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The definition of the lowlink in Tarjan's algorithm is the smallest index of a vertex that is reachable with at most one back-edge in SCC. This is not useful for a cross-edge. If we start traversing from A in the following graph, the final lowlink of D is 3. The cross-edge here is one between D and C. A -> B -> D D = (4, 3) (index, lowlink) ^ | | C = (3, 1) | V | B = (2, 1) `--- C <--' A = (1, 1) This is because the lowlink of D is updated with the index of C. In the following patch, we detect a dead SCC by checking two conditions for each vertex. 1) vertex has no edge directed to another SCC (no bridge) 2) vertex's out_degree is the same as the refcount of its file If 1) is false, there is a receiver of all fds of the SCC and its ancestor SCC. To evaluate 1), we need to assign a unique index to each SCC and assign it to all vertices in the SCC. This patch changes the lowlink update logic for cross-edge so that in the example above, the lowlink of D is updated with the lowlink of C. A -> B -> D D = (4, 1) (index, lowlink) ^ | | C = (3, 1) | V | B = (2, 1) `--- C <--' A = (1, 1) Then, all vertices in the same SCC have the same lowlink, and we can quickly find the bridge connecting to different SCC if exists. However, it is no longer called lowlink, so we rename it to scc_index. (It's sometimes called lowpoint.) Also, we add a global variable to hold the last index used in DFS so that we do not reset the initial index in each DFS. This patch can be squashed to the SCC detection patch but is split deliberately for anyone wondering why lowlink is not used as used in the original Tarjan's algorithm and many reference implementations. Signed-off-by: Kuniyuki Iwashima <kuniyu@amazon.com> Acked-by: Paolo Abeni <pabeni@redhat.com> Link: https://lore.kernel.org/r/20240325202425.60930-13-kuniyu@amazon.com Signed-off-by: Jakub Kicinski <kuba@kernel.org> |
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Linux kernel
============
There are several guides for kernel developers and users. These guides can
be rendered in a number of formats, like HTML and PDF. Please read
Documentation/admin-guide/README.rst first.
In order to build the documentation, use ``make htmldocs`` or
``make pdfdocs``. The formatted documentation can also be read online at:
https://www.kernel.org/doc/html/latest/
There are various text files in the Documentation/ subdirectory,
several of them using the reStructuredText markup notation.
Please read the Documentation/process/changes.rst file, as it contains the
requirements for building and running the kernel, and information about
the problems which may result by upgrading your kernel.