Breadth First search and Depth First search Finding connected components

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1 Breadth First search and Depth First search Finding connected components Biswadeep Ghosh March 15,

2 Contents 1 Binary Image 4 2 Graphs Definition Example Adjacency Matrix Path Chain Connected Component Cycle,Tree Tree,Forest Spanning Tree Queue 11 4 stack 12 5 Breadth-first search algorithm algorithm Depth-first search algorithm Properties of BFS and DFS 14 8 Neighbourhood of a pixel neighbour Diagonal-neighbour neighbour Adjacency adjacent adjacent Path Connected Components Connected Component Labeling Connected Component Labeling Algorithms Connected Component Labelling by DFS Connected Component Labelling by BFS Runtime 22 2

3 13 Implementation Application 24 3

1 Binary Image Binary image is a digital image which has only two values for each pixel. Typically two colours used for binary images are black and white.

4 1 Binary Image Binary image is a digital image which has only two values for each pixel. Typically two colours used for binary images are black and white. The colour used for objects in a binary image is foreground while the rest of the image are background. Binary images are also known as two-level of bi-level. This means each image is stored in a single bit - i.e. a 0 or 1. The names black-white,monochromatic,monochrome are used for this concept. Figure 1:Example of a binary image:monalisa Binary images arises in certain applications. 2 Graphs Graphs are used to give representation to the problems where the item and there is interrelation between them.in graph theory,items and relations are represented by vertices and edges respectively. 2.1 Definition A graph G = (V, E) is a set of vertices V and there interrelationships are given by the set of edges E. If there is an orientation with respect to any edge then the graph is said to be directed otherwise undirected. An edge connecting two vertices x and y of a graph G is denoted by (x, y).if (x, y) is a directed edge, then it is oriented from vertex x to vertex y.if (x, y) is an undirected edge, then (x, y) = (y, x). 4

5 2.2 Example Consider the set of five vertices V = 1, 2, 3, 4, 5 and consider the direct adjacency relationships where a location is visible from one another.the summary is given in the following table. Location V isiblelocation 1 2, 3, , 3 The graph representation of this data is shown in the figure below.here the set of vertices are given by V = 1, 2, 3, 4, 5. These are the set of locations. Vertices are represented by the circles in the figure. The set of edges corresponds to the interrelationships between the set of visibility vertices. An edge (u, v) exists if vertex v is visible from the vertex u. The set of edges are given by (1, 2), (1, 3), (1, 4), (2, 3), (3, 5), (4, 1), (4, 3). The Directed and undirected graphs are shown below. Figure 2: Graph (a) Directed Version 5

6 2.3 Adjacency Matrix Figure 3: Graph (a) Undirected Version Definition: A graph G = (V, E) where V = v 0, v 1,, v n 1 can be represented by an adjacency matrix M (G). = (m ij ) i=0,,n 1,j=0,,n 1 defined by, 1, if (v i, v j ) ɛ E. m ij = (1) 0, otherwise. It may be noted that the adjacency matrix of an undirected graph is symmetric. Continuing with the above example the adjacency matrix shown in the figure 2 of the directed graph is given below, and the adjacency matrix shown in figure 3 for the undirected graph is given below,

7 2.4 Path Definition:Given a graph G = (V, E), Let u ɛ V and v ɛ V, then a path from u to v is a sequence of vertices P uv = a 0, a 1, a 2,, a n where a 0 = u and a n = v, such that (a i, a i+1 ) ɛ E, for i = 0,, n 1. In the specific instances where each vertex is visited exactly once, the path is called a simple path.i.e, 2.5 Chain a i a j,for any pair of distinct vertices u i, u j P uv If the edges in this path are undirected, then this undirected path is known as a chain. 2.6 Connected Component Definition: A connected component is a graph G = (V, E) such that there exist a chain between any two vertices in Ḡ = ( V, Ē ). A special type of path is a cycle. 2.7 Cycle,Tree Definition: A cycle is path from a vertex v of a graph G = (V, E) to itself.clearly, a cycle is a closed path. Tree: A tree is a graph without a cycle. For example, in the above graph, there is a cycle at the vertex Tree,Forest A directed tree T = (V, E) rooted at vertex r ɛ V is a directed connected component such that there exist a unique directed path from root r to any other vertex ɛ V. A undirected tree T = (V, E) is a undirected connected component without any cycle. In an undirected tree,each vertex can be considered as a root. A set of trees is called a forest. Some examples of Tree and Forest is shown in the following figures. 7

8 Fig 4: Example of a Tree Fig 5: Example of a Forest 8

9 2.9 Spanning Tree Definition: Given a undirected graph G = (V, E), the spanning tree T is a subgraph which contain the minimum possible no of edges of G. A graph can have several spanning trees. A spanning tree of a graph G can also be defined as the subgraph of G which has maximal set of edges which don t contain any cycle. Spanning trees of a graph may not be unique. Consider the following the graph Fig 6: Another example The graph has four spanning trees shown in the following images. Fig 7: 1st spanning Tree 9

10 Fig 8: 2nd spanning Tree Fig 9: 3rd spanning Tree 10

11 Fig 10: 4th spanning Tree Here we have shown examples of spanning tree for undirected graph, but spanning tree of directed graphs also exist. There are two algorithms namely Depth First Search and Breadth First Search to find the connected components of a given graph. 3 Queue Queue is a special type of data structure. In a Queue, the element deleted always is that has been longest time in the Queue. The Queue implements the FIFO property first-in,first-out. The Insert operation in a Queue is known as ENQUEUE and the delete operation is known as DEQUEUE. The FIFO property resembles the Queue like a line of customers waiting to enter a shop. Likewise, when a new customer comes, he stands at the end of the line and when a customer comes out of the shop, He exits from the beginning of the line, when a new element is added in the Queue,it places at the tail of the queue and when an element is dequeued from the Queue, then the element is deleted from the head of the Queue. We will use the terms front and rear for start and end of a Queue. The algorithm for ENQUEUE and DEQUEUE on the array Q[1, 2,, n]is given below, ENQUEUE(Q, x) if is. full (Q) == " true " return " Overflow " else rear = rear + 1 Q[ rear ] = x DEQUEUE(Q) 11

12 if is. empty (Q) == " true " return " Underflow " else data = Q[ front ] front = front + 1 return data is.empty(q) if front == rear return " true " else return " false " is.full(q) if rear == length ( Q) return " true " else return " false " 4 stack The stack is a type of dynamic set where the insert and delete operation is prespecified. The stack is based of LIFO property,i.e, Last in first out. The insert and delete operations are known as push and pop respectively.it resembles like putting a new book on a heap of books and if we want to remove a book we have to first remove the last book put in the heap. The algorithms for push pop are given below, where we implement the stack operation on an array S [1, 2,, n]. The stack consists of elements S [1, 2,, S.top],where S [S.top] is the element at the top. is.empty(s) if S. top == 0 return " true " else return " false " is.full(s) if S. top == n return " true " else return " false " push(s, x) if is. full (S) == " true " error " overflow " else S. top = S. top + 1 S[S. top ] = x pop(s, x) 12

13 if is. empty (S) == " true " error " underflow " else S. top = S. top - 1 return (S[S. top +1]) 5 Breadth-first search algorithm Breadth-first Search is one of graph search algorithms. Given a Graph G = (V, E) and a source vertex s, BFS explores the all reachable vertex starting from the source vertex s in a systematic manner.it computes the distance (smallest no of edges) of any reachable vertex from the sourse vertex s. Also, it produces the Breadth-first tree with root s including all possible vertex reachable from s. BFS starts from the initial node s and then check all of it s neighbours until all of neighbours of the source node s has been visited. Then it moves down to the first neighbour and visits all of it s neighbours and so on. If we imagine that the source node is at level 0, then it s immediate neighbours at level 1 and so on, then we observe that the BFS visits all the vertices of level 1 and then all vertices of level 2 and so on. This is the reason why the algorithm is called so. 5.1 algorithm The Breadth-first search algorithm for graphs is written below The input for the algorithm is a graph G and a source vertex s. BFS(G,s) for each vertex v \ in G. V visited [v] = " false " Q = EmptyQueue visited [s] = " true " Enqueue (Q, s) While ( not empty Q) u = Dequeue ( Q) for each vertex w in adj [ u] if not visited [ w] visited [w] = " true " Enqueue (Q, w) 6 Depth-first search Depth-first Search is also another graph search algorithm like Breadth-first Search. As the name suggests, It visits deeper to search the vertices. DFS explores edges out of most recently discovered vertex v that has unexplored edges leaving it. Once all of the edges of the v has been explored, the search backtracks to explore the edges leaving the vertex from which v is discovered. This process continues until there are vertices reachable from the source vertex of the given graph. 13

14 Like BFS, DFS also produces a spanning tree of the input graph known as Depth-first search Tree. 6.1 algorithm The Depth-first search algorithm for graphs is written below The input for the algorithm is a graph G and a source vertex s. Following is the non-recursive version. DFS(G,s) for each vertex v \ in G. V visited [v] = " false " S = EmptyStack visited [s] = " true " push (S,s) While ( not empty S) u = pop (S) for each vertex w in adj [ u] if not visited [ w] visited [w] = " true " push (Q,w) Recursive version is given below. DFS(G,s) label v as discovered for all vertices w belonging to adjacent ( v) if w is undiscovered DFS (G,w) 7 Properties of BFS and DFS Both BFS and DFS are used to find all vertices reachable from a given source vertex.but, they have some different characteristics noted below. BFS starts traversal from the root node and search in the level by level manner i.e, as close as possible from the root node. on the other hand,dfs starts traversal from the root node and explore the search as far deep as possible. BFS can be implemented using a Queue. on the other hand,dfs can be implemented by a Stack. BFS works in a single stage as visited vertices are removed from the queue and displayed once. But DFS works in two stages. In first stage the vertices discovered are put into the stack and in the second stage, there is no vertex to visit which are popped. Consider the following example, 14

The set of vertices visited by BFS is 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 The set of vertices visited by DFS is 0, 1, 2, 3, 6, 7, 8, 9, 4, 5 8 Neighbourhood of a pixel Given a pixel of a binary image, there

15 The set of vertices visited by BFS is 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 The set of vertices visited by DFS is 0, 1, 2, 3, 6, 7, 8, 9, 4, 5 8 Neighbourhood of a pixel Given a pixel of a binary image, there can be different types of neighbourhoods neighbour Given a binary image, let an arbitary pixel be denoted by tuple (x, y). Then, the 4- neighbours of this pixel is denoted by N 4 (p) given by (x + 1, y),(x 1, y),(x, y 1),(x, y + 1). 8.2 Diagonal-neighbour Figure 11:4-neighbour of a pixel (x, y) The no of Diagonal-neighbour of pixel (x, y) is 4,denoted by N d (p) given by the following (x 1, y 1),(x 1, y + 1),(x + 1, y 1),(x + 1, y + 1). 15

8.3 8-neighbour Figure 12:Diagonal-neighbour of a pixel (x, y) The union of 4-neighbour and Diagonal-neighbour of the pixel (x, y) is known as 8-neighbour of

Figure 13:8-neighbour of a pixel (x, y) 9 Adjacency Two pixels p and q are said to be Adjacent if one is present in the set of any type of neighbours of

16 8.3 8-neighbour Figure 12:Diagonal-neighbour of a pixel (x, y) The union of 4-neighbour and Diagonal-neighbour of the pixel (x, y) is known as 8-neighbour of the given pixel, denoted by N 8 (p) These are the following, (x + 1, y),(x 1, y),(x, y 1),(x, y + 1),(x 1, y 1),(x 1, y + 1),(x + 1, y 1),(x + 1, y + 1) Figure 13:8-neighbour of a pixel (x, y) 9 Adjacency Two pixels p and q are said to be Adjacent if one is present in the set of any type of neighbours of pixel p adjacent : If q is present in the set N 4 p, then q is called 4-adjacent adjacent : If q is present in the set N 8 p q is called 8-adjacent. 16

17 9.3 Path Analogous to the graph theory, Path can also be defined in context of binary images. Definition: A Path from pixel p with co-ordinates (x, y) to pixel q with co-ordinates (s, t) is a sequence (a 0, b 0 ), (a 1, b 2 ),, (a n, b n ) where (a 0, b 0 ) = (x, y) and (a n, b n ) = (s, t), where co-ordinates (a i, b i ) is adjacent to (a i+1, b i+1 ). 10 Connected Components Definition: Given a binary image B, a Connected Component with value v is a set of pixels C such that each pixel has value v and there is a path from a pixel to another pixel in the set C. Fig 14 shows four connected components of 1 s in a binary image fig 14: A binary image with four connected components 10.1 Connected Component Labeling Definition: A Connected Component Labeling of a binary image B is a image where value of each pixel is label of the component associated with it. For example, the Connected Component Labeling of the previous binary image is the following, fig 15: A binary image with Connected Component Labeling A label is a symbol that uniquely determines an entity. The Connected Component Labeling can be done using different characters but positive integers are mostly used to label connected components.for the display of images, the different labels is replaced by different colours of different graylevels. Now, let us see the corresponding binary image and the labeled image shown in following images. 17

fig 16: corresponding binary image fig 17: Labeled image 11 Connected Component Labeling Algorithms There are several Connected Component Labeling algorithms of a binary image is available like One

18 fig 16: corresponding binary image fig 17: Labeled image 11 Connected Component Labeling Algorithms There are several Connected Component Labeling algorithms of a binary image is available like One Pass,Two Pass and Depth-first Search,Breadth-first Search etc. Here, we will concentrate only the Depth-first Search and Breadth-first Search algorithms. Here, our algorithms are based on 8-neighbour of a pixel and input binary matrix is M. 18

19 11.1 Connected Component Labelling by DFS neighour(k, l) x = -1,0,1,1,1,0, -1, -1 y = 1,1,1,0, -1, -1, -1,0 for (i = 1,...,8) n_k = k + x[ i] n_l = l + y[ i] PixelDFS(M, i, j, c, visited) M[i, j] = c # label the (i, j) th pixel visited [i, j] = " true " #( i, j) th pixel is visited for ( any (n_k, n_l ) belonging to neighbour (i,j)) if(m[n_k, n_l ] == 1) && ( visited [n_k, n_l ] == " false ") # visiting if a neighbour is black M[n_k, n_l ] = c # label the (n_k, n_l )th pixel visited [n_k, n_l ] = " true " #( n_k, n_l )th pixel is visited ConncomplabelDFS(M) set = 2 visited = " false " # create logical matrix of same dimension as M for (i = 1,2,.., no of row of M) for (j = 1,2,.., no of column of M) if ( M[i, j] == 1) & ( visited [i, j] == " false ") # visiting if (i, j) th pixel is black PixelDFS (M,i,j, set ++, visited ) # labelling the connected components for (i = 1,2,.., no of row of M) for (j = 1,2,.., no of column of M) if (M[i,j]!= 0) M[i, j] = M[i, j] - 1 return M 19

20 11.2 Connected Component Labelling by BFS PixelBFS(M, i, j, c, visited) Q_row = EmptyQueue # two empty queues for storing row and column indexes Q_col = EmptyQueue visited = " false " # Logical matrix of same dimenson as M, initially all element false M[i, j] = c # label the (i, j) th pixel visited [i, j] = " true " #( i, j) th pixel is visited Enqueue ( Q_row, i) # enqueueing indexes i and j Enqueue ( Q_col,j) while (( Q_row is empty )&( Q_col is empty )) k = Dequeue ( Q_row ) l = Dequeue ( Q_col ) for ( any (n_k, n_l ) belonging to neighbour (k,l)) if(m[n_k, n_l ] == 1) && ( visited [n_k, n_l ] == " false ")# visiting a black neighbour M[n_k, n_l ] = c # label the (n_k, n_l )th pixel visited [n_k, n_l ] = " true " #( n_k, n_l )th pixel is visited Enqueue ( Q_row, n_k ) Enqueue ( Q_col, n_l ) ConncomplabelBFS(M) visited = " false " # create logical matrix of same dimension as M set = 2 for (i = 1,2,.., no of row of M) for (j = 1,2,.., no of column of M) if ( M[i, j] == 1) & ( visited [i, j] == " false ") # visiting if (i, j) th pixel is black PixelBFS (M,i,j, set ++, visited ) # labelling the connected component # label increase by 1 for (i = 1,2,.., no of row of M) for (j = 1,2,.., no of column of M) if (M[i,j]!= 0) M[i, j] = M[i, j] - 1 return M 20

21 To find the no of connected components we use following algorithms NoCompDFS(M) set = 2 visited = " false " # create logical matrix of same dimension as M for (i = 1,2,.., no of row of M) for (j = 1,2,.., no of column of M) if ( M[i, j] == 1) & ( visited [i, j] == " false ") # visiting if (i, j) th pixel is black PixelDFS (M,i,j, set ++, visited ) # labelling the connected components count = set - 2 return count NoCompBFS(M) set = 2 visited = " false " # create logical matrix of same dimension as M for (i = 1,2,.., no of row of M) for (j = 1,2,.., no of column of M) if ( M[i, j] == 1) & ( visited [i, j] == " false ") # visiting if (i, j) th pixel is black PixelBFS (M,i,j, set ++, visited ) # labelling the connected components count = set - 2 return count 21

22 12 Runtime the runtime graph on connected component by DFS,BFS,ConnCompLabel function in R as follows, Runtime graph of three methods 13 Implementation Three algorithms discussed above is based on 8-neighbour of a pixel. We can implement any of the three algorithms in following successive steps written below, Given a binary image,say B,of dimension m n It is scanned to get a binary matrix,say M of same dimensions, where the 1 denotes the black pixel of the image and 0 indicates the white pixel present of the image. Then, the matrix M is augmented by two zero rows and two zero columns, this is done to make so that we can apply any of the two algorithms which is based on 8-neighbour. Call this new matrix as A. Now, apply any one of the three connected component labeling algorithm. Call the output labelled matrix as L, which is of dimension (m + 2) (n + 2). Now, select the submatrix of L, where the set of rows and set of columns are denoted by 2,3,,(m + 1) and 2,3,,(n + 1).Call this L M. 22

Now, from the final Labeled image L M, we can easily get the coloured image where the connected components present in the input image is identified by separate colours.

23 Now, from the final Labeled image L M, we can easily get the coloured image where the connected components present in the input image is identified by separate colours.here is an little application using ConnCompLabelDFS algorithm. Consider the image of monalisa, Fig 18: Monalisa image Binary The corresponding Labelled image(graylablled) is given below, Fig 19: Monalisa image labelled(graylabelled) 23

24 This is done by following R-code library ( EBImage ) library ( Rcpp ) library ( bmp ) sourcecpp (" dfs _ ccl. cpp ") B <- read. bmp (" monalisa. bmp ") #to scan the bitmap image C_1 <- B[,,1] C <- binary ( C_ 1) # to make image matrix binary x <- ncol (C) y <- nrow (C) C <- rbind ( rep (0, x),c) # augmenting two zero rows C <- rbind (C,rep (0,x)) C <- cbind ( rep (0, y +2),C) # augmenting two zero columns C <- cbind (C,rep (0,y +2) ) display ( t( C)) # to display the binary image D <- connlabel _ 1( C) # Labelling the connected components of augmented matrix E <- D [2:( nrow (D) -1),2:( ncol (D) -1)] # taking the proper submatrix display ( t( E)/ max ( E) # display the graylabelled image 14 Application Binary image connected component labelling has several applications. One of important application is to count the no of objects present in the image or distinguish the components separately from other. Consider the following example, consider the image,

25 consider the labelled image, So, there are 8 different components peresent in the image References [1] [2] Binary Digital Image Processing- S. Marchand-Maillet,Y.M. Sharaiha [3] Introduction to Algorithms- Thomas H.Cormen et al. 25

Graphs. Graph G = (V, E) Types of graphs E = O( V 2 ) V = set of vertices E = set of edges (V V)

Graphs. Graph G = (V, E) Types of graphs E = O( V 2 ) V = set of vertices E = set of edges (V V) Graph Algorithms Graphs Graph G = (V, E) V = set of vertices E = set of edges (V V) Types of graphs Undirected: edge (u, v) = (v, u); for all v, (v, v) E (No self loops.) Directed: (u, v) is edge from