Propagation down the dual segment tree

• In DSL 2 D Range Update Query I wrote
  • How should the “overwrite value” range action be defined?
  • The range action doesn’t know which is later, whether the node is above or below.
  • So I made the value timestamped, and propagation downwards is a binary operation that “takes the one with the newer timestamp”.
• This again implies the following
  • The implementation unintentionally assumed that the action was commutative.
  • Incorrect operation in case of non commutative “overwrite action
  • Converted to a commutative max operation by adding a timestamp.
• the problem is that you are not doing the propagation down correctly before the range action.
  • Half-delayed segment tree] omits the value segment tree when only a point acquisition is needed.
  • Furthermore, when the composition of actions is commutative, skipping propagation down dual segment tree.
  • I correctly constructed a dual segment tree, but overlooked the “action composition is commutative”.

Demo with two range actions and f and g overlapping out of order.

• If the propagation downwards is commutative, it doesn’t matter if the order is switched here. python

>>> range_update(table, 3, 10, lambda x: "f")
>>> range_update(table, 5, 15, lambda x: "g")
>>> debugprint(table)
|               0               |
|       0       |       0       |
|   0   |   f   |   g   |   0   |
| 0 | 0 | 0 | g | f | 0 | g | 0 |
|0|0|0|f|0|g|0|0|0|0|0|0|0|0|g|0|

The right way to do it when it is not commutative

• Propagate down before the second range action.
• It is guaranteed that the cell above the cell to be rewritten in the next range action is down.
• The range operator lambda x: "g" is the lower propagation binary operation lambda x, y: x if x else y with “g” in x. python

>>> table = [0] * SEGTREE_SIZE
>>> range_update(table, 3, 10, lambda x: "f")
>>> down_propagate(table, up(5), lambda x, y: x if x else y, 0)
>>> down_propagate(table, up(15), lambda x, y: x if x else y, 0)
>>> debugprint(table)
|               0               |
|       0       |       0       |
|   0   |   0   |   0   |   0   |
| 0 | 0 | 0 | f | f | 0 | 0 | 0 |
|0|0|0|f|f|f|0|0|0|0|0|0|0|0|0|0|

>>> range_update(table, 5, 15, lambda x: "g")
>>> debugprint(table)
|               0               |
|       0       |       0       |
|   0   |   0   |   g   |   0   |
| 0 | 0 | 0 | g | f | 0 | g | 0 |
|0|0|0|f|f|g|0|0|0|0|0|0|0|0|g|0|

Appendix: Range of influence of propagation downwards.

• dual segment tree
• Segment tree visualization

---

This page is auto-translated from /nishio/双対セグメント木の下への伝搬 using DeepL. If you looks something interesting but the auto-translated English is not good enough to understand it, feel free to let me know at @nishio_en. I’m very happy to spread my thought to non-Japanese readers.