CSE 373: More on graphs; DFS and BFS Michael Lee Wednesday, Feb 14, 2018 1
Warmup Warmup: Discuss with your neighbor: A simple graph is a graph that has no self-loops and no parallel edges. is the maximum number of edges it can have, in terms of x ? Each vertex can connect to x other vertices, so x x . with y edges. What is the maximum number of vertices it can have, in terms of y ? Infjnite: just keep adding nodes with no edges attached. 2 ◮ Remind your neighbor: what is a simple graph ? ◮ Suppose we have a simple, directed graph with x nodes. What ◮ Now, suppose we have a difgerent simple, undirected graph
Warmup Warmup: Discuss with your neighbor: A simple graph is a graph that has no self-loops and no parallel edges. is the maximum number of edges it can have, in terms of x ? with y edges. What is the maximum number of vertices it can have, in terms of y ? Infjnite: just keep adding nodes with no edges attached. 2 ◮ Remind your neighbor: what is a simple graph ? ◮ Suppose we have a simple, directed graph with x nodes. What Each vertex can connect to x − 1 other vertices, so x ( x − 1) . ◮ Now, suppose we have a difgerent simple, undirected graph
of what it would be if the graph were directed. So, x x Warmup: can just keep adding more and more self-loops. Note that if previously. Either way, it’s still infjnite, for the same reasons given have? What if the graph is not simple? with y edges. What is the maximum number of vertices it can , there can’t be any edges at all. x , we Some follow-up questions: If the graph is not simple, it’s infjnite: assuming x . If the graph is simple, the max number of edges is exactly half the graph is not simple? What is the maximum number of edges it can have? What if 3 ◮ Suppose we have a simple, undirected graph with x nodes. ◮ Now, suppose we have a difgerent simple, undirected graph
Warmup: Some follow-up questions: previously. Either way, it’s still infjnite, for the same reasons given have? What if the graph is not simple? with y edges. What is the maximum number of vertices it can can just keep adding more and more self-loops. Note that if 3 . If the graph is simple, the max number of edges is exactly half the graph is not simple? What is the maximum number of edges it can have? What if ◮ Suppose we have a simple, undirected graph with x nodes. of what it would be if the graph were directed. So, x ( x − 1) 2 If the graph is not simple, it’s infjnite: assuming x > 0 , we x = 0 , there can’t be any edges at all. ◮ Now, suppose we have a difgerent simple, undirected graph
Summary V To put it another way, sparse graphs have “few” edges. , we sau the graph is sparse . V If E Sparse graph To put it another way, dense graphs have “lots of edges” , we say the graph is dense . If E What did we learn? Dense graph , for both directed and undirected graphs. V know E In simple graphs, if we know V is some fjxed value, we also vertices are independent. 4 ◮ In graphs with no restrictions, number of edges and number of
Summary V To put it another way, sparse graphs have “few” edges. , we sau the graph is sparse . V If E Sparse graph To put it another way, dense graphs have “lots of edges” , we say the graph is dense . If E What did we learn? Dense graph , for both directed and undirected graphs. vertices are independent. 4 ◮ In graphs with no restrictions, number of edges and number of ◮ In simple graphs, if we know | V | is some fjxed value, we also � | V | 2 � know | E | ∈ O
Summary What did we learn? To put it another way, sparse graphs have “few” edges. , we sau the graph is sparse . V If E Sparse graph To put it another way, dense graphs have “lots of edges” , we say the graph is dense . 4 Dense graph , for both directed and undirected graphs. vertices are independent. ◮ In graphs with no restrictions, number of edges and number of ◮ In simple graphs, if we know | V | is some fjxed value, we also � | V | 2 � know | E | ∈ O � | V | 2 � If | E | ∈ Θ
Summary , for both directed and undirected graphs. To put it another way, sparse graphs have “few” edges. Sparse graph To put it another way, dense graphs have “lots of edges” , we say the graph is dense . What did we learn? Dense graph 4 vertices are independent. ◮ In graphs with no restrictions, number of edges and number of ◮ In simple graphs, if we know | V | is some fjxed value, we also � | V | 2 � know | E | ∈ O � | V | 2 � If | E | ∈ Θ If | E | ∈ O ( | V | ) , we sau the graph is sparse .
How do we represent graphs in code? So, how do we actually represent graphs in code? Two common approaches, with difgerent tradeofgs: Adjacency matrix Adjacency list 5
How do we represent graphs in code? So, how do we actually represent graphs in code? Two common approaches, with difgerent tradeofgs: 5 ◮ Adjacency matrix ◮ Adjacency list
Adjacency matrix b a b c d d c a Core idea: d c b a 6 ◮ Assign each node a number from 0 to | V | − 1 ◮ Create a | V | × | V | nested array of booleans or ints ◮ If ( x , y ) ∈ E , then nestedArray[x][y] == true
Adjacency matrix b a b c d d c a Core idea: d c b a 6 ◮ Assign each node a number from 0 to | V | − 1 ◮ Create a | V | × | V | nested array of booleans or ints ◮ If ( x , y ) ∈ E , then nestedArray[x][y] == true
Adjacency list How much space do we use? edges, not easily Self-loops yes, parallel Can we handle self-loops and parallel edges? Dense ones Is this better for sparse or dense graphs? V 7 What is the worst-case runtime to: V V ◮ Get out-edges: ◮ Get in-edges: ◮ Decide if an edge exists: ◮ Insert an edge: ◮ Delete an edge:
Adjacency list What is the worst-case runtime to: Is this better for sparse or dense graphs? Dense ones Can we handle self-loops and parallel edges? Self-loops yes, parallel edges, not easily 7 ◮ Get out-edges: O ( | V | ) ◮ Get in-edges: O ( | V | ) ◮ Decide if an edge exists: O (1) ◮ Insert an edge: O (1) ◮ Delete an edge: O (1) � | V | 2 � How much space do we use? O
On a higher level: represent as IDictionary<Vertex, Edges> . Adjacency list Core idea: a b c d 8 ◮ Assign each node a number from 0 to | V | − 1 ◮ Create an array of size | V | ◮ Each element in the array stores its out edges in a list or set
Adjacency list Core idea: a b c d 8 ◮ Assign each node a number from 0 to | V | − 1 ◮ Create an array of size | V | ◮ Each element in the array stores its out edges in a list or set ◮ On a higher level: represent as IDictionary<Vertex, Edges> .
Adjacency list Core idea: a b c d 8 ◮ Assign each node a number from 0 to | V | − 1 ◮ Create an array of size | V | ◮ Each element in the array stores its out edges in a list or set ◮ On a higher level: represent as IDictionary<Vertex, Edges> .
Adjacency list We can store edges using either sets or lists. Answer these questions for both. What is the worst-case runtime to: How much space do we use? Is this better for sparse or dense graphs? Can we handle self-loops and parallel edges? 9 ◮ Get out-edges: ◮ Get in-edges: ◮ Decide if an edge exists: ◮ Insert an edge: ◮ Delete an edge:
Which do we pick? So which do we pick? Observations: Most graphs are sparse If we implement adjacency lists using sets, we can get comparable worst-case performance So by default, pick adjacency lists. 10
Which do we pick? So which do we pick? Observations: comparable worst-case performance So by default, pick adjacency lists. 10 ◮ Most graphs are sparse ◮ If we implement adjacency lists using sets, we can get
Which do we pick? So which do we pick? Observations: comparable worst-case performance So by default, pick adjacency lists. 10 ◮ Most graphs are sparse ◮ If we implement adjacency lists using sets, we can get
Walks and paths d e d c b a Path e c Walk b a Walk A path is a walk that never visits the same vertex twice. Path More intuitively, a walk is one continous line following the edges. 11 A walk is a list of vertices v 0 , v 1 , v 2 , . . . , v n where if i is some int where 0 ≤ i < v n , every pair ( v i , v i +1 ) ∈ E is true.
Walks and paths d e d c b a Path e c Walk b a Walk A path is a walk that never visits the same vertex twice. Path More intuitively, a walk is one continous line following the edges. 11 A walk is a list of vertices v 0 , v 1 , v 2 , . . . , v n where if i is some int where 0 ≤ i < v n , every pair ( v i , v i +1 ) ∈ E is true.
Walks and paths d e d c b a Path Path or walk? e c Walk b a Walk Path or walk? A path is a walk that never visits the same vertex twice. Path More intuitively, a walk is one continous line following the edges. 11 A walk is a list of vertices v 0 , v 1 , v 2 , . . . , v n where if i is some int where 0 ≤ i < v n , every pair ( v i , v i +1 ) ∈ E is true.
Walks and paths d e d c b a Path e c Walk b a Walk A path is a walk that never visits the same vertex twice. Path More intuitively, a walk is one continous line following the edges. 11 A walk is a list of vertices v 0 , v 1 , v 2 , . . . , v n where if i is some int where 0 ≤ i < v n , every pair ( v i , v i +1 ) ∈ E is true.
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