Lecture-12-CS345A-2023 of Design and Analysis

Design and Analysis of Algorithms
Lecture 12
• Algorithms
1
CS345A
for Directed Acyclic Graphs

Problem Definition
Input: A directed graph with and a source vertex
Aim:
• Compute the shortest path to for all
Theorem:
There exists an time algorithm to build size data structure
that compactly stores shortest path to for all
3
Longest path to each vertex ?
No polynomial time algorithm
till date !

Imagine there were no research in Algorithms. And you are asked to write a program
to compute shortest paths from source s to every vertex. Your first attempt would
have been to enumerate all paths from source and then report the path which is
shortest. But the number of paths from source to any vertex can be exponential. So
you would have felt the hardness of the shortest paths problem. May be (if you were
like Dijkstra) you would have worked hard and with perseverance and designed the
algorithm which we call Dijkstra’s algorithm today.
But consider the situation now when the area of algorithm is quite developed ( very
active for more than 50 years). But the best algorithm the researchers have designed
till date for the longest paths problem takes exponential time ! What would be your
reaction to this reality ?
• Frustration, feeling down ? (not good for a person with true scientific spirit)
• Curiosity (natural)
There are hundreds of such algorithmic problems in computer science for which the
fastest algorithm takes only exponential time.
Towards the end of this course, we shall discuss a theory of such a family of problems.
“Theory of NP complete problems”
With this note, we begin today’s lecture.
4

Directed Acyclic Graphs
Question: what is a cycle in ?
Answer: A sequence , ,…, such that (,) for all
and (,) .
6
𝒗 𝟏 𝒗 𝟐 𝒗 𝟑 𝒗𝒌 −𝟏 𝒗 𝒌
…

Directed Acyclic Graphs
Definition:
A directed graph is said to be acyclic if there is no cycle present in it.
Example:
7
z
u
b v
x w a
y
DAG

Topological ordering
Definition: a mapping : []
such that for each edge
() < ()
Theorem: There exists a topological ordering for every DAG.
1 2 3 4 5 6 7 8 9 10 11
Does there exist a
topological ordering for
every directed graph ?
Certainly No if there is any cycle
< < <
< < <
>
Is it possible to arrange vertices in a line so that each edge goes from left to right ?
How to formalize it ?

Example:
This is indeed a valid topological ordering.
9
z
u
b v
x w a
y
u v a b w
x z y
1 2 3 4 5 6 7 8
How efficiently can we
determine if a given mapping
is indeed a topological
ordering ?
O() time

Three questions
Question 1: Why does a topological ordering exist for every DAG ?
Question 2: How efficiently can we compute a topological ordering ?
Question 3: What is the use of topological ordering ?
10

APPLICATIONS OF
TOPOLOGICAL ORDERING
Question 3
11

Applications of Topological ordering
Almost every algorithmic problem on DAG exploits Topological ordering.
Examples:
• Single source shortest paths
(No need of Dijkstra’s algorithm)
• Single source longest paths
• Count no. of paths from a source to a destination
(If the no. is a polynomial of )
All these problems have a simple time algorithm for DAG
12

WHY DOES
TOPOLOGICAL ORDERING EXIST FOR EVERY DAG?
Question 1
13

Why does Topological ordering exist ?
Example:
14
z
u
b v
x w a
y
Is there any vertex for
which you can be sure of
its place in the ordering?
Any vertex of indegree=0 can
be given number 1 in a
topological ordering.
What is the guarantee
that such a vertex always
exists in a DAG ?

Lemma 1: Every DAG has at least one vertex with in-degree = 0.
Proof:
(an algorithmic proof)
Pick any vertex .
While in-degree 0 do
{ Let be an edge
 ;
}
return ;
Observation: This algorithm, if terminates will output a vertex of indegree =0.
15
a
c
x
c …
b y
cycle
But what is the guarantee that it
will terminate.?
The algorithm does indeed terminate.
In fact, no vertex will be processed twice in
while loop.

Lemma 1: Every DAG has at least one vertex with in-degree = 0.
16
z
u
b v
x w a
y
How to use
Lemma 1 to show
existence of a valid
?

17
z
w
y
u
a
x
v
u v a
b
b w
x z y
1 2 3 4 5 6 7 8

Time complexity of the algorithm: ?
18
Is empty ?
 a vertex with
in-degree = 0
 num;
num  num + 1;
Remove and all its
outgoing edges;
NO
Yes
Input ;
num 1;
A valid
time

HOW EFFICIENTLY CAN WE COMPUTE
TOPOLOGICAL ORDERING ?
19

Important Questions
Aim:
To design a more efficient implementation of the algorithm.
Question:
What is the most time consuming step of the algorithm ?
Answer: Searching a vertex of in-degree = 0 ?
Question:
Do we need to compute from scratch the next vertex of in-degree 0 every time ?
Answer: Perhaps not !
20
Let us explore closely the algorithm to see
“how and when the vertices with in-degree=0 get created” ?

Revisiting the example
The new vertices with indegree=0 are created during ?
21
z
w
y
u
a
x
v
u v a
b
b w
x z y
1 2 3 4 5 6 7 8
..but slowly this time…
the processing of the current vertex of indegree=0

Algorithm for Topological ordering ?
Question:
How to design a more efficient implementation of the algorithm ?
Main step of the current algorithm:
• Searching a vertex of in-degree = 0
Hints:
1. Maintain a set of vertices of In-degree=0
2. Keep an array In-degree[] that stores for each vertex the number of edges
entering it at any stage during the algorithm.
22
time.
Attempt as a Homework.

APPLICATIONS OF
TOPOLOGICAL ORDERING ?
23

Example: Single source shortest paths
: the distance from source .
=0;
=
A correct formulation for distances from in a directed graph.
But difficult to translate to algorithms for general graphs.
Ponder over it.
24
But, this formulation leads to a very simple and
efficient algorithm for distance in DAGs.
Now ponder over it.

A mapping : [] such that
For each edge , () < ()
1 2 3 4 5 6 7 8 9 10 11

I
Let be a function on that we wish to compute
If can be expressed in terms of ONLY
We can compute by processing vertices in increasing order of .
1 2 3 4 5 6 7 8 9 10 11
𝒚
It is useful to compute in this order

I
Example: Single source shortest paths
: the distance from source .
=0;
=
1 2 3 4 5 6 7 8 9 10 11
𝒚
time algo

II
Let be a function on that we wish to compute
If can be expressed in terms of ONLY
We can compute by processing vertices in decreasing order of .
1 2 3 4 5 6 7 8 9 10 11
𝒚

II
Example: Number of paths to a vertex
: the number of paths to .
=1;
=
1 2 3 4 5 6 7 8 9 10 11
𝒚
time algo

Homework
Design an time algorithm for the following problem:
Given a directed acyclic graph , and a sequence of vertices ,
does there exist a path in which looks like :
30

Moral of the story
Think of a problem on DAG
 Think of topological ordering

31

Lecture-12-CS345A-2023 of Design and Analysis

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