Lecture 6: Dynamic Programming
0/1 Knapsack
Making Change (any coin set)
What is Dynamic Programming?
Dynamic programming is similar to divide-and-conquer in that the problem is
broken down into smaller subproblems. In this approach we solve the small
instances first, save the results, and look them up later when we needed, rather
than recompute them.
1. establish a recursive property that gives the solution to an instance of the
problem
2. solve instances of the problem in a bottom-up fashion starting with the
smaller instances first.
Dynamic programming can sometimes provide an efficient solution to a problem
for which divide-and-conquer produces an exponential run-time. Occasionally we
find that we do not need to save all subproblem solutions. In these cases we can
revise the algorithm greatly reducing the memory space requirements for the
algorithm.
Binomial Theorem
The binomial theorem gives a closed-form expression for the coefficient of
any term in the expansion of a binomial raised to the nth power.
(a+b)0 = 1
(a+b)1 = a+b
(a+b)2 = a2+2ab+b2
(a+b)3 = a3+3a2b+3ab2+b3
(a+b)4 = a4+4a3b+6a2b2+4ab3+b4
a  b
n
n
n!
a k b nk
k 0 k!n  k !

Counting Combinations & Pascal's Triangle
The binomial coefficient is also the number of combinations of n items taken
k at a time, sometimes called n-choose-k.
1
1
1
1
1
1
2
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1
1
3
6
10
1
4
10
1
5
1
n  1 n  1

for 0  k  n
n  


k   k  1 k 
  
1
k  0 or k  n

Binomial Coefficient D&C Version
function bin(n,k : integer) return integer is
begin
if k=0 or k=n then
return 1;
else
return bin(n-1,k-1) + bin(n-1,k);
end if;
end bin;
This version of bin requires that the subproblems are recalculated many
times for each recursive call.
Binomial Coefficient Dynamic Programming
function bin2(n,k:integer) return integer s
B : array(0..n,o..k) of integer;
begin
for i in 0..n loop
for j in 0..minimum(i,k) loop
if j=0 or j=i then
B(i,j)=1;
else
B(i,j):=B(i-1,j-1) + B(i-1,j);
end if;
end loop;
end loop;
return B(n,k);
end bin2;
In bin2 the smallest instances of the problem are solved first and then used
to compute values for the larger subproblems. Compare the computational
complexities of bin and bin2.
Floyd's Algorithm for Shortest Paths
procedure floyd(W,D:matype) is
2
begin
V5
D:=W;
5
3 V
2
for k in 1..n loop
2
for i in 1..n loop
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1
1
for j in 1..n loop
V3
V4
D(i,j):=min(D(i,j),D(i,k)+D(k,j));
1
end loop;
end loop;
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end loop;
1 0 2 - - end floyd;
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0
Floyd's algorithm is very simple to implement. The fact
that it works at all is not obvious. Be sure to work
through the proof of algorithm correctness in the text.
Dynamic Programming - The Coin Changing Problem
Northeastern University - Javed A. Aslam - CSG713 Advanced Algorithms
Defining the Recurrence Relation
Northeastern University - Javed A. Aslam - CSG713 Advanced Algorithms
Implementing the Bottom-Up Algorithm
Northeastern University - Javed A. Aslam - CSG713 Advanced Algorithms
Generating an Optimal Solution
Northeastern University - Javed A. Aslam - CSG713 Advanced Algorithms
Algorithm/Program Analysis
Northeastern University - Javed A. Aslam - CSG713 Advanced Algorithms
Summary
Dynamic Programming solves problems that can be expressed as
recurrence relations
Rather than recursive programming (e.g. D&C) DynPro solves these
problem in a bottom-up fashion
It is good practice to compare complexity for D&C vs DynPro
DynPro works when sub-problems are solved by sub-solutions. That is,
optimal partial solutions are parts of the optimal solution.