View
128
Download
0
Category
Preview:
Citation preview
POSCAT Seminar 9 :Graph 1
yougatup @ POSCAT
1
Topic
Topic today
− Basic concept
− Graph representation
• Adjacency matrix
• Adjacency list
• Tradeoffs
− Graph Traversal
• Depth First Search
• Breadth First Search
• Flood Fill
• Connected Component
• Strongly Connected Component
POSCAT Seminar 1-214 July 2014
yougatup
Graph
Basic Concept
− 𝐺 = (𝑉, 𝐸)
− A set of vertices and edges
POSCAT Seminar 1-314 July 2014
yougatup
1 2
3
4
6
5
7
8
Graph
Terminology
− Nodes
POSCAT Seminar 1-414 July 2014
yougatup
1 2
3
4
6
5
7
8
Graph
Terminology
− Edges
POSCAT Seminar 1-514 July 2014
yougatup
1 2
3
4
6
5
7
8
Graph
Basic Concept
− 𝐷𝑒𝑔𝑟𝑒𝑒(𝑣) : the number of adjacent edges for a vertex 𝑣
POSCAT Seminar 1-614 July 2014
yougatup
1 2
3
4
6
5
7
8
32
2
3 2
32
1
Graph
Basic Concept
− 𝐶𝑜𝑛𝑛𝑒𝑐𝑡𝑒𝑑𝑛𝑒𝑠𝑠 : a graph is connected if there exist always a
path which connects 𝑣 and 𝑤 for all 𝑣,𝑤
POSCAT Seminar 1-714 July 2014
yougatup
1 2
3
4
6
5
7
8
Graph Representation
Representation
− How can we represent a graph in the computer ?
− Adjacency matrix & list
POSCAT Seminar 1-814 July 2014
yougatup
1 2
3
4
6
5
7
8
Graph Representation
Adjacency Matrix
− Use matrix ! ( you may already know this )
− 𝐺𝑖𝑗 = 1 𝑖𝑓 𝑡ℎ𝑒𝑟𝑒 𝑖𝑠 𝑎 𝑒𝑑𝑔𝑒 𝑏𝑒𝑡𝑤𝑒𝑒𝑛 𝑣𝑒𝑟𝑡𝑒𝑥 𝑖 𝑎𝑛𝑑 𝑗
0 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒
POSCAT Seminar 1-914 July 2014
yougatup
1 2
3
4
6
5
7
8
Graph Representation
Adjacency Matrix
− Use matrix ! ( you may already know this )
− 𝐺𝑖𝑗 = 1 𝑖𝑓 𝑡ℎ𝑒𝑟𝑒 𝑖𝑠 𝑎 𝑒𝑑𝑔𝑒 𝑏𝑒𝑡𝑤𝑒𝑒𝑛 𝑣𝑒𝑟𝑡𝑒𝑥 𝑖 𝑎𝑛𝑑 𝑗
0 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒
POSCAT Seminar 1-1014 July 2014
yougatup
1 2
3
4
6
5
7
8
1 2 3 4 5 6 7 8
1 0 1 1 0 0 0 0 0
2 1 0 1 1 0 0 0 0
3 1 1 0 0 0 0 0 0
4 0 1 0 0 1 1 0 0
5 0 0 0 1 0 0 1 0
6 0 0 0 1 0 0 1 0
7 0 0 0 0 1 1 0 1
8 0 0 0 0 0 0 1 0
Graph Representation
Adjacency Matrix
− Use matrix ! ( you may already know this )
− 𝐺𝑖𝑗 = 1 𝑖𝑓 𝑡ℎ𝑒𝑟𝑒 𝑖𝑠 𝑎 𝑒𝑑𝑔𝑒 𝑏𝑒𝑡𝑤𝑒𝑒𝑛 𝑣𝑒𝑟𝑡𝑒𝑥 𝑖 𝑎𝑛𝑑 𝑗
0 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒
POSCAT Seminar 1-1114 July 2014
yougatup
1 2
3
4
6
5
7
8
1 2 3 4 5 6 7 8
1 0 1 1 0 0 0 0 0
2 1 0 1 1 0 0 0 0
3 1 1 0 0 0 0 0 0
4 0 1 0 0 1 1 0 0
5 0 0 0 1 0 0 1 0
6 0 0 0 1 0 0 1 0
7 0 0 0 0 1 1 0 1
8 0 0 0 0 0 0 1 0
Is it good ?
Graph Representation
Adjacency Matrix
− We can determine whether there is a edge between i and j directly
POSCAT Seminar 1-1214 July 2014
yougatup
1 2
3
4
6
5
7
8
1 2 3 4 5 6 7 8
1 0 1 1 0 0 0 0 0
2 1 0 1 1 0 0 0 0
3 1 1 0 0 0 0 0 0
4 0 1 0 0 1 1 0 0
5 0 0 0 1 0 0 1 0
6 0 0 0 1 0 0 1 0
7 0 0 0 0 1 1 0 1
8 0 0 0 0 0 0 1 0
Graph Representation
Adjacency Matrix
− We can determine whether there is a edge between i and j directly
POSCAT Seminar 1-1314 July 2014
yougatup
1 2
3
4
6
5
7
8
1 2 3 4 5 6 7 8
1 0 1 1 0 0 0 0 0
2 1 0 1 1 0 0 0 0
3 1 1 0 0 0 0 0 0
4 0 1 0 0 1 1 0 0
5 0 0 0 1 0 0 1 0
6 0 0 0 1 0 0 1 0
7 0 0 0 0 1 1 0 1
8 0 0 0 0 0 0 1 0Is there a edge between 3 and 4 ?
Graph Representation
Adjacency Matrix
− We can determine whether there is a edge between i and j directly
POSCAT Seminar 1-1414 July 2014
yougatup
1 2
3
4
6
5
7
8
1 2 3 4 5 6 7 8
1 0 1 1 0 0 0 0 0
2 1 0 1 1 0 0 0 0
3 1 1 0 0 0 0 0 0
4 0 1 0 0 1 1 0 0
5 0 0 0 1 0 0 1 0
6 0 0 0 1 0 0 1 0
7 0 0 0 0 1 1 0 1
8 0 0 0 0 0 0 1 0Is there a edge between 3 and 4 ? No!
Graph Representation
Adjacency Matrix
− We can determine whether there is a edge between i and j directly
− We have to maintain whole matrix to save a graph
POSCAT Seminar 1-1514 July 2014
yougatup
1 2
3
4
6
5
7
8
1 2 3 4 5 6 7 8
1 0 1 1 0 0 0 0 0
2 1 0 1 1 0 0 0 0
3 1 1 0 0 0 0 0 0
4 0 1 0 0 1 1 0 0
5 0 0 0 1 0 0 1 0
6 0 0 0 1 0 0 1 0
7 0 0 0 0 1 1 0 1
8 0 0 0 0 0 0 1 0
Graph Representation
Adjacency Matrix
− We can determine whether there is a edge between i and j directly
− We have to maintain whole matrix to save a graph
POSCAT Seminar 1-1614 July 2014
yougatup
1 2
3
4
6
5
7
8
1 2 3 4 5 6 7 8
1 0 1 0 0 0 0 0 0
2 1 0 1 1 0 0 0 0
3 0 1 0 0 0 0 0 0
4 0 1 0 0 1 0 0 0
5 0 0 0 1 0 0 1 0
6 0 0 0 0 0 0 1 0
7 0 0 0 0 1 1 0 1
8 0 0 0 0 0 0 1 0How about this ?
Graph Representation
Adjacency Matrix
− We can determine whether there is a edge between i and j directly
− We have to maintain whole matrix to save a graph
POSCAT Seminar 1-1714 July 2014
yougatup
1 2
3
4
6
5
7
8
1 2 3 4 5 6 7 8
1 0 1 0 0 0 0 0 0
2 1 0 1 1 0 0 0 0
3 0 1 0 0 0 0 0 0
4 0 1 0 0 1 0 0 0
5 0 0 0 1 0 0 1 0
6 0 0 0 0 0 0 1 0
7 0 0 0 0 1 1 0 1
8 0 0 0 0 0 0 1 0Compare the number of 1 and 0
Graph Representation
Adjacency Matrix
− We can determine whether there is a edge between i and j directly
− We have to maintain whole matrix to save a graph
POSCAT Seminar 1-1814 July 2014
yougatup
1 2
3
4
6
5
7
8
1 2 3 4 5 6 7 8
1 0 1 0 0 0 0 0 0
2 1 0 1 1 0 0 0 0
3 0 1 0 0 0 0 0 0
4 0 1 0 0 1 0 0 0
5 0 0 0 1 0 0 1 0
6 0 0 0 0 0 0 1 0
7 0 0 0 0 1 1 0 1
8 0 0 0 0 0 0 1 0Compare the number of 1 and 0Can we handle a graph with 100,000 vertex ?
Graph Representation
Adjacency Matrix
− We can determine whether there is a edge between i and j directly
− We have to maintain whole matrix to save a graph
− It takes 𝑂(𝑉) to find all the adjacent vertices for a vertex 𝑣
POSCAT Seminar 1-1914 July 2014
yougatup
1 2
3
4
6
5
7
8
1 2 3 4 5 6 7 8
1 0 1 1 0 0 0 0 0
2 1 0 1 1 0 0 0 0
3 1 1 0 0 0 0 0 0
4 0 1 0 0 1 1 0 0
5 0 0 0 1 0 0 1 0
6 0 0 0 1 0 0 1 0
7 0 0 0 0 1 1 0 1
8 0 0 0 0 0 0 1 0
Graph Representation
Adjacency Matrix
− We can determine whether there is a edge between i and j directly
− We have to maintain whole matrix to save a graph
− It takes 𝑂(𝑉) to find all the adjacent vertices for a vertex 𝑣
POSCAT Seminar 1-2014 July 2014
yougatup
1 2
3
4
6
5
7
8
1 2 3 4 5 6 7 8
1 0 1 1 0 0 0 0 0
2 1 0 1 1 0 0 0 0
3 1 1 0 0 0 0 0 0
4 0 1 0 0 1 1 0 0
5 0 0 0 1 0 0 1 0
6 0 0 0 1 0 0 1 0
7 0 0 0 0 1 1 0 1
8 0 0 0 0 0 0 1 0Give me all the adjacent vertices for vertex 3
Graph Representation
Adjacency Matrix
− We can determine whether there is a edge between i and j directly
− We have to maintain whole matrix to save a graph
− It takes 𝑂(𝑉) to find all the adjacent vertices for a vertex 𝑣
POSCAT Seminar 1-2114 July 2014
yougatup
1 2
3
4
6
5
7
8
1 2 3 4 5 6 7 8
1 0 1 1 0 0 0 0 0
2 1 0 1 1 0 0 0 0
3 1 1 0 0 0 0 0 0
4 0 1 0 0 1 1 0 0
5 0 0 0 1 0 0 1 0
6 0 0 0 1 0 0 1 0
7 0 0 0 0 1 1 0 1
8 0 0 0 0 0 0 1 0Give me all the adjacent vertices for vertex 3 It takes 𝑂(𝑉)
Graph Representation
Adjacency List
− Handle just adjacent vertices for all vertices
POSCAT Seminar 1-2214 July 2014
yougatup
1 2
3
4
6
5
7
8
Graph Representation
Adjacency List
− Handle just adjacent vertices for all vertices
POSCAT Seminar 1-2314 July 2014
yougatup
1 2
3
4
6
5
7
8
2 3
1 3 4
1 2
2 5 6
4 7
4 7
5 6 8
1
2
3
4
5
6
7
Graph Representation
Adjacency List
− Handle just adjacent vertices for all vertices
− Optimal space complexity (i.e. no redundant space)
POSCAT Seminar 1-2414 July 2014
yougatup
1 2
3
4
6
5
7
8
2 3
1 3 4
1 2
2 5 6
4 7
4 7
5 6 8
1
2
3
4
5
6
7
Graph Representation
Adjacency List
− Handle just adjacent vertices for all vertices
− Optimal space complexity (i.e. no redundant space)
− It takes O(𝑑𝑒𝑔𝑟𝑒𝑒(𝑣)) to find all the adjacent vertices for vertex 𝑣
POSCAT Seminar 1-2514 July 2014
yougatup
1 2
3
4
6
5
7
8
2 3
1 3 4
1 2
2 5 6
4 7
4 7
5 6 8
1
2
3
4
5
6
7
Graph Representation
Adjacency List
− Handle just adjacent vertices for all vertices
− Optimal space complexity (i.e. no redundant space)
− It takes O(𝑑𝑒𝑔𝑟𝑒𝑒(𝑣)) to find all the adjacent vertices for vertex 𝑣
− It take also O(𝑑𝑒𝑔𝑟𝑒𝑒(𝑣)) to determine the existence of edge
POSCAT Seminar 1-2614 July 2014
yougatup
1 2
3
4
6
5
7
8
2 3
1 3 4
1 2
2 5 6
4 7
4 7
5 6 8
1
2
3
4
5
6
7
Graph Representation
Adjacency List
− Handle just adjacent vertices for all vertices
− Optimal space complexity (i.e. no redundant space)
− It takes O(𝑑𝑒𝑔𝑟𝑒𝑒(𝑣)) to find all the adjacent vertices for vertex 𝑣
− It take also O(𝑑𝑒𝑔𝑟𝑒𝑒(𝑣)) to determine the existence of edge
POSCAT Seminar 1-2714 July 2014
yougatup
1 2
3
4
6
5
7
8
2 3
1 3 4
1 2
2 5 6
4 7
4 7
5 6 8
1
2
3
4
5
6
7Is there edge between vertex 1 and 3 ?
Graph Representation
Adjacency List
− Handle just adjacent vertices for all vertices
− Optimal space complexity (i.e. no redundant space)
− It takes O(𝑑𝑒𝑔𝑟𝑒𝑒(𝑣)) to find all the adjacent vertices for vertex 𝑣
− It take also O(𝑑𝑒𝑔𝑟𝑒𝑒(𝑣)) to determine the existence of edge
POSCAT Seminar 1-2814 July 2014
yougatup
1 2
3
4
6
5
7
8
2 3
1 3 4
1 2
2 5 6
4 7
4 7
5 6 8
1
2
3
4
5
6
7Is there edge between vertex 1 and 3 ?
We have to consider all of these
Graph Representation
Adjacency List
By the way, is it possible to manage this kind of structure ?
POSCAT Seminar 1-2914 July 2014
yougatup
1 2
3
4
6
5
7
8
2 3
1 3 4
1 2
2 5 6
4 7
4 7
5 6 8
1
2
3
4
5
6
7
Graph Representation
Adjacency List
By the way, is it possible to manage this kind of structure ?
Yes we can! Use STL vector ( I’ll give some simple lecture for you today )
POSCAT Seminar 1-3014 July 2014
yougatup
1 2
3
4
6
5
7
8
2 3
1 3 4
1 2
2 5 6
4 7
4 7
5 6 8
1
2
3
4
5
6
7
Graph Representation
Summary
− Adjacency matrix VS Adjacency list
− Existence of edge ?
− Find all the neighboring vertices ?
− Space ?
POSCAT Seminar 1-3114 July 2014
yougatup
1 2
3
4
6
5
7
8
Graph Representation
Summary
− Adjacency matrix VS Adjacency list
− Existence of edge ?
− Find all the neighboring vertices ?
− Space ?
POSCAT Seminar 1-3214 July 2014
yougatup
1 2
3
4
6
5
7
8
Which one is better ?
Graph Representation
Summary
− Adjacency matrix VS Adjacency list
− Existence of edge ?
− Find all the neighboring vertices ?
− Space ?
POSCAT Seminar 1-3314 July 2014
yougatup
1 2
3
4
6
5
7
8
Which one is better ?It depends, but usually we prefer adjacency list
Graph Representation
Summary
− Adjacency matrix VS Adjacency list
− Existence of edge ?
− Find all the neighboring vertices ?
− Space ?
POSCAT Seminar 1-3414 July 2014
yougatup
1 2
3
4
6
5
7
8
Graph Traversal
Traversal paradigm
− Basically, graph is also data structure
− In other words, we want to contain some information into a graph
− Therefore, we have to be able to traverse whole graph to find datas
− Depth First Search VS Breadth First Search
POSCAT Seminar 1-3514 July 2014
yougatup
1 2
3
4
6
5
7
8
Graph Traversal
Depth First Search
− Use stack to traverse
− Invarient : The current vertex is pointed by the top of the stack
( Discuss it later. Forget it )
POSCAT Seminar 1-3614 July 2014
yougatup
1 2
3
4
6
5
7
8
Graph Traversal
Depth First Search
− Procedure
1. Select a start vertex
2. Move to adjacent vertex we don’t visit yet
3. If there is no such a vertex, move backward
4. Go to step 2.
POSCAT Seminar 1-3714 July 2014
yougatup
1 2
3
4
6
5
7
8
Graph Traversal
Depth First Search
− Procedure
1. Select a start vertex
2. Move to adjacent vertex we don’t visit yet
3. If there is no such a vertex, move backward
4. Go to step 2.
POSCAT Seminar 1-3814 July 2014
yougatup
1 2
3
4
6
5
7
8
Graph Traversal
Depth First Search
− Procedure
1. Select a start vertex
2. Move to adjacent vertex we don’t visit yet
3. If there is no such a vertex, move backward
4. Go to step 2.
POSCAT Seminar 1-3914 July 2014
yougatup
1 2
3
4
6
5
7
8
Graph Traversal
Depth First Search
− Procedure
1. Select a start vertex
2. Move to adjacent vertex we don’t visit yet
3. If there is no such a vertex, move backward
4. Go to step 2.
POSCAT Seminar 1-4014 July 2014
yougatup
1 2
3
4
6
5
7
8
Graph Traversal
Depth First Search
− Procedure
1. Select a start vertex
2. Move to adjacent vertex we don’t visit yet
3. If there is no such a vertex, move backward
4. Go to step 2.
POSCAT Seminar 1-4114 July 2014
yougatup
1 2
3
4
6
5
7
8
Graph Traversal
Depth First Search
− Procedure
1. Select a start vertex
2. Move to adjacent vertex we don’t visit yet
3. If there is no such a vertex, move backward
4. Go to step 2.
POSCAT Seminar 1-4214 July 2014
yougatup
1 2
3
4
6
5
7
8
Graph Traversal
Depth First Search
− Procedure
1. Select a start vertex
2. Move to adjacent vertex we don’t visit yet
3. If there is no such a vertex, move backward
4. Go to step 2.
POSCAT Seminar 1-4314 July 2014
yougatup
1 2
3
4
6
5
7
8
Graph Traversal
Depth First Search
− Procedure
1. Select a start vertex
2. Move to adjacent vertex we don’t visit yet
3. If there is no such a vertex, move backward
4. Go to step 2.
POSCAT Seminar 1-4414 July 2014
yougatup
1 2
3
4
6
5
7
8
Graph Traversal
Depth First Search
− Procedure
1. Select a start vertex
2. Move to adjacent vertex we don’t visit yet
3. If there is no such a vertex, move backward
4. Go to step 2.
POSCAT Seminar 1-4514 July 2014
yougatup
1 2
3
4
6
5
7
8
Graph Traversal
Depth First Search
− Procedure
1. Select a start vertex
2. Move to adjacent vertex we don’t visit yet
3. If there is no such a vertex, move backward
4. Go to step 2.
POSCAT Seminar 1-4614 July 2014
yougatup
1 2
3
4
6
5
7
8
Graph Traversal
Depth First Search
− Procedure
1. Select a start vertex
2. Move to adjacent vertex we don’t visit yet
3. If there is no such a vertex, move backward
4. Go to step 2.
POSCAT Seminar 1-4714 July 2014
yougatup
1 2
3
4
6
5
7
8
Graph Traversal
Depth First Search
− Procedure
1. Select a start vertex
2. Move to adjacent vertex we don’t visit yet
3. If there is no such a vertex, move backward
4. Go to step 2.
POSCAT Seminar 1-4814 July 2014
yougatup
1 2
3
4
6
5
7
8
Graph Traversal
Depth First Search
− Procedure
1. Select a start vertex
2. Move to adjacent vertex we don’t visit yet
3. If there is no such a vertex, move backward
4. Go to step 2.
POSCAT Seminar 1-4914 July 2014
yougatup
1 2
3
4
6
5
7
8
Graph Traversal
Depth First Search
− Procedure
1. Select a start vertex
2. Move to adjacent vertex we don’t visit yet
3. If there is no such a vertex, move backward
4. Go to step 2.
POSCAT Seminar 1-5014 July 2014
yougatup
1 2
3
4
6
5
7
8
Graph Traversal
Depth First Search
− Procedure
1. Select a start vertex
2. Move to adjacent vertex we don’t visit yet
3. If there is no such a vertex, move backward
4. Go to step 2.
POSCAT Seminar 1-5114 July 2014
yougatup
1 2
3
4
6
5
7
8
Graph Traversal
Depth First Search
− Procedure
1. Select a start vertex
2. Move to adjacent vertex we don’t visit yet
3. If there is no such a vertex, move backward
4. Go to step 2.
POSCAT Seminar 1-5214 July 2014
yougatup
1 2
3
4
6
5
7
8
Graph Traversal
Depth First Search
− Procedure
1. Select a start vertex
2. Move to adjacent vertex we don’t visit yet
3. If there is no such a vertex, move backward
4. Go to step 2.
POSCAT Seminar 1-5314 July 2014
yougatup
1 2
3
4
6
5
7
8
Graph Traversal
Depth First Search
− Procedure
1. Select a start vertex
2. Move to adjacent vertex we don’t visit yet
3. If there is no such a vertex, move backward
4. Go to step 2.
POSCAT Seminar 1-5414 July 2014
yougatup
1 2
3
4
6
5
7
8Done ! What is the visiting sequence ?
Graph Traversal
Depth First Search
− Procedure
1. Select a start vertex
2. Move to adjacent vertex we don’t visit yet
3. If there is no such a vertex, move backward
4. Go to step 2.
POSCAT Seminar 1-5514 July 2014
yougatup
1 2
3
4
6
5
7
8Well, how can we implement it ?
Graph Traversal
Depth First Search
− Procedure
1. Select a start vertex
2. Move to adjacent vertex we don’t visit yet
3. If there is no such a vertex, move backward
4. Go to step 2.
POSCAT Seminar 1-5614 July 2014
yougatup
1 2
3
4
6
5
7
8Well, how can we implement it ?We have to implement Step 2. and 3.
Graph Traversal
Depth First Search
− Procedure
1. Select a start vertex
2. Move to adjacent vertex we don’t visit yet
3. If there is no such a vertex, move backward
4. Go to step 2.
POSCAT Seminar 1-5714 July 2014
yougatup
1 2
3
4
6
5
7
8Well, how can we implement it ?We have to implement Step 2. and 3.Step 2. is quite simple… but what about Step 3 ?We should move backward.
Graph Traversal
Depth First Search
− Procedure
1. Select a start vertex
2. Move to adjacent vertex we don’t visit yet
3. If there is no such a vertex, move backward
4. Go to step 2.
POSCAT Seminar 1-5814 July 2014
yougatup
1 2
3
4
6
5
7
8Well, how can we implement it ?We have to implement Step 2. and 3.Step 2. is quite simple… but what about Step 3 ?We should move backward.How can we remember the past vertices ?
Graph Traversal
Depth First Search
− Procedure
1. Select a start vertex
2. Move to adjacent vertex we don’t visit yet
3. If there is no such a vertex, move backward
4. Go to step 2.
POSCAT Seminar 1-5914 July 2014
yougatup
1 2
3
4
6
5
7
8Well, how can we implement it ?We have to implement Step 2. and 3.Step 2. is quite simple… but what about Step 3 ?We should move backward.How can we remember the past vertices ? STACK !
Graph Traversal
Depth First Search
− Procedure
1. Select a start vertex
2. Move to adjacent vertex we don’t visit yet
3. If there is no such a vertex, move backward
4. Go to step 2.
POSCAT Seminar 1-6014 July 2014
yougatup
1 2
3
4
6
5
7
8
Graph Traversal
Depth First Search
− Procedure
1. Select a start vertex
2. Move to adjacent vertex we don’t visit yet
3. If there is no such a vertex, move backward
4. Go to step 2.
POSCAT Seminar 1-6114 July 2014
yougatup
1 2
3
4
6
5
7
8
1
Graph Traversal
Depth First Search
− Procedure
1. Select a start vertex
2. Move to adjacent vertex we don’t visit yet
3. If there is no such a vertex, move backward
4. Go to step 2.
POSCAT Seminar 1-6214 July 2014
yougatup
1 2
3
4
6
5
7
8
1
2
Graph Traversal
Depth First Search
− Procedure
1. Select a start vertex
2. Move to adjacent vertex we don’t visit yet
3. If there is no such a vertex, move backward
4. Go to step 2.
POSCAT Seminar 1-6314 July 2014
yougatup
1 2
3
4
6
5
7
8
1
2
3
Graph Traversal
Depth First Search
− Procedure
1. Select a start vertex
2. Move to adjacent vertex we don’t visit yet
3. If there is no such a vertex, move backward
4. Go to step 2.
POSCAT Seminar 1-6414 July 2014
yougatup
1 2
3
4
6
5
7
8
1
2
Simply pop ! Then we get a vertex just before !
Graph Traversal
Depth First Search
− Procedure
1. Select a start vertex
2. Move to adjacent vertex we don’t visit yet
3. If there is no such a vertex, move backward
4. Go to step 2.
POSCAT Seminar 1-6514 July 2014
yougatup
1 2
3
4
6
5
7
8
1
2
4
Graph Traversal
Depth First Search
− Procedure
1. Select a start vertex
2. Move to adjacent vertex we don’t visit yet
3. If there is no such a vertex, move backward
4. Go to step 2.
POSCAT Seminar 1-6614 July 2014
yougatup
1 2
3
4
6
5
7
8
1
2
4
5
Graph Traversal
Depth First Search
− Procedure
1. Select a start vertex
2. Move to adjacent vertex we don’t visit yet
3. If there is no such a vertex, move backward
4. Go to step 2.
POSCAT Seminar 1-6714 July 2014
yougatup
1 2
3
4
6
5
7
8
1
2
4
5
Please do it yourself
Graph Traversal
Breadth First Search
− Use queue to traverse
− Invarient : The current vertex is pointed by front pointer
POSCAT Seminar 1-6814 July 2014
yougatup
1 2
3
4
6
5
7
8
Graph Traversal
Breadth First Search
− Procedure
1. Select a start vertex
2. Push all the adjacent vertices into queue
3. Pop to get a new vertex
4. Go to step 2.
POSCAT Seminar 1-6914 July 2014
yougatup
1 2
3
4
6
5
7
8
front
rear
Graph Traversal
Breadth First Search
− Procedure
1. Select a start vertex
2. Push all the adjacent vertices into queue
3. Pop to get a new vertex
4. Go to step 2.
POSCAT Seminar 1-7014 July 2014
yougatup
1 2
3
4
6
5
7
81
front
rear
Graph Traversal
Breadth First Search
− Procedure
1. Select a start vertex
2. Push all the adjacent vertices into queue
3. Pop to get a new vertex
4. Go to step 2.
POSCAT Seminar 1-7114 July 2014
yougatup
1 2
3
4
6
5
7
81 2 3
front
rear
Graph Traversal
Breadth First Search
− Procedure
1. Select a start vertex
2. Push all the adjacent vertices into queue
3. Pop to get a new vertex
4. Go to step 2.
POSCAT Seminar 1-7214 July 2014
yougatup
1 2
3
4
6
5
7
82 3
front
rear
Graph Traversal
Breadth First Search
− Procedure
1. Select a start vertex
2. Push all the adjacent vertices into queue
3. Pop to get a new vertex
4. Go to step 2.
POSCAT Seminar 1-7314 July 2014
yougatup
1 2
3
4
6
5
7
82 3
front
rear
4
Graph Traversal
Breadth First Search
− Procedure
1. Select a start vertex
2. Push all the adjacent vertices into queue
3. Pop to get a new vertex
4. Go to step 2.
POSCAT Seminar 1-7414 July 2014
yougatup
1 2
3
4
6
5
7
83
front
rear
4
Graph Traversal
Breadth First Search
− Procedure
1. Select a start vertex
2. Push all the adjacent vertices into queue
3. Pop to get a new vertex
4. Go to step 2.
POSCAT Seminar 1-7514 July 2014
yougatup
1 2
3
4
6
5
7
8
front
rear
4
Graph Traversal
Breadth First Search
− Procedure
1. Select a start vertex
2. Push all the adjacent vertices into queue
3. Pop to get a new vertex
4. Go to step 2.
POSCAT Seminar 1-7614 July 2014
yougatup
1 2
3
4
6
5
7
8
front
rear
4 5 6
Graph Traversal
Breadth First Search
− Procedure
1. Select a start vertex
2. Push all the adjacent vertices into queue
3. Pop to get a new vertex
4. Go to step 2.
POSCAT Seminar 1-7714 July 2014
yougatup
1 2
3
4
6
5
7
8
front
rear
5 6
Graph Traversal
Breadth First Search
− Procedure
1. Select a start vertex
2. Push all the adjacent vertices into queue
3. Pop to get a new vertex
4. Go to step 2.
POSCAT Seminar 1-7814 July 2014
yougatup
1 2
3
4
6
5
7
8
front
rear
5 6 7
Graph Traversal
Breadth First Search
− Procedure
1. Select a start vertex
2. Push all the adjacent vertices into queue
3. Pop to get a new vertex
4. Go to step 2.
POSCAT Seminar 1-7914 July 2014
yougatup
1 2
3
4
6
5
7
8
front
rear
6 7
Graph Traversal
Breadth First Search
− Procedure
1. Select a start vertex
2. Push all the adjacent vertices into queue
3. Pop to get a new vertex
4. Go to step 2.
POSCAT Seminar 1-8014 July 2014
yougatup
1 2
3
4
6
5
7
8
front
rear
7
Graph Traversal
Breadth First Search
− Procedure
1. Select a start vertex
2. Push all the adjacent vertices into queue
3. Pop to get a new vertex
4. Go to step 2.
POSCAT Seminar 1-8114 July 2014
yougatup
1 2
3
4
6
5
7
8
front
rear
7 8
Graph Traversal
Breadth First Search
− Procedure
1. Select a start vertex
2. Push all the adjacent vertices into queue
3. Pop to get a new vertex
4. Go to step 2.
POSCAT Seminar 1-8214 July 2014
yougatup
1 2
3
4
6
5
7
8
front
rear
8
Graph Traversal
Breadth First Search
− Procedure
1. Select a start vertex
2. Push all the adjacent vertices into queue
3. Pop to get a new vertex
4. Go to step 2.
POSCAT Seminar 1-8314 July 2014
yougatup
1 2
3
4
6
5
7
8
front
rear
Graph Traversal
Breadth First Search
− Procedure
1. Select a start vertex
2. Push all the adjacent vertices into queue
3. Pop to get a new vertex
4. Go to step 2.
POSCAT Seminar 1-8414 July 2014
yougatup
1 2
3
4
6
5
7
8
front
rear
Done. Queue is empty.
Graph Traversal
DFS vs BFS
− We have to choose what search technique to use
− Example ?
POSCAT Seminar 1-8514 July 2014
yougatup
1 2
3
4
6
5
7
8
Flood Fill
Problem
Given a map, categorize each cell as inner or outer cell
Suppose that the outer-most cell is always 0
POSCAT Seminar 1-8614 July 2014
yougatup
0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0
0 0 1 1 1 1 0 0 0 0
0 0 1 0 0 0 1 0 0 0
0 0 1 0 0 0 0 1 0 0
0 0 1 0 0 0 0 0 1 0
0 0 1 0 0 0 0 0 1 0
0 0 0 1 0 1 0 0 1 0
0 0 0 0 1 0 1 1 1 0
0 0 0 0 0 0 0 0 0 0
Flood Fill
Problem
Given a map, categorize each cell as inner or outer cell
Suppose that the outer-most cell is always 0
POSCAT Seminar 1-8714 July 2014
yougatup
0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0
0 0 1 1 1 1 0 0 0 0
0 0 1 0 0 0 1 0 0 0
0 0 1 0 0 0 0 1 0 0
0 0 1 0 0 0 0 0 1 0
0 0 1 0 0 0 0 0 1 0
0 0 0 1 0 1 0 0 1 0
0 0 0 0 1 0 1 1 1 0
0 0 0 0 0 0 0 0 0 0
Flood Fill
Problem
Any idea ?
POSCAT Seminar 1-8814 July 2014
yougatup
0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0
0 0 1 1 1 1 0 0 0 0
0 0 1 0 0 0 1 0 0 0
0 0 1 0 0 0 0 1 0 0
0 0 1 0 0 0 0 0 1 0
0 0 1 0 0 0 0 0 1 0
0 0 0 1 0 1 0 0 1 0
0 0 0 0 1 0 1 1 1 0
0 0 0 0 0 0 0 0 0 0
Flood Fill
Problem
Any idea ? Modeling it as a graph !
POSCAT Seminar 1-8914 July 2014
yougatup
0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0
0 0 1 1 1 1 0 0 0 0
0 0 1 0 0 0 1 0 0 0
0 0 1 0 0 0 0 1 0 0
0 0 1 0 0 0 0 0 1 0
0 0 1 0 0 0 0 0 1 0
0 0 0 1 0 1 0 0 1 0
0 0 0 0 1 0 1 1 1 0
0 0 0 0 0 0 0 0 0 0
Flood Fill
Problem
Any idea ? Modeling it as a graph !
Each cell is considered as a vertex. How about edge ?
POSCAT Seminar 1-9014 July 2014
yougatup
0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0
0 0 1 1 1 1 0 0 0 0
0 0 1 0 0 0 1 0 0 0
0 0 1 0 0 0 0 1 0 0
0 0 1 0 0 0 0 0 1 0
0 0 1 0 0 0 0 0 1 0
0 0 0 1 0 1 0 0 1 0
0 0 0 0 1 0 1 1 1 0
0 0 0 0 0 0 0 0 0 0
Flood Fill
Problem
Any idea ? Modeling it as a graph !
there is a edge if two vertices are adjacent in the map
POSCAT Seminar 1-9114 July 2014
yougatup
0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0
0 0 1 1 1 1 0 0 0 0
0 0 1 0 0 0 1 0 0 0
0 0 1 0 0 0 0 1 0 0
0 0 1 0 0 0 0 0 1 0
0 0 1 0 0 0 0 0 1 0
0 0 0 1 0 1 0 0 1 0
0 0 0 0 1 0 1 1 1 0
0 0 0 0 0 0 0 0 0 0
Flood Fill
Problem
Perform DFS or BFS starting from the outermost cell !
However, we never move to the ‘1’ cell
POSCAT Seminar 1-9214 July 2014
yougatup
0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0
0 0 1 1 1 1 0 0 0 0
0 0 1 0 0 0 1 0 0 0
0 0 1 0 0 0 0 1 0 0
0 0 1 0 0 0 0 0 1 0
0 0 1 0 0 0 0 0 1 0
0 0 0 1 0 1 0 0 1 0
0 0 0 0 1 0 1 1 1 0
0 0 0 0 0 0 0 0 0 0
Flood Fill
Problem
BFS ?
POSCAT Seminar 1-9314 July 2014
yougatup
0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0
0 0 1 1 1 1 0 0 0 0
0 0 1 0 0 0 1 0 0 0
0 0 1 0 0 0 0 1 0 0
0 0 1 0 0 0 0 0 1 0
0 0 1 0 0 0 0 0 1 0
0 0 0 1 0 1 0 0 1 0
0 0 0 0 1 0 1 1 1 0
0 0 0 0 0 0 0 0 0 0
Flood Fill
Problem
BFS ?
POSCAT Seminar 1-9414 July 2014
yougatup
0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0
0 0 1 1 1 1 0 0 0 0
0 0 1 0 0 0 1 0 0 0
0 0 1 0 0 0 0 1 0 0
0 0 1 0 0 0 0 0 1 0
0 0 1 0 0 0 0 0 1 0
0 0 0 1 0 1 0 0 1 0
0 0 0 0 1 0 1 1 1 0
0 0 0 0 0 0 0 0 0 0
Flood Fill
Problem
BFS ?
POSCAT Seminar 1-9514 July 2014
yougatup
0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0
0 0 1 1 1 1 0 0 0 0
0 0 1 0 0 0 1 0 0 0
0 0 1 0 0 0 0 1 0 0
0 0 1 0 0 0 0 0 1 0
0 0 1 0 0 0 0 0 1 0
0 0 0 1 0 1 0 0 1 0
0 0 0 0 1 0 1 1 1 0
0 0 0 0 0 0 0 0 0 0
Flood Fill
Problem
BFS ?
POSCAT Seminar 1-9614 July 2014
yougatup
0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0
0 0 1 1 1 1 0 0 0 0
0 0 1 0 0 0 1 0 0 0
0 0 1 0 0 0 0 1 0 0
0 0 1 0 0 0 0 0 1 0
0 0 1 0 0 0 0 0 1 0
0 0 0 1 0 1 0 0 1 0
0 0 0 0 1 0 1 1 1 0
0 0 0 0 0 0 0 0 0 0
Flood Fill
Problem
BFS ?
POSCAT Seminar 1-9714 July 2014
yougatup
0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0
0 0 1 1 1 1 0 0 0 0
0 0 1 0 0 0 1 0 0 0
0 0 1 0 0 0 0 1 0 0
0 0 1 0 0 0 0 0 1 0
0 0 1 0 0 0 0 0 1 0
0 0 0 1 0 1 0 0 1 0
0 0 0 0 1 0 1 1 1 0
0 0 0 0 0 0 0 0 0 0
Flood Fill
Problem
BFS ?
POSCAT Seminar 1-9814 July 2014
yougatup
0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0
0 0 1 1 1 1 0 0 0 0
0 0 1 0 0 0 1 0 0 0
0 0 1 0 0 0 0 1 0 0
0 0 1 0 0 0 0 0 1 0
0 0 1 0 0 0 0 0 1 0
0 0 0 1 0 1 0 0 1 0
0 0 0 0 1 0 1 1 1 0
0 0 0 0 0 0 0 0 0 0
Flood Fill
Problem
BFS ?
POSCAT Seminar 1-9914 July 2014
yougatup
0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0
0 0 1 1 1 1 0 0 0 0
0 0 1 0 0 0 1 0 0 0
0 0 1 0 0 0 0 1 0 0
0 0 1 0 0 0 0 0 1 0
0 0 1 0 0 0 0 0 1 0
0 0 0 1 0 1 0 0 1 0
0 0 0 0 1 0 1 1 1 0
0 0 0 0 0 0 0 0 0 0
Flood Fill
Problem
BFS ?
POSCAT Seminar 1-10014 July 2014
yougatup
0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0
0 0 1 1 1 1 0 0 0 0
0 0 1 0 0 0 1 0 0 0
0 0 1 0 0 0 0 1 0 0
0 0 1 0 0 0 0 0 1 0
0 0 1 0 0 0 0 0 1 0
0 0 0 1 0 1 0 0 1 0
0 0 0 0 1 0 1 1 1 0
0 0 0 0 0 0 0 0 0 0
Flood Fill
Problem
BFS ?
POSCAT Seminar 1-10114 July 2014
yougatup
0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0
0 0 1 1 1 1 0 0 0 0
0 0 1 0 0 0 1 0 0 0
0 0 1 0 0 0 0 1 0 0
0 0 1 0 0 0 0 0 1 0
0 0 1 0 0 0 0 0 1 0
0 0 0 1 0 1 0 0 1 0
0 0 0 0 1 0 1 1 1 0
0 0 0 0 0 0 0 0 0 0
Flood Fill
Problem
BFS ?
POSCAT Seminar 1-10214 July 2014
yougatup
0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0
0 0 1 1 1 1 0 0 0 0
0 0 1 0 0 0 1 0 0 0
0 0 1 0 0 0 0 1 0 0
0 0 1 0 0 0 0 0 1 0
0 0 1 0 0 0 0 0 1 0
0 0 0 1 0 1 0 0 1 0
0 0 0 0 1 0 1 1 1 0
0 0 0 0 0 0 0 0 0 0
Flood Fill
Problem
BFS ?
POSCAT Seminar 1-10314 July 2014
yougatup
0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0
0 0 1 1 1 1 0 0 0 0
0 0 1 0 0 0 1 0 0 0
0 0 1 0 0 0 0 1 0 0
0 0 1 0 0 0 0 0 1 0
0 0 1 0 0 0 0 0 1 0
0 0 0 1 0 1 0 0 1 0
0 0 0 0 1 0 1 1 1 0
0 0 0 0 0 0 0 0 0 0
Flood Fill
Problem
BFS ?
POSCAT Seminar 1-10414 July 2014
yougatup
0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0
0 0 1 1 1 1 0 0 0 0
0 0 1 0 0 0 1 0 0 0
0 0 1 0 0 0 0 1 0 0
0 0 1 0 0 0 0 0 1 0
0 0 1 0 0 0 0 0 1 0
0 0 0 1 0 1 0 0 1 0
0 0 0 0 1 0 1 1 1 0
0 0 0 0 0 0 0 0 0 0
Flood Fill
Problem
Can we apply this flood fill algorithm to solve a problem ?
POSCAT Seminar 1-10514 July 2014
yougatup
0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0
0 0 1 1 1 1 0 0 0 0
0 0 1 0 0 0 1 0 0 0
0 0 1 0 0 0 0 1 0 0
0 0 1 0 0 0 0 0 1 0
0 0 1 0 0 0 0 0 1 0
0 0 0 1 0 1 0 0 1 0
0 0 0 0 1 0 1 1 1 0
0 0 0 0 0 0 0 0 0 0
Flood Fill
Problem
Find the minimum length from the start cell and the end cell
POSCAT Seminar 1-10614 July 2014
yougatup
0 0 0 0 0 0 0 0 0 0
1 1 0 0 0 1 0 0 0 0
0 1 1 1 1 0 0 0 1 0
0 1 1 0 0 0 1 0 1 0
0 1 1 0 0 0 1 1 0 0
0 1 1 0 1 0 0 0 1 0
0 1 1 0 1 0 1 0 1 0
0 1 0 0 1 0 1 0 0 0
0 1 1 0 1 0 1 1 1 0
0 0 0 0 0 0 0 1 0 0
Flood Fill
Problem
Find the minimum length from the start cell and the end cell
Here !
POSCAT Seminar 1-10714 July 2014
yougatup
0 0 0 0 0 0 0 0 0 0
1 1 0 0 0 1 0 0 0 0
0 1 1 1 1 0 0 0 1 0
0 1 1 0 0 0 1 0 1 0
0 1 1 0 0 0 1 1 0 0
0 1 1 0 1 0 0 0 1 0
0 1 1 0 1 0 1 0 1 0
0 1 0 0 1 0 1 0 0 0
0 1 1 0 1 0 1 1 1 0
0 0 0 0 0 0 0 1 0 0
Flood Fill
Problem
Find the minimum length from the start cell and the end cell
However, there is another path which is longer !
POSCAT Seminar 1-10814 July 2014
yougatup
0 0 0 0 0 0 0 0 0 0
1 1 0 0 0 1 0 0 0 0
0 1 1 1 1 0 0 0 1 0
0 1 1 0 0 0 1 0 1 0
0 1 1 0 0 0 1 1 0 0
0 1 1 0 1 0 0 0 1 0
0 1 1 0 1 0 1 0 1 0
0 1 0 0 1 0 1 0 0 0
0 1 1 0 1 0 1 1 1 0
0 0 0 0 0 0 0 1 0 0
Flood Fill
Problem
How can we find the optimal path ?
POSCAT Seminar 1-10914 July 2014
yougatup
0 0 0 0 0 0 0 0 0 0
1 1 0 0 0 1 0 0 0 0
0 1 1 1 1 0 0 0 1 0
0 1 1 0 0 0 1 0 1 0
0 1 1 0 0 0 1 1 0 0
0 1 1 0 1 0 0 0 1 0
0 1 1 0 1 0 1 0 1 0
0 1 0 0 1 0 1 0 0 0
0 1 1 0 1 0 1 1 1 0
0 0 0 0 0 0 0 1 0 0
Flood Fill
Problem
How can we find the optimal path ?
Let 𝑃 𝑖, 𝑗 = 𝑡ℎ𝑒 𝑠ℎ𝑜𝑟𝑡𝑒𝑠𝑡 𝑙𝑒𝑛𝑔𝑡ℎ 𝑡𝑜 (𝑖, 𝑗), and fill it by using flood fill
POSCAT Seminar 1-11014 July 2014
yougatup
0 0 0 0 0 0 0 0 0 0
1 1 0 0 0 1 0 0 0 0
0 1 1 1 1 0 0 0 1 0
0 1 1 0 0 0 1 0 1 0
0 1 1 0 0 0 1 1 0 0
0 1 1 0 1 0 0 0 1 0
0 1 1 0 1 0 1 0 1 0
0 1 0 0 1 0 1 0 0 0
0 1 1 0 1 0 1 1 1 0
0 0 0 0 0 0 0 1 0 0
Flood Fill
Problem
Which traversal is better? DFS or BFS ?
POSCAT Seminar 1-11114 July 2014
yougatup
0 0 0 0 0 0 0 0 0 0
1 1 0 0 0 1 0 0 0 0
0 1 1 1 1 0 0 0 1 0
0 1 1 0 0 0 1 0 1 0
0 1 1 0 0 0 1 1 0 0
0 1 1 0 1 0 0 0 1 0
0 1 1 0 1 0 1 0 1 0
0 1 0 0 1 0 1 0 0 0
0 1 1 0 1 0 1 1 1 0
0 0 0 0 0 0 0 1 0 0
Flood Fill
Problem
Which traversal is better? DFS or BFS ?
Definitely, BFS is better in this time. Think about the reason
POSCAT Seminar 1-11214 July 2014
yougatup
0 0 0 0 0 0 0 0 0 0
1 1 0 0 0 1 0 0 0 0
0 1 1 1 1 0 0 0 1 0
0 1 1 0 0 0 1 0 1 0
0 1 1 0 0 0 1 1 0 0
0 1 1 0 1 0 0 0 1 0
0 1 1 0 1 0 1 0 1 0
0 1 0 0 1 0 1 0 0 0
0 1 1 0 1 0 1 1 1 0
0 0 0 0 0 0 0 1 0 0
Flood Fill
Problem
How can we find the optimal path ?
Let 𝑃 𝑖, 𝑗 = 𝑡ℎ𝑒 𝑠ℎ𝑜𝑟𝑡𝑒𝑠𝑡 𝑙𝑒𝑛𝑔𝑡ℎ 𝑡𝑜 (𝑖, 𝑗), and fill it by using flood fill
POSCAT Seminar 1-11314 July 2014
yougatup
0 0 0 0 0 0 0 0 0 0
1 1 0 0 0 1 0 0 0 0
0 1 1 1 1 0 0 0 1 0
0 1 1 0 0 0 1 0 1 0
0 1 1 0 0 0 1 1 0 0
0 1 1 0 1 0 0 0 1 0
0 1 1 0 1 0 1 0 1 0
0 1 0 0 1 0 1 0 0 0
0 1 1 0 1 0 1 1 1 0
0 0 0 0 0 0 0 1 0 0
Flood Fill
Problem
How can we find the optimal path ?
Let 𝑃 𝑖, 𝑗 = 𝑡ℎ𝑒 𝑠ℎ𝑜𝑟𝑡𝑒𝑠𝑡 𝑙𝑒𝑛𝑔𝑡ℎ 𝑡𝑜 (𝑖, 𝑗), and fill it by using flood fill
POSCAT Seminar 1-11414 July 2014
yougatup
0 0 0 0 0 0 0 0 0 0
1 1 0 0 0 1 0 0 0 0
0 1 1 1 1 0 0 0 1 0
0 1 1 0 0 0 1 0 1 0
0 1 1 0 0 0 1 1 0 0
0 1 1 0 1 0 0 0 1 0
0 1 1 0 1 0 1 0 1 0
0 1 0 0 1 0 1 0 0 0
1 1 1 0 1 0 1 1 1 0
0 1 0 0 0 0 0 1 0 0
Flood Fill
Problem
How can we find the optimal path ?
Let 𝑃 𝑖, 𝑗 = 𝑡ℎ𝑒 𝑠ℎ𝑜𝑟𝑡𝑒𝑠𝑡 𝑙𝑒𝑛𝑔𝑡ℎ 𝑡𝑜 (𝑖, 𝑗), and fill it by using flood fill
POSCAT Seminar 1-11514 July 2014
yougatup
0 0 0 0 0 0 0 0 0 0
1 1 0 0 0 1 0 0 0 0
0 1 1 1 1 0 0 0 1 0
0 1 1 0 0 0 1 0 1 0
0 1 1 0 0 0 1 1 0 0
0 1 1 0 1 0 0 0 1 0
0 1 1 0 1 0 1 0 1 0
2 1 0 0 1 0 1 0 0 0
1 1 1 0 1 0 1 1 1 0
0 1 2 0 0 0 0 1 0 0
Flood Fill
Problem
How can we find the optimal path ?
Let 𝑃 𝑖, 𝑗 = 𝑡ℎ𝑒 𝑠ℎ𝑜𝑟𝑡𝑒𝑠𝑡 𝑙𝑒𝑛𝑔𝑡ℎ 𝑡𝑜 (𝑖, 𝑗), and fill it by using flood fill
POSCAT Seminar 1-11614 July 2014
yougatup
0 0 0 0 0 0 0 0 0 0
1 1 0 0 0 1 0 0 0 0
0 1 1 1 1 0 0 0 1 0
0 1 1 0 0 0 1 0 1 0
0 1 1 0 0 0 1 1 0 0
0 1 1 0 1 0 0 0 1 0
3 1 1 0 1 0 1 0 1 0
2 1 0 0 1 0 1 0 0 0
1 1 1 0 1 0 1 1 1 0
0 1 2 3 0 0 0 1 0 0
Flood Fill
Problem
How can we find the optimal path ?
Let 𝑃 𝑖, 𝑗 = 𝑡ℎ𝑒 𝑠ℎ𝑜𝑟𝑡𝑒𝑠𝑡 𝑙𝑒𝑛𝑔𝑡ℎ 𝑡𝑜 (𝑖, 𝑗), and fill it by using flood fill
POSCAT Seminar 1-11714 July 2014
yougatup
0 0 0 0 0 0 0 0 0 0
1 1 0 0 0 1 0 0 0 0
0 1 1 1 1 0 0 0 1 0
0 1 1 0 0 0 1 0 1 0
0 1 1 0 0 0 1 1 0 0
4 1 1 0 1 0 0 0 1 0
3 1 1 0 1 0 1 0 1 0
2 1 0 0 1 0 1 0 0 0
1 1 1 4 1 0 1 1 1 0
0 1 2 3 4 0 0 1 0 0
Flood Fill
Problem
How can we find the optimal path ?
Let 𝑃 𝑖, 𝑗 = 𝑡ℎ𝑒 𝑠ℎ𝑜𝑟𝑡𝑒𝑠𝑡 𝑙𝑒𝑛𝑔𝑡ℎ 𝑡𝑜 (𝑖, 𝑗), and fill it by using flood fill
POSCAT Seminar 1-11814 July 2014
yougatup
0 0 0 0 0 0 0 0 0 0
1 1 0 0 0 1 0 0 0 0
0 1 1 1 1 0 0 0 1 0
0 1 1 0 0 0 1 0 1 0
5 1 1 0 0 0 1 1 0 0
4 1 1 0 1 0 0 0 1 0
3 1 1 0 1 0 1 0 1 0
2 1 0 5 1 0 1 0 0 0
1 1 1 4 1 0 1 1 1 0
0 1 2 3 4 5 0 1 0 0
Flood Fill
Problem
How can we find the optimal path ?
Let 𝑃 𝑖, 𝑗 = 𝑡ℎ𝑒 𝑠ℎ𝑜𝑟𝑡𝑒𝑠𝑡 𝑙𝑒𝑛𝑔𝑡ℎ 𝑡𝑜 (𝑖, 𝑗), and fill it by using flood fill
POSCAT Seminar 1-11914 July 2014
yougatup
0 0 0 0 0 0 0 0 0 0
1 1 0 0 0 1 0 0 0 0
0 1 1 1 1 0 0 0 1 0
6 1 1 0 0 0 1 0 1 0
5 1 1 0 0 0 1 1 0 0
4 1 1 0 1 0 0 0 1 0
3 1 1 6 1 0 1 0 1 0
2 1 6 5 1 0 1 0 0 0
1 1 1 4 1 6 1 1 1 0
0 1 2 3 4 5 6 1 0 0
Flood Fill
Problem
How can we find the optimal path ?
Let 𝑃 𝑖, 𝑗 = 𝑡ℎ𝑒 𝑠ℎ𝑜𝑟𝑡𝑒𝑠𝑡 𝑙𝑒𝑛𝑔𝑡ℎ 𝑡𝑜 (𝑖, 𝑗), and fill it by using flood fill
POSCAT Seminar 1-12014 July 2014
yougatup
0 0 0 0 0 0 0 0 0 0
1 1 0 0 0 1 0 0 0 0
7 1 1 1 1 0 0 0 1 0
6 1 1 0 0 0 1 0 1 0
5 1 1 8 0 0 1 1 0 0
4 1 1 7 1 0 0 0 1 0
3 1 1 6 1 8 1 0 1 0
2 1 6 5 1 7 1 0 0 0
1 1 1 4 1 6 1 1 1 0
0 1 2 3 4 5 6 1 0 0
Flood Fill
Problem
How can we find the optimal path ?
Let 𝑃 𝑖, 𝑗 = 𝑡ℎ𝑒 𝑠ℎ𝑜𝑟𝑡𝑒𝑠𝑡 𝑙𝑒𝑛𝑔𝑡ℎ 𝑡𝑜 (𝑖, 𝑗), and fill it by using flood fill
POSCAT Seminar 1-12114 July 2014
yougatup
0 0 0 0 0 0 0 0 0 0
1 1 0 0 0 1 0 0 0 0
7 1 1 1 1 0 0 0 1 0
6 1 1 9 0 0 1 0 1 0
5 1 1 8 9 0 1 1 0 0
4 1 1 7 1 9 0 0 1 0
3 1 1 6 1 8 1 0 1 0
2 1 6 5 1 7 1 0 0 0
1 1 1 4 1 6 1 1 1 0
0 1 2 3 4 5 6 1 0 0
Fill it yourself
Connected Component
Problem
Given a graph, determine whether this graph is connected or not
POSCAT Seminar 1-12214 July 2014
yougatup
1 2
3
4
6
5
7
8
Connected Component
Problem
Given a graph, determine whether this graph is connected or not
POSCAT Seminar 1-12314 July 2014
yougatup
1 2
3
4
6
5
7
8
Is it connected ?
Connected Component
Problem
Given a graph, determine whether this graph is connected or not
POSCAT Seminar 1-12414 July 2014
yougatup
1 2
3
4
6
5
7
8
Is it connected ? NO!
Connected Component
Problem
Given a graph, determine whether this graph is connected or not
One traversal from any start vertex is enough. Easy.
POSCAT Seminar 1-12514 July 2014
yougatup
1 2
3
4
6
5
7
8
Connected Component
Problem
Given a graph, determine whether this graph is connected or not
One traversal from any start vertex is enough. Easy.
What if this graph is directed graph ?
POSCAT Seminar 1-12614 July 2014
yougatup
1 2
3
4
6
5
7
8
Connected Component
Problem
Given a graph, determine whether this graph is connected or not
One traversal from any start vertex is enough. Easy.
What if this graph is directed graph ?
POSCAT Seminar 1-12714 July 2014
yougatup
1 2
3
4
6
5
7
8
Strongly Connected Component
Problem
Given a direct graph, Find Strongly Connected Component
SCC : a graph is strongly connected if every vertex is reachable from every other vertex
POSCAT Seminar 1-12814 July 2014
yougatup
1 2
3
4
6
5
7
8
Strongly Connected Component
Problem
Given a direct graph, Find Strongly Connected Component
SCC : a graph is strongly connected if every vertex is reachable from every other vertex
POSCAT Seminar 1-12914 July 2014
yougatup
1 2
3
4
6
5
7
8
Strongly Connected Component
Problem
Suppose that there is strongly connected component. ( just imagine )
POSCAT Seminar 1-13014 July 2014
yougatup
Strongly Connected Component
Problem
Suppose that there is strongly connected component. ( just imagine )
Then we can make a graph consisting of “big” vertex
( big vertex makes a strongly connected component )
Let me call this graph as a meta graph
POSCAT Seminar 1-13114 July 2014
yougatup
Strongly Connected Component
Problem
Trivially, this meta graph has no cycle.
If not, they had to shrink into a vertex
POSCAT Seminar 1-13214 July 2014
yougatup
Strongly Connected Component
Problem
Trivially, this meta graph has no cycle.
If not, they had to shrink into a vertex
POSCAT Seminar 1-13314 July 2014
yougatup
Strongly Connected Component
Problem
Trivially, this meta graph has no cycle.
If not, they had to shrink into a vertex
POSCAT Seminar 1-13414 July 2014
yougatup
Strongly Connected Component
Problem
Think about the leaf big node. (a big node which has no outer edge)
Choose one vertex from the leaf big node.
POSCAT Seminar 1-13514 July 2014
yougatup
Strongly Connected Component
Problem
If we perform traverse started from the vertex, then it will traverse
whole big node because the big node is SCC
If not, contradiction.
POSCAT Seminar 1-13614 July 2014
yougatup
We can find all the nodes
Strongly Connected Component
Problem
Therefore, we can derive a beautiful algorithm !
1. Find a vertex within the leaf big node
2. Traverse started from the vertex we get a SCC
3. Remove that big node
4. Repeat it
POSCAT Seminar 1-13714 July 2014
yougatup
Strongly Connected Component
Problem
Therefore, we can derive a beautiful algorithm !
1. Find a vertex within the leaf big node
2. Traverse started from the vertex
3. Remove that big node
4. Repeat it
POSCAT Seminar 1-13814 July 2014
yougatup
Strongly Connected Component
Problem
Therefore, we can derive a beautiful algorithm !
1. Find a vertex within the leaf big node
2. Traverse started from the vertex
3. Remove that big node
4. Repeat it
POSCAT Seminar 1-13914 July 2014
yougatup
Strongly Connected Component
Problem
Therefore, we can derive a beautiful algorithm !
1. Find a vertex within the leaf big node
2. Traverse started from the vertex
3. Remove that big node
4. Repeat it
POSCAT Seminar 1-14014 July 2014
yougatup
Strongly Connected Component
Problem
Therefore, we can derive a beautiful algorithm !
1. Find a vertex within the leaf big node
2. Traverse started from the vertex
3. Remove that big node
4. Repeat it
POSCAT Seminar 1-14114 July 2014
yougatup
Strongly Connected Component
Problem
Therefore, we can derive a beautiful algorithm !
1. Find a vertex within the leaf big node
2. Traverse started from the vertex
3. Remove that big node
4. Repeat it
POSCAT Seminar 1-14214 July 2014
yougatup
Strongly Connected Component
Problem
Therefore, we can derive a beautiful algorithm !
1. Find a vertex within the leaf big node
2. Traverse started from the vertex
3. Remove that big node
4. Repeat it
POSCAT Seminar 1-14314 July 2014
yougatup
Strongly Connected Component
Problem
Therefore, we can derive a beautiful algorithm !
1. Find a vertex within the leaf big node
2. Traverse started from the vertex
3. Remove that big node
4. Repeat it
POSCAT Seminar 1-14414 July 2014
yougatup
Strongly Connected Component
Problem
Therefore, we can derive a beautiful algorithm !
1. Find a vertex within the leaf big node
2. Traverse started from the vertex
3. Remove that big node
4. Repeat it
POSCAT Seminar 1-14514 July 2014
yougatup
Strongly Connected Component
Problem
Therefore, we can derive a beautiful algorithm !
1. Find a vertex within the leaf big node
2. Traverse started from the vertex
3. Remove that big node
4. Repeat it
POSCAT Seminar 1-14614 July 2014
yougatup
Strongly Connected Component
Problem
Therefore, we can derive a beautiful algorithm !
1. Find a vertex within the leaf big node
2. Traverse started from the vertex
3. Remove that big node
4. Repeat it
POSCAT Seminar 1-14714 July 2014
yougatup
Strongly Connected Component
Problem
Therefore, we can derive a beautiful algorithm !
1. Find a vertex within the leaf big node
2. Traverse started from the vertex
3. Remove that big node
4. Repeat it
Is it correct ?
POSCAT Seminar 1-14814 July 2014
yougatup
Strongly Connected Component
Problem
Therefore, we can derive a beautiful algorithm !
1. Find a vertex within the leaf big node
2. Traverse started from the vertex
3. Remove that big node
4. Repeat it
Is it correct ? Sure
POSCAT Seminar 1-14914 July 2014
yougatup
Strongly Connected Component
Problem
Therefore, we can derive a beautiful algorithm !
1. Find a vertex within the leaf big node
2. Traverse started from the vertex
3. Remove that big node
4. Repeat it
However, how can we find a vertex in step 1 ?
We don’t know whether a vertex is contained in the leaf big node or not
Actually, it is impossible
POSCAT Seminar 1-15014 July 2014
yougatup
Strongly Connected Component
Problem
OTL …
POSCAT Seminar 1-15114 July 2014
yougatup
Strongly Connected Component
Problem
OTL …
We can choose a vertex, but we don’t know whether it is contained in
the leaf big node
POSCAT Seminar 1-15214 July 2014
yougatup
Strongly Connected Component
Problem
OTL …
Then, just traverse it ! Then it will traverse all the big node reached
from the vertex
POSCAT Seminar 1-15314 July 2014
yougatup
Strongly Connected Component
Problem
OTL …
And reverse the whole graph !
POSCAT Seminar 1-15414 July 2014
yougatup
Strongly Connected Component
Problem
OTL …
And reverse the whole graph !
Then our start vertex becomes the vertex contained in the leaf node !
POSCAT Seminar 1-15514 July 2014
yougatup
We don’t know its existencebecause we don’t traverse it
Strongly Connected Component
Problem
OTL …
And reverse the whole graph !
Then our start vertex becomes the vertex contained in the leaf node !
It is strong evidence that we can perform our algorithm
POSCAT Seminar 1-15614 July 2014
yougatup
We don’t know its existencebecause we don’t traverse it
Strongly Connected Component
Overall procedure
1. Choose a start vertex
2. Perform DFS, and write the exit time for all the vertices
3. After doing DFS, reverse whole graph
4. Perform DFS or BFS starting from a vertex whose exit time is the
largest
5. All the vertices gathering in step 4 performs a SCC
POSCAT Seminar 1-15714 July 2014
yougatup
Strongly Connected Component
Overall procedure
POSCAT Seminar 1-15814 July 2014
yougatup
1 2
3
4
6
5
7
8
Strongly Connected Component
Overall procedure
Pick a start vertex
POSCAT Seminar 1-15914 July 2014
yougatup
1 2
3
4
6
5
7
8
Strongly Connected Component
Overall procedure
Pick a start vertex
Perform DFS and write the exit time
POSCAT Seminar 1-16014 July 2014
yougatup
1 2
3
4
6
5
7
8
Strongly Connected Component
Overall procedure
Pick a start vertex
Perform DFS and write the exit time
POSCAT Seminar 1-16114 July 2014
yougatup
1 2
3
4
6
5
7
8
Strongly Connected Component
Overall procedure
Pick a start vertex
Perform DFS and write the exit time
POSCAT Seminar 1-16214 July 2014
yougatup
1 2
3
4
6
5
7
8
Strongly Connected Component
Overall procedure
Pick a start vertex
Perform DFS and write the exit time
POSCAT Seminar 1-16314 July 2014
yougatup
1 2
3
4
6
5
7
8
I can’t traverse further. Exit !
Strongly Connected Component
Overall procedure
Pick a start vertex
Perform DFS and write the exit time
POSCAT Seminar 1-16414 July 2014
yougatup
1 2
3
4
6
5
7
8
1
Strongly Connected Component
Overall procedure
Pick a start vertex
Perform DFS and write the exit time
POSCAT Seminar 1-16514 July 2014
yougatup
1 2
3
4
6
5
7
8
1
Strongly Connected Component
Overall procedure
Pick a start vertex
Perform DFS and write the exit time
POSCAT Seminar 1-16614 July 2014
yougatup
1 2
3
4
6
5
7
8
1
2
Strongly Connected Component
Overall procedure
Pick a start vertex
Perform DFS and write the exit time
POSCAT Seminar 1-16714 July 2014
yougatup
1 2
3
4
6
5
7
8
1
2
3
Strongly Connected Component
Overall procedure
Pick a start vertex
Perform DFS and write the exit time
POSCAT Seminar 1-16814 July 2014
yougatup
1 2
3
4
6
5
7
8
1
2
3
Strongly Connected Component
Overall procedure
Pick a start vertex
Perform DFS and write the exit time
POSCAT Seminar 1-16914 July 2014
yougatup
1 2
3
4
6
5
7
8
1
2
34
Strongly Connected Component
Overall procedure
Pick a start vertex
Perform DFS and write the exit time
POSCAT Seminar 1-17014 July 2014
yougatup
1 2
3
4
6
5
7
8
1
2
34
5
Strongly Connected Component
Overall procedure
Pick a start vertex
Perform DFS and write the exit time
POSCAT Seminar 1-17114 July 2014
yougatup
1 2
3
4
6
5
7
8
1
2
34
5
Strongly Connected Component
Overall procedure
Pick a start vertex
Perform DFS and write the exit time
POSCAT Seminar 1-17214 July 2014
yougatup
1 2
3
4
6
5
7
8
1
2
34
5
Strongly Connected Component
Overall procedure
Pick a start vertex
Perform DFS and write the exit time
POSCAT Seminar 1-17314 July 2014
yougatup
1 2
3
4
6
5
7
8
1
2
34
5
Strongly Connected Component
Overall procedure
Pick a start vertex
Perform DFS and write the exit time
POSCAT Seminar 1-17414 July 2014
yougatup
1 2
3
4
6
5
7
8
1
2
34
56
Strongly Connected Component
Overall procedure
Pick a start vertex
Perform DFS and write the exit time
POSCAT Seminar 1-17514 July 2014
yougatup
1 2
3
4
6
5
7
8
1
2
34
567
Strongly Connected Component
Overall procedure
Pick a start vertex
Perform DFS and write the exit time
POSCAT Seminar 1-17614 July 2014
yougatup
1 2
3
4
6
5
7
8
1
2
34
567
8
Strongly Connected Component
Overall procedure
After writing all the exit time, reverse whole graph
POSCAT Seminar 1-17714 July 2014
yougatup
1 2
3
4
6
5
7
8
1
2
34
567
8
Strongly Connected Component
Overall procedure
After writing all the exit time, reverse whole graph
POSCAT Seminar 1-17814 July 2014
yougatup
1 2
3
4
6
5
7
8
1
2
34
567
8
Strongly Connected Component
Overall procedure
Perform DFS or BFS started from a vertex with highest exit time
All the vertices reached from the start vertex makes a SCC
POSCAT Seminar 1-17914 July 2014
yougatup
1 2
3
4
6
5
7
8
1
2
34
567
8
Strongly Connected Component
Overall procedure
Perform DFS or BFS started from a vertex with highest exit time
All the vertices reached from the start vertex makes a SCC
POSCAT Seminar 1-18014 July 2014
yougatup
1 2
3
4
6
5
7
8
1
2
34
567
8
Strongly Connected Component
Overall procedure
Perform DFS or BFS started from a vertex with highest exit time
All the vertices reached from the start vertex makes a SCC
POSCAT Seminar 1-18114 July 2014
yougatup
1 2
3
4
6
5
7
8
1
2
34
567
8
Strongly Connected Component
Overall procedure
Perform DFS or BFS started from a vertex with highest exit time
All the vertices reached from the start vertex makes a SCC
POSCAT Seminar 1-18214 July 2014
yougatup
1 2
3
4
6
5
7
8
1
2
34
567
8
Strongly Connected Component
Overall procedure
Perform DFS or BFS started from a vertex with highest exit time
All the vertices reached from the start vertex makes a SCC
POSCAT Seminar 1-18314 July 2014
yougatup
1 2
3
4
6
5
7
8
1
2
34
567
8
Strongly Connected Component
Overall procedure
Perform DFS or BFS started from a vertex with highest exit time
All the vertices reached from the start vertex makes a SCC
POSCAT Seminar 1-18414 July 2014
yougatup
1 2
3
4
6
5
7
8
1
2
34
567
8
Strongly Connected Component
Overall procedure
Perform DFS or BFS started from a vertex with highest exit time
All the vertices reached from the start vertex makes a SCC
POSCAT Seminar 1-18514 July 2014
yougatup
1 2
3
4
6
5
7
8
1
2
34
567
8
Strongly Connected Component
Overall procedure
Perform DFS or BFS started from a vertex with highest exit time
All the vertices reached from the start vertex makes a SCC
POSCAT Seminar 1-18614 July 2014
yougatup
1 2
3
4
6
5
7
8
1
2
34
567
8
Strongly Connected Component
Overall procedure
Perform DFS or BFS started from a vertex with highest exit time
All the vertices reached from the start vertex makes a SCC
POSCAT Seminar 1-18714 July 2014
yougatup
1 2
3
4
6
5
7
8
1
2
34
567
8
Strongly Connected Component
Overall procedure
Perform DFS or BFS started from a vertex with highest exit time
All the vertices reached from the start vertex makes a SCC
POSCAT Seminar 1-18814 July 2014
yougatup
1 2
3
4
6
5
7
8
1
2
34
567
8
Done !
Strongly Connected Component
Analysis
It is correct ?
How long does it takes ?
POSCAT Seminar 1-18914 July 2014
yougatup
1 2
3
4
6
5
7
8
1
2
34
567
8
Done !
Strongly Connected Component
Analysis
It is correct ? We need some detail proof
How long does it takes ? We need 2 DFS (or 1 DFS + 1 BFS)
POSCAT Seminar 1-19014 July 2014
yougatup
1 2
3
4
6
5
7
8
1
2
34
567
8
Done !
Strongly Connected Component
Analysis
It is correct ? We need some detail proof
How long does it takes ? O(𝑽 + 𝑬)
POSCAT Seminar 1-19114 July 2014
yougatup
1 2
3
4
6
5
7
8
1
2
34
567
8
Done !
Recommended