Perform ( |V| ) iterations of Bellman-Ford; save nodes relaxed on the ( V )-th iteration to set ( A ).

Place nodes from ( A ) in queue ( Q ).

Use BFS with queue ( Q ) to find nodes reachable from ( A ). These nodes indicate possible infinite arbitrage.

Reconstruct Infinite Arbitrage

During BFS, track the parent of each visited node.
Reconstruct the path to ( u ) from node ( w ) relaxed on iteration ( V ).
Trace back from ( w ) to find the negative cycle reachable from ( w ).
Use this cycle to achieve infinite arbitrage from ( S ) to ( u ).

Steps:

We can take following steps:

Log Transformation:
- Convert exchange rates to log values and negate them. This way, we transform the problem into finding negative cycles.
Bellman-Ford Algorithm:
- The Bellman-Ford algorithm is used to find the shortest path from a single source to all other vertices in a graph. By performing up to (V-1) relaxations (where (V) is the number of vertices), we ensure the shortest paths are correctly calculated. If further relaxations are possible, it indicates a negative cycle (an arbitrage opportunity).
Checking for Negative Cycles:
- After running the (V-1) relaxations, if you can still relax any edge, it confirms the presence of a negative-weight cycle.

Code

[{"language":"Java","code":"public class ArbitrageDetection {\n\n\tpublic boolean detectArbitrage(double[][] exchangeRates) {\n\t\tint n = exchangeRates.length;\n\t\tdouble[][] logRates = new double[n][n];\n\n\t\t// Step 1: Transform the exchange rates to a log graph\n\t\tfor (int i = 0; i < n; i++) {\n\t\t\tfor (int j = 0; j < n; j++) {\n\t\t\t\tlogRates[i][j] = -Math.log(exchangeRates[i][j]);\n\t\t\t}\n\t\t}\n\n\t\t// Step 2: Use Bellman-Ford to detect negative weight cycles\n\t\tfor (int source = 0; source < n; source++) {\n\t\t\tif (hasNegativeCycle(logRates, source)) {\n\t\t\t\treturn true;\n\t\t\t}\n\t\t}\n\n\t\treturn false;\n\t}\n\n\tprivate boolean hasNegativeCycle(double[][] graph, int source) {\n\t\tint n = graph.length;\n\t\tdouble[] dist = new double[n];\n\t\tArrays.fill(dist, Double.POSITIVE_INFINITY);\n\t\tdist[source] = 0;\n\n\t\t// Step 3: Relax edges up to n-1 times\n\t\tfor (int i = 0; i < n - 1; i++) {\n\t\t\tfor (int u = 0; u < n; u++) {\n\t\t\t\tfor (int v = 0; v < n; v++) {\n\t\t\t\t\tif (dist[u] != Double.POSITIVE_INFINITY && dist[u] + graph[u][v] < dist[v]) {\n\t\t\t\t\t\tdist[v] = dist[u] + graph[u][v];\n\t\t\t\t\t}\n\t\t\t\t}\n\t\t\t}\n\t\t}\n\n\t\t// Step 4: Check for negative cycles\n\t\tfor (int u = 0; u < n; u++) {\n\t\t\tfor (int v = 0; v < n; v++) {\n\t\t\t\tif (dist[u] != Double.POSITIVE_INFINITY && dist[u] + graph[u][v] < dist[v]) {\n\t\t\t\t\treturn true;\n\t\t\t\t}\n\t\t\t}\n\t\t}\n\n\t\treturn false;\n\t}\n}","lang":"java","html":"<pre class=\"shiki shiki-themes github-light github-dark\" style=\"background-color:#fff;--shiki-dark-bg:#24292e;color:#24292e;--shiki-dark:#e1e4e8\" tabindex=\"0\"><code><span class=\"line\"><span style=\"color:#D73A49;--shiki-dark:#F97583\">public</span><span style=\"color:#D73A49;--shiki-dark:#F97583\"> class</span><span style=\"color:#6F42C1;--shiki-dark:#B392F0\"> ArbitrageDetection</span><span style=\"color:#24292E;--shiki-dark:#E1E4E8\"> {</span></span>\n<span class=\"line\"></span>\n<span class=\"line\"><span style=\"color:#D73A49;--shiki-dark:#F97583\">\tpublic</span><span style=\"color:#D73A49;--shiki-dark:#F97583\"> boolean</span><span style=\"color:#6F42C1;--shiki-dark:#B392F0\"> detectArbitrage</span><span style=\"color:#24292E;--shiki-dark:#E1E4E8\">(</span><span style=\"color:#D73A49;--shiki-dark:#F97583\">double</span><span s

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