popleft() for neighbour in graph[vertex]: if neighbour not in visited: visited.add(neighbour) queue.append(neighbour)
if __name__ == '__main__': graph = 0: [1, 2], 1: [2], 2: [3], 3: [1,2] breadth_first_search(graph, 0)
```

## Data Structure & Algorithms

Bubble Sort Algorithm |

Insertion Sort Algorithm |

Selection Sort Algorithm |

Heap Sort Algorithm |

Merge Sort Algorithm |

Stack |

Queue |

Circular Queue |

Linked List |

Types of Linked List – Singly linked, doubly linked and circular |

Linked List Operations |

Tree Data Structure |

Tree Traversal – inorder, preorder and postorder |

Binary Search Tree(BST) |

Graph Data Stucture |

DFS algorithm |

Adjacency List |

Adjacency Matrix |

Breadth first search |

Kruskals Algorithm |

Prims Algorithm |

Dynamic Programming |

Dijkstras Algorithm |

Bellman Fords Algorithm |

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