DYNAMIC PROGRAMMING AND GREEDY TECHNIQUE

DYNAMIC PROGRAMMING AND
GREEDY TECHNIQUE

Dynamic Programming
 Dynamic Programming is mainly an optimization
over plain recursion.
 Wherever we see a recursive solution that has
repeated calls for same inputs, we can optimize it
using Dynamic Programming.
 The idea is to simply store the results of
subproblems, so that we do not have to re-
compute them when needed later.

Principle of Optimality
 An optimal policy (or a set of decisions) has the
property that whatever the initial state and initial
decision are, the remaining decisions must constitute
an optimal policy with regard to the state resulting
from the first decision

Tabulation vs Memoization
 There are two different ways to store the values
so that the values of a sub-problem can be
reused. Here, will discuss two patterns of solving
dynamic programming (DP) problems:
1. Tabulation: Bottom Up
2. Memoization: Top Down

Coin changing problem
 Give change for amount n using the minimum number of
coins of denominations d1<d2 < . . .<dm.
 Let F(n) be the minimum number of coins whose values add
up to n; it is convenient to define F(0) = 0.
 The amount n can only be obtained by adding one coin of
denomination dj to the amount n dj for j = 1, 2, . . . , m
−
such that n ≥ dj
 Therefore, consider all such denominations and select the
one minimizing F(n dj ) + 1
−
F(n) = min {F(n dj )} + 1 for n > 0,
−
j : n≥dj
F(0) = 0.

 Use backtracking for finding optimal solution
 For the instance considered, the last application of the
formula (for n = 6), the minimum was produced by d2 = 3.
 The second minimum (for n = 6 3) was also
− produced for
a coin of that denomination.
 Thus, the minimum-coin set for n = 6 is two 3’s.

Computing a Binomial Coefficient
 Binomial coefficient represented by c(n,k) is the number
of combinations(subsets) of k elements from an n-element
set (0≤k≤n)
 Binomial coefficients are derived from the participations
of these numbers in the binomial formula:
 The recursion relation is defined by:
 C(n,k)=1 if k=0/n=k

 General formula for binomial coefficient is
 can prove the equation 2 with general equation
 For n=k=5

Analysis
 The basic operation is addition
 First k+1 rows of the table form a triangle and remaining
n-k rows form a rectangle, so the sum is split into two
parts:

Floyd‘s algorithm ( All Pairs Shortest
Path)
 A weighted, directed graph is a collection vertices connected
by weighted edges (where the weight is some real number).
 One of the most common examples of a graph in the real world is
a road map.
 Each location is a vertex and each road connecting locations is an edge.
 We can think of the distance traveled on a road from one location to
another as the weight of that edge.
Tampa Orlando
Jacksonville
3.5 4
1.5
1.7
2.5
Tampa Orlando Jaxville
Tampa 0 1.7 3.5
Orlando 1.5 0 ∞
Jax 4 2.5 0

Storing a Weighted, Directed Graph
 Adjacency Matrix:
 Let D be an edge-weighted graph in adjacency-matrix form
• D(i,j) is the weight of edge (i, j), or ¥ if there is no such edge.
• Update matrix D, with the shortest path through
immediate vertices.

 Given a weighted graph, we want to know the shortest path from
one vertex in the graph to another.
 The Floyd-Warshall algorithm determines the shortest path between all
pairs of vertices in a graph.
 Let’s go over the premise of how Floyd-Warshall algorithm works…
 Let the vertices in a graph be numbered from 1 … n.
 Consider the subset {1,2,…, k} of these n vertices.
 Imagine finding the shortest path from vertex i to vertex j that uses
vertices in the set {1,2,…,k} only.
 There are two situations:
1) k is an intermediate vertex on the shortest path.
2) k is not an intermediate vertex on the shortest path.

Floyd Warshall Algorithm - Example
ConsiderVertex 3:
Nothing changes.
ConsiderVertex 2:
D(1,3) = D(1,2) + D(2,3)
ConsiderVertex 1:
D(3,2) = D(3,1) + D(1,2)
Original weights.

 Looking at this example, we can come up with the following
algorithm:
 Let D store the matrix with the initial graph edge information initially,
and update D with the calculated shortest paths.
For k=1 to n {
For i=1 to n {
For j=1 to n
D[i,j] = min(D[i,j],D[i,k]+D[k,j])
}
}
 The final D matrix will store all the shortest paths.
 Analysis - ?

 Let path(i, j) be some specification of the minimal
path from vertex j in setVi to vertex t;
 Cost(i, j) is the cost of this path; c(j, t) is the weight of
the edge from j to t.

Optimal Binary Search Tree
 applications is to implement a dictionary, a set of
elements with the operations of searching, insertion, and
deletion.
 the total number of binary search trees with n keys is
equal to the nth Catalan number,

Knapsack using Memory functions

Greedy Technique
 The greedy approach suggests constructing a solution
through a sequence of steps, each expanding a partially
constructed solution obtained so far, until a complete
solution to the problem is reached. On each step—and
this is the central point of this technique—the choice
made must be:
 feasible, i.e., it has to satisfy the problem’s constraints
 locally optimal, i.e., it has to be the best local choice among all
feasible choices available on that step
 irrevocable, i.e., once made, it cannot be changed on subsequent
steps of the algorithm

Prim’s Algorithm (Minimum Spanning
Tree)
 Given n points, connect them in the cheapest possible way
so that there will be a path between every pair of points.
 It has direct applications to the design of all kinds of
networks— including communication, computer,
transportation, and electrical—by providing the cheapest
way to achieve connectivity.
 It identifies clusters of points in data sets.
 It has been used for classification purposes in archeology,
biology, sociology, and other sciences.
 It is also helpful for constructing approximate solutions to
more difficult problems such the traveling salesman problem

 A spanning tree of an undirected connected graph is
its connected acyclic subgraph (i.e., a tree) that contains
all the vertices of the graph.
 If such a graph has weights assigned to its edges, a
minimum spanning tree is its spanning tree of the
smallest weight, where the weight of a tree is defined
as the sum of the weights on all its edges.
 The minimum spanning tree problem is the problem
of finding a minimum spanning tree for a given weighted
connected graph.

 Exhaustive Search
 The number of spanning trees grows exponentially with the
graph size (at least for dense graphs).
 Generating all spanning trees for a given graph is not easy; in
fact, it is more difficult than finding a minimum spanning tree for
a weighted graph by using one of several efficient algorithms
available for this problem.

Kruskal’s Algorithm
 Kruskal’s algorithm looks at a minimum spanning tree of a
weighted connected graph G =V, E as an acyclic subgraph
with |V| 1 edges for which the sum of the edge weights is
−
the smallest.
 the algorithm constructs a minimum spanning tree as an
expanding sequence of subgraphs that are always acyclic
but are not necessarily connected on the intermediate
stages of the algorithm.

 The algorithm begins by sorting the graph’s edges in non-
decreasing order of their weights.
 Then, starting with the empty subgraph, it scans this
sorted list, adding the next edge on the list to the current
subgraph if such an inclusion does not create a cycle and
simply skipping the edge otherwise.

 Analysis: with an efficient sorting algorithm, the time
efficiency of Kruskal’s algorithm will be in O(|E| log |E|).

DYNAMIC PROGRAMMING AND GREEDY TECHNIQUE

DYNAMIC PROGRAMMING AND GREEDY TECHNIQUE

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