{ "slug": "heap-sort", "title": "Heap Sort", "category": "Sorting", "difficulty": "hard", "complexity": { "time": "O(n log n)", "space": "O(1)" }, "idea": "Treat the array as a binary heap (children of i live at 2i+1 and 2i+2). Build a max-heap so the biggest value is at the front, swap it to the end, shrink the heap, and sift the new root down. Repeat.", "code": [ "build: for i = n/2-1 down to 0: siftDown(i, n)", "for end = n-1 down to 1:", " swap(0, end) # max to the end", " siftDown(0, end) # restore the heap", "siftDown(i, size): sink a[i] past its larger child" ], "example": { "array": [ 4, 10, 3, 5, 1, 8, 2, 7 ] }, "invariant": "Each step is the real data structure at that step — the picture is a pure function of it.", "stepCount": 37, "steps": [ { "i": 0, "op": "say", "caption": "Heap sort: first arrange the array into a max-heap.", "note": null, "codeLine": 1, "stats": null, "array": [ 4, 10, 3, 5, 1, 8, 2, 7 ], "focus": [ null, null, null, null, null, null, null, null ], "markers": [], "regions": [] }, { "i": 1, "op": "compare", "caption": "Parent 5 vs larger child 7.", "note": null, "codeLine": 5, "stats": null, "array": [ 4, 10, 3, 5, 1, 8, 2, 7 ], "focus": [ null, null, null, "compare", null, null, null, "compare" ], "markers": [ { "name": "node", "index": 3, "tone": "pivot" } ], "regions": [ { "label": "heap", "from": 0, "to": 7, "tone": "window" } ] }, { "i": 2, "op": "swap", "caption": "Parent is smaller — sink it down.", "note": null, "codeLine": 5, "stats": null, "array": [ 4, 10, 3, 7, 1, 8, 2, 5 ], "focus": [ null, null, null, "swap", null, null, null, "swap" ], "markers": [ { "name": "node", "index": 3, "tone": "pivot" } ], "regions": [ { "label": "heap", "from": 0, "to": 7, "tone": "window" } ] }, { "i": 3, "op": "compare", "caption": "Parent 3 vs larger child 8.", "note": null, "codeLine": 5, "stats": null, "array": [ 4, 10, 3, 7, 1, 8, 2, 5 ], "focus": [ null, null, "compare", null, null, "compare", null, null ], "markers": [ { "name": "node", "index": 2, "tone": "pivot" } ], "regions": [ { "label": "heap", "from": 0, "to": 7, "tone": "window" } ] }, { "i": 4, "op": "swap", "caption": "Parent is smaller — sink it down.", "note": null, "codeLine": 5, "stats": null, "array": [ 4, 10, 8, 7, 1, 3, 2, 5 ], "focus": [ null, null, "swap", null, null, "swap", null, null ], "markers": [ { "name": "node", "index": 2, "tone": "pivot" } ], "regions": [ { "label": "heap", "from": 0, "to": 7, "tone": "window" } ] }, { "i": 5, "op": "compare", "caption": "Parent 10 vs larger child 7.", "note": null, "codeLine": 5, "stats": null, "array": [ 4, 10, 8, 7, 1, 3, 2, 5 ], "focus": [ null, "compare", null, "compare", null, null, null, null ], "markers": [ { "name": "node", "index": 1, "tone": "pivot" } ], "regions": [ { "label": "heap", "from": 0, "to": 7, "tone": "window" } ] }, { "i": 6, "op": "compare", "caption": "Parent 4 vs larger child 10.", "note": null, "codeLine": 5, "stats": null, "array": [ 4, 10, 8, 7, 1, 3, 2, 5 ], "focus": [ "compare", "compare", null, null, null, null, null, null ], "markers": [ { "name": "node", "index": 0, "tone": "pivot" } ], "regions": [ { "label": "heap", "from": 0, "to": 7, "tone": "window" } ] }, { "i": 7, "op": "swap", "caption": "Parent is smaller — sink it down.", "note": null, "codeLine": 5, "stats": null, "array": [ 10, 4, 8, 7, 1, 3, 2, 5 ], "focus": [ "swap", "swap", null, null, null, null, null, null ], "markers": [ { "name": "node", "index": 0, "tone": "pivot" } ], "regions": [ { "label": "heap", "from": 0, "to": 7, "tone": "window" } ] }, { "i": 8, "op": "compare", "caption": "Parent 4 vs larger child 7.", "note": null, "codeLine": 5, "stats": null, "array": [ 10, 4, 8, 7, 1, 3, 2, 5 ], "focus": [ null, "compare", null, "compare", null, null, null, null ], "markers": [ { "name": "node", "index": 1, "tone": "pivot" } ], "regions": [ { "label": "heap", "from": 0, "to": 7, "tone": "window" } ] }, { "i": 9, "op": "swap", "caption": "Parent is smaller — sink it down.", "note": null, "codeLine": 5, "stats": null, "array": [ 10, 7, 8, 4, 1, 3, 2, 5 ], "focus": [ null, "swap", null, "swap", null, null, null, null ], "markers": [ { "name": "node", "index": 1, "tone": "pivot" } ], "regions": [ { "label": "heap", "from": 0, "to": 7, "tone": "window" } ] }, { "i": 10, "op": "compare", "caption": "Parent 4 vs larger child 5.", "note": null, "codeLine": 5, "stats": null, "array": [ 10, 7, 8, 4, 1, 3, 2, 5 ], "focus": [ null, null, null, "compare", null, null, null, "compare" ], "markers": [ { "name": "node", "index": 3, "tone": "pivot" } ], "regions": [ { "label": "heap", "from": 0, "to": 7, "tone": "window" } ] }, { "i": 11, "op": "swap", "caption": "Parent is smaller — sink it down.", "note": null, "codeLine": 5, "stats": null, "array": [ 10, 7, 8, 5, 1, 3, 2, 4 ], "focus": [ null, null, null, "swap", null, null, null, "swap" ], "markers": [ { "name": "node", "index": 3, "tone": "pivot" } ], "regions": [ { "label": "heap", "from": 0, "to": 7, "tone": "window" } ] }, { "i": 12, "op": "say", "caption": "Max-heap ready — the largest value sits at the front.", "note": null, "codeLine": 2, "stats": null, "array": [ 10, 7, 8, 5, 1, 3, 2, 4 ], "focus": [ null, null, null, null, null, null, null, null ], "markers": [], "regions": [ { "label": "heap", "from": 0, "to": 7, "tone": "window" } ] }, { "i": 13, "op": "swap", "caption": "Move the max 10 to index 7.", "note": null, "codeLine": 3, "stats": null, "array": [ 4, 7, 8, 5, 1, 3, 2, 10 ], "focus": [ "swap", null, null, null, null, null, null, "swap" ], "markers": [], "regions": [ { "label": "heap", "from": 0, "to": 7, "tone": "window" } ] }, { "i": 14, "op": "compare", "caption": "Parent 4 vs larger child 8.", "note": null, "codeLine": 5, "stats": null, "array": [ 4, 7, 8, 5, 1, 3, 2, 10 ], "focus": [ "compare", null, "compare", null, null, null, null, null ], "markers": [ { "name": "node", "index": 0, "tone": "pivot" } ], "regions": [ { "label": "heap", "from": 0, "to": 6, "tone": "window" }, { "label": "sorted", "from": 7, "to": 7, "tone": "settled" } ] }, { "i": 15, "op": "swap", "caption": "Parent is smaller — sink it down.", "note": null, "codeLine": 5, "stats": null, "array": [ 8, 7, 4, 5, 1, 3, 2, 10 ], "focus": [ "swap", null, "swap", null, null, null, null, null ], "markers": [ { "name": "node", "index": 0, "tone": "pivot" } ], "regions": [ { "label": "heap", "from": 0, "to": 6, "tone": "window" }, { "label": "sorted", "from": 7, "to": 7, "tone": "settled" } ] }, { "i": 16, "op": "compare", "caption": "Parent 4 vs larger child 3.", "note": null, "codeLine": 5, "stats": null, "array": [ 8, 7, 4, 5, 1, 3, 2, 10 ], "focus": [ null, null, "compare", null, null, "compare", null, null ], "markers": [ { "name": "node", "index": 2, "tone": "pivot" } ], "regions": [ { "label": "heap", "from": 0, "to": 6, "tone": "window" }, { "label": "sorted", "from": 7, "to": 7, "tone": "settled" } ] }, { "i": 17, "op": "swap", "caption": "Move the max 8 to index 6.", "note": null, "codeLine": 3, "stats": null, "array": [ 2, 7, 4, 5, 1, 3, 8, 10 ], "focus": [ "swap", null, null, null, null, null, "swap", null ], "markers": [ { "name": "node", "index": 2, "tone": "pivot" } ], "regions": [ { "label": "heap", "from": 0, "to": 6, "tone": "window" }, { "label": "sorted", "from": 7, "to": 7, "tone": "settled" } ] }, { "i": 18, "op": "compare", "caption": "Parent 2 vs larger child 7.", "note": null, "codeLine": 5, "stats": null, "array": [ 2, 7, 4, 5, 1, 3, 8, 10 ], "focus": [ "compare", "compare", null, null, null, null, null, null ], "markers": [ { "name": "node", "index": 0, "tone": "pivot" } ], "regions": [ { "label": "heap", "from": 0, "to": 5, "tone": "window" }, { "label": "sorted", "from": 6, "to": 7, "tone": "settled" } ] }, { "i": 19, "op": "swap", "caption": "Parent is smaller — sink it down.", "note": null, "codeLine": 5, "stats": null, "array": [ 7, 2, 4, 5, 1, 3, 8, 10 ], "focus": [ "swap", "swap", null, null, null, null, null, null ], "markers": [ { "name": "node", "index": 0, "tone": "pivot" } ], "regions": [ { "label": "heap", "from": 0, "to": 5, "tone": "window" }, { "label": "sorted", "from": 6, "to": 7, "tone": "settled" } ] }, { "i": 20, "op": "compare", "caption": "Parent 2 vs larger child 5.", "note": null, "codeLine": 5, "stats": null, "array": [ 7, 2, 4, 5, 1, 3, 8, 10 ], "focus": [ null, "compare", null, "compare", null, null, null, null ], "markers": [ { "name": "node", "index": 1, "tone": "pivot" } ], "regions": [ { "label": "heap", "from": 0, "to": 5, "tone": "window" }, { "label": "sorted", "from": 6, "to": 7, "tone": "settled" } ] }, { "i": 21, "op": "swap", "caption": "Parent is smaller — sink it down.", "note": null, "codeLine": 5, "stats": null, "array": [ 7, 5, 4, 2, 1, 3, 8, 10 ], "focus": [ null, "swap", null, "swap", null, null, null, null ], "markers": [ { "name": "node", "index": 1, "tone": "pivot" } ], "regions": [ { "label": "heap", "from": 0, "to": 5, "tone": "window" }, { "label": "sorted", "from": 6, "to": 7, "tone": "settled" } ] }, { "i": 22, "op": "swap", "caption": "Move the max 7 to index 5.", "note": null, "codeLine": 3, "stats": null, "array": [ 3, 5, 4, 2, 1, 7, 8, 10 ], "focus": [ "swap", null, null, null, null, "swap", null, null ], "markers": [ { "name": "node", "index": 1, "tone": "pivot" } ], "regions": [ { "label": "heap", "from": 0, "to": 5, "tone": "window" }, { "label": "sorted", "from": 6, "to": 7, "tone": "settled" } ] }, { "i": 23, "op": "compare", "caption": "Parent 3 vs larger child 5.", "note": null, "codeLine": 5, "stats": null, "array": [ 3, 5, 4, 2, 1, 7, 8, 10 ], "focus": [ "compare", "compare", null, null, null, null, null, null ], "markers": [ { "name": "node", "index": 0, "tone": "pivot" } ], "regions": [ { "label": "heap", "from": 0, "to": 4, "tone": "window" }, { "label": "sorted", "from": 5, "to": 7, "tone": "settled" } ] }, { "i": 24, "op": "swap", "caption": "Parent is smaller — sink it down.", "note": null, "codeLine": 5, "stats": null, "array": [ 5, 3, 4, 2, 1, 7, 8, 10 ], "focus": [ "swap", "swap", null, null, null, null, null, null ], "markers": [ { "name": "node", "index": 0, "tone": "pivot" } ], "regions": [ { "label": "heap", "from": 0, "to": 4, "tone": "window" }, { "label": "sorted", "from": 5, "to": 7, "tone": "settled" } ] }, { "i": 25, "op": "compare", "caption": "Parent 3 vs larger child 2.", "note": null, "codeLine": 5, "stats": null, "array": [ 5, 3, 4, 2, 1, 7, 8, 10 ], "focus": [ null, "compare", null, "compare", null, null, null, null ], "markers": [ { "name": "node", "index": 1, "tone": "pivot" } ], "regions": [ { "label": "heap", "from": 0, "to": 4, "tone": "window" }, { "label": "sorted", "from": 5, "to": 7, "tone": "settled" } ] }, { "i": 26, "op": "swap", "caption": "Move the max 5 to index 4.", "note": null, "codeLine": 3, "stats": null, "array": [ 1, 3, 4, 2, 5, 7, 8, 10 ], "focus": [ "swap", null, null, null, "swap", null, null, null ], "markers": [ { "name": "node", "index": 1, "tone": "pivot" } ], "regions": [ { "label": "heap", "from": 0, "to": 4, "tone": "window" }, { "label": "sorted", "from": 5, "to": 7, "tone": "settled" } ] }, { "i": 27, "op": "compare", "caption": "Parent 1 vs larger child 4.", "note": null, "codeLine": 5, "stats": null, "array": [ 1, 3, 4, 2, 5, 7, 8, 10 ], "focus": [ "compare", null, "compare", null, null, null, null, null ], "markers": [ { "name": "node", "index": 0, "tone": "pivot" } ], "regions": [ { "label": "heap", "from": 0, "to": 3, "tone": "window" }, { "label": "sorted", "from": 4, "to": 7, "tone": "settled" } ] }, { "i": 28, "op": "swap", "caption": "Parent is smaller — sink it down.", "note": null, "codeLine": 5, "stats": null, "array": [ 4, 3, 1, 2, 5, 7, 8, 10 ], "focus": [ "swap", null, "swap", null, null, null, null, null ], "markers": [ { "name": "node", "index": 0, "tone": "pivot" } ], "regions": [ { "label": "heap", "from": 0, "to": 3, "tone": "window" }, { "label": "sorted", "from": 4, "to": 7, "tone": "settled" } ] }, { "i": 29, "op": "swap", "caption": "Move the max 4 to index 3.", "note": null, "codeLine": 3, "stats": null, "array": [ 2, 3, 1, 4, 5, 7, 8, 10 ], "focus": [ "swap", null, null, "swap", null, null, null, null ], "markers": [ { "name": "node", "index": 0, "tone": "pivot" } ], "regions": [ { "label": "heap", "from": 0, "to": 3, "tone": "window" }, { "label": "sorted", "from": 4, "to": 7, "tone": "settled" } ] }, { "i": 30, "op": "compare", "caption": "Parent 2 vs larger child 3.", "note": null, "codeLine": 5, "stats": null, "array": [ 2, 3, 1, 4, 5, 7, 8, 10 ], "focus": [ "compare", "compare", null, null, null, null, null, null ], "markers": [ { "name": "node", "index": 0, "tone": "pivot" } ], "regions": [ { "label": "heap", "from": 0, "to": 2, "tone": "window" }, { "label": "sorted", "from": 3, "to": 7, "tone": "settled" } ] }, { "i": 31, "op": "swap", "caption": "Parent is smaller — sink it down.", "note": null, "codeLine": 5, "stats": null, "array": [ 3, 2, 1, 4, 5, 7, 8, 10 ], "focus": [ "swap", "swap", null, null, null, null, null, null ], "markers": [ { "name": "node", "index": 0, "tone": "pivot" } ], "regions": [ { "label": "heap", "from": 0, "to": 2, "tone": "window" }, { "label": "sorted", "from": 3, "to": 7, "tone": "settled" } ] }, { "i": 32, "op": "swap", "caption": "Move the max 3 to index 2.", "note": null, "codeLine": 3, "stats": null, "array": [ 1, 2, 3, 4, 5, 7, 8, 10 ], "focus": [ "swap", null, "swap", null, null, null, null, null ], "markers": [ { "name": "node", "index": 0, "tone": "pivot" } ], "regions": [ { "label": "heap", "from": 0, "to": 2, "tone": "window" }, { "label": "sorted", "from": 3, "to": 7, "tone": "settled" } ] }, { "i": 33, "op": "compare", "caption": "Parent 1 vs larger child 2.", "note": null, "codeLine": 5, "stats": null, "array": [ 1, 2, 3, 4, 5, 7, 8, 10 ], "focus": [ "compare", "compare", null, null, null, null, null, null ], "markers": [ { "name": "node", "index": 0, "tone": "pivot" } ], "regions": [ { "label": "heap", "from": 0, "to": 1, "tone": "window" }, { "label": "sorted", "from": 2, "to": 7, "tone": "settled" } ] }, { "i": 34, "op": "swap", "caption": "Parent is smaller — sink it down.", "note": null, "codeLine": 5, "stats": null, "array": [ 2, 1, 3, 4, 5, 7, 8, 10 ], "focus": [ "swap", "swap", null, null, null, null, null, null ], "markers": [ { "name": "node", "index": 0, "tone": "pivot" } ], "regions": [ { "label": "heap", "from": 0, "to": 1, "tone": "window" }, { "label": "sorted", "from": 2, "to": 7, "tone": "settled" } ] }, { "i": 35, "op": "swap", "caption": "Move the max 2 to index 1.", "note": null, "codeLine": 3, "stats": null, "array": [ 1, 2, 3, 4, 5, 7, 8, 10 ], "focus": [ "swap", "swap", null, null, null, null, null, null ], "markers": [ { "name": "node", "index": 0, "tone": "pivot" } ], "regions": [ { "label": "heap", "from": 0, "to": 1, "tone": "window" }, { "label": "sorted", "from": 2, "to": 7, "tone": "settled" } ] }, { "i": 36, "op": "settle", "caption": "Sorted.", "note": null, "codeLine": 3, "stats": null, "array": [ 1, 2, 3, 4, 5, 7, 8, 10 ], "focus": [ "settled", "settled", "settled", "settled", "settled", "settled", "settled", "settled" ], "markers": [], "regions": [ { "label": "sorted", "from": 0, "to": 7, "tone": "settled" } ] } ] }