Receiver Operating Characteristic Curve Area Computation

In the evaluation of binary classification models, the Receiver Operating Characteristic (ROC) Curve provides a comprehensive visualization of a classifier's performance across all possible decision thresholds. The area under this curve (AUC) distills this rich information into a single scalar value that quantifies the classifier's ability to discriminate between positive and negative classes.

The ROC curve is constructed by plotting the True Positive Rate (TPR) against the False Positive Rate (FPR) at various threshold settings:

$$TPR = \frac{\text{True Positives}}{\text{True Positives} + \text{False Negatives}}$$

$$FPR = \frac{\text{False Positives}}{\text{False Positives} + \text{True Negatives}}$$

Interpreting the ROC-AUC Score:

• AUC = 1.0: Perfect classifier that achieves 100% true positive rate with 0% false positives
• AUC = 0.5: Random classifier that provides no discriminative power (equivalent to random guessing)
• AUC < 0.5: Classifier performing worse than random (predictions are inversely correlated with actual labels)
• AUC = 0.0: Perfectly inverse classifier (always predicts the opposite of the true label)

Algorithm Overview: To compute the AUC, you must:

1. Sort the samples by their predicted scores in descending order
2. Traverse the sorted predictions, computing TPR and FPR at each unique threshold
3. Calculate the area under the resulting ROC curve using numerical integration (e.g., trapezoidal rule)

Your Task: Implement a function that takes binary ground truth labels and corresponding prediction probability scores, then returns the Area Under the ROC Curve. If all labels belong to a single class (all 0s or all 1s), return 0.0 since the ROC curve is undefined in this degenerate case.