Podcast
Questions and Answers
What is the first step in finding the maximum flow in a network?
What is the first step in finding the maximum flow in a network?
- Find an open path from source to sink. (correct)
- Subtract the flow amount from all paths.
- Calculate the edge weights.
- Draw a minimum cut through the edges.
In the context of the Hungarian Algorithm, when should you return to Step 3?
In the context of the Hungarian Algorithm, when should you return to Step 3?
- When the number of lines is less than the number of rows. (correct)
- When at least one entry has been circled.
- After completing Step 6.
- When all entries have been paired.
What is a minimum cut in a flow network?
What is a minimum cut in a flow network?
- The total flow in all paths chosen.
- The maximum flow available from source to sink.
- A line that separates the source from the sink minimizing flow. (correct)
- The greatest edge weight within the network.
Which of the following describes the role of the source in a flow network?
Which of the following describes the role of the source in a flow network?
During Step 5 of the Hungarian algorithm, what action is taken with the smallest uncovered entry?
During Step 5 of the Hungarian algorithm, what action is taken with the smallest uncovered entry?
Which statement is true regarding the outcome of the Hungarian Algorithm?
Which statement is true regarding the outcome of the Hungarian Algorithm?
What does maximum flow refer to in the context of a flow network?
What does maximum flow refer to in the context of a flow network?
What happens when all possible open paths are chosen in the maximum flow algorithm?
What happens when all possible open paths are chosen in the maximum flow algorithm?
What is the correct definition of a circuit in the context of Eulerian graphs?
What is the correct definition of a circuit in the context of Eulerian graphs?
What condition must be met for a graph to be classified as Eulerian?
What condition must be met for a graph to be classified as Eulerian?
Which of the following statements is true regarding minimum cut and maximum flow?
Which of the following statements is true regarding minimum cut and maximum flow?
In a semi-Eulerian graph, how many vertices can have an odd degree?
In a semi-Eulerian graph, how many vertices can have an odd degree?
What is the capacity of the cut that includes edges with flows measuring 5, 0, 1, and 6?
What is the capacity of the cut that includes edges with flows measuring 5, 0, 1, and 6?
What kind of path characterizes Hamiltonian graphs?
What kind of path characterizes Hamiltonian graphs?
The maximum flow through a network can be verified through what process?
The maximum flow through a network can be verified through what process?
In a closed trail within Eulerian graphs, what condition must hold for the trail's vertices?
In a closed trail within Eulerian graphs, what condition must hold for the trail's vertices?
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Study Notes
Hungarian Algorithm
- The algorithm aims to find minimum cost assignments in a cost matrix.
- Steps include covering lines, finding the smallest uncovered entry, and manipulating the matrix.
- The process involves circling zero entries to match rows and columns effectively.
- Multiple solutions can exist for a given cost matrix.
Example of Minimum Cost Assignment
- Initial cost matrix:
- [8, 8, 6]
- [2, 3, 7]
- [4, 9, 3]
- Through several steps, the algorithm transforms the matrix to find the optimal zero entries leading to minimum costs.
Maximum Cost Assignment
- For non-square matrices, the algorithm adjusts entries for calculations.
- Example provided:
- Original cost matrix:
- [3, 5, 2]
- [6, 4, 1]
- A maximum assignment is derived through similar steps as in the minimum cost assignment.
- Original cost matrix:
Flow Network Concepts
- Source: Starting point for flow.
- Sink: Endpoint where flow concludes.
- Flow: Balance of flow into and out of a node; respects edge weights.
- Maximum Flow: Greatest feasible flow under given constraints.
- Cut: Represents a division in the network that halts flow.
- Minimum Cut: The smallest cut quantifying the minimum sum of edge weights.
Maximum Flow Steps
- Identify accessible paths from source to sink.
- Calculate the maximum flow along chosen paths and adjust edge values accordingly.
- Repeat until no further flow can be achieved.
Drawing a Cut
- Edges that direct flow towards the source are included in the cut.
- Cut capacity is determined by summing the weights of connecting edges.
Max Flow-Min Cut Theorem
- The capacity of the maximum flow equals the capacity of the minimum cut, confirming flow limits in a network.
Walks, Paths, and Trails
- Walk: Sequence of vertices where repeats are allowed.
- Path: Sequence without repeated vertices; open and closed types exist based on endpoints.
- Trail: Similar to paths but allows edge repeats; can also be open or closed.
Eulerian Graphs
- Defined by having a closed trail that visits every edge once.
- All vertices must have even degrees for a graph to be Eulerian.
- Semi-Eulerian graphs allow one pair of vertices with an odd degree.
Hamiltonian Graphs
- Feature closed paths visiting all vertices once, excluding start and end revisits.
- Semi-Hamiltonian graphs contain open paths that cover all vertices exactly once.
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