Parallel Algorithm

Searching is one of the fundamental operations in computer science. It is used in all applications where we need to find if an element is in the given list or not. In this chapter, we will discuss the following search algorithms −

• Divide and Conquer
• Depth-First Search
• Breadth-First Search
• Best-First Search

## Divide and Conquer

In divide and conquer approach, the problem is divided into several small sub-problems. Then the sub-problems are solved recursively and combined to get the solution of the original problem.

The divide and conquer approach involves the following steps at each level −

• Divide − The original problem is divided into sub-problems.

• Conquer − The sub-problems are solved recursively.

• Combine − The solutions of the sub-problems are combined to get the solution of the original problem.

Binary search is an example of divide and conquer algorithm.

### Pseudocode

```Binarysearch(a, b, low, high)

if low < high then
return NOT FOUND
else
mid ← (low+high) / 2
if b = key(mid) then
return key(mid)
else if b < key(mid) then
return BinarySearch(a, b, low, mid−1)
else

return BinarySearch(a, b, mid+1, high)
```

## Depth-First Search

Depth-First Search (or DFS) is an algorithm for searching a tree or an undirected graph data structure. Here, the concept is to start from the starting node known as the root and traverse as far as possible in the same branch. If we get a node with no successor node, we return and continue with the vertex, which is yet to be visited.

### Steps of Depth-First Search

• Consider a node (root) that is not visited previously and mark it visited.

• Visit the first adjacent successor node and mark it visited.

• If all the successors nodes of the considered node are already visited or it doesn’t have any more successor node, return to its parent node.

### Pseudocode

Let v be the vertex where the search starts in Graph G.

```DFS(G,v)

Stack S := {};

for each vertex u, set visited[u] := false;
push S, v;
while (S is not empty) do
u := pop S;

if (not visited[u]) then
visited[u] := true;
for each unvisited neighbour w of u
push S, w;
end if

end while

END DFS()
```

## Breadth-First Search

Breadth-First Search (or BFS) is an algorithm for searching a tree or an undirected graph data structure. Here, we start with a node and then visit all the adjacent nodes in the same level and then move to the adjacent successor node in the next level. This is also known as level-by-level search.

### Steps of Breadth-First Search

• Start with the root node, mark it visited.
• As the root node has no node in the same level, go to the next level.
• Visit all adjacent nodes and mark them visited.
• Go to the next level and visit all the unvisited adjacent nodes.
• Continue this process until all the nodes are visited.

### Pseudocode

Let v be the vertex where the search starts in Graph G.

```BFS(G,v)

Queue Q := {};

for each vertex u, set visited[u] := false;
insert Q, v;
while (Q is not empty) do
u := delete Q;

if (not visited[u]) then
visited[u] := true;
for each unvisited neighbor w of u
insert Q, w;
end if

end while

END BFS()
```

## Best-First Search

Best-First Search is an algorithm that traverses a graph to reach a target in the shortest possible path. Unlike BFS and DFS, Best-First Search follows an evaluation function to determine which node is the most appropriate to traverse next.

### Steps of Best-First Search

• Start with the root node, mark it visited.
• Find the next appropriate node and mark it visited.
• Go to the next level and find the appropriate node and mark it visited.
• Continue this process until the target is reached.

### Pseudocode

```BFS( m )

Insert( m.StartNode )
Until PriorityQueue is empty
c ← PriorityQueue.DeleteMin
If c is the goal
Exit
Else

Foreach neighbor n of c
If n "Unvisited"
Mark n "Visited"
Insert( n )
Mark c "Examined"

End procedure
```
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