Algorithm

Getting Started with AlgorithmWhat is an Algorithm?

Characteristics of Algorithm1 Topic

Analysis Framework

Performance Analysis3 Topics

Mathematical Analysis2 Topics

Sorting AlgorithmSorting Algorithm10 Topics

Searching Algorithm6 Topics

Fundamental of Data StructuresStacks

Queues

Graphs

Trees

Sets

Dictionaries

Divide and ConquerGeneral Method

Binary Search

Recurrence Equation for Divide and Conquer

Finding the Maximum and Minimum

Merge Sort

Quick Sort

Stassen’s Matrix Multiplication

Advantages and Disadvantages of Divide and Conquer

Decrease and ConquerInsertion Sort

Topological Sort

Greedy MethodGeneral Method

Coin Change Problem

Knapsack Problem

Job Sequencing with Deadlines

Minimum Cost Spanning Trees2 Topics

Single Source Shortest Paths1 Topic

Optimal Tree Problem1 Topic

Transform and Conquer Approach1 Topic

Dynamic ProgrammingGeneral Method with Examples

Multistage Graphs

Transitive Closure1 Topic

All Pairs Shortest Paths6 Topics

BacktrackingGeneral Method

NQueens Problem

Sum of Subsets problem

Graph Coloring

Hamiltonian Cycles

Branch and Bound2 Topics

0/1 Knapsack problem2 Topics

NPComplete and NPHard Problems1 Topic
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Multistage Graphs
A multistage graph G = (V, E) is a directed graph where vertices are partitioned into k (where k > 1) number of disjoint subsets S = {s_{1},s_{2},…,s_{k}} such that edge (u, v) is in E, then u Є s_{i} and v Є s_{1 + 1} for some subsets in the partition and s_{1} = s_{k} = 1.
The vertex s Є s_{1} is called the source and the vertex t Є s_{k} 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.
In multistage graph problem we have to find the shortest path from source to sink.
The cost of a path from source (denoted by S) to sink (denoted by T) is the sum of the costs of edges on the path. In multistage graph problem we have to find the path from S to T. there is set of vertices in each stage. The multistage graph can be solved using forward and backward approach. Let us solve multistage problem for both the approaches with the help of example.
Consider the graph G as shown in the figure.
There is a single vertex in stage 1, then 3 vertices in stage 2, then 2 vertices in stage 3 and only one vertex in stage 4 (this is a target stage).
Backward Approach
d(S, T)=min {1+d(A, T),2+d(B,T),7+d(C,T)} …(1)
We will compute d(A,T), d(B,T) and d(C,T).
d(A,T)=min{3+d(D,T),6+d(E,T)} …(2)
d(B,T)=min{4+d(D,T),10+d(E,T)} …(3)
d(C,T)=min{3+d(E,T),d(C,T)} …(4)
Now let us compute d(D,T) and d(E,T).
d(D,T)=8
d(E,T)=2 backward vertex=E
Let us put these values in equations (2), (3) and (4)
d(A,T)=min{3+8, 6+2}
d(A,T)=8 AET
d(B,T)=min{4+8,10+2}
d{B,T}=12 ADT
d(C,T)=min(3+2,10)
d(C,T)=5 CET
d(S,T)=min{1+d(A,T), 2+d(B,T), 7+d(C,T)}
=min{1+8, 2+12,7+5}
=min{9,14,12}
d(S,T)=9 SAET
The path with minimum cost is SAET with the cost 9.
Algorithm for Backward Approach
1. Algorithm BGraph (G, K, n, p)
2. // some function as FGraph
3. {
4. B cost [1] = 0.0;
5. For j = 2 to n do
6. {
7. // compute b cost [j].
8. Let r be such that is an edge of
9. G and b cost [r] + c [r, j];
10. D [j] = r;
11. }
12. // find a minimum cost path
13. P [1] = 1; p [k] = n;
14. For j = k1 to 2 do p[j] = d [p (j+1)];
15. }
Forward Approach
d(S,A)=1
d(S,B)=2
d(S,C)=7
d(S,D)=min{1+d(A,D),2+d(B,D)}
=min{1+3,2+4}
d(S,D)=4
d(S,E)=min{1+d(A,E), 2+d(B,E),7+d(C,E)}
=min {1+6,2+10,7+3}
=min {7,12,10}
d(S,E)=7 i.e. Path SAE is chosen.
d(S,T)=min{d(S,D)+d(D,T),d(S,E),d(E,T),d(S,C)+d(C,T)}
=min {4+8,7+2,7+10}
d(S,T)=9 i.e. Path SE, ET is chosen.
The minimum cost=9 with the path SAET.
Algorithm for Forward Approach
1. F graph (graph G, int K, int n, int p[])
2. {
3. Float cost [max size], int d [max size], r;
4. Cost [n] = 0.0
5. For (int j = n1; j>= 1; j)
6. {
7. Let r be a vertex such that is an edge of G and C[j][r] + cost[r]
is minimum;
8. Cost [j] = C[j][r] + Cost[r]
9. D [j] = r
10. }
11. P [1] = 1 , P[k] = n
12. For (j = 2 ; j <= K1; j++)
13. P[j] = d[P(j1)];
14. }
Using dynamic approach programming strategy, the multistage graph problem is solved. This is because in multistage graph problem we obtain the minimum path at each current stage by considering the path length of each vertex obtained in earlier stage.
Thus the sequence of decisions is taken by considering overlapping solutions. In dynamic programming, we may get any number of solutions for given problem. From all these solutions we seek for optimal solution, finally solution becomes the solution to given problem. Multistage graph problem is solved using this same approach.