1. CSE 2331/5331 Topic 9: Basic Graph Alg.
Representations Basic traversal algorithms
Topological sort
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2. What Is A Graph Graph G = (V, E) V: set of nodes E: set of edgesb a f c d e
Example:
V ={ a, b, c, d, e, f }
E ={(a, b), (a, d), (a, e), (b, c), (b, d), (b, e), (c, e), (e,f)}
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3. Un-directed graph Directed graph 𝐸≤ 𝑉 2 𝐸≤ 𝑉 2 b a f c d e
𝐸≤ 𝑉 2
Directed graph
𝐸≤ 𝑉 2 b a f c d e
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4. Un-directed graphs
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5. Vertex Degree deg(v) = degree of v = # edges incident on vb a f c d e
Lemma: 𝑣 𝑖 ∈𝑉(𝐺) deg 𝑣 𝑖 = 2|𝐸|
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6. Some Special Graphs Complete graph Path Cycle Planar graph Tree
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7. Representations of Graphs
Adjacency lists
Each vertex u has a list, recording its neighbors
i.e., all v’s such that (u, v)  E
An array of V lists
V[i].degree = size of adj list for node 𝑣 𝑖
V[i].AdjList = adjacency list for node 𝑣 𝑖
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8. Adjacency Lists For vertex v  V, its adjacency list
has size: deg(v)
decide whether (v, u)  E or not in time Θ (deg(v))
Size of data structure (space complexity):
Θ(|V| + |E|) = Θ (V+E)
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9. Adjacency Matrix 𝑉×𝑉 matrix 𝐴 𝐴 𝑖,𝑗 =1 if 𝑣 𝑖 , 𝑣 𝑗 is an edge
Otherwise, 𝐴 𝑖,𝑗 =0
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10. Adjacency Matrix Size of data structure:
Θ ( V  V)
Time to determine if (v, u)  E :
Θ(1)
Though larger, it is simpler compared to adjacency list.
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11. Sample Graph Algorithm
Input: Graph G represented by adjacency lists
Running time:
Θ(V + E)
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12. Connectivity
A path in a graph is a sequence of vertices 𝑢 1 , 𝑢 2 ,…, 𝑢 𝑘 such that there is an edge 𝑢 𝑖 , 𝑢 𝑖+1 between any two adjacent vertices in the sequence
Two vertices 𝑢, 𝑤∈𝑉 𝐺 are connected if there is a path in G from u to w.
We also say that w is reachable from u.
A graph G is connected if every pair of nodes 𝑢, 𝑤∈𝑉 𝐺 are connected.
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13. Connectivity Checking
How to check if the graph is connected?
One approach: Graph traversal
BFS: breadth-first search
DFS: depth-first search
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14. BFS: Breadth-first search
Input:
Given graph G = (V, E), and a source node s  V
Output:
Will visit all nodes in V reachable from s
For each 𝑣∈𝑉, output a value 𝑣.𝑑
𝑣.𝑑= distance (smallest # of edges) from s to v.
𝑣.𝑑= ∞ if v is not reachable from s.
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15. Intuition Starting from source node s, Need a data-structure
Spread a wavefront to visit other nodes
First visit all nodes one edge away from s
Then all nodes two edges away from s …
Need a data-structure
to store nodes
to be explored.
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16. Intuition cont. A node can be: 𝑣.𝑑 :
un-discovered
discovered, but not explored
explored (finished)
𝑣.𝑑 :
is set when node v is first discovered.
Need a data structure to store discovered but un-explored nodes
FIFO ! Queue
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17. Pseudo-code Use adjacency list representation Time complexity: Θ(V+E)
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18. Correctness of Algorithm
A node not reachable from s will not be visited
A node reachable from s will be visited
𝑣.𝑑 computed is correct:
Intuitively, if all nodes k distance away from s are in level k, and no other nodes are in level k,
Then all nodes (k+1)-distance away from s must be in level (k+1).
Rigorous proof by induction
Consider the breadth-first tree generated by algorithm:
Any node u from k’th level: d[u] = k
Claim: all nodes v with  (s, v) = k are in level k
and all nodes from level k has  (s, v) = k
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19. BFS tree A node 𝑣 is the parent of 𝑢 if A BFS tree 𝑇
𝑢 was first discovered when exploring 𝑣
A BFS tree 𝑇
Root: source node 𝑠
Nodes in level 𝑘 of 𝑇 are distance 𝑘 away from 𝑠
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20. Connectivity-Checking
Time complexity:
Θ(V+E)
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21. Summary for BFS
Starting from source node s, visits remaining nodes of graph from small distance to large distance
This is one way to traverse an input graph
With some special property where nodes are visited in non-decreasing distance to the source node s.
Return distance between s to any reachable node in time Θ (|V| + |E|)
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22. DFS: Depth-First Search
Another graph traversal algorithm
BFS:
Go as broad as possible in the algorithm
DFS:
Go as deep as possible in the algorithm
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23. Example Perform DFS starting from 𝑣 1 What if we add edge 𝑣 1 , 𝑣 8
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24. DFS Time complexity Θ(V+E) CSE 2331/5331
Intuitively, instead of using a FIFO Queue, now we use a LIFO stack.
Time complexity
Θ(V+E)
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25. Depth First Search Tree
If v is discovered when exploring u
Set 𝑣.𝑝𝑎𝑟𝑒𝑛𝑡=𝑢
The collection of edges
𝑣.𝑝𝑎𝑟𝑒𝑛𝑡, 𝑣 form a tree,
called Depth-first search tree.
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26. DFS with DFS Tree
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27. Example
Perform DFS starting from 𝑣 1
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28. Another Connectivity Test
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29. Traverse Entire Graph
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30. Remarks DFS(G, k) Same time complexity as BFS
Another way to compute all nodes reachable to the node 𝑣 𝑘
Same time complexity as BFS
There are nice properties of DFS and DFS tree that we are not reviewing in this class.
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31. Directed Graphs
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32. Un-directed graph Directed graph 𝐸≤ 𝑉 2
𝐸≤ 𝑉 2
Directed graph
Each edge 𝑢,𝑣 is directed from u to v
𝐸≤ 𝑉 2 b a f c d e
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33. Vertex Degree indeg(v) = # edges of the form 𝑢, 𝑣
outdeg(v) = # edges of the form 𝑣, 𝑢 b a f c d e
Lemma: 𝑣 𝑖 ∈𝑉(𝐺) indeg 𝑣 𝑖 = |𝐸|
Lemma: 𝑣 𝑖 ∈𝑉(𝐺) outdeg 𝑣 𝑖 = |𝐸|
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34. Representations of Graphs
Adjacency lists
Each vertex u has a list, recording its neighbors
i.e., all v’s such that (u, v)  E
An array of V lists
V[i].degree = size of adj list for node 𝑣 𝑖
V[i].AdjList = adjacency list for node 𝑣 𝑖
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35. Adjacency Lists For vertex v  V, its adjacency list
has size: outdeg(v)
decide whether (v, u)  E or not in time O(outdeg(v))
Size of data structure (space complexity):
Θ(V+E)
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36. Adjacency Matrix 𝑉×𝑉 matrix 𝐴 𝐴 𝑖,𝑗 =1 if 𝑣 𝑖 , 𝑣 𝑗 is an edge
Otherwise, 𝐴 𝑖,𝑗 =0
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37. Adjacency Matrix Size of data structure:
Θ ( V  V)
Time to determine if (v, u)  E :
O(1)
Though larger, it is simpler compared to adjacency list.
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38. Sample Graph Algorithm
Input: Directed graph G represented by adjacency list
Running time:
O(V + E)
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39. Connectivity
A path in a graph is a sequence of vertices 𝑢 1 , 𝑢 2 ,…, 𝑢 𝑘 such that there is an edge 𝑢 𝑖 , 𝑢 𝑖+1 between any two adjacent vertices in the sequence
Given two vertices 𝑢, 𝑤∈𝑉 𝐺 , we say that w is reachable from u if there is a path in G from u to w.
Note: w is reachable from u DOES NOT necessarily mean that u is reachable from w.
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40. Reachability Test
How many (or which) vertices are reachable from a source node, say 𝑣 1 ?
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41. BFS and DFS The algorithms for BFS and DFS remain the same
Each edge is now understood as a directed edge
BFS(V,E, s) :
visits all nodes reachable from s in non-decreasing order
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42. BFS Starting from source node s,
Spread a wavefront to visit other nodes
First visit all nodes one edge away from s
Then all nodes two edges away from s …
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43. Pseudo-code Use adjacency list representation Time complexity: Θ(V+E)
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44. Number of Reachable Nodes
Compute # nodes
reachable from 𝑣 𝑘
Time complexity:
Θ(V+E)
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45. DFS: Depth-First Search
Similarly, DFS remains the same
Each edge is now a directed edge
If we start with all nodes unvisited,
Then DFS(G, 𝑘) visits all nodes reachable to node 𝑣 𝑘
BFS from previous NumReachable() procedure can be replaced with DFS.
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46. More Example Is 𝑣 1 reachable from 𝑣 12 ? Is 𝑣 12 reachable from 𝑣 1 ?
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47. DFS above can be replaced with BFS
DFS(G, k);
DFS above can be replaced with BFS
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48. Topological Sort
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49. Directed Acyclic Graph
A directed cycle is a sequence 𝑢 1 , 𝑢 2 ,…, 𝑢 𝑘 , 𝑢 1 such that there is a directed edge between any two consecutive nodes.
DAG: directed acyclic graph
Is a directed graph with no directed cycles.
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50. Topological Sort A topological sort of a DAG G = (V, E)
A linear ordering A of all vertices from V
If edge (u,v)  E => A[u] < A[v]
undershorts
pants
belt
shirt
tie
jacket
shoes
socks
watch
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51. Another Example Why requires DAG? Is the sorting order unique?
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52. A topological sorted order of graph G exists if and only if G is a directed acyclic graph (DAG).
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53. Question How to topologically sort a given DAG? Intuition:
Which node can be the first node in the topological sort order?
A node with in-degree 0 !
After we remove this, the process can be repeated.
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54. Example undershorts pants belt shirt tie jacket shoes socks watch
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55. CSE 2331/5331

56. Topological Sort – Simplified Implementation
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57. Time complexity Correctness: Θ(V+E)
What if the algorithm terminates before we finish visiting all nodes?
Procedure TopologicalSort(G) outputs a sorted list of all nodes if and only if the input graph G is a DAG
If G is not DAG, the algorithm outputs only a partial list of vertices.
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58. Remarks Other topological sort algorithm by using properties of DFS
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59. Last Analyzing graph algorithms Adjacency list representation or
Adjacency matrix representation
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60. Edge-weighted un-directed graph G = (V, E) and edge weight function 𝑤:𝐸→𝑅
E.g, road network, where each node is a city, and each edge is a road, and weight is the length of this road. 3 5 b a f c d e 1 1 2 1 1 2 4
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61. Example 1 Assume G is represented by adjacency list
Q is priority-queue implemented by min-heap
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62. Example 2 Assume G is represented by adjacency matrix
What if move line 10 to above line 8?
What if move line 10 to above line 9?
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