149 lines
3.8 KiB
Markdown
149 lines
3.8 KiB
Markdown
---
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title: 最小送金回数で精算する割り勘アルゴリズム
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date: 2021-02-14
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---
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大人数でキャンプを楽しんだあとに待っているのは耐え難き送金処理です。
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次回から楽をするために、送金回数を最小化する制約で精算表を作る方法を考えてみます。
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# tl;dr
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アイディアは「最も支払わなかった人が最も支払った人に払えるだけ払う ⇢ 債権を再計算して繰り返す」です。
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1. 全員の出費を算出（払い過ぎは正、払わなさすぎは負の数）
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2. 降順でソート（出費過多が先頭）
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3. リストの最後（最大債務者, 出費=L）がリストの最初（最大債権者, F）に $\min(F, |L|)$ を支払ってバランスを再計算
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4. 全員のバランスが 0 になるまで 2-3 を繰り返す
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# 実験
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実際にコードを書いて本当に望んでいる結果が得られるのかを検証します。
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```5js
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const history = [
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{
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amount: 121,
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payer: "A",
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involves: ["A", "B", "C"],
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},
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{
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amount: 98,
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payer: "B",
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involves: ["A", "B", "C"],
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},
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{
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amount: 10,
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payer: "C",
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involves: ["A", "B", "C"],
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},
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{
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amount: 10,
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payer: "C",
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involves: ["A", "B"],
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},
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{
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amount: 50,
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payer: "C",
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involves: ["A"], // meaning C lent A 50
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},
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];
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// calculate balance sheet
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const init = { balance: 0, consumption: 0 };
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Map.prototype.fetch = function (id) {
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return (
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this.get(id) || this.set(id, Object.assign({ name: id }, init)).get(id)
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);
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};
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const data = new Map();
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for (const { payer, amount, involves } of history) {
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const record = data.fetch(payer);
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record.balance += amount;
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const dept = Math.ceil(amount / involves.length);
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// actual payer should not owe extra dept coming from rounded up numbers
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const payerDept = amount - dept * (involves.length - 1);
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for (const deptor of involves.map((i) => data.fetch(i))) {
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const cost = Math.round(amount / involves.length);
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deptor.balance -= cost;
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deptor.consumption += cost;
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}
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}
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console.log(data);
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// calculate transaction table
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const transaction = [];
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let paidTooMuch, paidLess;
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while (true) {
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for (const [_, tbl] of data) {
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if (tbl.balance >= (paidTooMuch?.balance || 0)) {
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paidTooMuch = tbl;
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}
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if (tbl.balance <= (paidLess?.balance || 0)) {
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paidLess = tbl;
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}
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}
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if (paidLess.balance == 0 || paidTooMuch.balance == 0) break;
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const amount = Math.min(paidTooMuch.balance, Math.abs(paidLess.balance));
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transaction.push({
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sender: paidLess.name,
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receiver: paidTooMuch.name,
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amount,
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});
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paidTooMuch.balance -= amount;
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paidLess.balance += amount;
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}
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console.log("Settled");
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console.log("\n# Transaction table");
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for (const ev of transaction) {
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console.log(`${ev.sender} owes ${ev.receiver} ¥${ev.amount}`);
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}
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console.log("\n# History");
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for (const { payer, amount, involves } of history) {
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if (involves.length === 1) {
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console.log(`${payer} lent ¥${amount} to ${involves[0]}`);
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} else {
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console.log(`${payer} paid ¥${amount} for ${involves.join(", ")}`);
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}
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}
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console.log("\n# Expenses");
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for (const [_, { name, consumption }] of data) {
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console.log(`${name} virtually paid ¥${consumption} in total`);
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}
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```
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`history`に支払い履歴を書き込んでから実行すると、「送金表」「履歴」「実質支払総額」が得られます。
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```md
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# Transaction table
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A owes B ¥10
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C owes B ¥6
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# History
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A paid ¥121 for A, B, C
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B paid ¥98 for A, B, C
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C paid ¥10 for A, B, C
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C paid ¥10 for A, B
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C lent ¥50 to A
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# Expenses
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A virtually paid ¥131 in total
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B virtually paid ¥81 in total
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C virtually paid ¥76 in total
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```
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プログラムに落とし込めたら、あとはアプリを作るなりスプレッドシートのマクロにするなり自由です。面倒なことは全部コンピューターにやらせよう！
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