Chapter 2: Fundamentals of the Analysis of Algorithm Efficiency

1
Dynamic Programming Dr. M. Sakalli, Marmara
University Matrix Chain Problem
Assembly-line scheduling Elements of dynamic
programming
Picture reference to http//www.flickr.com/photos/
7271221_at_N04/3408234040/sizes/l/in/photostream/.
Crane strokes
2
Dynamic Programming (pro-gram)

• Like Divide and Conquer, DP solves a problem by
  partitioning the problem into sub-problems and
  combines solutions. The differences are that
• DC is top-down while DP is bottom to top
  approach. But memoization will allow top to down
  method.
• The sub-problems are independent from each other
  in the former case, while they are not
  independent in the dynamic programming.
• Therefore, DP algorithm solves every sub-problem
  just ONCE and saves its answer in a TABLE and
  then reuse it. Memoization.
• Optimization problems Many solutions possible
  solutions and each has a value.
• A solution with the optimal sub-solutions. Such a
  solution is called an optimal solution to the
  problem. Not the optimum. Shortest path example
• The development of a dp algorithm can be in four
  steps.
• Characterize the structure of an optimal
  solution.
• Recursively define the value of an optimal
  solution.
• Compute the value of an optimal solution in a
  bottom-up fashion.
• Construct an optimal solution from computed
  information.

Dynamic Programming
DivideConquer
3
Assembly-Line Scheduling

• ei time to enter
• xi time to exit assembly lines
• tj time to transfer from assembly
• aj processing time in each station.
• Brute-force approach
• Enumerate all possible sequences through lines i
  ?1, 2,
• For each sequence of n stations Sj j ?1, n,
  compute the passing time. (the computation takes
  ?(n) time.)
• Record the sequence with smaller passing time.
• However, there are too many possible sequences
  totaling 2n

4
Assembly-Line Scheduling

• DP Step 1 Analyze the structure of the fastest
  way through the paths
• Seeking an optimal substructure The fastest
  possible way (minf) through a station Si,j
  contains the fastest way from start to the
  station Si,j trough either assembly line S1, j-1
  or S2, j-1.
• For j1, there is only one possibility
• For j2,3,,n, two possibilities from S1, j-1 or
  S2, j-1
• from S1, j-1, additional time a1, j
• from S2, j-1, additional time t2, j-1 a1,j
• suppose the fastest way through S1, j is through
  S1, j-1, then the chassis must have taken a
  fastest way from starting point through S1,j-1.
  Why???
• Similar rendering for S2, j-1.
• An optimal solution to a problem contains within
  it an optimal solution to sub-prbls.
• the fastest way through station Si,j contains
  within it the fastest way through station S1,j-1
  or S2,j-1 .
• Thus can construct an optimal solution to a
  problem from the optimal solutions to
  sub-problems.

5

• DP Step 2 A recursive solution
• DP Step 3 Computing the fastest times in T(n)
  time.
• Problem ri (j) 2n-j. So f11 is referred to
  2n-1 times.
• Total references to all fij is ?(2n).

6
Running time O(n).

• Step 4 Construct the fastest way through the
  factory

7

• Determining the fastest way through the factory

8
Matrix-chain Multiplication

• Problem definition Given a chain of matrices A1,
  A2, ..., An, where matrix Ai has dimension
  pi-1pi, find the order of matrix multiplications
  minimizing the number of the total scalar
  multiplications to compute the final product.
• Let A be a p, q matrix, and B be a q, r
  matrix. Then the complexity is pqr.
• In the matrix-chain multiplication problem, the
  actually matrices are not multiplied, the aim is
  to determine an order for multiplying matrices
  that has the lowest cost.
• Then, the time invested in determining optimal
  order must worth more than paid for by the time
  saved later on when actually performing the
  matrix multiplications.

• C(p,r) A(p,q) B(q,r)
• for i?1 to p for j?1 to r
• Ci,j0
• for i1 to p
• for j1 to r
• for k1 to q Ci,j Ci,j Ai,kBk,j

9
Example given in class
2x3 3x5 5x7 7x2 A1 A2
A3 A4
Suppose we want to multiply a sequence of
matrices, A1A4 with dimensions. Remember
Matrix multiplication is not commutative. 1-
Total of multiplication for this method is 30
70 20 120 2- Above the total of
multiplications is 30 70 28 128 3- Below
the total of multiplications is 70 30 12
112
10
Parenthesization

• The aim as to fully parenthesize the product of
  matrices minimizing scalar multiplications.
• For example, for the product A1 A2 A3 A4, a fully
  parenthesization is ((A1 A2) A3) A4.
• A product of matrices is fully parenthesized if
  it is either a single matrix, or a product of two
  fully parenthesized matrix product, surrounded by
  parentheses.
• Brute-force approach
• Enumerate all possible parenthesizations.
• Compute the number of scalar multiplications of
  each parenthesization.
• Select the parenthesization needing the least
  number of scalar multiplications.
• The number of enumerated parenthesizations of a
  product of n matrices, denoted by P(n), is the
  sequence of Catalan number growing as O(4n/n3/2)
  and solution to recurrence is O(2n).

The Brute-force approach is inefficient.
11
Catalan numbers the number of ways in which
parentheses can be placed in a sequence of
numbers to be multiplied, two at a time

• 3 numbers
• (1 (2 3)), ((1 2) 3)
• 4 numbers
• (1 (2 (3 4))), (1 ((2 3) 4)), ((1 2) (3 4)), ((1
  (2 3)) 4), (((1 2) 3) 4)
• 5 numbers
• (1 (2 (3 (4 5)))), (1 (2 ((3 4) 5))), (1 ((2 3)
  (4 5))), (1 ((2 (3 4)) 5)),
• (1 (((2 3) 4) 5)), ((1 2) (3 (4 5))), ((1 2) ((3
  4) 5)), ((1 (2 3)) (4 5)),
• ((1 (2 (3 4))) 5), ((1 ((2 3) 4)) 5), (((1 2) 3)
  (4 5)), (((1 2) (3 4)) 5),
• (((1 (2 3)) 4) 5) ((((1 2) 3) 4) 5)

12
With DP

• DP Step 1 structure of an optimal
  parenthesization
• Let Ai..j (i?j) denote the matrix resulting from
  Ai?Ai1??Aj
• Any parenthesization of Ai?Ai1??Aj must split
  the product between Ak and Ak1 for some k,
  (i?kltj). The cost of computing Ai..k of
  computing Ak1..j Ai..k ? Ak1..j.
• If k is the position for an optimal
  parenthesization, the parenthesization of
  prefix subchain Ai?Ai1??Ak within this
  optimal parenthesization of Ai?Ai1??Aj must be
  an optimal parenthesization of Ai?Ai1??Ak.
• Ai?Ai1??Ak ? Ak1??Aj

13
Step 2 Recursively define the value of an
optimal solution

• DP Step 2 a recursive relation
• Let mi,j be the minimum number of
  multiplications needed to compute the matrix
  Ai?Ai1??Aj
• The lowest cost to compute A1 A2 An would be
  m1,n
• Recurrence
• 0 if i j
• mi,j
• mini?kltj mi,k
  mk1,j pi-1pkpj if iltj
• ( (Ai Ak) (Ak1 Aj) )
  (Split at k)

14
Recursive (top-down) solution

• using the formula for mi,j
• RECURSIVE-MATRIX-CHAIN(p, i, j)
• if ij then return 0
• mi, j ?
• for k ? 1 to j - 1
• q ? RECURSIVE-MATRIX-CHAIN (p, i , k)
• RECURSIVE-MATRIX-CHAIN (p, k1 , j)
• pi-1 pk pj
• if q lt mi, j then mi, j ? q
• return mi, j

Line 1
Line 6
Line 4
Line 5
Line 6
Complexity
for n gt 1
15
Complexity of the recursive solution

• Using inductive method to prove by using the
  substitution method we guess a solution and
  then prove by using mathematical induction that
  it is correct.
• Prove that T(n) ?(2n) that is T(n) 2n-1 for
  all n 1.
• Induction Base T(1) 120
• Induction Assumption assume T(k) 2k-1 for all
  1 k lt n
• Induction Step

16

• Step 3, Computing the optimal cost
• If by recursive algorithm is exponential in n,
  ?(2n), no better than brute-force.
• But only have n ?(n2)
  subproblems.
• Recursive behavior will encounter to revisit the
  same overlapping subproblems many times.
• If tabling the answers for subproblems, each
  subproblem is only solved once.
• The second hallmark of DP overlapping
  subproblems and solve every subproblem just once.

17

• Step 3 Compute the value of an optimal solution
  bottom-up
• Input n an array p0n containing matrix
  dimensions
• State m1..n, 1..n for storing mi, j
• s1..n, 1..n for storing the optimal k that
  was used to calculate mi, j
• Result Minimum-cost table m and split table s
• MATRIX-CHAIN-TABLE(p, n)
• for i ? 1 to n
• mi, i ? 0
• for l ? 2 to n
• for i ? 1 to n-l1
• j ? il-1
• mi, j ? ?
• for k ? i to j-1
• q ? mi, k mk1, j pi-1 pk pj
• if q lt mi, j
• mi, j ? q
• si, j ? k
• return m and s

Takes O(n3) time Requires ?(n2) space
18
chains of length 1
A1 301
A2 140
A3 4010
A4 1025
j
1 2 3 4
1 0
2 0
3 0
4 0
i
19
chains of length 2
j
A1 301
A2 140
A3 4010
A4 1025
1 2 3 4
1 0 1200 1
2 0 400 2
3 0 10000 3
4 0
i
20
chains of length 3
j
1 2 3 4
1 0 1200 1 700 1
2 0 400 2 650 3
3 0 10000 3
4 0
i
A1 301
A2 140
A3 4010
A4 1025
21
chains of length 4
j
1 2 3 4
1 0 1200 1 700 1 1400 1
2 0 400 2 650 3
3 0 10000 3
4 0
i
A1 301
A2 140
A3 4010
A4 1025
22
Printing the solution
j
2 3 4
1 1 1 1
2 2 3
3 3
A1 301
A2 140
A3 4010
A4 1025
i
PRINT(s, 1, 4)
PRINT (s, 1, 1)
PRINT (s, 2, 4)
PRINT (s, 2, 3)
PRINT (s, 4, 4)
Output (A1((A2A3)A4))
PRINT (s, 2, 2)
PRINT (s, 3, 3)
23

• Step 4 Constructing an optimal solution
• Each entry si, j k shows where to split the
  product Ai Ai1 Aj for the minimum cost
• A1 An ( (A1 Asi, n) (Asi, n1 An) )
  
• To print the solution invoke the following
  function with (s, 1, n) as the parameter
• PRINT-OPTIMAL-PARENS(s, i, j)
• if ij then print Ai
• else print (
• PRINT-OPTIMAL-PARENS(s, i, si,
  j)
• PRINT-OPTIMAL-PARENS(s, si,
  j1, j)
• print )

24

• Suppose
• A1 A2 Ai.Ar
• P1xP2 P2xP3 PixPi1 . PrxPr1
• Assume
• mij the of multiplication needed to multiply
  Ai A i1......Aj
• Initial value mii mjj 0
• Final Value m1r
• A1..AiAk Ak1..Aj
• mij mik mk1, j Pi Pk1 Pj1
• k could be i lt k lt j-1
• We know the range of k but dont know the exact
  value of k

25

• Thus mij min(mik mk1, j Pi Pk1 Pj1) for
  i lt k lt j-1
• recurrence index (j-1) - i 1(j-i)
• Example Calculate m14 for
• A1 A2 A3 A4
• 2x5 5x3 3x7 7x2
• P1 2, P2 5, P3 3, P4 7, P5 2
• j-i 0 j-i 1 j-i 2
  j-i 3
• m11 0 m12 30 m13 72 m14
  84
• m220 m23105 m2472
• m330 m3442
  m440
• mij min(mik mk1,i PiPk1Pj1) for i lt k
  lt j-1
• m12 min(m11 m22 P1P2P3) for 1 lt k lt 1
• min( 0 0 2x5x3)

m(1,1) m(1,2) m(1,3) m(1,4) m(1,5) m(1,6)
m(2,2) m(2,3) m(2,4) m(2,5) m(2,6)
m(3,3) m(3,4) m(3,5)
m(3,6) m(4,4)
m(4,5) m(4,6)
m(5,5) m(5,6)
m(6,6)
26

• mij min(mik mk1, j PiPk1Pj1) for ilt klt
  j-1
• m13 min(m1k mk1,3 P1Pk1P4)for 1ltklt2
• min(m11 m23 P1P2P4 , m12 m33 P1P3P4
  )
• min( 01052x5x7, 30 0 2x3x7)
• min( 10570, 30 42) 72
• m24 min(m2k mk1,3 P2Pk1P4)for 2ltklt3
• min(m22 m34 P2P3P5 , m23 m44 P2P4P5
  )
• min( 0425x3x2, 105 0 5x7x2)
• min( 42 30, .) 72
• m14 min(m1k mk1,4 P1Pk1P5)for 1ltklt3
• min(m11 m24 P1P2P5 , m12 m34 P1P3P5 ,
  m13 m44 P1P4P5 )
• min( 722x5x2, 3042 2x3x2, 722x7x2)
• min( 7220, 304212, 72 28) min( 92, 84,
  100) 84

27
(No Transcript)
28
Memoized Matrix Chain

• LOOKUP-CHAIN(p,i,j)
• if mi,jlt? then return mi,j
• if ij then mi,j ?0
• else for k?i to j-1
• do q? LOOKUP-CHAIN(p,i,k)
  
• 
  LOOKUP-CHAIN(p,k1,j)pi-1pkpj
• if qlt
  mi,j then mi,j ?q
• return mi,j

29
For a DP to be applicable an optmztn prbl must
have

• Optimal substructure
• An optimal solution to the problem contains
  within it optimal solutions to subproblems.
• Overlapping (dependencies) subproblems
• The space of subproblems must be small i.e., the
  same subproblems are encountered over and over.

DP step3. Memoization T(n)O(n3),
PSpace(n)?(n2)

• A top-down variation of dynamic programming
• Idea remember the solution to subproblems as
  they are solved in the simple recursive algorithm
  but may be quite costly
• DP is considered better when all subproblems must
  be calculated, because there is no overhead for
  recursion. Lookup-Chain(p, i, j)
• LookUp-Table(p,i,j)
• Initialize all mi,j to ?
• if mi,j lt ? then return mi,j
• if i j then mi,j ? 0
• else for k ? 1 to j - 1
• q ? LookUp-Table (p, i , k)
• LookUp-Table (p, k1 ,
  j) pi-1 pk pj
• if q lt mi, j then mi, j ? q
• return mi, j

30
Elements of DP

• Optimal substructure
• A problem exhibits optimal substructure if an
  optimal solution to the problem contains within
  its optimal solution to subproblems.
• Overlapping subproblems
• When a recursive algorithm revisits the same
  problem over and over again, that is the
  optimization problem has overlapping subproblems.
• Subtleties
• Better to not assume that optimal substructure
  applies in general. Two examples in a directed
  graph G (V, E) and vertices u, v ? V.
• Unweighted shortest path
• Find a path from u to v consisting of the fewest
  edges. Good for Dynamic programming.
• Unweighted longest simple path
• Find a simple path from u to v consisting of the
  most edges. Not good for Dynamic programming.

31

• The running time of a dynamic-programming
  algorithm depends on the product of two factors.
• The number of subproblems overall the number
  of choices for each subproblem. Sum of entire
  choices.
• Assembly line scheduling
• T(n) subproblems 2 choices T(n)
• Matrix chain multiplication
• T(n2) subproblems (n-1) choices O(n3)

32
Principle of Optimality (Optimal Substructure)

• The principle of optimality applies to a problem
  (not an algorithm)
• A large number of optimization problems satisfy
  this principle.
• Principle of optimality Given an optimal
  sequence of decisions or choices, each
  subsequence must also be optimal.
• Principle of optimality - shortest path problem
• Problem Given a graph G and vertices s and t,
  find a shortest path in G from s to t
• Theorem A subpath P (from s to t) of a
  shortest path P is a shortest path from s to t
  of the subgraph G induced by P. Subpaths are
  paths that start or end at an intermediate vertex
  of P.
• Proof If P was not a shortest path from s to
  t in G, we can substitute the subpath from s
  to t in P, by the shortest path in G from s
  to t. The result is a shorter path from s to t
  than P. This contradicts our assumption that P
  is a shortest path from s to t.

33
Principle of Optimality

• P must be a shortest path from c to e in G,
  otherwise P cannot be a shortest path from a to e
  in G.

• Problem What is the longest simple route
  between City A and B?
• Simple never visit the same spot twice.
• The longest simple route (solid line) has city
  C as an intermediate city.
• Does not consist of the longest simple route
  from A to C and the longest simple route from C
  to B. Therefore does not satisfy the Principle of
  Optimality