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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 𝐸𝐸 ≤ 𝑉𝑉

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Directed graph 𝐸𝐸 ≤ 𝑉𝑉2

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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: ∑𝑣𝑣𝑖𝑖∈𝑉𝑉(𝐺𝐺) deg 𝑣𝑣𝑖𝑖 = 2|𝐸𝐸|

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 v’s such that (u, v) ∈ E

An array of V lists V[i].degree = size of adj list for node 𝑣𝑣𝑖𝑖 V[i].AdjList = adjacency list for node 𝑣𝑣𝑖𝑖

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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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Adjacency Matrix

𝑉𝑉 × 𝑉𝑉 matrix 𝐴𝐴 𝐴𝐴 𝑖𝑖, 𝑗𝑗 = 1 if 𝑣𝑣𝑖𝑖 ,𝑣𝑣𝑗𝑗 is an edge Otherwise, 𝐴𝐴 𝑖𝑖, 𝑗𝑗 = 0

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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.

Sample Graph Algorithm

Input: Graph G represented by adjacency lists

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Running time:Θ(V + E)

Connectivity

A path in a graph is a sequence of vertices 𝑢𝑢1,𝑢𝑢2, … ,𝑢𝑢𝑘𝑘 such that there is an edge 𝑢𝑢𝑖𝑖 ,𝑢𝑢𝑖𝑖+1 between any two adjacent vertices in the

sequence Two vertices 𝑢𝑢,𝑤𝑤 ∈ 𝑉𝑉 𝐺𝐺 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 𝑢𝑢,𝑤𝑤 ∈𝑉𝑉 𝐺𝐺 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 ∈ V

Output: Will visit all nodes in V reachable from s For each 𝑣𝑣 ∈ 𝑉𝑉, output a value 𝑣𝑣.𝑑𝑑

𝑣𝑣.𝑑𝑑 = distance (smallest # of edges) from s to v. 𝑣𝑣.𝑑𝑑 = ∞ 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 …

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)

𝑣𝑣.𝑑𝑑 : 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:Θ(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 𝑣𝑣.𝑑𝑑 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 𝑣𝑣 is the parent of 𝑢𝑢 if 𝑢𝑢 was first discovered when exploring 𝑣𝑣

A BFS tree 𝑇𝑇 Root: source node 𝑠𝑠 Nodes in level 𝑘𝑘 of 𝑇𝑇 are distance 𝑘𝑘 away from 𝑠𝑠

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Connectivity-Checking

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Time complexity:Θ(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 Θ (|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 𝑣𝑣1 What if we add edge 𝑣𝑣1, 𝑣𝑣8

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DFS

Time complexity Θ(V+E)

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Depth First Search Tree

If v is discovered when exploring u Set 𝑣𝑣.𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝 = 𝑢𝑢

The collection of edges 𝑣𝑣.𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝, 𝑣𝑣 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 𝑣𝑣1

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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 𝑣𝑣𝑘𝑘 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 𝐸𝐸 ≤ 𝑉𝑉

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Directed graph Each edge 𝑢𝑢, 𝑣𝑣 is directed from u to v 𝐸𝐸 ≤ 𝑉𝑉2

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Vertex Degree

indeg(v) = # edges of the form 𝑢𝑢, 𝑣𝑣 outdeg(v) = # edges of the form 𝑣𝑣,𝑢𝑢

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Lemma: ∑𝑣𝑣𝑖𝑖∈𝑉𝑉(𝐺𝐺) indeg 𝑣𝑣𝑖𝑖 = |𝐸𝐸| Lemma: ∑𝑣𝑣𝑖𝑖∈𝑉𝑉(𝐺𝐺) outdeg 𝑣𝑣𝑖𝑖 = |𝐸𝐸|

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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 𝑣𝑣𝑖𝑖 V[i].AdjList = adjacency list for node 𝑣𝑣𝑖𝑖

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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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Adjacency Matrix

𝑉𝑉 × 𝑉𝑉 matrix 𝐴𝐴 𝐴𝐴 𝑖𝑖, 𝑗𝑗 = 1 if 𝑣𝑣𝑖𝑖 ,𝑣𝑣𝑗𝑗 is an edge Otherwise, 𝐴𝐴 𝑖𝑖, 𝑗𝑗 = 0

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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.

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 𝑢𝑢1,𝑢𝑢2, … ,𝑢𝑢𝑘𝑘 such that there is an edge 𝑢𝑢𝑖𝑖 ,𝑢𝑢𝑖𝑖+1 between any two adjacent vertices in the

sequence Given two vertices 𝑢𝑢,𝑤𝑤 ∈ 𝑉𝑉 𝐺𝐺 , 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 𝑣𝑣1?

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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 …

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Pseudo-code

Time complexity:Θ(V+E)

Use adjacency listrepresentation

Number of Reachable Nodes

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Time complexity:Θ(V+E)

Compute # nodesreachable from 𝑣𝑣𝑘𝑘

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, 𝑘𝑘) visits all nodes reachable to node 𝑣𝑣𝑘𝑘

BFS from previous NumReachable() procedure can be replaced with DFS.

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More Example

Is 𝑣𝑣1 reachable from 𝑣𝑣12 ? Is 𝑣𝑣12 reachable from 𝑣𝑣1 ?

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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 𝑢𝑢1,𝑢𝑢2, … ,𝑢𝑢𝑘𝑘 ,𝑢𝑢1such 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) ∈ E => A[u] < A[v]

undershorts

pants

beltshirt

tie

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 – Simplified Implementation

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Time complexity Θ(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 𝑤𝑤:𝐸𝐸 → 𝑅𝑅 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?