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David Luebke 1 04/21/23
CS 332: Algorithms
Graph Algorithms
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David Luebke 2 04/21/23
Review: Graphs
A graph G = (V, E) V = set of vertices, E = set of edges Dense graph: |E| |V|2; Sparse graph: |E| |V| Undirected graph:
Edge (u,v) = edge (v,u) No self-loops
Directed graph: Edge (u,v) goes from vertex u to vertex v, notated uv
A weighted graph associates weights with either the edges or the vertices
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David Luebke 3 04/21/23
Review: Representing Graphs
Assume V = {1, 2, …, n} An adjacency matrix represents the graph as a n x n
matrix A: A[i, j] = 1 if edge (i, j) E (or weight of edge)
= 0 if edge (i, j) E Storage requirements: O(V2)
A dense representation But, can be very efficient for small graphs
Especially if store just one bit/edge Undirected graph: only need one diagonal of matrix
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David Luebke 4 04/21/23
Review: Adjacency Matrix
Example:
1
2 4
3
a
d
b c
A 1 2 3 4
1 0 1 1 0
2 0 0 1 0
3 0 0 0 0
4 0 0 1 0
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David Luebke 5 04/21/23
Review: Adjacency List
Adjacency list: for each vertex v V, store a list of vertices adjacent to v
Example: Adj[1] = {2,3} Adj[2] = {3} Adj[3] = {} Adj[4] = {3}
Storage: O(V+E) Good for large, sparse graphs (e.g., planar maps)
1
2 4
3
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David Luebke 6 04/21/23
Graph Searching
Given: a graph G = (V, E), directed or undirected Goal: methodically explore every vertex and
every edge Ultimately: build a tree on the graph
Pick a vertex as the root Choose certain edges to produce a tree Note: might also build a forest if graph is not
connected
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David Luebke 7 04/21/23
Breadth-First Search
“Explore” a graph, turning it into a tree One vertex at a time Expand frontier of explored vertices across the
breadth of the frontier Builds a tree over the graph
Pick a source vertex to be the root Find (“discover”) its children, then their children,
etc.
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David Luebke 8 04/21/23
Breadth-First Search
Again will associate vertex “colors” to guide the algorithm White vertices have not been discovered
All vertices start out white Grey vertices are discovered but not fully explored
They may be adjacent to white vertices Black vertices are discovered and fully explored
They are adjacent only to black and gray vertices
Explore vertices by scanning adjacency list of grey vertices
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David Luebke 9 04/21/23
Breadth-First Search
BFS(G, s) {
initialize vertices;
Q = {s}; // Q is a queue (duh); initialize to s
while (Q not empty) {
u = RemoveTop(Q);
for each v u->adj { if (v->color == WHITE)
v->color = GREY;
v->d = u->d + 1;
v->p = u;
Enqueue(Q, v);
}
u->color = BLACK;
}
}
What does v->p represent?
What does v->d represent?
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David Luebke 10 04/21/23
Breadth-First Search: Example
r s t u
v w x y
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David Luebke 11 04/21/23
Breadth-First Search: Example
0
r s t u
v w x y
sQ:
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David Luebke 12 04/21/23
Breadth-First Search: Example
1
0
1
r s t u
v w x y
wQ: r
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David Luebke 13 04/21/23
Breadth-First Search: Example
1
0
1
2
2
r s t u
v w x y
rQ: t x
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David Luebke 14 04/21/23
Breadth-First Search: Example
1
2
0
1
2
2
r s t u
v w x y
Q: t x v
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David Luebke 15 04/21/23
Breadth-First Search: Example
1
2
0
1
2
2
3
r s t u
v w x y
Q: x v u
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David Luebke 16 04/21/23
Breadth-First Search: Example
1
2
0
1
2
2
3
3
r s t u
v w x y
Q: v u y
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David Luebke 17 04/21/23
Breadth-First Search: Example
1
2
0
1
2
2
3
3
r s t u
v w x y
Q: u y
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David Luebke 18 04/21/23
Breadth-First Search: Example
1
2
0
1
2
2
3
3
r s t u
v w x y
Q: y
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David Luebke 19 04/21/23
Breadth-First Search: Example
1
2
0
1
2
2
3
3
r s t u
v w x y
Q: Ø
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David Luebke 20 04/21/23
BFS: The Code Again
BFS(G, s) {
initialize vertices;
Q = {s};
while (Q not empty) {
u = RemoveTop(Q);
for each v u->adj { if (v->color == WHITE)
v->color = GREY;
v->d = u->d + 1;
v->p = u;
Enqueue(Q, v);
}
u->color = BLACK;
}
}What will be the running time?
Touch every vertex: O(V)
u = every vertex, but only once (Why?)
So v = every vertex that appears in some other vert’s adjacency list
Total running time: O(V+E)
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David Luebke 21 04/21/23
BFS: The Code Again
BFS(G, s) {
initialize vertices;
Q = {s};
while (Q not empty) {
u = RemoveTop(Q);
for each v u->adj { if (v->color == WHITE)
v->color = GREY;
v->d = u->d + 1;
v->p = u;
Enqueue(Q, v);
}
u->color = BLACK;
}
}
What will be the storage cost in addition to storing the tree?Total space used: O(max(degree(v))) = O(E)
![Page 22: David Luebke 1 1/6/2016 CS 332: Algorithms Graph Algorithms](https://reader036.vdocuments.us/reader036/viewer/2022062423/5697c00d1a28abf838cc93dc/html5/thumbnails/22.jpg)
David Luebke 22 04/21/23
Breadth-First Search: Properties
BFS calculates the shortest-path distance to the source node Shortest-path distance (s,v) = minimum number of
edges from s to v, or if v not reachable from s Proof given in the book (p. 472-5)
BFS builds breadth-first tree, in which paths to root represent shortest paths in G Thus can use BFS to calculate shortest path from
one vertex to another in O(V+E) time
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David Luebke 23 04/21/23
Depth-First Search
Depth-first search is another strategy for exploring a graph Explore “deeper” in the graph whenever possible Edges are explored out of the most recently
discovered vertex v that still has unexplored edges When all of v’s edges have been explored,
backtrack to the vertex from which v was discovered
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David Luebke 24 04/21/23
Depth-First Search
Vertices initially colored white Then colored gray when discovered Then black when finished
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David Luebke 25 04/21/23
Depth-First Search: The Code
DFS(G)
{
for each vertex u G->V {
u->color = WHITE;
}
time = 0;
for each vertex u G->V {
if (u->color == WHITE)
DFS_Visit(u);
}
}
DFS_Visit(u)
{
u->color = GREY;
time = time+1;
u->d = time;
for each v u->Adj[] {
if (v->color == WHITE)
DFS_Visit(v);
}
u->color = BLACK;
time = time+1;
u->f = time;
}
![Page 26: David Luebke 1 1/6/2016 CS 332: Algorithms Graph Algorithms](https://reader036.vdocuments.us/reader036/viewer/2022062423/5697c00d1a28abf838cc93dc/html5/thumbnails/26.jpg)
David Luebke 26 04/21/23
Depth-First Search: The Code
DFS(G)
{
for each vertex u G->V {
u->color = WHITE;
}
time = 0;
for each vertex u G->V {
if (u->color == WHITE)
DFS_Visit(u);
}
}
DFS_Visit(u)
{
u->color = GREY;
time = time+1;
u->d = time;
for each v u->Adj[] {
if (v->color == WHITE)
DFS_Visit(v);
}
u->color = BLACK;
time = time+1;
u->f = time;
}
What does u->d represent?
![Page 27: David Luebke 1 1/6/2016 CS 332: Algorithms Graph Algorithms](https://reader036.vdocuments.us/reader036/viewer/2022062423/5697c00d1a28abf838cc93dc/html5/thumbnails/27.jpg)
David Luebke 27 04/21/23
Depth-First Search: The Code
DFS(G)
{
for each vertex u G->V {
u->color = WHITE;
}
time = 0;
for each vertex u G->V {
if (u->color == WHITE)
DFS_Visit(u);
}
}
DFS_Visit(u)
{
u->color = GREY;
time = time+1;
u->d = time;
for each v u->Adj[] {
if (v->color == WHITE)
DFS_Visit(v);
}
u->color = BLACK;
time = time+1;
u->f = time;
}
What does u->f represent?
![Page 28: David Luebke 1 1/6/2016 CS 332: Algorithms Graph Algorithms](https://reader036.vdocuments.us/reader036/viewer/2022062423/5697c00d1a28abf838cc93dc/html5/thumbnails/28.jpg)
David Luebke 28 04/21/23
Depth-First Search: The Code
DFS(G)
{
for each vertex u G->V {
u->color = WHITE;
}
time = 0;
for each vertex u G->V {
if (u->color == WHITE)
DFS_Visit(u);
}
}
DFS_Visit(u)
{
u->color = GREY;
time = time+1;
u->d = time;
for each v u->Adj[] {
if (v->color == WHITE)
DFS_Visit(v);
}
u->color = BLACK;
time = time+1;
u->f = time;
}
Will all vertices eventually be colored black?
![Page 29: David Luebke 1 1/6/2016 CS 332: Algorithms Graph Algorithms](https://reader036.vdocuments.us/reader036/viewer/2022062423/5697c00d1a28abf838cc93dc/html5/thumbnails/29.jpg)
David Luebke 29 04/21/23
Depth-First Search: The Code
DFS(G)
{
for each vertex u G->V {
u->color = WHITE;
}
time = 0;
for each vertex u G->V {
if (u->color == WHITE)
DFS_Visit(u);
}
}
DFS_Visit(u)
{
u->color = GREY;
time = time+1;
u->d = time;
for each v u->Adj[] {
if (v->color == WHITE)
DFS_Visit(v);
}
u->color = BLACK;
time = time+1;
u->f = time;
}
What will be the running time?
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David Luebke 30 04/21/23
Depth-First Search: The Code
DFS(G)
{
for each vertex u G->V {
u->color = WHITE;
}
time = 0;
for each vertex u G->V {
if (u->color == WHITE)
DFS_Visit(u);
}
}
DFS_Visit(u)
{
u->color = GREY;
time = time+1;
u->d = time;
for each v u->Adj[] {
if (v->color == WHITE)
DFS_Visit(v);
}
u->color = BLACK;
time = time+1;
u->f = time;
}Running time: O(n2) because call DFS_Visit on each vertex,
and the loop over Adj[] can run as many as |V| times
![Page 31: David Luebke 1 1/6/2016 CS 332: Algorithms Graph Algorithms](https://reader036.vdocuments.us/reader036/viewer/2022062423/5697c00d1a28abf838cc93dc/html5/thumbnails/31.jpg)
David Luebke 31 04/21/23
Depth-First Search: The Code
DFS(G)
{
for each vertex u G->V {
u->color = WHITE;
}
time = 0;
for each vertex u G->V {
if (u->color == WHITE)
DFS_Visit(u);
}
}
DFS_Visit(u)
{
u->color = GREY;
time = time+1;
u->d = time;
for each v u->Adj[] {
if (v->color == WHITE)
DFS_Visit(v);
}
u->color = BLACK;
time = time+1;
u->f = time;
}BUT, there is actually a tighter bound.
How many times will DFS_Visit() actually be called?
![Page 32: David Luebke 1 1/6/2016 CS 332: Algorithms Graph Algorithms](https://reader036.vdocuments.us/reader036/viewer/2022062423/5697c00d1a28abf838cc93dc/html5/thumbnails/32.jpg)
David Luebke 32 04/21/23
Depth-First Search: The Code
DFS(G)
{
for each vertex u G->V {
u->color = WHITE;
}
time = 0;
for each vertex u G->V {
if (u->color == WHITE)
DFS_Visit(u);
}
}
DFS_Visit(u)
{
u->color = GREY;
time = time+1;
u->d = time;
for each v u->Adj[] {
if (v->color == WHITE)
DFS_Visit(v);
}
u->color = BLACK;
time = time+1;
u->f = time;
}
So, running time of DFS = O(V+E)
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David Luebke 33 04/21/23
Depth-First Sort Analysis
This running time argument is an informal example of amortized analysis “Charge” the exploration of edge to the edge:
Each loop in DFS_Visit can be attributed to an edge in the graph
Runs once/edge if directed graph, twice if undirected Thus loop will run in O(E) time, algorithm O(V+E)
Considered linear for graph, b/c adj list requires O(V+E) storage
Important to be comfortable with this kind of reasoning and analysis
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David Luebke 34 04/21/23
DFS Example
sourcevertex
![Page 35: David Luebke 1 1/6/2016 CS 332: Algorithms Graph Algorithms](https://reader036.vdocuments.us/reader036/viewer/2022062423/5697c00d1a28abf838cc93dc/html5/thumbnails/35.jpg)
David Luebke 35 04/21/23
DFS Example
1 | | |
| | |
| |
sourcevertex
d f
![Page 36: David Luebke 1 1/6/2016 CS 332: Algorithms Graph Algorithms](https://reader036.vdocuments.us/reader036/viewer/2022062423/5697c00d1a28abf838cc93dc/html5/thumbnails/36.jpg)
David Luebke 36 04/21/23
DFS Example
1 | | |
| | |
2 | |
sourcevertex
d f
![Page 37: David Luebke 1 1/6/2016 CS 332: Algorithms Graph Algorithms](https://reader036.vdocuments.us/reader036/viewer/2022062423/5697c00d1a28abf838cc93dc/html5/thumbnails/37.jpg)
David Luebke 37 04/21/23
DFS Example
1 | | |
| | 3 |
2 | |
sourcevertex
d f
![Page 38: David Luebke 1 1/6/2016 CS 332: Algorithms Graph Algorithms](https://reader036.vdocuments.us/reader036/viewer/2022062423/5697c00d1a28abf838cc93dc/html5/thumbnails/38.jpg)
David Luebke 38 04/21/23
DFS Example
1 | | |
| | 3 | 4
2 | |
sourcevertex
d f
![Page 39: David Luebke 1 1/6/2016 CS 332: Algorithms Graph Algorithms](https://reader036.vdocuments.us/reader036/viewer/2022062423/5697c00d1a28abf838cc93dc/html5/thumbnails/39.jpg)
David Luebke 39 04/21/23
DFS Example
1 | | |
| 5 | 3 | 4
2 | |
sourcevertex
d f
![Page 40: David Luebke 1 1/6/2016 CS 332: Algorithms Graph Algorithms](https://reader036.vdocuments.us/reader036/viewer/2022062423/5697c00d1a28abf838cc93dc/html5/thumbnails/40.jpg)
David Luebke 40 04/21/23
DFS Example
1 | | |
| 5 | 63 | 4
2 | |
sourcevertex
d f
![Page 41: David Luebke 1 1/6/2016 CS 332: Algorithms Graph Algorithms](https://reader036.vdocuments.us/reader036/viewer/2022062423/5697c00d1a28abf838cc93dc/html5/thumbnails/41.jpg)
David Luebke 41 04/21/23
DFS Example
1 | 8 | |
| 5 | 63 | 4
2 | 7 |
sourcevertex
d f
![Page 42: David Luebke 1 1/6/2016 CS 332: Algorithms Graph Algorithms](https://reader036.vdocuments.us/reader036/viewer/2022062423/5697c00d1a28abf838cc93dc/html5/thumbnails/42.jpg)
David Luebke 42 04/21/23
DFS Example
1 | 8 | |
| 5 | 63 | 4
2 | 7 |
sourcevertex
d f
![Page 43: David Luebke 1 1/6/2016 CS 332: Algorithms Graph Algorithms](https://reader036.vdocuments.us/reader036/viewer/2022062423/5697c00d1a28abf838cc93dc/html5/thumbnails/43.jpg)
David Luebke 43 04/21/23
DFS Example
1 | 8 | |
| 5 | 63 | 4
2 | 7 9 |
sourcevertex
d f
What is the structure of the grey vertices? What do they represent?
![Page 44: David Luebke 1 1/6/2016 CS 332: Algorithms Graph Algorithms](https://reader036.vdocuments.us/reader036/viewer/2022062423/5697c00d1a28abf838cc93dc/html5/thumbnails/44.jpg)
David Luebke 44 04/21/23
DFS Example
1 | 8 | |
| 5 | 63 | 4
2 | 7 9 |10
sourcevertex
d f
![Page 45: David Luebke 1 1/6/2016 CS 332: Algorithms Graph Algorithms](https://reader036.vdocuments.us/reader036/viewer/2022062423/5697c00d1a28abf838cc93dc/html5/thumbnails/45.jpg)
David Luebke 45 04/21/23
DFS Example
1 | 8 |11 |
| 5 | 63 | 4
2 | 7 9 |10
sourcevertex
d f
![Page 46: David Luebke 1 1/6/2016 CS 332: Algorithms Graph Algorithms](https://reader036.vdocuments.us/reader036/viewer/2022062423/5697c00d1a28abf838cc93dc/html5/thumbnails/46.jpg)
David Luebke 46 04/21/23
DFS Example
1 |12 8 |11 |
| 5 | 63 | 4
2 | 7 9 |10
sourcevertex
d f
![Page 47: David Luebke 1 1/6/2016 CS 332: Algorithms Graph Algorithms](https://reader036.vdocuments.us/reader036/viewer/2022062423/5697c00d1a28abf838cc93dc/html5/thumbnails/47.jpg)
David Luebke 47 04/21/23
DFS Example
1 |12 8 |11 13|
| 5 | 63 | 4
2 | 7 9 |10
sourcevertex
d f
![Page 48: David Luebke 1 1/6/2016 CS 332: Algorithms Graph Algorithms](https://reader036.vdocuments.us/reader036/viewer/2022062423/5697c00d1a28abf838cc93dc/html5/thumbnails/48.jpg)
David Luebke 48 04/21/23
DFS Example
1 |12 8 |11 13|
14| 5 | 63 | 4
2 | 7 9 |10
sourcevertex
d f
![Page 49: David Luebke 1 1/6/2016 CS 332: Algorithms Graph Algorithms](https://reader036.vdocuments.us/reader036/viewer/2022062423/5697c00d1a28abf838cc93dc/html5/thumbnails/49.jpg)
David Luebke 49 04/21/23
DFS Example
1 |12 8 |11 13|
14|155 | 63 | 4
2 | 7 9 |10
sourcevertex
d f
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David Luebke 50 04/21/23
DFS Example
1 |12 8 |11 13|16
14|155 | 63 | 4
2 | 7 9 |10
sourcevertex
d f
![Page 51: David Luebke 1 1/6/2016 CS 332: Algorithms Graph Algorithms](https://reader036.vdocuments.us/reader036/viewer/2022062423/5697c00d1a28abf838cc93dc/html5/thumbnails/51.jpg)
David Luebke 51 04/21/23
DFS: Kinds of edges
DFS introduces an important distinction among edges in the original graph: Tree edge: encounter new (white) vertex
The tree edges form a spanning forest Can tree edges form cycles? Why or why not?
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David Luebke 52 04/21/23
DFS Example
1 |12 8 |11 13|16
14|155 | 63 | 4
2 | 7 9 |10
sourcevertex
d f
Tree edges
![Page 53: David Luebke 1 1/6/2016 CS 332: Algorithms Graph Algorithms](https://reader036.vdocuments.us/reader036/viewer/2022062423/5697c00d1a28abf838cc93dc/html5/thumbnails/53.jpg)
David Luebke 53 04/21/23
DFS: Kinds of edges
DFS introduces an important distinction among edges in the original graph: Tree edge: encounter new (white) vertex Back edge: from descendent to ancestor
Encounter a grey vertex (grey to grey)
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David Luebke 54 04/21/23
DFS Example
1 |12 8 |11 13|16
14|155 | 63 | 4
2 | 7 9 |10
sourcevertex
d f
Tree edges Back edges
![Page 55: David Luebke 1 1/6/2016 CS 332: Algorithms Graph Algorithms](https://reader036.vdocuments.us/reader036/viewer/2022062423/5697c00d1a28abf838cc93dc/html5/thumbnails/55.jpg)
David Luebke 55 04/21/23
DFS: Kinds of edges
DFS introduces an important distinction among edges in the original graph: Tree edge: encounter new (white) vertex Back edge: from descendent to ancestor Forward edge: from ancestor to descendent
Not a tree edge, though From grey node to black node
![Page 56: David Luebke 1 1/6/2016 CS 332: Algorithms Graph Algorithms](https://reader036.vdocuments.us/reader036/viewer/2022062423/5697c00d1a28abf838cc93dc/html5/thumbnails/56.jpg)
David Luebke 56 04/21/23
DFS Example
1 |12 8 |11 13|16
14|155 | 63 | 4
2 | 7 9 |10
sourcevertex
d f
Tree edges Back edges Forward edges
![Page 57: David Luebke 1 1/6/2016 CS 332: Algorithms Graph Algorithms](https://reader036.vdocuments.us/reader036/viewer/2022062423/5697c00d1a28abf838cc93dc/html5/thumbnails/57.jpg)
David Luebke 57 04/21/23
DFS: Kinds of edges
DFS introduces an important distinction among edges in the original graph: Tree edge: encounter new (white) vertex Back edge: from descendent to ancestor Forward edge: from ancestor to descendent Cross edge: between a tree or subtrees
From a grey node to a black node
![Page 58: David Luebke 1 1/6/2016 CS 332: Algorithms Graph Algorithms](https://reader036.vdocuments.us/reader036/viewer/2022062423/5697c00d1a28abf838cc93dc/html5/thumbnails/58.jpg)
David Luebke 58 04/21/23
DFS Example
1 |12 8 |11 13|16
14|155 | 63 | 4
2 | 7 9 |10
sourcevertex
d f
Tree edges Back edges Forward edges Cross edges
![Page 59: David Luebke 1 1/6/2016 CS 332: Algorithms Graph Algorithms](https://reader036.vdocuments.us/reader036/viewer/2022062423/5697c00d1a28abf838cc93dc/html5/thumbnails/59.jpg)
David Luebke 59 04/21/23
DFS: Kinds of edges
DFS introduces an important distinction among edges in the original graph: Tree edge: encounter new (white) vertex Back edge: from descendent to ancestor Forward edge: from ancestor to descendent Cross edge: between a tree or subtrees
Note: tree & back edges are important; most algorithms don’t distinguish forward & cross
![Page 60: David Luebke 1 1/6/2016 CS 332: Algorithms Graph Algorithms](https://reader036.vdocuments.us/reader036/viewer/2022062423/5697c00d1a28abf838cc93dc/html5/thumbnails/60.jpg)
David Luebke 60 04/21/23
DFS: Kinds Of Edges
Thm 23.9: If G is undirected, a DFS produces only tree and back edges
Proof by contradiction: Assume there’s a forward edge
But F? edge must actually be a back edge (why?)
sourceF?
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David Luebke 61 04/21/23
DFS: Kinds Of Edges
Thm 23.9: If G is undirected, a DFS produces only tree and back edges
Proof by contradiction: Assume there’s a cross edge
But C? edge cannot be cross: must be explored from one of the
vertices it connects, becoming a treevertex, before other vertex is explored
So in fact the picture is wrong…bothlower tree edges cannot in fact betree edges
source
C?
![Page 62: David Luebke 1 1/6/2016 CS 332: Algorithms Graph Algorithms](https://reader036.vdocuments.us/reader036/viewer/2022062423/5697c00d1a28abf838cc93dc/html5/thumbnails/62.jpg)
David Luebke 62 04/21/23
DFS And Graph Cycles
Thm: An undirected graph is acyclic iff a DFS yields no back edges If acyclic, no back edges (because a back edge implies a
cycle If no back edges, acyclic
No back edges implies only tree edges (Why?) Only tree edges implies we have a tree or a forest Which by definition is acyclic
Thus, can run DFS to find whether a graph has a cycle
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David Luebke 63 04/21/23
DFS And Cycles
How would you modify the code to detect cycles?DFS(G)
{
for each vertex u G->V {
u->color = WHITE;
}
time = 0;
for each vertex u G->V {
if (u->color == WHITE)
DFS_Visit(u);
}
}
DFS_Visit(u)
{
u->color = GREY;
time = time+1;
u->d = time;
for each v u->Adj[] {
if (v->color == WHITE)
DFS_Visit(v);
}
u->color = BLACK;
time = time+1;
u->f = time;
}
![Page 64: David Luebke 1 1/6/2016 CS 332: Algorithms Graph Algorithms](https://reader036.vdocuments.us/reader036/viewer/2022062423/5697c00d1a28abf838cc93dc/html5/thumbnails/64.jpg)
David Luebke 64 04/21/23
DFS And Cycles
What will be the running time?DFS(G)
{
for each vertex u G->V {
u->color = WHITE;
}
time = 0;
for each vertex u G->V {
if (u->color == WHITE)
DFS_Visit(u);
}
}
DFS_Visit(u)
{
u->color = GREY;
time = time+1;
u->d = time;
for each v u->Adj[] {
if (v->color == WHITE)
DFS_Visit(v);
}
u->color = BLACK;
time = time+1;
u->f = time;
}
![Page 65: David Luebke 1 1/6/2016 CS 332: Algorithms Graph Algorithms](https://reader036.vdocuments.us/reader036/viewer/2022062423/5697c00d1a28abf838cc93dc/html5/thumbnails/65.jpg)
David Luebke 65 04/21/23
DFS And Cycles
What will be the running time? A: O(V+E) We can actually determine if cycles exist in
O(V) time: In an undirected acyclic forest, |E| |V| - 1 So count the edges: if ever see |V| distinct edges,
must have seen a back edge along the way
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David Luebke 66 04/21/23
The End
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David Luebke 67 04/21/23
Exercise 1 Feedback
First, my apologies… Harder than I thought Too late to help with midterm
Proof by substitution: T(n) = T(n/2 + n) + n Most people assumed it was O(n lg n)…why?
Resembled proof from class: T(n) = 2T(n/2 + 17) + n The correct intuition: n/2 dominates n term, so it resembles
T(n) = T(n/2) + n, which is O(n) by m.t. Still, if it’s O(n) it’s O(n lg n), right?
![Page 68: David Luebke 1 1/6/2016 CS 332: Algorithms Graph Algorithms](https://reader036.vdocuments.us/reader036/viewer/2022062423/5697c00d1a28abf838cc93dc/html5/thumbnails/68.jpg)
David Luebke 68 04/21/23
Exercise 1: Feedback
So, prove by substitution thatT(n) = T(n/2 + n) + n = O(n lg n)
Assume T(n) cn lg n Then T(n) c(n/2 + n) lg (n/2 + n)
c(n/2 + n) lg (n/2 + n) c(n/2 + n) lg (3n/2)
…