AAD- Dynamic Programming Introduction and General Method multistage graphs with example Part 15

Ali Seron Jason

2 weeks ago

https://www.gyanodhan.com/video/7B1.%20GATE%20CSEIT/Algorith%20Analysis%20and%20Design/626.%20AAD-%20Dynamic%20Programming%20%20%20Introduction%20and%20%20General%20Method%20multistage%20graphs%20with%20example%20Part%2015.mp4

Dynamic Programming (DP) is a method for solving complex problems by breaking them down into simpler subproblems. It is particularly effective for optimization problems where decisions are made in stages, and each decision affects subsequent choices. A classic application of DP is in solving multistage graph problems, where the goal is to find the shortest or minimum-cost path from a source to a destination node across multiple stages.

Contents

1 Dynamic Programming: General Method
2 Multistage Graphs
- 2.1 What is a Multistage Graph?
3 Approaches to Solve Multistage Graph Problems
- 3.1 1. Forward Approach
- 3.2 2. Backward Approach
4 Example: Solving a Multistage Graph Problem

Dynamic Programming: General Method

Dynamic Programming is applicable when a problem exhibits overlapping subproblems and optimal substructure. The general steps involved are:

Characterize the Structure of an Optimal Solution: Understand how a solution to the problem can be constructed from solutions to subproblems.
Define the Value of an Optimal Solution Recursively: Express the solution in terms of solutions to smaller subproblems.
Compute the Value of an Optimal Solution: Typically done in a bottom-up manner, solving all subproblems and storing their solutions.
Construct an Optimal Solution: If required, build the solution to the original problem from the computed information.

This method ensures that each subproblem is solved only once, and its solution is stored for future reference, thereby reducing the computational effort.

Multistage Graphs

What is a Multistage Graph?

A multistage graph is a directed graph partitioned into multiple stages, where:

Vertices are divided into k stages: $V_1 \cup V_2 \cup \dots \cup V_k$ .
Edges connect vertices from stage $i$ to stage $i + 1$ , i.e., if $\in E$ , then $\in V_i$ and $\in V_{i+1}$ .
There is a source node in stage 1 and a destination node in stage $k$ .

The objective is to find the shortest or minimum-cost path from the source to the destination.

Approaches to Solve Multistage Graph Problems

1. Forward Approach

This method starts from the source node and progresses towards the destination.

Algorithm Steps:

Initialize the cost of the source node to 0 and all others to infinity.
For each stage $i$ from 1 to $k - 1$ :
- For each node $u$ in stage $i$ :
  - For each edge $(u, v)$ :
    - If $cos t [v] > cos t [u] + w e i g h t (u, v)$ , then update $cos t [v] = cos t [u] + w e i g h t (u, v)$ .
After processing all stages, $cos t [d es t ina t i o n]$ will hold the minimum cost.

2. Backward Approach

This method starts from the destination node and works backward towards the source.

Algorithm Steps:

Initialize the cost of the destination node to 0 and all others to infinity.
For each stage $i$ from $k$ down to 1:
- For each node $v$ in stage $i$ :
  - For each edge $(u, v)$ :
    - If $cos t [u] > cos t [v] + w e i g h t (u, v)$ , then update $cos t [u] = cos t [v] + w e i g h t (u, v)$ .
After processing all stages, $cos t [so u rce]$ will hold the minimum cost.

Example: Solving a Multistage Graph Problem

Consider a multistage graph with 4 stages:

Stage 1: Node 1 (Source)
Stage 2: Nodes 2, 3
Stage 3: Nodes 4, 5
Stage 4: Node 6 (Destination)

Edges and their weights:

(1,2): 2
(1,3): 1
(2,4): 2
(2,5): 3
(3,4): 3
(3,5): 6
(4,6): 1
(5,6): 2

Using the Forward Approach:

Initialize: cost[1]=0, cost[others]=∞
From Node 1:
- cost[2] = min(∞, 0+2) = 2
- cost[3] = min(∞, 0+1) = 1
From Node 2:
- cost[4] = min(∞, 2+2) = 4
- cost[5] = min(∞, 2+3) = 5
From Node 3:
- cost[4] = min(4, 1+3) = 4
- cost[5] = min(5, 1+6) = 5
From Node 4:
- cost[6] = min(∞, 4+1) = 5
From Node 5:
- cost[6] = min(5, 5+2) = 5

Result: The minimum cost from Node 1 to Node 6 is 5.

For a visual and detailed explanation, you might find this video helpful:

If you need further clarification or assistance with specific problems related to dynamic programming and multistage graphs, feel free to ask!

AAD- Dynamic Programming Introduction and General Method multistage graphs with example Part 15

Dynamic Programming: General Method