| Mr. Fox sure loves his rocks! In fact, when he's not in a hurry, he often |
| looks at the rocks lying around near him to decide where to wander in his |
| forest. |
|
|
| Mr. Fox lives in a forest with **N** clearings, numbered from 0 to **N**-1, |
| with **P** one-way trails initially running amongst them. The **i**th trail |
| runs from clearing **Ai** to a different clearing **Bi**, and is littered with |
| **Ri** rocks. No two clearings are connected by multiple trails running in the |
| same direction, though they could be connected by 2 trails running in opposite |
| directions. Additionally, an interesting property of this forest is that a |
| trail from clearing **a** to clearing **b** may only exist if 0 ≤ |
| floor(**b**/4) - floor(**a**/4) ≤ 1. |
|
|
| To entertain himself over a period of **D** days, Mr. Fox will cause one event |
| to occur on each day. The **i**th event may be one of 3 types, determined by |
| the value of **Ei**: |
|
|
| 1. Given 3 integers **Xi**, **Yi**, and **Zi**, Mr. Fox will create a new trail from clearing **Xi** to a different clearing **Yi**, and drop **Zi** rocks onto it. It's guaranteed that no trail from **Xi** to **Yi** will exist at the start of the **i**th day, and that 0 ≤ floor(**Yi**/4) - floor(**Xi**/4) ≤ 1 will hold. |
|
|
| 2. Given 2 distinct integers **Xi** and **Yi**, Mr. Fox will completely destroy the trail from clearing **Xi** to clearing **Yi** (which is guaranteed to exist at the start of the **i**th day). Note that, once such a trail is destroyed, a new trail from **Xi** to **Yi** may be created in the future. |
|
|
| 3. Given 2 distinct integers **Xi** and **Yi**, Mr. Fox will take a "random stroll" starting at clearing **Xi**, and would like to determine the probability that he'll visit clearing **Yi** at least once during it. |
|
|
| A "random stroll" consists of repeating the following process potentially |
| infinitely: If Mr. Fox is currently in some clearing **c**, and there are no |
| outgoing trails from **c**, then the stroll ends immediately. Otherwise, he'll |
| consider all of the rocks on all of the outgoing trails from **c**, choose one |
| of these rocks uniformly at random, follow the trail on which that rock lies |
| to its destination clearing (without removing any rocks), and repeat the |
| process from his new clearing. |
|
|
| For each event of type 3, output the requested probability. |
|
|
| ### Input |
|
|
| Input begins with an integer **T**, the number of test cases. For each test |
| case, there is first a line containing the space-separated integers **N**, |
| **P**, and **D**. |
|
|
| Then, **P** lines follow, the **i**th of which contains the space-separated |
| integers **Ai**, **Bi**, and **Ri**. |
|
|
| Then, **D** lines follow, the **i**th of which contains the space-separated |
| integers **Ei**, **Xi**, and **Yi**. If **Ei** = 1, this line additionally |
| contains the integer **Zi**. |
|
|
| ### Output |
|
|
| For the **i**th test case, print a line containing "Case #**i**: " followed by |
| the computed probabilities for each stroll that Mr. Fox takes. These |
| probabilities should be space-separated, and rounded to 6 decimal places. |
|
|
| Absolute errors of up to 2 * 10-6 will be ignored. |
|
|
| ### Constraints |
|
|
| 1 ≤ **T** ≤ 20 |
| 1 ≤ **N** ≤ 50,000 |
| 0 ≤ **P** ≤ 100,000 |
| 1 ≤ **D** ≤ 20,000 |
| 0 ≤ **Ai**, **Bi**, **Xi**, **Yi** < **N** |
| 1 ≤ **Ei** ≤ 3 |
| 1 ≤ **Ri**, **Zi** ≤ 5 |
|
|
| ### Explanation of Sample |
|
|
| In the first test case, Mr. Fox does multiple strolls from clearing 0 while |
| looking out for clearing 3. His first stroll has probability 1 as he must |
| always end up at clearing 3. His second stroll has probability 1/2 as there's |
| a 50% chance he gets stuck in clearing 4. His third stroll has probability 2/3 |
| as he now only goes to clearing 4 1/3 of the time. His fourth stroll has |
| probability 1/3 as he gets stuck in clearing 4 1/3 of the time, and in |
| clearing 1 1/3 of the time. |
|
|
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|
