tests and docs

explanations/Min_Binary_Heap.md

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+# Explanations by AI using my code as example
+
+## Min Binary Heap
+
+> Source: `trees/heap/minHeap.go`
+
+A binary heap is a complete binary tree flattened into a single slice
+(`MinHeap.Array`). No node pointers — the tree shape lives in the index math.
+
+### The invariant
+
+Every parent is **≤** both its children. The smallest value therefore always sits
+at the root (`Array[0]`). This is a *partial* order: siblings are unordered, so the
+array is "loosely sorted", which is cheaper to maintain than a fully sorted array.
+
+### Index math (implicit tree)
+
+For a node at index `i`:
+
+| Relation     | Index       |
+| ------------ | ----------- |
+| left child   | `2*i + 1`   |
+| right child  | `2*i + 2`   |
+| parent       | `(i-1) / 2` |
+
+### The two repair operations
+
+Both walk one root-to-leaf path, so both are **O(log n)**.
+
+- **`siftUp(i)`** — used by `Insert`. A value that may be *too small* for its position
+  bubbles **up**, swapping with its parent while it's smaller than the parent.
+- **`siftDown(i)`** — used by `PopMin` and `Heapify`. A value that may be *too large*
+  sinks **down**, repeatedly swapping with its **smallest** child until both children
+  are ≥ it (or it hits a leaf).
+
+### Methods
+
+| Method       | What it does                                                    | Cost         |
+| ------------ | -------------------------------------------------------------- | ------------ |
+| `NewMinHeap` | returns an empty heap                                           | O(1)         |
+| `Insert`     | appends to the end, then `siftUp` to restore the invariant     | O(log n)     |
+| `PopMin`     | returns the root; see the swap-and-sink dance below            | O(log n)     |
+| `Heapify`    | bulk-builds a heap from an arbitrary slice (empty heap only)   | O(n)         |
+
+### PopMin, step by step
+
+You can't just delete `Array[0]` — that leaves a hole. Instead:
+
+1. Save the root (`Array[0]`) as the return value.
+2. Move the **last** element into the root slot.
+3. Zero the old last slot and shrink the slice by one (order matters — clear the slot
+   *before* reslicing, or the index is out of range).
+4. `siftDown(0)` to push that moved-up value back to its rightful depth.
+
+Returns `(zero, false)` when the heap is empty.
+
+> The zeroing (`Array[last] = zero`) only matters when `T` holds a pointer
+> (e.g. `string`, which `cmp.Ordered` allows) — it lets the GC reclaim the dropped
+> element instead of keeping it alive in the backing array.
+
+### Heapify: why it's O(n), not O(n log n)
+
+Inserting `n` items one by one would be O(n log n). `Heapify` is faster: it copies the
+slice, then calls `siftDown` on every **non-leaf** node, from the last parent up to the
+root:
+
+```go
+for i := len(h.Array)/2 - 1; i >= 0; i-- {
+    h.siftDown(i)
+}
+```
+
+Starting at `len/2 - 1` skips the leaves (they're already trivially valid heaps of
+size 1). Working bottom-up means each `siftDown` sinks into subtrees that are *already*
+valid heaps. Most nodes are near the bottom and barely move, so the total work sums to
+O(n), not O(n log n).
+
+It refuses to run on a non-empty heap (returns an error) to avoid clobbering existing
+data.
+
+### Summary
+
+- **Array + index math** = a tree with no pointers and cache-friendly memory.
+- **Invariant (parent ≤ children)** = the min is always `Array[0]`, peek is O(1).
+- **`siftUp` / `siftDown`** = the O(log n) repairs that keep the invariant after an
+  insert or a pop.
+- **`Heapify`** = build the whole heap in O(n) by sinking non-leaves bottom-up.