J. Jenny and the Batteries

Codeforces

IDCF10162J

Time2000ms

Memory64MB

Difficulty—

English · Original

Formal · Original

Jenny works at BBB (Battery Bank of Brazil). She's in charge of optimizing the batteries' displacements, but can't solve the problem. Knowing you are a programmer, she asked for your help. At BBB, the batteries are arranged in piles. In a bank, the batteries can't be discharged. So, when recharging then, the bank spends M energy units, where M is the size of the greatest pile. Each pile initially has ai batteries. There is a machine the can move a battery from pile i to pile j with cost bi + cj. Jenny has a limited budget of K for moving the batteries and the bank wants the minimum M possible at the end. The first line of input consists of the integers N and K (1 ≤ N ≤ 105, 1 ≤ K ≤ 1018). The following N lines contain ai, bi and ci (0 ≤ ai, bi, ci ≤ 106). Print the minumum M possible. ## Input The first line of input consists of the integers N and K (1 ≤ N ≤ 105, 1 ≤ K ≤ 1018). The following N lines contain ai, bi and ci (0 ≤ ai, bi, ci ≤ 106). ## Output Print the minumum M possible. [samples]

**Definitions** Let $ N \in \mathbb{Z}^+ $ be the number of piles. Let $ K \in \mathbb{Z}^+ $ be the budget for battery movements. For each pile $ i \in \{1, \dots, N\} $: - $ a_i \in \mathbb{Z}_{\geq 0} $: initial number of batteries. - $ b_i \in \mathbb{Z}_{\geq 0} $: cost to remove one battery from pile $ i $. - $ c_i \in \mathbb{Z}_{\geq 0} $: cost to add one battery to pile $ i $. Let $ x_i \in \mathbb{Z}_{\geq 0} $ be the final number of batteries in pile $ i $. Let $ M = \max_{i} x_i $ be the maximum pile size after movements. **Constraints** 1. $ \sum_{i=1}^N x_i = \sum_{i=1}^N a_i $ (total batteries conserved). 2. The total movement cost satisfies: $$ \sum_{i=1}^N b_i (a_i - x_i)^+ + \sum_{i=1}^N c_i (x_i - a_i)^+ \leq K $$ where $ (z)^+ = \max(0, z) $. 3. $ x_i \geq 0 $ for all $ i $. **Objective** Minimize $ M = \max_{i} x_i $.

API Response (JSON)

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