- Design and Analysis of Algorithms
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- Basics of Algorithms
- DAA - Introduction
- DAA - Analysis of Algorithms
- DAA - Methodology of Analysis
- Asymptotic Notations & Apriori Analysis
- Time Complexity
- Master’s Theorem
- DAA - Space Complexities
- Divide & Conquer
- DAA - Divide & Conquer
- DAA - Max-Min Problem
- DAA - Merge Sort
- DAA - Binary Search
- Strassen’s Matrix Multiplication
- Karatsuba Algorithm
- Towers of Hanoi
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- DAA - Greedy Method
- Travelling Salesman Problem
- Prim's Minimal Spanning Tree
- Kruskal’s Minimal Spanning Tree
- Dijkstra’s Shortest Path Algorithm
- Map Colouring Algorithm
- DAA - Fractional Knapsack
- DAA - Job Sequencing with Deadline
- DAA - Optimal Merge Pattern
- Dynamic Programming
- DAA - Dynamic Programming
- Matrix Chain Multiplication
- Floyd Warshall Algorithm
- DAA - 0-1 Knapsack
- Longest Common Subsequence
- Travelling Salesman Problem | Dynamic Programming
- Randomized Algorithms
- Randomized Algorithms
- Randomized Quick Sort
- Karger’s Minimum Cut
- Fisher-Yates Shuffle
- Approximation Algorithms
- Approximation Algorithms
- Vertex Cover Problem
- Set Cover Problem
- Travelling Salesperson Approximation Algorithm
- Graph Theory
- DAA - Spanning Tree
- DAA - Shortest Paths
- DAA - Multistage Graph
- Optimal Cost Binary Search Trees
- Heap Algorithms
- DAA - Binary Heap
- DAA - Insert Method
- DAA - Heapify Method
- DAA - Extract Method
- Sorting Techniques
- DAA - Bubble Sort
- DAA - Insertion Sort
- DAA - Selection Sort
- DAA - Shell Sort
- DAA - Heap Sort
- DAA - Bucket Sort
- DAA - Counting Sort
- DAA - Radix Sort
- Searching Techniques
- Searching Techniques Introduction
- DAA - Linear Search
- DAA - Binary Search
- DAA - Interpolation Search
- DAA - Jump Search
- DAA - Exponential Search
- DAA - Fibonacci Search
- DAA - Sublist Search
- Complexity Theory
- Deterministic vs. Nondeterministic Computations
- DAA - Max Cliques
- DAA - Vertex Cover
- DAA - P and NP Class
- DAA - Cook’s Theorem
- NP Hard & NP-Complete Classes
- DAA - Hill Climbing Algorithm
- DAA Useful Resources
- DAA - Quick Guide
- DAA - Useful Resources
- DAA - Discussion
Design and Analysis Multistage Graph
A multistage graph G = (V, E) is a directed graph where vertices are partitioned into k (where k > 1) number of disjoint subsets S = {s1,s2,…,sk} such that edge (u, v) is in E, then u Є si and v Є s1 + 1 for some subsets in the partition and |s1| = |sk| = 1.
The vertex s Є s1 is called the source and the vertex t Є sk is called sink.
G is usually assumed to be a weighted graph. In this graph, cost of an edge (i, j) is represented by c(i, j). Hence, the cost of path from source s to sink t is the sum of costs of each edges in this path.
The multistage graph problem is finding the path with minimum cost from source s to sink t.
Example
Consider the following example to understand the concept of multistage graph.
According to the formula, we have to calculate the cost (i, j) using the following steps
Step 1: Cost (K-2, j)
In this step, three nodes (node 4, 5. 6) are selected as j. Hence, we have three options to choose the minimum cost at this step.
Cost(3, 4) = min {c(4, 7) + Cost(7, 9),c(4, 8) + Cost(8, 9)} = 7
Cost(3, 5) = min {c(5, 7) + Cost(7, 9),c(5, 8) + Cost(8, 9)} = 5
Cost(3, 6) = min {c(6, 7) + Cost(7, 9),c(6, 8) + Cost(8, 9)} = 5
Step 2: Cost (K-3, j)
Two nodes are selected as j because at stage k - 3 = 2 there are two nodes, 2 and 3. So, the value i = 2 and j = 2 and 3.
Cost(2, 2) = min {c(2, 4) + Cost(4, 8) + Cost(8, 9),c(2, 6) +
Cost(6, 8) + Cost(8, 9)} = 8
Cost(2, 3) = {c(3, 4) + Cost(4, 8) + Cost(8, 9), c(3, 5) + Cost(5, 8)+ Cost(8, 9), c(3, 6) + Cost(6, 8) + Cost(8, 9)} = 10
Step 3: Cost (K-4, j)
Cost (1, 1) = {c(1, 2) + Cost(2, 6) + Cost(6, 8) + Cost(8, 9), c(1, 3) + Cost(3, 5) + Cost(5, 8) + Cost(8, 9))} = 12
c(1, 3) + Cost(3, 6) + Cost(6, 8 + Cost(8, 9))} = 13
Hence, the path having the minimum cost is 1→ 3→ 5→ 8→ 9.
