Question

6. Single-source shortest paths, continued (a) Algorithm DAG-SHORTEST-PATHS finds shortest paths from a source s by re- laxing edges in some order. Consider a modification of the algorithm that relaxes edges in a reversed order. • Statement: If each vertex is reachable by an edge from the source then the modified algorithm correctly finds the shortest paths to each vertex. Is this statement true? Circle your answer: yes no • Statement: If the modified algorithm correctly finds the shortest paths to each vertex then each vertex is reachable by an edge from the source. Is this statement true? Circle your answer: yes no (b) Algorithm DIJKSTRA uses a priority queue on which we can perform the opera- tions insert, extract minimum, and decrease key. Statement: In an execution of DIJKSTRA, at least one decrease-key operation is always performed between any two consecutive extract-min operations. Is this statement true? Circle your answer: yes no (c) Algorithm DIJKSTRA is always correct on input weighted graphs with non-negative weights. Statement: Algorithm DIJKSTRA is always correct on input weighted graphs with at most one edge of negative weight. Is this statement true? Circle your answer: yes no

          6. Single-source shortest paths, continued
(a) Algorithm DAG-SHORTEST-PATHS finds shortest paths from a source s by re-
laxing edges in some order. Consider a modification of the algorithm that relaxes
edges in a reversed order.
• Statement: If each vertex is reachable by an edge from the source then the
modified algorithm correctly finds the shortest paths to each vertex.
Is this statement true?
Circle your answer:
yes
no
• Statement: If the modified algorithm correctly finds the shortest paths to
each vertex then each vertex is reachable by an edge from the source.
Is this statement true?
Circle your answer:
yes
no
(b) Algorithm DIJKSTRA uses a priority queue on which we can perform the opera-
tions insert, extract minimum, and decrease key.
Statement: In an execution of DIJKSTRA, at least one decrease-key operation is
always performed between any two consecutive extract-min operations.
Is this statement true?
Circle your answer:
yes
no
(c) Algorithm DIJKSTRA is always correct on input weighted graphs with non-negative
weights.
Statement: Algorithm DIJKSTRA is always correct on input weighted graphs with
at most one edge of negative weight.
Is this statement true?
Circle your answer:
yes
no

6. Single-source shortest paths, continued
(a) Algorithm DAG-SHORTEST-PATHS finds shortest paths from a source s by re-
laxing edges in some order. Consider a modification of the algorithm that relaxes
edges in a reversed order.
• Statement: If each vertex is reachable by an edge from the source then the
modified algorithm correctly finds the shortest paths to each vertex.
Is this statement true?
Circle your answer:
yes
no
• Statement: If the modified algorithm correctly finds the shortest paths to
each vertex then each vertex is reachable by an edge from the source.
Is this statement true?
Circle your answer:
yes
no
(b) Algorithm DIJKSTRA uses a priority queue on which we can perform the opera-
tions insert, extract minimum, and decrease key.
Statement: In an execution of DIJKSTRA, at least one decrease-key operation is
always performed between any two consecutive extract-min operations.
Is this statement true?
Circle your answer:
yes
no
(c) Algorithm DIJKSTRA is always correct on input weighted graphs with non-negative
weights.
Statement: Algorithm DIJKSTRA is always correct on input weighted graphs with
at most one edge of negative weight.
Is this statement true?
Circle your answer:
yes
no

Added by Trevor S.

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