| You're given an undirected, connected graph with **N** nodes (numbered from 1 |
| to **N**) and **N** edges. The **i**th edge connects distinct nodes **Ai** and |
| **Bi**, and has a capacity of **Ci**. No two edges directly connect the same |
| pair of nodes. |
|
|
| **M** operations will be performed on this graph, one after another. The nature of the **i**th operation is described by the value of **Oi**: |
|
|
| \- If **Oi** = 1, then the **i**th operation is an update, in which the |
| capacity of the **Xi**th edge is changed to be **Yi**. |
|
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| \- Otherwise, if **Oi** = 2, then the **i**th operation is a query, in which |
| you must determine the maximinimax flow in the graph after **Zi** edge |
| augmentations. |
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| What do any of those terms mean? Let's define them: |
|
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| \- An edge augmentation is a temporary increase of a certain edge's capacity |
| by 1 for the current query. |
|
|
| \- The max flow from node **u** to a different node **v** is the usual |
| definition of maximum flow in computer science (hopefully you're familiar with |
| it!), with node **u** being the source and node **v** being the sink. Each |
| edge may transport flow in either direction, so it may be thought of as two |
| directed edges (one in each direction), both with the same capacity. |
|
|
| \- The minimax flow in the graph is the smallest max flow value across all |
| pairs of distinct nodes. In other words, min{1 ≤ **u**, **v** ≤ **N**, **u** ≠ |
| **v**} (F(**u**, **v**)), where F(**u**, **v**) is the max flow from node |
| **u** to node **v**. |
|
|
| \- The maximinimax flow in the graph after **x** edge augmentations is the |
| largest possible minimax flow which the graph can have after **x** optimal |
| edge augmentations are applied. Note that each edge can be augmented any non- |
| negative number of times (as long as the total number of augmentations in the |
| graph is **x**), and that the chosen edge augmentations are temporary — they |
| do not change the graph for future operations. |
|
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| To reduce the size of the output, you should simply output one integer, the |
| sum of the answers to all of the queries. |
|
|
| ### Input |
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|
| Input begins with an integer **T**, the number of graphs. For each graph, |
| there is a first a line containing the space-separated integers **N** and |
| **M**. Then, **N** lines follow, the **i**th of which contains the space- |
| separated integers **Ai**, **Bi**, and **Ci**. Then, **M** lines follow, the |
| **i**th of which contains the space-separated integers **Oi**, **Xi**, and |
| **Yi** (if **Oi** = 1) or **Oi** and **Zi** (if **Oi** = 2). |
|
|
| ### Output |
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| For the **i**th graph, print a line containing "Case #**i**: " followed by the |
| sum of the answers to all queries on that graph. |
|
|
| ### Constraints |
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| 1 ≤ **T** ≤ 85 |
| 3 ≤ **N** ≤ 500,000 |
| 1 ≤ **M** ≤ 500,000 |
| 1 ≤ **Ai**, **Bi**, **Xi** ≤ **N** |
| 1 ≤ **Ci**, **Yi** ≤ 1,000,000 |
| 0 ≤ **Zi** ≤ 1,000,000,000,000 |
| 1 ≤ **Oi** ≤ 2 |
|
|
| ### Explanation of Sample |
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| In the first graph, the max flow between any two nodes is 10, so the minimax |
| flow is also 10. If we do no edge augmentations, then the maximinimax flow is |
| still 10. In the second graph, the maximinimax flow is initially 3, but is |
| then increased to 5 before the second query for a total of 8. In the third |
| graph, the maximinimax flow is initially 7 (between node 2 and any other |
| node). If we augment the edge from node 3 to node 2 twice, then the max flow |
| between node 2 and any other node is now 9. The max flow between any other |
| pair of nodes was already greater than 9, so the minimax flow is now 9. We |
| can't do any better than that, so the maximinimax flow is also 9. |
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