Optimal Account Balancing 最优账户平衡

A group of friends went on holiday and sometimes lent each other money. For example, Alice paid for Bill's lunch for 10.ThenlaterChrisgaveAlice10.ThenlaterChrisgaveAlice5 for a taxi ride. We can model each transaction as a tuple (x, y, z) which means person x gave person y $z. Assuming Alice, Bill, and Chris are person 0, 1, and 2 respectively (0, 1, 2 are the person's ID), the transactions can be represented as [[0, 1, 10], [2, 0, 5]].

Given a list of transactions between a group of people, return the minimum number of transactions required to settle the debt.

Note:

  1. A transaction will be given as a tuple (x, y, z). Note that x ≠ y and z > 0.
  2. Person's IDs may not be linear, e.g. we could have the persons 0, 1, 2 or we could also have the persons 0, 2, 6.

 

Example 1:

Input:
[[0,1,10], [2,0,5]]

Output:
2

Explanation:
Person #0 gave person #1 $10.
Person #2 gave person #0 $5.

Two transactions are needed. One way to settle the debt is person #1 pays person #0 and #2 $5 each.

 

Example 2:

Input:
[[0,1,10], [1,0,1], [1,2,5], [2,0,5]]

Output:
1

Explanation:
Person #0 gave person #1 $10.
Person #1 gave person #0 $1.
Person #1 gave person #2 $5.
Person #2 gave person #0 $5.

Therefore, person #1 only need to give person #0 $4, and all debt is settled.

 

class Solution {
public:
    int minTransfers(vector<vector<int>>& transactions) {
        unordered_map<int, int> m;
        for (auto t : transactions) {
            m[t[0]] -= t[2];
            m[t[1]] += t[2];
        }
        vector<int> accnt(m.size());
        int cnt = 0;
        for (auto a : m) {
            if (a.second != 0) accnt[cnt++] = a.second;
        }
        return helper(accnt, 0, cnt, 0);
    }
    int helper(vector<int>& accnt, int start, int n, int num) {
        int res = INT_MAX;
        while (start < n && accnt[start] == 0) ++start;
        for (int i = start + 1; i < n; ++i) {
            if ((accnt[i] < 0 && accnt[start] > 0) || (accnt[i] > 0 && accnt[start] < 0)) {
                accnt[i] += accnt[start];
                res = min(res, helper(accnt, start + 1, n, num + 1));
                accnt[i] -= accnt[start];
            }
        }
        return res == INT_MAX ? num : res;
    }
};

 

转载于:https://www.cnblogs.com/jxr041100/p/8433967.html

### 最优传输理论概述 最优传输(Optimal Transport, OT)是一种用于衡量概率分布之间距离的方法,在计算机科学和机器学习领域有着广泛应用。该理论旨在找到一种方法,使得在一个成本函数下,将一个分布转换成另一个分布的成本小化[^1]。 OT问题的经典形式可以描述为:给定两个离散的概率分布 \(\mu\) 和 \(\nu\) ,以及定义在这两组点之间的运输成本矩阵 \(C\) 。目标是在所有可能的联合分布中寻找使期望运输成本小化的那个联合分布 \(\pi^*\)[^2]。 ```python import numpy as np from ot import emd2 # 定义源分布和支持集 a = [.7, .3] # 源分布权重向量 b = [.4, .6] # 目标分布权重向量 M = [[0., 1.], [1., 0.]] # 成本矩阵 # 计算Wasserstein距离平方 w_dist_sq = emd2(a, b, M) print(f"Wasserstein distance squared: {w_dist_sq}") ``` ### 应用场景 #### 图像处理生成对抗网络 (GANs) 在图像处理方面,通过引入基于OT的距离度量,能够更有效地训练生成模型。具体来说,利用Wassertein GAN(W-GAN),即采用Wassertein距离作为损失函数来替代传统的Jensen-Shannon散度,从而解决了模式崩溃等问题并提高了样本质量[^3]。 #### 自然语言处理(NLP) 对于NLP任务而言,词嵌入空间中的句子表示可以通过OT技术实现更好的语义相似性匹配。例如,Sinkhorn算法被用来计算不同文档间词语频率直方图间的Earth Mover's Distance(EMD), 进而评估它们的内容差异程度[^4]。 #### 数据增强迁移学习 当面临数据不足的情况时,可以从已有的大规模标注数据集中抽取特征,并借助OT框架将其适配到新的未标记或少量标签的数据上;这有助于提高下游分类器性能的同时减少人工标注工作量[^5]。
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