Alpha-Beta Pruning

Alpha-beta pruning gives the same answer as minimax while skipping branches that cannot change the result. It tracks two bounds: alpha, the best score MAX is already sure of, and beta, the best score MIN is already sure of.

The game tree is complete: every internal node has branching children, and all leaves are at the same depth. leaves lists the leaf scores from left to right. The root is a MAX node and the levels alternate MAX, MIN, MAX, …

Search children from left to right, starting with alpha = -∞ and beta = +∞:

At a MAX node, after each child: best = max(best, value), then alpha = max(alpha, best).
At a MIN node, after each child: best = min(best, value), then beta = min(beta, best).
As soon as alpha ≥ beta, skip the node's remaining children.

Implement alpha_beta(leaves, branching).

Input

leaves: The leaf scores, left to right; the length is a power of branching.

branching: Children per internal node.

Output

Return [value, evaluated]: the minimax value of the root and how many leaves were evaluated.

Constraints

2 ≤ branching ≤ 5, length of leaves = branching^depth with 1 ≤ depth ≤ 8, at most 10,000 leaves

-1,000 ≤ each leaf ≤ 1,000

Examples

Example 1

Input:
leaves = [3, 12, 8, 2, 4, 6, 14, 5, 2]
branching = 3

Expected output:
[3, 7]

Explanation

The first MIN node sees 3, 12, 8 and returns 3, so alpha = 3. The second MIN node sees 2: beta = 2 ≤ alpha, so 4 and 6 are skipped. The third sees 14, 5, 2 and returns 2. The root value is max(3, 2, 2) = 3 after 7 of the 9 leaves.

Example 2

Input:
leaves = [5, 6, 7, 4]
branching = 2

Expected output:
[5, 4]

Explanation

The left MIN node evaluates 5 and 6 and returns 5, so alpha = 5. The right MIN node evaluates 7 (beta = 7), then 4 (beta = 4 ≤ 5). All 4 leaves are evaluated and the value is 5.

References

Run the sample tests to check your code without submitting.

Example 1 Not run

Input:

leaves = [3, 12, 8, 2, 4, 6, 14, 5, 2]
branching = 3

Expected output:

[3, 7]

Your output

Example 2 Not run
Input:
```
leaves = [5, 6, 7, 4]
branching = 2
```
Expected output:
```
[5, 4]
```
Your output

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