Classic arbitrage detection but the transaction cost modifier threw me for a second.
Model the exchange rates as a directed graph where edge weights are the negative logarithms of the rates, so that finding a profitable cycle reduces to detecting a negative-weight cycle. Apply the 0.01% transaction cost by adjusting each edge weight (e.g., subtract log(0.9999)) and then run the Bellman-Ford algorithm to check for negative cycles.
Pro tip: Mention that you would first clarify whether the transaction cost is applied per trade or per completed cycle, and whether the cycle must return to the starting currency. This shows attention to detail and avoids incorrect assumptions.
Confirm the exact meaning of the transaction cost (e.g., 0.01% deducted once per cycle) and whether the cycle must start and end with the same currency. Also check if the matrix is guaranteed to be consistent (e.g., R[i][j] * R[j][i] <= 1).
Convert each exchange rate R[i][j] to an edge weight w(i,j) = -log(R[i][j]). A profitable cycle exists if the sum of weights around a cycle is negative after accounting for the transaction cost.
Adjust each edge weight by adding -log(1 - 0.0001) (since 0.01% cost reduces the effective rate). Alternatively, subtract log(0.9999) from each edge weight to reflect the cost per trade, but ensure it's applied once per cycle.
Run the Bellman-Ford algorithm from any node (or a super-source connected to all nodes) to detect a negative-weight cycle. If found, a profitable arbitrage opportunity exists.
Discuss time complexity O(K^3) and space O(K^2). Mention alternative approaches like Floyd-Warshall for all-pairs shortest paths, but note Bellman-Ford is sufficient for cycle detection.
AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.