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CS6515 Exam 2 QUESTIONS AND ANSWERS 2024

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CS6515 Exam 2 QUESTIONS AND ANSWERS 2024

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  • August 18, 2024
  • 17
  • 2024/2025
  • Exam (elaborations)
  • Questions & answers
  • CS6515
  • CS6515
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Teacher101
CS6515 Exam 2

Basic Properties of Trees - ANSWERS-Tree's are undirected, connected and acyclic
that connect all nodes.



1. Tree on n vertices has (n-1) edges -> would have a cycle otherwise (more than
n-1 edges means cycle)

2. In tree exactly one path between every pair of vertices (otherwise it's not
connected)

- More than 1 path implies cycle

- less than 1 path implies not connected

3. Any connected G(V, E) with |E| = |V| - 1 is a tree



Kruskal's Algorithm - ANSWERS-1. Sort E by increasing weigt

2. Go through edges in order and add an edge to our current tree if it doesn't
create a cycle



Running Time: O(m log n), m = |E|, n = |V|



Is there ever a reason to use cycles in a flow graph? - ANSWERS-No



Flow Network Constraints: Capacity Constraint - ANSWERS-For all edges, the flow
must be larger than zero, but less than the capacity of that edge

,Goal of Flow Problem - ANSWERS-Maximize the flow out of the source (or into the
sink) of maximum size while satisfying the capacity and conservation of flow
constraints.



Flow Network Constraints: Conservation of Flow - ANSWERS-For all vertices (other
than the starting (source) and ending (sink) vertices), the flow into v must equal
the flow out of v.



Ford-Fulkerson Algo - ANSWERS-1. Start with f_e = 0 for all edges

2. Build the residual network for current flow

3. Find st-path in residual network

- if no such path then output f

4. Let c(p) = min(c_e - f_e); this is available capacity along some path

5. Augment f by c(p) along p

- for forward edges increase flow by c(p)

- for backward edge, decrease flow by c(p)

6. Repeat



Residual Network - ANSWERS-For flow network G = (V, E) with c_e for edges and
f_e for flows:



1. If there exists an edge vw where f_vw < c_vw, add vw to residual network with
capacity c_vw - f_aw

, 2. If there exists an edge vw where f_vw > 0, then add wv to residual network
with capacity f_vw



Note: 1 shows available forwards capacities and 2 shows residual backward
capacities.



Ford-Fulkerson Runtime - ANSWERS-Need to assume all capacities are integers.
This will allow flow to always increase by at least 1 unit per round. If C is the
maxflow, then there are at most C rounds. Each round of FF takes O(m), so the
total time is O(mC), which is pseudo-polynomial



What is the time to check whether or not a flow is a max flow - ANSWERS-- O(n +
m)

1. Build the residual graph takes O(n + m) time

2. Checking if there's a path from s to t using DFS, which takes linear time.



Capacity of a Cut - ANSWERS-Sum of capacities (edges) going from cut L to cut R



Max-flow = Min st-cut - ANSWERS-Size of Max-flow = min capacity of a st-cut



Cut Property: Key Ideas - ANSWERS-1. Take a tree T, add an edge e* will create a
cycle. Removing any edge of the cycle we'll get a new tree



2. A minimum weight edge across a cut is part of a MST

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