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title TARGET DECK FILE TAGS tags
Depth-First Search Obsidian::STEM algorithm::dfs data_structure::graph
dfs
graph

Overview

Depth-first search operates on a graph G = \langle V, E \rangle and a source vertex s.

!dfs.gif

To keep track of progress, DFS colors each vertex white, gray, or black. All vertices start out white. They are colored gray upon discovery. They are painted black once all edges have been explored.

Vertices also typically have two timestamps recorded: on discovery and on finish.

%%ANKI Basic What is DFS an acronym for? Back: Depth-first search. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Cloze Depth-first search is characterized by a graph and a {source vertex}. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic Which of undirected and directed graphs is DFS applicable to? Back: Both. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic What ADT is typically used to manage the set of most recently discovered DFS vertices? Back: A stack. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Cloze A {1:queue} is to {2:BFS} whereas a {2:stack} is to {1:DFS}. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022). Tags: algorithm::bfs

END%%

%%ANKI Basic Which vertices are not discovered during a graph DFS? Back: Those not reachable from the source vertex. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic What basic graph algorithm is the following a demonstration of? !dfs.gif Back: Depth-first search. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022). Tags: algorithm::bfs

END%%

%%ANKI Basic Which standard graph representation has worst-case DFS running time of O(\lvert V \rvert + \lvert E \rvert)? Back: The adjacency-list representation. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic Given graph \langle V, E \rangle with adjacency-list representation, what is the worst-case run time of DFS? Back: O(\lvert V \rvert + \lvert E \rvert) Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic Which standard graph representation has worst-case DFS running time of O(\lvert V \rvert^2)? Back: The adjacency-matrix representation. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic Given graph \langle V, E \rangle with adjacency-matrix representation, what is the worst-case run time of DFS? Back: O(\lvert V \rvert^2) Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic Why is DFS of an adjacency-list representation O(\lvert V \rvert + \lvert E \rvert)? Back: For each vertex being analyzed, we examine all of its adjacent vertices. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic Why is DFS of an adjacency-matrix representation O(\lvert V \rvert^2)? Back: For each vertex being analyzed, we must examine \lvert V \rvert entries for adjacent vertices. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic What does a white vertex typically represent? Back: A vertex that has not been discovered. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic When is a white vertex painted gray? Back: Upon discovery. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic When is a white vertex painted black? Back: N/A. It must be painted gray before it's painted black. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic What does a gray vertex typically represent? Back: A vertex that is in the stack, i.e. the frontier discovery happens against. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic When is a gray vertex painted white? Back: N/A. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic When is a gray vertex painted black? Back: After all of its edges have been examined. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic What does a black vertex typically represent? Back: A vertex whose edges have all been explored. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic What two timestamps are associated with each vertex? Back: A timestamp on discovery and a timestamp when finished processing. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic What range of values can a timestamp t take on? Back: 1 \leq t \leq 2\lvert V \rvert Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic Suppose vertex v has discovery time d and finish time f. When was v colored white? Back: At timestamps < d. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic Suppose vertex v has discovery time d and finish time f. When was v colored gray? Back: At timestamps \geq d and < f. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic Suppose vertex v has discovery time d and finish time f. When was v colored black? Back: At timestamps \geq f. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

Parenthesis Theorem

In any depth-first search of a graph, for any two vertices u and v, exactly one of the following three conditions holds:

  1. The intervals [u{.}d, u{.}f] and [v{.}d, v{.}f] are disjoint.
    • No ancestor-descendant relation exists between u and v.
  2. The interval [u{.}d, u{.}f] is contained entirely within [v{.}d, v{.}f].
    • u is a descendant of v.
  3. The interval [v{.}d, v{.}f] is contained entirely within [u{.}d, u{.}f].
    • v is a descendant of u.

%%ANKI Basic What aspect of DFS has parenthesis structure? Back: The discovery and finish timestamps of vertices. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic Let u be a vertex. What does u{.}d refer to? Back: When vertex u was discovered during DFS. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic Let u be a vertex. What does u{.}f refer to? Back: When vertex u was finished processing during DFS. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic Given vertices a and b, what does the parenthesis theorem state? Back: Either [a{.}d, a{.}f] and [b{.}d, b{.}f] are disjoint or one interval is contained entirely within the other. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic If a is a descendant of b, what can be said about their discovery and finish times? Back: b{.}d < a{.}d < a{.}f < b{.}f Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic If b is a descendant of a, what can be said about their discovery and finish times? Back: a{.}d < b{.}d < b{.}f < a{.}f Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic If a and b have no ancestor-descendant relation, what can be said about their discovery and finish times? Back: [a{.}d, a{.}f] and [b{.}d, b{.}f] are disjoint. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic Let a and b be vertices with timestamps satisfying a{.}d < b{.}d < b{.}f < a{.}f. What ancestor-descendant relation exists? Back: b is a descendant of a. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic Let a and b be vertices with timestamps satisfying a{.}d < b{.}d < a{.}f < b{.}f. What ancestor-descendant relation exists? Back: N/A. This is an impossible chain of inequalities. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic Let a and b be vertices with timestamps satisfying b{.}d < a{.}d < b{.}f < a{.}f. What ancestor-descendant relation exists? Back: N/A. This is an impossible chain of inequalities. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic Let a and b be vertices with timestamps satisfying b{.}d < a{.}d < a{.}f < b{.}f. What ancestor-descendant relation exists? Back: a is a descendant of b. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

Depth-First Forests

To color an entire graph black, BFS may need to be invoked multiple times. After each invocation of BFS, a new invocation can be run with any remaining white vertex as the source. Each invocation yields a depth-first tree. Multiple invocations yield a depth-first forest.

%%ANKI Basic With respect to depth-first trees, what does the predecessor of a node N refer to? Back: The node from which N was discovered. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic When might white vertices remain after DFS is invoked? Back: When there exist vertices unreachable from the last used source vertex. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic How many invocations of DFS are required to color a graph black? Back: One or more. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Cloze The {1:source} of a depth-first {2:search} is the {2:root} of a depth-first {1:tree}. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic With respect to depth-first trees, what does the predecessor of a node N refer to? Back: The node from which N was discovered. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022). END%%

%%ANKI Basic With respect to depth-first trees, what does the parent of a node N refer to? Back: The node from which N was discovered. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic With respect to depth-first trees, the predecessor of a node is also known as what? Back: The parent of the node. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic With respect to depth-first trees, the parent of a node is also known as what? Back: The predecessor of the node. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

Edge Classification

A depth-first forest can contain four different types of edges:

  1. A tree edge is an edge \langle u, v \rangle such that v was first discovered by exploring edge \langle u, v \rangle.
  2. A back edge is an edge \langle u, v \rangle connecting vertex u to an ancestor v.
    1. Self-loops are considered back edges.
  3. A forward edge is a non-tree edge \langle u, v \rangle connecting vertex u to a proper descendant v.
  4. A cross edge is any other edge.

%%ANKI Basic In a depth-first forest, edges are classified in how many ways? Back: Four. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic In a depth-first forest, what are the four edge classifications? Back: Tree, forward, back, and cross. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic Let \langle u, v \rangle be an edge in a depth-first forest. When is \langle u, v \rangle a tree edge? Back: When v was first discovered along edge \langle u, v \rangle. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic Let \langle u, v \rangle be an edge in a depth-first forest. When is \langle u, v \rangle a back edge? Back: When v is an ancestor of u. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic Let \langle u, v \rangle be an edge in a depth-first forest. When is \langle u, v \rangle a forward edge? Back: When \langle u, v \rangle is a non-tree edge such that v is a proper descendant of u. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Let \langle u, v \rangle be an edge in a depth-first forest. When is \langle u, v \rangle a cross edge? Back: When u and v have no ancestor-descendant relation. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022). END%%

%%ANKI Basic What kind of edge is a self-loop in a depth-first forest classified as? Back: A back edge. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic Consider depth-first forest with edge \langle u, v \rangle. What kind of edge is it if:

  • \langle u, v \rangle is not a tree edge; and
  • v is a descendant of u. Back: A forward edge. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic Consider depth-first forest with edge \langle u, v \rangle. What kind of edge is it if:

  • \langle u, v \rangle is not a tree edge; and
  • v is not a descendant of u. Back: Either a back edge or cross edge. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic Consider depth-first forest with edge \langle u, v \rangle. What kind of edge is it if:

  • \langle u, v \rangle is not a tree edge; and
  • v is an ancestor of u. Back: A back edge. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic Consider depth-first forest with edge \langle u, v \rangle. What kind of edge is it if:

  • \langle u, v \rangle is not a tree edge; and
  • v is not an ancestor of u. Back: Either a forward edge or cross edge. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic Consider depth-first forest with edge \langle u, v \rangle. What kind of edge is it if:

  • u and v share no ancestor-descendant relation. Back: A cross edge. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic Assuming solid lines are tree edges, how is edge \langle s, c \rangle classified? !dfs-edge-classification.png Back: As a forward edge. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic Assuming solid lines are tree edges, why is edge \langle s, c \rangle a forward edge? !dfs-edge-classification.png Back: Because c is a descendant of s. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic Assuming solid lines are tree edges, how is edge \langle c, s \rangle classified? !dfs-edge-classification.png Back: N/A. This edge does not exist. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic Assuming solid lines are tree edges, how is edge \langle b, s \rangle classified? !dfs-edge-classification.png Back: As a back edge. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic !dfs-edge-classification.png Back: Because s is an ancestor of b. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic Assuming solid lines are tree edges, how is edge \langle d, c \rangle classified? !dfs-edge-classification.png Back: As a cross edge. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic Assuming solid lines are tree edges, why is edge \langle d, c \rangle a cross edge? !dfs-edge-classification.png Back: Because d and c have no ancestor-descendant relation between them. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic Let v be white when \langle u, v \rangle is explored. What kind of edge is \langle u, v \rangle classified as? Back: A tree edge. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic Let v be gray when \langle u, v \rangle is explored. What kind of edge is \langle u, v \rangle classified as? Back: A back edge. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic Let v be black when \langle u, v \rangle is explored. What kind of edge is \langle u, v \rangle classified as? Back: Either a forward edge or cross edge. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic Let v be black when \langle u, v \rangle is explored. If u{.}d < v{.}d, what kind of edge is \langle u, v \rangle classified as? Back: A forward edge. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic Let v be black when \langle u, v \rangle is explored. If v{.}d < u{.}d, what kind of edge is \langle u, v \rangle classified as? Back: A cross edge. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic When performing DFS on a directed graph, what possible edge classifications are there? Back: Tree, forward, back, and cross. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic When performing DFS on an undirected graph, what possible edge classifications are there? Back: Tree and back. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

Bibliography

  • Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).