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Chapter 1 : Dynamic programming
ENSIIE - Operations Research Module
Dimitri Watel ([email protected])
2016-2017
Dimitri Watel
MRO Chap 01 Dyn Prog
Examples
Generalization
When does it not work ?
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Shortest paths
The Partition problem
What is the shortest path from A to E ?
50
40
25
A
10
B
20
20
C
15
D
10
20
35
Dimitri Watel
MRO Chap 01 Dyn Prog
E
Examples
Generalization
When does it not work ?
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Shortest paths
The Partition problem
What is the shortest path from A to E ?
50
40
25
A
10
B
20
20
C
15
D
10
20
35
Dimitri Watel
MRO Chap 01 Dyn Prog
E
Examples
Generalization
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Shortest paths
The Partition problem
Brute force algorithm
If we enumerate all the paths :
(A - E) costs 50
50
(A - B - E) costs 45
40
25
A
10
B
(A - C - E) costs 45
20
20
C
15
D
10
(A - B - C - E) costs 50
E
20
(A - D - E) costs 50
35
...
Every path containing A and E
If there are n vertices : 2n−2 possible paths to enumerate.
Dimitri Watel
MRO Chap 01 Dyn Prog
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Shortest paths
The Partition problem
Interesting property
In order to reach E, there are 4 ways :
I go from A to E with the arc (A,E)
I go to B then I go to E with the arc (B,E)
I go to C then I go to E with the arc (C,E)
I go to D then I go to E with the arc (D,E)
Dimitri Watel
MRO Chap 01 Dyn Prog
Examples
Generalization
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Shortest paths
The Partition problem
Interesting property
In order to reach E, there are 4 ways :
I go to A then I go to E with the arc (A,E)
I go to B then I go to E with the arc (B,E)
I go to C then I go to E with the arc (C,E)
I go to D then I go to E with the arc (D,E)
Dimitri Watel
MRO Chap 01 Dyn Prog
Examples
Generalization
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Shortest paths
The Partition problem
Interesting property
In order to reach E, there are 4 ways :
I go to A from A with a shortest path then I go to E with
the arc (A,E)
I go to B from A with a shortest path then I go to E with
the arc (B,E)
I go to C from A with a shortest path then I go to E with
the arc (C,E)
I go to D from A with a shortest path then I go to E with
the arc (D,E)
Dimitri Watel
MRO Chap 01 Dyn Prog
Examples
Generalization
When does it not work ?
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Shortest paths
The Partition problem
Interesting property
In order to reach E, there are 4 ways :
I go to A from A with a shortest path then I go to E with
the arc (A,E)
I go to B from A with a shortest path then I go to E with
the arc (B,E)
I go to C from A with a shortest path then I go to E with
the arc (C,E)
I go to D from A with a shortest path then I go to E with
the arc (D,E)
We have to compute the shortest path from A to A, B, C et D in
order to compute the shortest path from A to E .
Dimitri Watel
MRO Chap 01 Dyn Prog
Examples
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Shortest paths
The Partition problem
Extending the property : dynamic programming
Let d(v ) be the shortest path from A to the node v .
Let ω(u, v 
) be the weight of the arc (u, v ).
d(A) + ω(A, E )



d(B) + ω(B, E )
d(A) + ω(A, C )
d(E ) = min
d(C ) = min
d(C ) + ω(C , E )


d(B)
+ ω(B, C )

d(D) + ω(D, E )
d(B) = d(A) + ω(A, B)

 d(A) + ω(A, D)
d(A) = 0
d(B) + ω(B, D)
d(D) = min

d(C ) + ω(C , D)
Complete the following table from left to right
v
A B C D E
d(v )
Dimitri Watel
MRO Chap 01 Dyn Prog
Examples
Generalization
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Shortest paths
The Partition problem
Dynamic programming
50
40
25
A
10
B
20
20
C
15
D
10
20
35
Dimitri Watel
MRO Chap 01 Dyn Prog
E
Examples
Generalization
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Shortest paths
The Partition problem
Who is the best ?
Dynamic programming : n·(n+1)
sums and comparisons
2
n−2
Brute force : 2
operations and comparisons.
Dimitri Watel
MRO Chap 01 Dyn Prog
Examples
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Shortest paths
The Partition problem
Partition the set
X = {1,5,6,7,9,10,16}
It is possible to partition X into two subsets such that the sum of
the elements in the two subsets is the same ?
!
P
P
∃?X1 , X2 | (X = X1 ] X2 ) ∧
x=
x
x∈X1
Dimitri Watel
x∈X2
MRO Chap 01 Dyn Prog
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Shortest paths
The Partition problem
Partition the set
X = {1,5,6,7,9,10,16}
It is possible to partition X into two subsets such that the sum of
the elements in the two subsets is the same ?
!
P
P
∃?X1 , X2 | (X = X1 ] X2 ) ∧
x=
x
x∈X1
Dimitri Watel
x∈X2
MRO Chap 01 Dyn Prog
Examples
Generalization
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Shortest paths
The Partition problem
Brute force algorithm
X = {1, 5, 6, 7, 9, 10, 16}
If we enumerate all the partitions ?
X1 = ∅ and X2 = {1, 5, 6, 7, 9, 10, 16} : No
X1 = {1} and X2 = {5, 6, 7, 9, 10, 16} : No
X1 = {5} and X2 = {1, 6, 7, 9, 10, 16} : No
X1 = {1, 5} and X2 = {6, 7, 9, 10, 16} : No
X1 = {6} and X2 = {1, 5, 7, 9, 10, 16} : No
...
X1 = {1, 7, 9, 10} and X2 = {5, 6, 16} : Yes
...
If |X | = n : 2n possible partitions to enumerate.
Dimitri Watel
MRO Chap 01 Dyn Prog
Examples
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Shortest paths
The Partition problem
A similar problem
X = {1, 5, 6, 7, 9, 10, 16}
Property
!
If
P
x∈X1
x=
P
x∈X2
x
then
P
x=
x∈X1
P
x /2 = 27.
x∈X
New problem
P
∃?X1 ⊂ X |
x = 27
x∈X1
Dimitri Watel
MRO Chap 01 Dyn Prog
Examples
Generalization
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Shortest paths
The Partition problem
Interesting property
X = {1, 5, 6, 7, 9, 10, 16}
If there exists a set X1 for which the sum of the elements is 27,
there are 7 ways to build such a set :
X1 contains 16 and there is a subset of X1 for which the sum
is 27 − 16 = 11
X1 contains 10, but does not contain 16 and there is a subset
of X1 for which the sum is 27 − 10 = 17
X1 contains 9, but contains neither 10 nor 16 and there is a
subset of X1 for which the sum is 27 − 9 = 18
...
Dimitri Watel
MRO Chap 01 Dyn Prog
Examples
Generalization
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Shortest paths
The Partition problem
Interesting property
X = {1, 5, 6, 7, 9, 10, 16}
Many new problems : is there a subset of X that does not contain
16 and such that the sum is 11 ?, a subset of X that does not
contain 10 and 16 and such that the sum is 17 ?, a subset of X that
does not contain 9, 10 and 16 and such that the sum is 18 ?, . . .
Dimitri Watel
MRO Chap 01 Dyn Prog
Examples
Generalization
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Shortest paths
The Partition problem
Dynamic programming
X = {1, 5, 6, 7, 9, 10, 16}
New new problem for every i ≤ 7 and B ≤ 27 :
Is there a subset X1 using only the first i elements of X and such
that the sum is B ?
Let f (i, B) be the answer to that question (true or false).
f (7, 27) = f (6, 11) ∨ f (5, 17) ∨ f (4, 18) ∨ f (3, 20) ∨ f (2, 21) ∨
f (1, 22) ∨ f (0, 26)
f (6, 11) = f (5, 1) ∨ f (4, 2) ∨ f (3, 4) ∨ f (2, 5) ∨ f (1, 6) ∨ f (0, 10)
f (5, 17) = f (4, 8) ∨ f (3, 10) ∨ f (2, 11) ∨ f (1, 12) ∨ f (0, 16)
...
Dimitri Watel
MRO Chap 01 Dyn Prog
Examples
Generalization
When does it not work ?
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Shortest paths
The Partition problem
Dynamic programming
i
X = {1, 5, 6, 7, 9, 10, 16}
Complete the following table from top to bottom with f (i, B)
B
0 1 2 3 4 5 6 7 8 9 10 11 . . . 27
0
1
2
3
4
5
6
7
Dimitri Watel
MRO Chap 01 Dyn Prog
Examples
Generalization
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Shortest paths
The Partition problem
Who is the best ?
Dynamic programming : n · B sums
Brute force : 2n operations and comparisons.
Dimitri Watel
MRO Chap 01 Dyn Prog
Examples
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Principle
Definition
Dynamic programming is a method for solving problems, it is not
an algorithm but a way to build an algorithm.
In order to solve a problem with dynamic programming, we need
to rewrite the problem as the computation of a recursive
function f
Initialize f
Recursively compute f (in a smart way)
(see the board for examples)
Dimitri Watel
MRO Chap 01 Dyn Prog
Examples
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Principle
Definition
Dynamic programming is a method for solving problems, it is not
an algorithm but a way to build an algorithm.
In order to solve a problem with dynamic programming, we need
to rewrite the problem as the computation of a recursive
function f
Initialize f
Recursively compute f (in a smart way)
(see the board for examples)
Hardest part
Dimitri Watel
MRO Chap 01 Dyn Prog
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In other words
Other definition
Dynamic programming consists in defining a problem of size n as a
bunch of subproblems such that
the nature of each subproblem and of the problem itself are
the same (recursion)
each subproblem is smaller, the size is at most n − 1 (the
recursive algorithm stops)
(see the board for examples)
Dimitri Watel
MRO Chap 01 Dyn Prog
Examples
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Problem
Warning ! The recursive calls may be done twice or more. We say
the subproblems overlap.
(see the board for examples)
The recursive computation must be smart : every
subproblem solution must be stored when it is computed
(iterative way (bottom-up) or memoization (top-down)).
Dimitri Watel
MRO Chap 01 Dyn Prog
Examples
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Recursive combinatorial explosion
We want to transform a binary number into another using
transformation rules. Each rule transform a number into a bigger
number.
For example :
(1)
0 −−→ 10
(2)
1 −−→ 01
(3)
101 −−→ 01001
(4)
010 −−→ 0111
We can now transform numbers
(1)
(2)
(4)
(2)
(3)
0 −−→ 10 −−→ 010 −−→ 0111 −−→ 01011 −−→ . . .
Question : we are given two binary numbers x and y and a set T of
m transformation rules, it is possible to transform x into y with the
rules of T ?
Dimitri Watel
MRO Chap 01 Dyn Prog
Examples
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Recursive combinatorial explosion
There is a dynamic programming algorithm to solve the problem.
But we need to store too many information (2|y |−|x| bits).
Dimitri Watel
MRO Chap 01 Dyn Prog
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Wrong recursion
Latin square problems
Input
An empty grid with n × n cells
Output
n cells such that
exactly one cell per line and
column
maximize the minimum
distance between two chosen
cells.
Dimitri Watel
MRO Chap 01 Dyn Prog
Examples
Generalization
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Wrong recursion
Latin square problems
Input
×
An empty grid with n × n cells
×
Output
×
n cells such that
×
exactly one cell per line and
column
maximize the minimum
distance between two chosen
cells.
×
Dimitri Watel
MRO Chap 01 Dyn Prog
Examples
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Wrong recursion
No obvious way to generate a recursive function.
Dimitri Watel
MRO Chap 01 Dyn Prog
Examples
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In the syllabus, to be seen in tutorials
How to find the solution in addition to the value of the
solution ? (for instance, how to find a shortest path in addition
to the cost of the shortest path ? ; how to find the partition
instead of only proving it exists, . . .)
The Roy-Floyd-Warshall algorithm : find all the pair of shortest
paths in a graph.
Dimitri Watel
MRO Chap 01 Dyn Prog