Plan
CS 312: Graphs: BFS and DFS
Dan Sheldon
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Gale-Shapley Running Time
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Graphs
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Motivation and definitions
Graph traversal: BFS and DFS
February 5, 2015
Running Time of Gale-Shapley?
Data Structures
Initially all colleges and students are free
while some college is free and hasn’t made o↵ers to every
student do
Choose such a college c
Let s be the highest ranked student to whom c has not made
an o↵er
if s is free then
c and s become engaged
else if s is engaged to c0 but prefers c to c0 then
c0 becomes free
c and s become engaged
else
c remains free
end if
end while
Running-time depends on implementation details and data
structures (e.g. how to “choose such a college c”).
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Q: How should we think about data structures when designing
algorithms?
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A: Most of the time, as black boxes with running-time
guarantees (e.g., “find an element in O(log n) time”).
Good news: don’t need to remember details of data structures
Bad news: they may seem opaque
O(n2 ) iterations. Are all statements inside the loop constant time?
Review: Lists and Arrays
What Data Structures to Use For G-S?
Array
List
Get ith entry
O(1)
O(i)
Find element
O(n)
O(log n) if sorted
O(n)
Insert/delete
O(n)
O(1)
Note: O(1) = constant number of steps
Need to do the following in O(1) time
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Find free college c
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Find next student s in preference list of c
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Find current college c0 of s
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Check if s likes c0 better than c
What Data Structures to Use For G-S?
Another Example: Heapsort
Input: prefence lists = 2D arrays, e.g.
CollegePref[c, i] = student in position i on c’s list
Operation
Data structure
Fnd a free college c
linked list: freeColleges
Find next student s in
preference list of c
array
i = Next[c]
s = CollegePref[c, i]
Find current college c0 of s
1-D array: Current[s]
Check if s likes
c0
better than c
Input unsorted array A[] of length n
Let Q be a heap-based priority queue
for i = 1 to n do
Insert(Q, A[i])
end for
for i = 1 to n do
A[i] = ExtractMin(Q)
end for
Running time: (n⇥ Insert) + (n⇥ ExtractMin)
2-D array: Ranking[s,c]
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O(n log n) if both operations are O(log n)
example on board
Graphs
Undirected Graph
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Motivation and Definitions
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Breadth-First Search (BFS) and Depth-First Search (DFS)
Undirected graph. G = (V, E)!
V = nodes (vertices)!
E = edges between pairs of nodes.!
Captures pairwise relationship between objects.!
Graph size parameters: n = |V|, m = |E|.
1
2
3
5
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V = {1, 2, 3, 4, 5}!
E = {(1,2), (1,4), (1,5), (2,3), (2,4), !
(3,5)}!
n=5!
m=6
Graphs
Four Degrees of Separation
Four Degrees of Separation
⇤ †
†
†
⇤
⇤
†
Lars† Backstrom
Boldi
Marco
Rosa
Johan
Ugander
Lars Backstrom⇤ Paolo Boldi
Marco RosaPaolo
Johan
Ugander
Sebastiano
Vigna
January 6, 2012
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Facebook: how many “degrees of separation” between me and
Barck Obama?
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And many more. . .
studying the distance distribution
of very large
graphs: distribution
HyAbstract
studying
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perANF [3]. Building on previousperANF
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1
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1 Introduction
ship links).
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Abstract
v:1111.4570v3 [cs.SI] 5 Jan 2012
Google Maps: what is the shortest driving route from South
Hadley to Florida?
Xiv:1111.4570v3 [cs.SI] 5 Jan 2012
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Sebastian
January 6, 2012
Terminology
If e = (u, v) is an edge, then:
(1) u is a neighbor of v
(2) u is adjacent to v
(3) e is incident on u and v
(4) u and v are the endpoints of e
Definitions
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2
3
5
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Path
Distance
The distance from u to v is the minimum number
of edges in any path from u to v
A path is a sequence P of nodes v1, v2, …, vk-1, vk
with the property that each consecutive pair vi,
vi+1 is joined by an edge in E.
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1
2
3
2
1-4-2 is a path.!
1-3-4 is NOT a path.
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5
5
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Connectivity
An undirected graph is connected if for every pair
of nodes u and v, there is a path between u and v.
A cycle is a path v1, v2, …, vk-1, vk in which v1 = vk, k
> 2, and the first k-1 nodes are all distinct.
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2
3
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distance(1,2) = 1!
distance(1,3) = 2
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Cycle
1
1 is adjacent to 2!
(1,2) is incident on 1 and 2
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1-2-4-1 is a cycle.!
1-2-4 is NOT a cycle.!
1-2-4-1-5 is NOT a cycle.!
1-2-4-1-5-3-2-1 is NOT a cycle.
2
5
4
1
2
5
4
3
is a connected graph.
3
is NOT a connected
graph.
Parents, descendants, ancestors?
(Upside-down) Trees
Trees
A tree is an undirected graph that is connected
and does not contain a cycle.
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2
3
1
2
5
3
5
4
1
2
is a tree
1
2
2
5
3
5
4
is NOT a tree
4
1
3
4
5
http://www.offbeattravel.com/MoCA.html
Review Definitions
Graph Traversal
Is a graph connected?
What to know: n, m, neighbor, incident, path, distance, cycle,
connected, tree!
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2
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Example on board
easy
Graph Traversal
hmmm...
Is a Graph Connected?
Algorithm 1: Breadth-first search (BFS)
Explore outward by distance
Is a graph connected?!
Approach: explore outward from arbitrary
starting node s to find all nodes reachable from
s (connected component)
a
c
b
d
Start at a:
e
Visit all nodes at
distance 1 from a:
a
c
b
d
Visit all nodes at
distance 2 from a:
a
c
b
d
e
e
4
3
Breadth-First Search
BFS Tree
Layers
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L0 = {s}
L1 = all neighbors of L0
L2 = all nodes with an edge to L1 that don’t belong to L0 or L1
...
Li+1 = nodes with an edge to Li that don’t belong to an earlier
layer:
If we keep only the edges traversed while doing a breadth-first
search, we will have a tree
Example on board
Li+1 = {v : 9(u, v) 2 E, u 2 Li , v 2
/ (L0 [ . . . [ Li )}
Observation: Li consists of all nodes at distance exactly i from s.
There is a path from s to t if and only if t appears in some layer.
A More General Strategy
BFS Tree
Property. Let T be a BFS tree of G = (V, E), and let (x,
y) be an edge of G. Then the layer of x and y differ by
at most 1.
a
Layer 0: {a}!
Layer 1: {b, c, d}!
Layer 2: {e}
c
e
b
d
To explore the connected component, add any node v for which
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(u, v) is an edge
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u is explored, but v is not
Picture on board
Proof on board
DFS Algorithm
Is a Graph Connected?
Algorithm 2: Depth-first search (DFS) - Keep exploring from
most recently added node until you have to backtrack
a
c
a
c
e
b
d
a
c
b
d
a
c
b
d
e
b
d
a
c
e
e
b
d
e
DFS(u)
Mark u as ”Explored”
for each edge (u, v) incident to u do
if v is not marked ”Explored” then
Recursively invoke DFS(v)
end if
end for
Summary
Depth First Search
Theorem: Let T be a depth-first search tree. Let
x and y be 2 nodes in the tree. Let (x, y) be an
edge that is in G but not in T. Then either x is an
ancestor of y or y is an ancestor of x in T.
a
a
b
e
d
b
Proof?
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G = (V, E), n = |V |, m = |E|
neighbor, incident, cycle, path, connected
BFS and DFS
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e
d
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c
c
Definitions
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Two ways to traverse a graph, each produces a tree
BFS tree: shallow and wide (“bushy”)
DFS tree: deep and narrow (“scraggly”)