Merging the final groups

For example, consider these two graphs, which are almost identical except that in the first, w6 targets shard D and in the second in targets B. In the second graph, w6 has been placed in its own group rather than merging it into the [w1] group, because according to the above discussion we should place w6 in a group whose depth is at least 1.

Shard │
      │        ┌───────────┐
    A │        │ w2     w3 │
      │        └/─────────\┘
      │        /           \
      │   ┌───/┐            \
    B │   │ w1 │             \
      │   └────┘              \
      │                        \
      │   ┌────┐               ┌\───┐
    C │   │ w5 │               │ w4 │
      │   └───\┘               └────┘
      │        \
      │        ┌\───┐
    D │        │ w6 │
      │        └────┘

Shard │
      │        ┌───────────┐
    A │        │ w2     w3 │
      │        └/─────────\┘
      │        /           \
      │   ┌───/┐    ┌────┐  \
    B │   │ w1 │    │ w6 │   \
      │   └────┘    └/───┘    \
      │             /          \
      │   ┌────┐   /           ┌\───┐
    C │   │ w5 ───'            │ w4 │
      │   └────┘               └────┘

In both these graphs, the [w5] and [w4] groups can be merged at the cost of increasing D from three to four. In the second graph it is possible to instead merge the [w1] and [w6] groups, which also increases D to four. That is, we could end up with either of these graphs after performing group merging on the second graph:

Shard │
      │        ┌───────────┐
    A │        │ w2     w3 │
      │        └/─────────\┘
      │        /           \
      │   ┌───/┐            \              ┌────┐
    B │   │ w1 │             \             │ w6 │
      │   └────┘              \            └/───┘
      │                        \           /
      │                        ┌\─────────/┐
    C │                        │ w4     w5 │
      │                        └───────────┘

Shard │
      │                     ┌───────────┐
    A │                     │ w2     w3 │
      │                     └/─────────\┘
      │                     /           \
      │        ┌───────────/┐            \
    B │        │ w6      w1 │             \
      │        └/───────────┘              \
      │        /                            \
      │   ┌───/┐                            ┌\───┐
    C │   │ w5 │                            │ w4 │
      │   └────┘                            └────┘

There’s no reason to prefer either of these as they have the same N and D values. What’s important to note is that the decision to merge [w1] and [w6], or to merge [w4] and [w5] are mutually exclusive; once one of these merges has been done, the other is illegal without violating causality.