2. 2
Topics
What is Dynamic Programming
Binomial Coefficient
Floyd’s Algorithm
Chained Matrix Multiplication
Optimal Binary Search Tree
Traveling Salesperson
3. 3
Divide-and-Conquer: a top-down approach.
Many smaller instances are computed more
than once.
Dynamic programming: a bottom-up approach.
Solutions for smaller instances are stored in a
table for later use.
Why Dynamic Programming?
4. 4
An Algorithm Design Technique
A framework to solve Optimization problems
Elements of Dynamic Programming
Dynamic programming version of a recursive
algorithm.
Developing a Dynamic Programming Algorithm
– Example: Multiplying a Sequence of Matrices
Dynamic Programming
5. 5
Why Dynamic Programming?
• It sometimes happens that the natural way of dividing anIt sometimes happens that the natural way of dividing an
instance suggested by the structure of the problem leads us toinstance suggested by the structure of the problem leads us to
consider several overlapping subinstances.consider several overlapping subinstances.
• If we solve each of these independently, they will in turnIf we solve each of these independently, they will in turn
create a large number of identical subinstances.create a large number of identical subinstances.
• If we pay no attention to this duplication, it is likely that weIf we pay no attention to this duplication, it is likely that we
will end up with an inefficient algorithm.will end up with an inefficient algorithm.
• If, on the other hand, we take advantage of the duplication andIf, on the other hand, we take advantage of the duplication and
solve each subinstance only once, saving the solution for latersolve each subinstance only once, saving the solution for later
use, then a more efficient algorithm will result.use, then a more efficient algorithm will result.
6. 6
Why Dynamic Programming? …
The underlying idea of dynamic programming isThe underlying idea of dynamic programming is
thus quite simple: avoid calculating the same thingthus quite simple: avoid calculating the same thing
twice, usually by keeping a table of known results,twice, usually by keeping a table of known results,
which we fill up as subinstances are solved.which we fill up as subinstances are solved.
• Dynamic programming is aDynamic programming is a bottom-upbottom-up technique.technique.
• Examples:Examples:
1) Fibonacci numbers1) Fibonacci numbers
2) Computing a Binomial coefficient2) Computing a Binomial coefficient
7. 7
Dynamic Programming
• Dynamic ProgrammingDynamic Programming is a general algorithm designis a general algorithm design
technique.technique.
• Invented by American mathematician Richard Bellman inInvented by American mathematician Richard Bellman in
the 1950s to solve optimization problems.the 1950s to solve optimization problems.
• ““Programming” here means “planning”.Programming” here means “planning”.
• Main idea:Main idea:
• solve several smaller (overlapping) subproblems.solve several smaller (overlapping) subproblems.
• record solutions in a table so that each subproblem isrecord solutions in a table so that each subproblem is
only solved once.only solved once.
• final state of the table will be (or contain) solution.final state of the table will be (or contain) solution.
8. 8
Dynamic Programming
Define a container to store intermediate
results
Access container versus recomputing results
Fibonacci numbers example (top down)
– Use vector to store results as calculated so they
are not re-calculated
10. 10
Example: Fibonacci numbers
• Recall definition of Fibonacci numbers:
f(0) = 0
f(1) = 1
f(n) = f(n-1) + f(n-2) for n ≥ 2
• Computing the nth
Fibonacci number recursively (top-
down): f(n)
f(n-1) + f(n-2)
f(n-2) + f(n-3) f(n-3) + f(n-4)
11. 11
Fib vs. fibDyn
int fib(int n) {
if (n <= 1) return n; // stopping conditions
else return fib(n-1) + fib(n-2); // recursive step
}
int fibDyn(int n, vector<int>& fibList) {
int fibValue;
if (fibList[n] >= 0) // check for a previously computed result and return
return fibList[n];
// otherwise execute the recursive algorithm to obtain the result
if (n <= 1) // stopping conditions
fibValue = n;
else // recursive step
fibValue = fibDyn(n-1, fibList) + fibDyn(n-2, fibList);
// store the result and return its value
fibList[n] = fibValue;
return fibValue;
}
16. 16
Top down vs. Bottom up
Top down dynamic programming moves
through recursive process and stores results
as algorithm computes
Bottom up dynamic programming evaluates
by computing all function values in order,
starting at lowest and using previously
computed values.
17. 17
Examples of Dynamic Programming Algorithms
• Computing binomial coefficientsComputing binomial coefficients
• Optimal chain matrix multiplicationOptimal chain matrix multiplication
• Floyd’s algorithms for all-pairs shortest pathsFloyd’s algorithms for all-pairs shortest paths
• Constructing an optimal binary search treeConstructing an optimal binary search tree
• Some instances of difficult discrete optimization problems:Some instances of difficult discrete optimization problems:
• travelling salesmantravelling salesman
• knapsackknapsack
18. 18
A framework to solve Optimization problems
For each current choice:
– Determine what subproblem(s) would remain if this
choice were made.
– Recursively find the optimal costs of those
subproblems.
– Combine those costs with the cost of the current
choice itself to obtain an overall cost for this choice
Select a current choice that produced the
minimum overall cost.
19. 19
Elements of Dynamic Programming
Constructing solution to a problem by building it up
dynamically from solutions to smaller (or simpler) sub-
problems
– sub-instances are combined to obtain sub-instances of
increasing size, until finally arriving at the solution of the
original instance.
– make a choice at each step, but the choice may depend on the
solutions to sub-problems.
20. 20
Elements of Dynamic Programming …
Principle of optimality
– the optimal solution to any nontrivial instance of a problem is a
combination of optimal solutions to some of its sub-instances.
Memorization (for overlapping sub-problems)
– avoid calculating the same thing twice,
– usually by keeping a table of know results that fills up as sub-
instances are solved.
21. 21
Development of a dynamic programming
algorithm
Characterize the structure of an optimal solution
– Breaking a problem into sub-problem
– whether principle of optimality apply
Recursively define the value of an optimal solution
– define the value of an optimal solution based on value of solutions
to sub-problems
Compute the value of an optimal solution in a bottom-up fashion
– compute in a bottom-up fashion and save the values along the
way
– later steps use the save values of pervious steps
Construct an optimal solution from computed information
22. 22
Binomial Coefficient
Binomial coefficient:
Cannot compute using this formula because of
n!
Instead, use the following formula:
nkfor
knk
n
k
n
≤≤
−
=
0
)!(!
!
23. 23
Binomial Using Divide & Conquer
Binomial formula:
==
<<
−
+
−
−
=
)
0
(01
0
1
1
1
n
n
or C
n
Cnor kk
nk
k
n
C
k
n
C
k
n
C
24. 24
Binomial using Dynamic Programming
Just like Fibonacci, that formula is very inefficient
Instead, we can use the following:
niinnn
bnnCbainCanCba ),(...),(...)0,()( ++++=+ −
27. 27
Binomial Coefficient
Record the values in a table of n+1 rows and k+1 columns
0 1 2 3 … k-1 k
0 1
1 1 1
2 1 2 1
3 1 3 3 1
...
k 1 1
…
n-1 1
n 1
k
n
C
−
−
1
1
k
n
C
−
k
n
C
1
28. 28
Binomial Coefficient
ALGORITHM Binomial(n,k)
//Computes C(n, k) by the dynamic programming algorithm
//Input: A pair of nonnegative integers n ≥ k ≥ 0
//Output: The value of C(n ,k)
for i 0 to n do
for j 0 to min (i ,k) do
if j = 0 or j = k
C [i , j] 1
else C [i , j] C[i-1, j-1] + C[i-1, j]
return C [n, k]
)()(
2
)1(
)1(11),(
1
1
1 1 1 1 1
nkknk
kk
kiknA
k
i
i
j
n
ki
k
j
k
i
n
Ki
Θ∈−+
−
=
+−=+= ∑∑ ∑∑ ∑ ∑=
−
= += = = +=
29. 29
Floyd’s Algorithm: All pairs shortest paths
•Find shortest path when direct path doesn’t existFind shortest path when direct path doesn’t exist
•In a weighted graph, find shortest paths between every pair ofIn a weighted graph, find shortest paths between every pair of
verticesvertices
• Same idea: construct solution through series of matricesSame idea: construct solution through series of matrices
D(0), D(1), … using an initial subset of the vertices asD(0), D(1), … using an initial subset of the vertices as
intermediaries.intermediaries.
• Example:Example:
3
4
2
1
4
1
6
1
5
3
30. 30
Shortest Path
Optimization problem – more than one candidate for
the solution
Solution is the candidate with optimal value
Solution 1 – brute force
– Find all possible paths, compute minimum
– Efficiency?
Solution 2 – dynamic programming
– Algorithm that determines only lengths of shortest paths
– Modify to produce shortest paths as well
Worse than O(n2
)
35. 35
Floyd’s Algorithm: All pairs shortest paths
• ALGORITHM Floyd (W[1 … n, 1… n])ALGORITHM Floyd (W[1 … n, 1… n])
•For kFor k ← 1 to n do← 1 to n do
•For iFor i ← 1 to n do← 1 to n do
•For j ← 1 to n doFor j ← 1 to n do
•W[i, j] ← min{W[i,j], W{i, k] + W[k, j]}W[i, j] ← min{W[i,j], W{i, k] + W[k, j]}
•Return WReturn W
•Efficiency = ?Efficiency = ?
Θ(n)
36. 36
Example: All-pairs shortest-path problem
ExampleExample: Apply Floyd’s algorithm to find the t All-: Apply Floyd’s algorithm to find the t All-
pairs shortest-path problem of the digraph defined bypairs shortest-path problem of the digraph defined by
the following weight matrixthe following weight matrix
0 20 2 ∞∞ 1 81 8
6 0 3 2 ∞6 0 3 2 ∞
∞ ∞∞ ∞ 0 4 ∞0 4 ∞
∞ ∞∞ ∞ 2 0 32 0 3
3 ∞ ∞ ∞3 ∞ ∞ ∞ 00
38. 38
Chained Matrix Multiplication
Problem: Matrix-chain multiplication
– a chain of <A1, A2, …, An> of n matrices
– find a way that minimizes the number of scalar multiplications to
compute the product A1A2…An
Strategy:
Breaking a problem into sub-problem
– A1A2...Ak, Ak+1Ak+2…An
Recursively define the value of an optimal solution
– m[i,j] = 0 if i = j
– m[i,j]= min{i<=k<j} (m[i,k]+m[k+1,j]+pi-1pkpj)
– for 1 <= i <= j <= n
39. 39
Example
Suppose we want to multiply a 2x2 matrix
with a 3x4 matrix
Result is a 2x4 matrix
In general, an i x j matrix times a j x k matrix
requires i x j x k elementary multiplications
40. 40
Example
Consider multiplication of four matrices:
A x B x C x D
(20 x 2) (2 x 30) (30 x 12) (12 x 8)
Matrix multiplication is associative
A(B (CD)) = (AB) (CD)
Five different orders for multiplying 4 matrices
1. A(B (CD)) = 30*12*8 + 2*30*8 + 20*2*3 = 3,680
2. (AB) (CD) = 20*2*30 + 30*12*8 + 20*30*8 = 8,880
3. A ((BC) D) = 2*30*12 + 2*12*3 + 20*2*8 = 1,232
4. ((AB) C) D = 20*2*30 + 20*30*12 + 20*12*8 = 10,320
5. (A (BC)) D = 2*30*12 + 20*2*12 + 20*12*8 = 3,120
41. 41
Algorithm
int minmult (int n, const ind d[], index P[ ] [ ])
{
index i, j, k, diagonal;
int M[1..n][1..n];
for (i = 1; i <= n; i++)
M[i][i] = 0;
for (diagonal = 1; diagonal <= n-1; diagonal++)
for (i = 1; i <= n-diagonal; i++)
{ j = i + diagonal;
M[i] [j] = minimum(M[i][k] + M[k+1][j] + d[i-1]*d[k]*d[j]);
// minimun (i <= k <= j-1)
P[i] [j] = a value of k that gave the minimum;
}
return M[1][n];
}
42. 42
Optimal Binary Trees
Optimal way of constructing a binary search
tree
Minimum depth, balanced (if all keys have
same probability of being the search key)
What if probability is not all the same?
Multiply probability of accessing that key by
number of links to get to that key
44. 44
Traveling Salesperson
The Traveling Salesman Problem (TSP) is a
deceptively simple combinatorial problem. It
can be stated very simply:
A salesman spends his time visiting n cities
(or nodes) cyclically. In one tour he visits
each city just once, and finishes up where he
started. In what order should he visit them to
minimize the distance traveled?
45. 45
Why study?
The problem has some direct importance, since
quite a lot of practical applications can be put in this
form.
It also has a theoretical importance in complexity
theory, since the TSP is one of the class of "NP
Complete" combinatorial problems.
NP Complete problems are intractable in the sense
that no one has found any really efficient way of
solving them for large n.
– They are also known to be more or less equivalent to each
other; if you knew how to solve one kind of NP Complete
problem you could solve the lot.
46. 46
Efficiency
The holy grail is to find a solution algorithm
that gives an optimal solution in a time that
has a polynomial variation with the size n of
the problem.
The best that people have been able to do,
however, is to solve it in a time that varies
exponentially with n.
47. 47
Later…
We’ll get back to the traveling salesperson
problem in the next chapter….
49. 49
Chapter Summary
• Dynamic programming is similar to divide-and-
conquer.
• Dynamic programming is a bottom-up approach.
• Dynamic programming stores the results (small
instances) in the table and reuses it instead of
recomputing it.
• Two steps in development of a dynamic
programming algorithm:
• Establish a recursive property
• Solve an instance of the problem in a bottom-up
51. 51
Rules of Sudoku
• Place a number (1-9) in each blank cell.
• Each row (nine lines from left to right), column (also
nine lines from top to bottom) and 3x3 block bounded
by bold line (nine blocks) contains number from 1
through 9.
52. 52
A Little Help Please…
Try this:
– http://www.ccs.neu.edu/jpt/sudoku/
