Design and Analysis of Algorithms

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DEPARTMENT OF COMPUTER SCIENCE UNIVERSITY OF
JOENSUU JOENSUU, FINLAND

• Design and Analysis of Algorithms
• Lecture
• Dynamic programming
• Alexander Kolesnikov
• 17.10.205

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List of sample problems

• Shortest path in trellis graph
• Optimal allocation of constrained resource
• Optimal sequence partition (k-link shortest
  path).
• to be continued ...

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Shortest path in trellis graph
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Stagecoach problem

A traveler wishes to minimize the length of a
journey from town A to J.
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Greedy algorithm

The length of the route A-B-F-I-J 243413.
Can we find shorter route?
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Exhaustive search try all

Route A-D-F-I-J 313411 The total number
of routes to be tested 3?3?2?118 Can
we avoid exhaustive search?
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Shortest path construction 1st stage (B)

?
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Shortest path construction 1st stage

S(A,B)2 S(A,C)4 S(A,D)3
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Shortest path construction 2nd stage (E)

1. (A..B)-E 279 2. (A..C)-E 437 3.
(A..D)-E 347 ----------------------
(A..C)-E 7
?
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Shortest path construction 2nd stage (E)

1. (A..B)-E 279 2. (A..C)-E 437 ) 3.
(A..D)-E 347 ----------------------
(A..C)-E 7
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Shortest path construction 2nd stage (F)

1. (A..B)-F 246 2. (A..C)-F 426 3.
(A..D)-F 314 ) ----------------------
(A..C)-F 4
?
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Shortest path construction 2nd stage (F)

1. (A..B)-F 246 2. (A..C)-F 426 3.
(A..D)-F 314 ) ----------------------
(A..D)-F 4
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Shortest path construction 2nd stage (G)

1. (A..B)-G 268 ) 2. (A..C)-G 4610 3.
(A..D)-G 358 ----------------------
(A..B)-G 8
?
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Shortest path construction 2nd stage (G)

1. (A..B)-G 268 ) 2. (A..C)-G 4610 3.
(A..D)-G 358 ----------------------
(A..B)-G 8
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Shortest path construction 3rd stage (H)

1. (A..E)-H 718 ) 2. (A..F)-H 4610 3.
(A..G)-H 538 ----------------------
(A..E)-H 5
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Shortest path construction 3rd stage (H)

1. (A..E)-H 718 ) 2. (A..F)-H 4610 3.
(A..G)-H 538 ----------------------
(A..E)-H 5
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Shortest path construction 3rd stage (I)

1. (A..E)-I 7411 2. (A..F)-I 437 ) 3.
(A..G)-I 538 ----------------------
(A..F)-I 7
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Shortest path construction 3rd stage (I)

1. (A..E)-I 7411 2. (A..F)-I 437 ) 3.
(A..G)-I 538 ----------------------
(A..F)-I 7
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Shortest path construction 4th stage (J)

1. (A..H)-J 8311 ) 2. (A..I) -J
7411 ---------------------- (A..H)-J 11
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Shortest path construction 4th stage (J)

1. (A..H)-J 8311 ) 2. (A..I) -J
7411 ---------------------- (A..H)-J 11
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Backtrack the shortest path

7
2
8
11
4
4
0
7
3
8
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The shortest path

7
2
8
11
4
4
0
7
3
8
Route A-C-E-H-J 431311
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Trellis graph
1 2 3 4
K-1 K
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Trellis graph

• Distance (weight) from point i1 at stage (j?1)
  to point i2
• at stage j

• The total value of cost function

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Principle of optimality of Bellman
An optimal path has the property that whatever
the initial conditions and control variables
(choices) over some initial period, the control
(or decision variables) chosen over the remaining
period must be optimal for the remaining problem,
with the state resulting from the early decisions
taken to be the initial condition.
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Dynamic programming

• Cost function

• Recursive eqution

• Initialization

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Complexity

• Exhaustive search O(nK)
• Dynamic programming algorithm O(Kn2)
• where K is the number of stages,
• n is the number of points in a stage

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Optimal allocation of constrained resource
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Problem formulation

• N units of a resource
• This resource must be distributed among K
  activities
• Functions fk(x) - profit for allocated resource
• Allocate N units of resource to K activities
  with given
• return functions so that the total profit is
  maximal

subject to
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Dynamic programming formulation

Optimal value function
Recursive equation
Initialization
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Allocate 3 mln euros into four projects
Profit fk(x), K3, N3.
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Trellis graph
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Solution
)
1 2 f1(2)8 2 1 f2(1)5 3 0
f3(0)0 4 0 f4(0)0 ----------------------
N3 G4(3)13
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Search in the state space
GK(N)
K
Gk(n)
fk(n-j)
k
k-1

Gk-1(j)
1
0
j
N
0
n
Start state
Gk(n) maxGk(0) fk(n),
Gk(1) fk(n-1), . . .
Gk(j-1) fk(j), . . .
Gk(n) fk(0)
Ak(n)jopt
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Optimal partition of data sequence
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Problem formulation

• Given a sequence of data Xx1, x2, ,xN
• Do partition of the sequence X into to K groups
  with
• given cost functions f(xi,xj) so that the total
  value of
• the cost function is minimal

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Partition into groups Example

• Data x 0 ?? lt x1 lt... lt xj lt ... lt xN
• Partition indices i0 0 lt i1 lt... lt ij lt ... lt
  iM N.

• Groups

1
2
3
(
(
(
















...
(x0??) x1 x2 x3 x4
x5 x6 x7x8 x9 x10 x11 x12 x13
x14 xN
(i00) i14
i210
iK N15
K3
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Problem formulation
Cost function
Recursive equation
Initialization
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Search in the state space ?
GK(N)
K
b
State space ?
Gk (n)
f(j,n)
k
k-1

Gk-1(j)
0
j
N
1
n
Start state
Gk(n) minGk(k) f(k, n,
Gk(k) f(k1,n, . . .
Gk(j-1) f(j, n), . .
. Gk(n-1) f(n, n
Ak(n)jopt
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Scheme of the DP algorithm

• // Initialization
• FOR n 1 TO N DO G1(n) f(1,n
• // Minimum search
• FOR k 2 TO K DO
• FOR n k TO N DO
• dmin ? ?
• FOR j k-1 TO n-1 DO
• c ? Gk-1(j) f(j,n
• IF(c lt cmin)
• cmin ? c
• jmin ? j
• ENDIF
• ENDFOR
• Gk (n) ? dmin
• Ak (n) ? jmin
• ENDFOR
• ENDFOR

Complexity O(KN2)
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Backtrack in the state space ?
AK(N)
K
b
State space ?
0
N
1
n
j
Start state
S(M1) N FOR m K1 TO 2 DO S(m?1)
A(S(m), m)) P GK(N)
N22, K8 S22,18,14,12,9,6,4,3,1 (x0,x3,
(x3,x4, (x4,x6, (x6,x9, (x9,x12, (x12,x14,
(x14,x18, (x18,x22