fix tarjan algorithm

[edogawashinichi]

• TarjanTest.test_4 failed #408
• when a neighbor node doesn't finish its dfs_helper, use its discoveryTime instead of lowestDisc to update lowestDisc of the current node.
• @ZigRazor @bbannier @Gonzalo-Mier @gitter-badger @nrkramer

All 300 unit tests passed.

To explain the correctness of the modified version:

For any node, discoveryTime[node] >= lowestDisc[node].
PROOF:
because lowestDisc is initialized with discoveryTime,
and may be updated by smaller values.

If a node finishes its dfs_helper, its lowestDisc is the true value,
otherwise its lowestDisc may be greater than the true value.
Therefore the first time to update lowestDisc[curNode] in file Tarjan_impl.hpp shall be unchanged.
PROOF:
because in the process of the node's dfs_helper,
lowestDisc[node] may be updated smaller by lowestDisc of any of its neighbor node,
after the neighbor node finishes its dfs_helper.

If we are in curNode's dfs_helper, and neighborNode we are traversing is already discovered, then
neighborNode doesn't finish its dfs_helper when the graph is undirected,
and hence lowestDisc[neighborNode] may be greater.
PROOF by contradiction:
suppose to the contrary that neighborNode finishes its dfs_helper.
Noting that neighborNode and curNode are adjacent in the undirected graph,
and that discoveryTime[neighborNode] < discoveryTime[curNode],
we can conclude that curNode must also finish its dfs_helper,
violating that we are in curNode's dfs_helper.

For the second time to update lowestDisc[curNode] in file Tarjan_impl.hpp,
replacing lowestDisc[neighborNode] with discoveryTime[neighborNode] is correct
when the graph is undirected.
PROOF:
we can verify each case.
case TARJAN_FIND_BRIDGE.
####if (discoveryTime[curNode] < lowestDisc[neighborNode]) then curNode-neighborNode is a bridge.
####if (discoveryTime[curNode] < lowestDisc[neighborNode]) is false for all curNode's neighbors, then
########curNode-neighborNode is not a bridge.
case TARJAN_FIND_CUTV.
####if (curNode not root node and discoveryTime[curNode] <= lowestDisc[neighborNode]), then
########curNode is a cut vertex. (This case needs discreetly consider the traversal order of nodes.)
####otherwise of course holds true.
case TARJAN_FIND_VBCC.
####similar to case TARJAN_FIND_CUTV.
case TARJAN_FIND_SCC or TARJAN_FIND_EBCC.
####similar to case TARJAN_FIND_CUTV.

The above pipeline only demonstrates the correctness of any TASK for UNDIRECTED graph.
For other TASKs with DIRECTED graph and neighborNode inStack,
it needs to be considered carefully,
and perhaps a better choice is to add more unit tests.

[edogawashinichi]

The following is some references.

• TARJAN_FIND_VBCC

https://www.geeksforgeeks.org/biconnected-components/
uses low[u] = min(low[u], disc[v]),
for the second time to update lowestDisc[curNode].

https://www.hackerearth.com/practice/algorithms/graphs/biconnected-components/tutorial/
uses an equivalent version
else if parent[vertex] != i AND disc[i] < low[vertex] low[vertex] = disc[i].

https://www.cs.cmu.edu/~15451-f23/lectures/lecture11-dfs-bicon.pdf
is the same solution.

• TARJAN_FIND_BRIDGE

https://cp-algorithms.com/graph/bridge-searching.html
uses if (visited[to]) { low[v] = min(low[v], tin[to]); }

• TARJAN_FIND_CUTV

https://cp-algorithms.com/graph/cutpoints.html
uses if (visited[to]) { low[v] = min(low[v], tin[to]); }

• TARJAN_FIND_SCC

https://en.wikipedia.org/wiki/Tarjan%27s_strongly_connected_components_algorithm
uses v.lowlink := min(v.lowlink, w.index)

• TARJAN_FIND_EBCC

[codecov]

Codecov Report

All modified and coverable lines are covered by tests ✅

Project coverage is 97.88%. Comparing base (1b31e63) to head (3983db4).
Report is 16 commits behind head on master.

Additional details and impacted files

@@            Coverage Diff             @@
##           master     #409      +/-   ##
==========================================
+ Coverage   97.58%   97.88%   +0.29%     
==========================================
  Files          87       87              
  Lines        9492    10061     +569     
  Branches        0      660     +660     
==========================================
+ Hits         9263     9848     +585     
+ Misses        229      213      -16     

Flag	Coverage Δ
unittests	97.88% <100.00%> (+0.29%)	⬆️

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[ZigRazor]

Well Done!