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CSE 2331/5331
CSE 2331/5331
Topic 9:Basic Graph Alg.
RepresentationsBasic traversal algorithmsTopological sort
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What Is A Graph
Graph G = (V, E) V: set of nodes E: set of edges
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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Un-directed graph πΈπΈ β€ ππ
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Directed graph πΈπΈ β€ ππ2
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Un-directed graphs
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Vertex Degree
deg(v) = degree of v = # edges incident on v
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Lemma: βπ£π£ππβππ(πΊπΊ) deg π£π£ππ = 2|πΈπΈ|
Some Special Graphs
Complete graph Path Cycle Planar graph Tree
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Representations of Graphs
Adjacency lists Each vertex u has a list, recording its neighbors
i.e., all vβs such that (u, v) β E
An array of V lists V[i].degree = size of adj list for node π£π£ππ V[i].AdjList = adjacency list for node π£π£ππ
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Adjacency Lists
For vertex v β V, its adjacency list has size: deg(v) decide whether (v, u) β E or not in time Ξ (deg(v))
Size of data structure (space complexity): Ξ(|V| + |E|) = Ξ (V+E)
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Adjacency Matrix
ππ Γ ππ matrix π΄π΄ π΄π΄ ππ, ππ = 1 if π£π£ππ ,π£π£ππ is an edge Otherwise, π΄π΄ ππ, ππ = 0
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Adjacency Matrix
Size of data structure: Ξ ( V Γ V)
Time to determine if (v, u) β E : Ξ(1)
Though larger, it is simpler compared to adjacency list.
Sample Graph Algorithm
Input: Graph G represented by adjacency lists
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Running time:Ξ(V + E)
Connectivity
A path in a graph is a sequence of vertices π’π’1,π’π’2, β¦ ,π’π’ππ such that there is an edge π’π’ππ ,π’π’ππ+1 between any two adjacent vertices in the
sequence Two vertices π’π’,π€π€ β ππ πΊπΊ 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 π’π’,π€π€ βππ πΊπΊ are connected.
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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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BFS: Breadth-first search
Input: Given graph G = (V, E), and a source node s β V
Output: Will visit all nodes in V reachable from s For each π£π£ β ππ, output a value π£π£.ππ
π£π£.ππ = distance (smallest # of edges) from s to v. π£π£.ππ = β if v is not reachable from s.
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Intuition
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 β¦
Need a data-structureto store nodes to be explored.
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Intuition cont.
A node can be: un-discovered discovered, but not explored explored (finished)
π£π£.ππ : 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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Pseudo-code
Time complexity:Ξ(V+E)
Use adjacency listrepresentation
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Correctness of Algorithm
A node not reachable from s will not be visited A node reachable from s will be visited π£π£.ππ 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
BFS tree
A node π£π£ is the parent of π’π’ if π’π’ was first discovered when exploring π£π£
A BFS tree ππ Root: source node π π Nodes in level ππ of ππ are distance ππ away from π π
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Connectivity-Checking
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Time complexity:Ξ(V+E)
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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 Ξ (|V| + |E|)
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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Example
Perform DFS starting from π£π£1 What if we add edge π£π£1, π£π£8
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DFS
Time complexity Ξ(V+E)
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Depth First Search Tree
If v is discovered when exploring u Set π£π£.ππππππππππππ = π’π’
The collection of edges π£π£.ππππππππππππ, π£π£ form a tree, called Depth-first search tree.
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DFS with DFS Tree
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Example
Perform DFS starting from π£π£1
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Another Connectivity Test
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Traverse Entire Graph
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Remarks
DFS(G, k) Another way to compute all nodes reachable to the
node π£π£ππ 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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Directed Graphs
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Un-directed graph πΈπΈ β€ ππ
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Directed graph Each edge π’π’, π£π£ is directed from u to v πΈπΈ β€ ππ2
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Vertex Degree
indeg(v) = # edges of the form π’π’, π£π£ outdeg(v) = # edges of the form π£π£,π’π’
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Lemma: βπ£π£ππβππ(πΊπΊ) indeg π£π£ππ = |πΈπΈ| Lemma: βπ£π£ππβππ(πΊπΊ) outdeg π£π£ππ = |πΈπΈ|
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Representations of Graphs
Adjacency lists Each vertex u has a list, recording its neighbors
i.e., all vβs such that (u, v) β E
An array of V lists V[i].degree = size of adj list for node π£π£ππ V[i].AdjList = adjacency list for node π£π£ππ
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Adjacency Lists
For vertex v β V, its adjacency list has size: outdeg(v) decide whether (v, u) β E or not in time O(outdeg(v))
Size of data structure (space complexity): Ξ(V+E)
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Adjacency Matrix
ππ Γ ππ matrix π΄π΄ π΄π΄ ππ, ππ = 1 if π£π£ππ ,π£π£ππ is an edge Otherwise, π΄π΄ ππ, ππ = 0
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Adjacency Matrix
Size of data structure: Ξ ( V Γ V)
Time to determine if (v, u) β E : O(1)
Though larger, it is simpler compared to adjacency list.
Sample Graph Algorithm
Input: Directed graph G represented by adjacency list
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Running time:O(V + E)
Connectivity
A path in a graph is a sequence of vertices π’π’1,π’π’2, β¦ ,π’π’ππ such that there is an edge π’π’ππ ,π’π’ππ+1 between any two adjacent vertices in the
sequence Given two vertices π’π’,π€π€ β ππ πΊπΊ , 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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Reachability Test
How many (or which) vertices are reachable from a source node, say π£π£1?
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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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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 β¦
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Pseudo-code
Time complexity:Ξ(V+E)
Use adjacency listrepresentation
Number of Reachable Nodes
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Time complexity:Ξ(V+E)
Compute # nodesreachable from π£π£ππ
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, ππ) visits all nodes reachable to node π£π£ππ
BFS from previous NumReachable() procedure can be replaced with DFS.
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More Example
Is π£π£1 reachable from π£π£12 ? Is π£π£12 reachable from π£π£1 ?
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DFS above can be replaced with BFS
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DFS(G, k);
Topological Sort
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Directed Acyclic Graph
A directed cycle is a sequence π’π’1,π’π’2, β¦ ,π’π’ππ ,π’π’1such 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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Topological Sort
A topological sort of a DAG G = (V, E) A linear ordering A of all vertices from V If edge (u,v) β E => A[u] < A[v]
undershorts
pants
beltshirt
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jacket
shoes
sockswatch
Another Example
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Is the sorting order unique?
Why requires DAG?
A topological sorted order of graph G exists if and only if G is a directed acyclic graph (DAG).
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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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Example
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undershorts
pants
beltshirt
tie
jacket
shoes
sockswatch
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Topological Sort β Simplified Implementation
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Time complexity Ξ(V+E)
Correctness: 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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Remarks
Other topological sort algorithm by using properties of DFS
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Last
Analyzing graph algorithms Adjacency list representation or Adjacency matrix representation
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Edge-weighted un-directed graph G = (V, E) and edge weight function π€π€:πΈπΈ β π π E.g, road network, where each node is a city, and each
edge is a road, and weight is the length of this road.
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Example 1
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Assume G is represented by adjacency list
Q is priority-queue implemented by min-heap
Example 2
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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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