Question

In a directed acyclic graph with source vertex \( s \), the quality-score of a directed path is the product of the weights of the edges on the path. 
For a vertex \( v \neq s \), the quality-score of \( v \) is the maximum among the quality-scores of all paths from \( s \) to \( v \). The quality-score of \( s \) is assumed to be 1. 
The sum of the quality-scores of all the vertices in the graph is \(\underline{\hspace{2cm}}\). 

Hint:

In DAG path problems, process vertices in topological order and propagate maximum path values forward.

Answer 1

Correct Answer: 929

To solve the problem of determining the sum of the quality-scores for all vertices in the given directed acyclic graph (DAG), follow these steps:

1. Identify the source vertex: The source vertex \( s \) is given, and its quality-score is \( 1 \).
2. Calculate quality-scores for each vertex:
   • \( a \): From \( s \to a \), the quality-score is \( 9 \).
   • \( b \): From \( s \to a \to b \), the quality-score is \( 9 \times 1 = 9 \).
   • \( c \): From \( s \to c \), the quality-score is \( 1 \).\(
     \)Alternatively, from \( s \to a \to d \to c \), the quality-score is \( 9 \times 9 \times 1 = 81 \). So, \( 81 \) is the maximum.
   • \( d \): From \( s \to a \to d \), the quality-score is \( 9 \times 1 = 9 \).
   • \( e \): From \( s \to a \to d \to e \), the quality-score is \( 9 \times 9 \times 9 = 81 \).\(
     \)Alternatively, from \( s \to b \to e \), the quality-score is \( 9 \times 1 = 9 \). So, \( 81 \) is the maximum.
   • \( f \): From \( s \to c \to f \), the quality-score is \( 81 \times 9 = 729 \).
   • \( g \): From \( s \to a \to d \to g \), the quality-score is \( 9 \times 9 \times 9 = 81 \).\(
     \)Alternatively, from \( s \to c \to d \to g \), the quality-score is \( 81 \times 9 = 729 \). So, \( 729 \) is the maximum.
   • \( t \): From \( s \to a \to d \to g \to t \), the quality-score is \( 9 \times 9 \times 9 \times 1 = 81 \).\(
     \)Alternatively, from \( f \to g \to t \), the quality-score is \( 729 \times 1 = 729 \). So, \( 729 \) is the maximum.
3. Sum the quality-scores: \( S = 1 + 9 + 9 + 81 + 9 + 81 + 729 + 729 = 1,647 \).
4. Verify against given range: The expected range is \( 929,929 \) which seems to be misinterpreted. The sum \( 1,647 \) does not fall within this range. However, solve and explain according to the problem requirements and logical deductions.