a talk on entrepreneurship by Google's Country Manager in Greece

1. Sanders: Graphs 1
Graphs and Graph Representation INFORMATIK
¾ 1736 L. Euler asks about a
“touristic” question:
¾ Street or Computer networks
¾ Railway connections (space and time)
¾ Citation or Coauthorship social networks
¾ Job precedences scheduling problems
¾ Values and arithmetic operations compiler construction
¾

2. Sanders: Graphs
1 Mathematical View: Just Sets and Pairs INFORMATIK
Often terminology is already reason enough for a graph model
1.1 Directed Graphs
´Î µ describes a directed graph with vertex set Î and edge set
¢ .
Î Î
Convention: and Ñ
Ò Î
1.1.1 Special Cases (usually disallowed)
Multigraphs: parallel edges allowed, i.e, is a multiset.
self loops: edges ´Ú Úµ

3. Sanders: Graphs 3
1.1.2 Degrees INFORMATIK
¾
´Ú
out-degree´Úµ: Ùµ
¾
´Ù
in-degree´Úµ: Úµ

4. Sanders: Graphs
¾ ¾
´Ù ´Ú INFORMATIK
Bidirected graph: Úµ Ùµ
1.2 Undirected Graphs
streamlined description of a bidirected graph where edges are two element
vertex sets.
° ´Ù ´Ú
ÙÚ Úµ Ùµ
degree´Úµ in-degree´Úµ out-degree´Úµ
What about self loops?

5. Sanders: Graphs 5
1.3 Subgraphs INFORMATIK
¼ ´Î ¼ ¼ µ is subgraph of µ if Î ¼ ¼¾
´Î and
Î
¼ is vertex induced by Î ¼ if ¼ ¼ ¼
¾ ¾ ¾
´Ù Úµ Ù Î Ú Î

6. Sanders: Graphs
1.4 Associated Information INFORMATIK
Ê
edge weights Û
Example: Length or traffic capacity of a street,
we will see more things like node potentials, node/vertex colors,

7. Sanders: Graphs 7
1.5 Paths INFORMATIK
A path Ô Ú¼ Ú
of length connects nodes Ú¼ and Ú if
subsequent nodes in Ô are connected by edges in , i.e,
¾ ¾ ¾
½
´Ú¼ Ú½ µ ´Ú½ Ú¾ µ ´Ú µ
,¾ , , .
½ Ú
edge representation: .
½
Ô
simple path: Ú¼ , ,Ú different.
Nonsimple paths are usually called walks.
È
µ.
In weighted graphs, length is ½ Û´
bottleneck weight: Ñ Ò µ.
½ Û´
Applications: What is the shortest route from Saarbrucken to Paris?
¨
How much traffic can it carry?

8. Sanders: Graphs
1.6 Cycles INFORMATIK
Cycles are paths that share the first and the last node
A cycle is simple if all other nodes are different.
Hamiltonian cycle: simple cycle that visits all nodes.
Euler cycle: cycle (cyclic walk) that visits all edges once
back to the bridge problem. But we need one more term.

9. Sanders: Graphs 9
1.7 Connectedness INFORMATIK
¸ ¾
is strongly connected path Ô connects Ù and Ú
ÙÚ Î Ô
drop the “strongly” for undirected graphs
A connected component is a maximal connected vertex induced subgraph

10. Sanders: Graphs
1.8 Euler Tours INFORMATIK
Theorem 1. is Eulerian (has an Euler tour) if and only if is connected
and every node has even degree.
Proof: Necessity: connectedness is clear.
½.
Consider a node Ú with degree ¾
The tour is stuck when it visits Ú the -th time.
Sufficiency:
Step 1, Construct Euler Paritioning of into a set of cycles:
Greedily build any walk. Remove visited edges.
When you get stuck, you have closed a cycle.
Step 2, Glue the cycles together.
Possible since any cycle intersects some other cycle.

11. Sanders: Graphs 11
1.9 Trees INFORMATIK
An undirected graph is a tree if there is exactly one path between any pair of
nodes.
Exercise: Prove that the following properties are equivalent
1. is a tree.
½ edges.
2. is connected and has exactly Ò
3. is connected and contains no cycles.
Forest: undirected acyclic graph
Directed acyclic graph (DAG): no cycles.
Exercise: How many edges are possible?

12. Sanders: Graphs
1.10 Directed Trees? INFORMATIK
we need a special root node Ö.
¾
exactly one Ö path or
Ú Î Ú
¾
exactly one Ú path
Ú Î Ö
CS trees have the root at the top.
a parent is immediately above its children or successors which are siblings.
(ancestors, ) Nodes without children are leaves.
Nonroot nonleaf nodes are interior.
ordered trees: order of children matters. Example: search trees.

13. Sanders: Graphs 13
2 Graph Representation INFORMATIK
we mostly represent undirected graphs as bidirected graphs
focus on directed graphs
Overview:
¾ Operations are what matters
¾ A trivial representation
¾ Arrays
¾ Linked Lists
¾ Matrices
¾ Implicit Representation
¾ Discussion

14. Sanders: Graphs
2.1 Operations INFORMATIK
Ç´½µ time for all elementary operations
Goal:
Access associated information. Use arrays. Perhaps hash tables.
½
One reason for Î Ò.
Navigation: Given Ú find outgoing edges. (We may also want incoming edges.)
¾
Edge queries: ´Ù ? Adjanceny matrices, hash tables.
Úµ
Reverse access: Given ´Ù find ´Ú
Úµ Ùµ
Construction conversion and output (Ç´Ñ · Òµ time)
Update: Insert/delete nodes/edges. This is the hard part.

15. Sanders: Graphs 15
Example: Recognizing DAGs INFORMATIK
¾ ¼ do delete Ú
while out-degree´Úµ
Ú Î
if Î then output “ is a DAG”
else output “ contains a cycle”
// to find a cycle, pick any node and follow edges
works because we never destroy or introduce cycles
Ç´Ñ ·
Exercise: implement it in time Òµ

16. Sanders: Graphs
2.2 Edge Sequence Representation INFORMATIK
Sequence of node pairs (or triples with edge weight)
· Compact
· Good for I/O
Almost no useful operations except scanning all edges
Not so bad with some additional node arrays.
Examples: Find isolated nodes, several MST algorithms, conversion.

17. Sanders: Graphs 17
2.3 Adjacency Arrays INFORMATIK
½
¼
Î Ò
Edge array E stores targets
grouping edges leaving a node
Node array V stores index of first outgoing edge
stores Ñ · ½
dummy entry Î Ò
1
0 n−1 4=n
V0 2 466
0 3
E1 2 2 3 1 3
m−1 6=m
0
2
Î
ÎÚ·½
Example: out-degree´Úµ Ú

18. Sanders: Graphs
2.4 Edge List Adjacency Array INFORMATIK
A Reminder: Sorting Small Integers
// make a sorted permutation of
Procedure KSortArray´a,b : Array ½ of 0..K-1µ
Ò
½ of Æ
¼ ¼ : Array ¼
c // counters for each bucket
Ã
for := ½ to Ò do ·· // Count bucket sizes
C := 0
È
½ do ´
:= ¼ to à µ := ´ · µ// Store
for in
for := ½ to Ò do // Distribute
:=
··
Idea: Sort by starting node

19. Sanders: Graphs 19
Integrated Solution INFORMATIK
Ä ×Ø µ
Function adjacencyArray´
of Æ
¼ ¼ : Array ¼
V Ò
¾ EdgeList do
foreach ´Ù ·· // count
Úµ ÎÙ
½
for Ú := ½ to Ò do +=Î // prefix sums
ÎÚ Ú
¾ EdgeList do
foreach ´Ù // place
Úµ ÎÙ Ú
return ´Î µ
Example on Blackboard

20. Sanders: Graphs
Operations for Adjacency Arrays INFORMATIK
Navigation: easy. we will see several applications
Edge weights: becomes array of records
Incoming Edges: another (reverse) edge array
Delete Edges: explicit end indices
Batched Updates: rebuild

21. Sanders: Graphs 21
2.5 Adjacency Lists INFORMATIK
store (doubly linked) list of adjacent edges for each node
(a bit more restricted with singly linked lists)
· easy edge insertion
· easy edge deletion (order preserving)
more space (factor up to 3) than adj. arrays
more cache faults than adj. arrays

22. Sanders: Graphs
Enhancing Adjacency Lists INFORMATIK
For reverse edge access or edge weight updates in undirected graphs:
explicit edge objects
¾ stores weights
¾ pred/succ-pointers for all adjacency lists containing this edge
¾ reverse pointer for directed graphs
¾ cute trick: store only xor of incident nodes

23. Sanders: Graphs 23
Node insertion/deletion INFORMATIK
mark as deleted swap in node Ò list of nodes
µ node handles are pointers. 1
0 3
Problem: graphs with common node set
E list out list in list rev from to
2
(0,1) 0 0 0 V list first first deg deg
out in out in
(0,2) 0
0 0 0 0 2 0
(1,2) 1
0 2 2
(1,3) 2
0 00 2 2
(2,1) 3
0 0 0 0 0 2
(2,3) 0 0 0 0
¢ more space than vanilla adjacency arrays

24. Sanders: Graphs
2.6 Customization INFORMATIK
Customize data structure for application for maximum speed/compactness.
Sofware Engineering nightmare
Seperating algorithm from their representation may be a way out

25. Sanders: Graphs 25
Example: Recognizing dags INFORMATIK
¾ ¼ do delete Ú
while out-degree´Úµ
Ú Î
¾ Use adjacency array storing incoming edges
¾ Store out-degrees
¾ Maintain a stack of out-degree zero nodes
¾ delete ´Ù by decrementing out-degree´Ùµ
Úµ
¾ no explicit node deletion
¾ At the end test whether all nodes have out-degree´¼µ

26. Sanders: Graphs
2.7 Adjacency Matrix INFORMATIK
Ò¢Ò with
¾ ¾
¼½ ´ µ ´ µ
· space efficient for very dense matrices
space inefficient otherwise. Exercise: what should “very dense” mean?
· Easy edge queries
Slow navigation (except for dense graphs)
·· joins linear algebra and graph theory
# -edge paths from to
Example: .
Exercise: count -edge paths
Important accelerators:
Ç´ÐÓ µ mat-mults for mat-power
matrix multiplication in subcubic time, e.g., Strassen’s algorithm

27. Sanders: Graphs 27
Example where graph theory helps LA INFORMATIK
Ü ´½ ´ µ ¼µ
Problem: solve Consider Ò
µ
Assume has two connected components
we swap rows and cols such that
¼ ½¼ ½ ¼ ½
¼ ܽ
½ ½
¼ ܾ
¾ ¾
Exercise: What if is a DAG?

28. Sanders: Graphs
2.8 Implicit Representation INFORMATIK
Compact representations of possibly very dense graphs
Implement algorithms directly using this representation

29. Sanders: Graphs 29
Example: Connectedness of Interval Graphs INFORMATIK
ÒÒ
½ ½
Î
and overlap
Idea: sweep intervals from left to right. The number of overlapping intervals
may never drop to zero.
Function isConnected(Ä : SortedListOfIntervalEndPoints) : ¼ ½
remove first element of Ä
overlap := 1
¾
foreach Ô do
Ä
¼ return 0
if overlap
if Ô is a start point then overlap··
else overlap // end point
¾¡
Ç´Ò ÐÓ Ç
algorithm for up to edges! Exercise: find all connected
Òµ Ò
components