Institute of Operating Systems
and Computer Networks
Algorithms Group
Network Algorithms
Tutorial 1: Flashback and Fast Forward
Christian Rieck—April 24, 2017
Staff
Lecture:
Dr. Christian Scheffer
scheff[email protected]
Room: IZ 331
Tutorial:
Christian Rieck
[email protected]
Room: IZ 314
Hiwi:
Jakob Keller
[email protected]
Network Algorithms—Tutorial 1: Flashback and Fast Forward
April 24, 2017
2
Homework and exam
There will be five homework assignment sheets.
You have to get at least 50% of the points in total.
You have to pass a written exam at the end
of the lecture, probably at August, 7th.
Network Algorithms—Tutorial 1: Flashback and Fast Forward
April 24, 2017
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Extra points
You can earn extra points for your homework!
‣ find a mistake in the slides
‣ improve the tutorial by a better example
‣ improve the tutorial by ideas on new interesting stuff
‣ …
Please send an email or something and I will think
about it. Usually the amount of extra points is in some way
related to the improvement and is a fraction of my room
number π.
Network Algorithms—Tutorial 1: Flashback and Fast Forward
April 24, 2017
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(preliminary) Schedule
Semesterplan Netzwerkalgorithmen Sommer’17
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
Woche
(ab 3.4.)
(ab 10.4.)
(ab 17.4.)
(ab 24.4.)
(ab 1.5.)
(ab 8.5.)
(ab 15.5.)
(ab 22.5.)
(ab 29.5.)
(ab 5.6.)
(ab 12.6.)
(ab 19.6.)
(ab 26.6.)
(ab 3.7.)
(ab 10.7.)
Vorlesung
(Do.)
1
2
3
4
5
6
Christi Himmelfahrt
7
Gr. Übung
(Mo.)
Kl. Übung
(Do.)
HA Rückgabe
HA Ausgabe
HA Abgabe
(Mo.)
(Mo. bis 11:30 Uhr) (in kl. Übung)
HA1
1
HA2
2
1
HA1
HA3
3
2*
HA1
HA2
Christi Himmelfahrt
8
4
9
10
5
11
12
6
*: Die zweite kleine Übung wird
HA4
Exkursionswoche
HA3
3
HA2*
HA3
HA5
HA4
4
HA4
HA5
5
auf den Slot der großen Übung verschoben!
Network Algorithms—Tutorial 1: Flashback and Fast Forward
April 24, 2017
HA5
5
(preliminary) Schedule
Semesterplan Netzwerkalgorithmen Sommer’17
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
Woche
(ab 3.4.)
(ab 10.4.)
(ab 17.4.)
(ab 24.4.)
(ab 1.5.)
(ab 8.5.)
(ab 15.5.)
(ab 22.5.)
(ab 29.5.)
(ab 5.6.)
(ab 12.6.)
(ab 19.6.)
(ab 26.6.)
(ab 3.7.)
(ab 10.7.)
Vorlesung
(Do.)
1
2
3
4
5
6
Gr. Übung
(Mo.)
Kl. Übung
(Do.)
HA Rückgabe
HA Ausgabe
HA Abgabe
(Mo.)
(Mo. bis 11:30 Uhr) (in kl. Übung)
HA1
1
HA2
HA1
2
1
There is a calendar
on
the webpage
which
will
HA3
HA2
3
be up-to-date!
7
2*
HA4
HA3
Christi Himmelfahrt
HA1
Christi Himmelfahrt
HA2*
Exkursionswoche
8
4
9
10
5
11
12
6
*: Die zweite kleine Übung wird
3
HA3
HA5
HA4
4
HA4
HA5
5
auf den Slot der großen Übung verschoben!
Network Algorithms—Tutorial 1: Flashback and Fast Forward
April 24, 2017
HA5
5
Mailing list
There is a mailing list for this lecture.
https://mail.ibr.cs.tu-bs.de/mailman/listinfo/na
We will distribute several announcements via this list.
So, please subscribe.
Network Algorithms—Tutorial 1: Flashback and Fast Forward
April 24, 2017
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Webpage
There is a webpage for this lecture.
https://www.ibr.cs.tu-bs.de/courses/ss17/na/index.html
There is all the stuff like the slides of the tutorials,
the homework assignment sheets, the schedule
of the lecture, informations regarding the exam,…
Network Algorithms—Tutorial 1: Flashback and Fast Forward
April 24, 2017
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Questions?
Network Algorithms—Tutorial 1: Flashback and Fast Forward
April 24, 2017
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Previously in *…
Network Algorithms—Tutorial 1: Flashback and Fast Forward
April 24, 2017
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Graphs
A graph is an ordered pair G=(V,E) comprising
a set V of vertices together with a set E of edges,
which are 2-element subsets of V.
We normally consider simple graphs, i.e.,
there is at most one edge between two
specific vertices and there are no edges
connecting a vertex with itself.
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Digraphs
A digraph is an ordered pair D=(V,A) comprising
a set V of vertices together with a set A of arcs (edges),
which are ordered pairs of vertices.
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Data structures
There are several ways to represent graphs, e.g.,
‣ adjacency list
‣ adjacency matrix
‣ incidence matrix
Please make yourself familiar with these data structures.
A good reference is the seventh lecture on algorithms and data structures.
https://www.ibr.cs.tu-bs.de/courses/ws1617/aud/webpages/skript/VL7a.pdf
Network Algorithms—Tutorial 1: Flashback and Fast Forward
April 24, 2017
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Data structures
Stack
‣ last in, first out (LIFO)
‣ push
‣ pop
Queue
‣ first in, first out (FIFO)
‣ enqueue
‣ dequeue
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Breadth-First-Search
v2
v1
procedure BFS(G,v)
let Q be a queue
Q.enqueue(v)
while Q is not empty
v = Q.dequeue()
for each edge e incident to v
let w be the other endpoint of e
if w is not marked
mark w
Q.enqueue(w)
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Depth-First-Search
v2
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procedure DFS(G,v)
let S be a stack
S.push(v)
while S is not empty
v = S.pop()
if v is not labeled as discovered
label v as discovered
for all edges from v to w
S.push(w)
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Walk, path, trail,…
Kantenfolge:
Alternating sequence of vertices and edges.
Weg:
Kantenfolge
+
no repeating edges
Pfad:
Kantenfolge
+
no repeating vertices
Tour:
Weg
+
same start and end vertex
Kreis:
Pfad
+
same start and end vertex
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Big O-Notation
‣
describes the limiting behavior of a function when the
argument tends towards infinity
‣
is used to classify algorithms according to their
running time
O( f ) = { g : N ! R + | 9c 2 R >0 9n0 2 N 8n 2 N
g(n) 2 O( f (n))
n0
: g(n)  c · f (n)}
f (n)
g(n)
n0
Network Algorithms—Tutorial 1: Flashback and Fast Forward
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Big O-Notation
f ( x ) = 6x4
2x3 + 42
Show: f ( x ) 2 O( x4 )
|6x4
2x3 + 42|  6x4 + |2x3 | + 42
 6x4 + 2x4 + 42x4
=
4
50x
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Other Landau symbols
There are some other notations for analyzing
the running time and space requirements of
algorithms, e.g., small-o, big-Omega,…
Please make yourself familiar with these notations.
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Soon in network algorithms…
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Minimum-cost problems
Minimum Spanning Tree
Given: A graph G=(V,E) with edge weights
Wanted: A tree T of minimum cost that spans the graph
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Minimum-cost problems
Minimum Spanning Tree
Given: A graph G=(V,E) with edge weights
Wanted: A tree T of minimum cost that spans the graph
A geometric generalisation is the Minimum Steiner Tree Problem.
One can also ask for a spanning tree where the longest edge is as short as
7possible—the Minimum Bottleneck Spanning
7
Tree!
6
6
7 spanning tree where each vertex degrees
7 are
Or we5want a 8minimum
5 1
5 1
bounded, or
11 satisfy some other constraints.
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6
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4them are really hard, e.g., the Steiner problems.
4
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of
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Shortest-path problems
Single source shortest paths
Given: A graph G=(V,E) with edge weights, and a vertex s
Wanted: A shortest path tree T routed in s
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Shortest-path problems
Single source shortest paths
Given: A graph G=(V,E) with edge weights, and a vertex s
Wanted: A shortest path tree T routed in s
There are many related problems, e.g., Longest Path, Hamiltonian Path,
7 tour problems like the Traveling Salesman
7 Problem, …
several
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Almost
all
of
them
are
NP-hard!
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Flows
Maximum Flow
Given: A digraph D=(V,A) with edge weights, and vertices s and t
Wanted: A maximum flow F starting from s to t
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Flows
Maximum Flow
Given: A digraph D=(V,A) with edge weights, and vertices s and t
Wanted: A maximum flow F starting from s to t
A fundamental theorem is the MaxFlow-MinCut-Theorem
of Ford and Fulkerson that shows that a MaxFlow is equal to a
MinCut
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There1 are polynomial-time algorithms for5solving
1 such flow problems.
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But there are many related problems,
and
some of them3 are NP-hard. 6
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Matchings
Maximum Matching
Given: A graph G=(V,E)
Wanted: A maximum set M of pairwise non-adjacent edges
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April 24, 2017
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Matchings
Maximum Matching
Given: A graph G=(V,E)
There
many different
of the matching
problem, e.g.,
Wanted:
A are
maximum
set variants
M of pairwise
non-adjacent
edges
Perfect Matching
Minimum Cost Perfect Matching
Maximal Matching
Matching in bipartite Graphs
…
G
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April 24, 2017
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Questions?
Network Algorithms—Tutorial 1: Flashback and Fast Forward
April 24, 2017
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