{ "slug": "merge-sort", "title": "Merge Sort", "category": "Sorting", "difficulty": "medium", "complexity": { "time": "O(n log n)", "space": "O(n)" }, "idea": "Recursively split the array in half until each piece is one element, then merge sorted pieces back together. Each merge fills a separate buffer one value at a time — comparing the fronts of the two halves — then copies the buffer back, so no value is ever lost or duplicated.", "code": [ "sort(lo, hi): if lo>=hi return; mid=(lo+hi)/2", " sort(lo,mid); sort(mid+1,hi); merge(lo,mid,hi)", "merge: compare the fronts of the two halves", " write the smaller into buffer[k++]", " copy the buffer back into the array" ], "example": { "array": [ 6, 2, 8, 4, 3, 7, 1, 5 ] }, "invariant": "Each step is the real data structure at that step — the picture is a pure function of it.", "stepCount": 40, "steps": [ { "i": 0, "op": "lanes", "caption": "Merge sort: split down to single elements, then merge through a buffer.", "note": null, "codeLine": 1, "stats": null, "lanes": [ { "id": "arr", "label": "array", "array": [ 6, 2, 8, 4, 3, 7, 1, 5 ], "focus": [ null, null, null, null, null, null, null, null ], "markers": [], "regions": [] }, { "id": "buf", "label": "merge buffer", "array": [ 0, 0, 0, 0, 0, 0, 0, 0 ], "focus": [ null, null, null, null, null, null, null, null ], "placeholders": [ true, true, true, true, true, true, true, true ], "markers": [], "regions": [] } ] }, { "i": 1, "op": "lanes", "caption": "Merge sorted halves [0..0] and [1..1].", "note": null, "codeLine": 3, "stats": null, "lanes": [ { "id": "arr", "label": "array", "array": [ 6, 2, 8, 4, 3, 7, 1, 5 ], "focus": [ null, null, null, null, null, null, null, null ], "markers": [], "regions": [ { "label": "range", "from": 0, "to": 1, "tone": "window" } ] }, { "id": "buf", "label": "merge buffer", "array": [ 0, 0, 0, 0, 0, 0, 0, 0 ], "focus": [ null, null, null, null, null, null, null, null ], "placeholders": [ true, true, true, true, true, true, true, true ], "markers": [], "regions": [] } ] }, { "i": 2, "op": "lanes", "caption": "6 vs 2 → take 2 into buffer[0].", "note": null, "codeLine": 4, "stats": null, "lanes": [ { "id": "arr", "label": "array", "array": [ 6, 2, 8, 4, 3, 7, 1, 5 ], "focus": [ null, "read", null, null, null, null, null, null ], "markers": [ { "name": "i", "index": 0, "tone": "compare" }, { "name": "j", "index": 1, "tone": "compare" } ], "regions": [ { "label": "range", "from": 0, "to": 1, "tone": "window" } ] }, { "id": "buf", "label": "merge buffer", "array": [ 2, 0, 0, 0, 0, 0, 0, 0 ], "focus": [ "insert", null, null, null, null, null, null, null ], "placeholders": [ false, true, true, true, true, true, true, true ], "markers": [ { "name": "k", "index": 0, "tone": "pivot" } ], "regions": [] } ] }, { "i": 3, "op": "lanes", "caption": "Copy remaining 6 → buffer[1].", "note": null, "codeLine": 4, "stats": null, "lanes": [ { "id": "arr", "label": "array", "array": [ 6, 2, 8, 4, 3, 7, 1, 5 ], "focus": [ "read", null, null, null, null, null, null, null ], "markers": [ { "name": "i", "index": 0, "tone": "compare" }, { "name": "j", "index": 1, "tone": "compare" } ], "regions": [ { "label": "range", "from": 0, "to": 1, "tone": "window" } ] }, { "id": "buf", "label": "merge buffer", "array": [ 2, 6, 0, 0, 0, 0, 0, 0 ], "focus": [ null, "insert", null, null, null, null, null, null ], "placeholders": [ false, false, true, true, true, true, true, true ], "markers": [ { "name": "k", "index": 1, "tone": "pivot" } ], "regions": [] } ] }, { "i": 4, "op": "lanes", "caption": "Copy the merged run back into the array [0..1].", "note": null, "codeLine": 5, "stats": null, "lanes": [ { "id": "arr", "label": "array", "array": [ 2, 6, 8, 4, 3, 7, 1, 5 ], "focus": [ "write", "write", null, null, null, null, null, null ], "markers": [], "regions": [ { "label": "range", "from": 0, "to": 1, "tone": "window" } ] }, { "id": "buf", "label": "merge buffer", "array": [ 2, 6, 0, 0, 0, 0, 0, 0 ], "focus": [ null, null, null, null, null, null, null, null ], "placeholders": [ false, false, true, true, true, true, true, true ], "markers": [], "regions": [] } ] }, { "i": 5, "op": "lanes", "caption": "Merge sorted halves [2..2] and [3..3].", "note": null, "codeLine": 3, "stats": null, "lanes": [ { "id": "arr", "label": "array", "array": [ 2, 6, 8, 4, 3, 7, 1, 5 ], "focus": [ null, null, null, null, null, null, null, null ], "markers": [], "regions": [ { "label": "range", "from": 2, "to": 3, "tone": "window" } ] }, { "id": "buf", "label": "merge buffer", "array": [ 0, 0, 0, 0, 0, 0, 0, 0 ], "focus": [ null, null, null, null, null, null, null, null ], "placeholders": [ true, true, true, true, true, true, true, true ], "markers": [], "regions": [] } ] }, { "i": 6, "op": "lanes", "caption": "8 vs 4 → take 4 into buffer[2].", "note": null, "codeLine": 4, "stats": null, "lanes": [ { "id": "arr", "label": "array", "array": [ 2, 6, 8, 4, 3, 7, 1, 5 ], "focus": [ null, null, null, "read", null, null, null, null ], "markers": [ { "name": "i", "index": 2, "tone": "compare" }, { "name": "j", "index": 3, "tone": "compare" } ], "regions": [ { "label": "range", "from": 2, "to": 3, "tone": "window" } ] }, { "id": "buf", "label": "merge buffer", "array": [ 0, 0, 4, 0, 0, 0, 0, 0 ], "focus": [ null, null, "insert", null, null, null, null, null ], "placeholders": [ true, true, false, true, true, true, true, true ], "markers": [ { "name": "k", "index": 2, "tone": "pivot" } ], "regions": [] } ] }, { "i": 7, "op": "lanes", "caption": "Copy remaining 8 → buffer[3].", "note": null, "codeLine": 4, "stats": null, "lanes": [ { "id": "arr", "label": "array", "array": [ 2, 6, 8, 4, 3, 7, 1, 5 ], "focus": [ null, null, "read", null, null, null, null, null ], "markers": [ { "name": "i", "index": 2, "tone": "compare" }, { "name": "j", "index": 3, "tone": "compare" } ], "regions": [ { "label": "range", "from": 2, "to": 3, "tone": "window" } ] }, { "id": "buf", "label": "merge buffer", "array": [ 0, 0, 4, 8, 0, 0, 0, 0 ], "focus": [ null, null, null, "insert", null, null, null, null ], "placeholders": [ true, true, false, false, true, true, true, true ], "markers": [ { "name": "k", "index": 3, "tone": "pivot" } ], "regions": [] } ] }, { "i": 8, "op": "lanes", "caption": "Copy the merged run back into the array [2..3].", "note": null, "codeLine": 5, "stats": null, "lanes": [ { "id": "arr", "label": "array", "array": [ 2, 6, 4, 8, 3, 7, 1, 5 ], "focus": [ null, null, "write", "write", null, null, null, null ], "markers": [], "regions": [ { "label": "range", "from": 2, "to": 3, "tone": "window" } ] }, { "id": "buf", "label": "merge buffer", "array": [ 0, 0, 4, 8, 0, 0, 0, 0 ], "focus": [ null, null, null, null, null, null, null, null ], "placeholders": [ true, true, false, false, true, true, true, true ], "markers": [], "regions": [] } ] }, { "i": 9, "op": "lanes", "caption": "Merge sorted halves [0..1] and [2..3].", "note": null, "codeLine": 3, "stats": null, "lanes": [ { "id": "arr", "label": "array", "array": [ 2, 6, 4, 8, 3, 7, 1, 5 ], "focus": [ null, null, null, null, null, null, null, null ], "markers": [], "regions": [ { "label": "range", "from": 0, "to": 3, "tone": "window" } ] }, { "id": "buf", "label": "merge buffer", "array": [ 0, 0, 0, 0, 0, 0, 0, 0 ], "focus": [ null, null, null, null, null, null, null, null ], "placeholders": [ true, true, true, true, true, true, true, true ], "markers": [], "regions": [] } ] }, { "i": 10, "op": "lanes", "caption": "2 vs 4 → take 2 into buffer[0].", "note": null, "codeLine": 4, "stats": null, "lanes": [ { "id": "arr", "label": "array", "array": [ 2, 6, 4, 8, 3, 7, 1, 5 ], "focus": [ "read", null, null, null, null, null, null, null ], "markers": [ { "name": "i", "index": 0, "tone": "compare" }, { "name": "j", "index": 2, "tone": "compare" } ], "regions": [ { "label": "range", "from": 0, "to": 3, "tone": "window" } ] }, { "id": "buf", "label": "merge buffer", "array": [ 2, 0, 0, 0, 0, 0, 0, 0 ], "focus": [ "insert", null, null, null, null, null, null, null ], "placeholders": [ false, true, true, true, true, true, true, true ], "markers": [ { "name": "k", "index": 0, "tone": "pivot" } ], "regions": [] } ] }, { "i": 11, "op": "lanes", "caption": "6 vs 4 → take 4 into buffer[1].", "note": null, "codeLine": 4, "stats": null, "lanes": [ { "id": "arr", "label": "array", "array": [ 2, 6, 4, 8, 3, 7, 1, 5 ], "focus": [ null, null, "read", null, null, null, null, null ], "markers": [ { "name": "i", "index": 1, "tone": "compare" }, { "name": "j", "index": 2, "tone": "compare" } ], "regions": [ { "label": "range", "from": 0, "to": 3, "tone": "window" } ] }, { "id": "buf", "label": "merge buffer", "array": [ 2, 4, 0, 0, 0, 0, 0, 0 ], "focus": [ null, "insert", null, null, null, null, null, null ], "placeholders": [ false, false, true, true, true, true, true, true ], "markers": [ { "name": "k", "index": 1, "tone": "pivot" } ], "regions": [] } ] }, { "i": 12, "op": "lanes", "caption": "6 vs 8 → take 6 into buffer[2].", "note": null, "codeLine": 4, "stats": null, "lanes": [ { "id": "arr", "label": "array", "array": [ 2, 6, 4, 8, 3, 7, 1, 5 ], "focus": [ null, "read", null, null, null, null, null, null ], "markers": [ { "name": "i", "index": 1, "tone": "compare" }, { "name": "j", "index": 3, "tone": "compare" } ], "regions": [ { "label": "range", "from": 0, "to": 3, "tone": "window" } ] }, { "id": "buf", "label": "merge buffer", "array": [ 2, 4, 6, 0, 0, 0, 0, 0 ], "focus": [ null, null, "insert", null, null, null, null, null ], "placeholders": [ false, false, false, true, true, true, true, true ], "markers": [ { "name": "k", "index": 2, "tone": "pivot" } ], "regions": [] } ] }, { "i": 13, "op": "lanes", "caption": "Copy remaining 8 → buffer[3].", "note": null, "codeLine": 4, "stats": null, "lanes": [ { "id": "arr", "label": "array", "array": [ 2, 6, 4, 8, 3, 7, 1, 5 ], "focus": [ null, null, null, "read", null, null, null, null ], "markers": [ { "name": "i", "index": 1, "tone": "compare" }, { "name": "j", "index": 3, "tone": "compare" } ], "regions": [ { "label": "range", "from": 0, "to": 3, "tone": "window" } ] }, { "id": "buf", "label": "merge buffer", "array": [ 2, 4, 6, 8, 0, 0, 0, 0 ], "focus": [ null, null, null, "insert", null, null, null, null ], "placeholders": [ false, false, false, false, true, true, true, true ], "markers": [ { "name": "k", "index": 3, "tone": "pivot" } ], "regions": [] } ] }, { "i": 14, "op": "lanes", "caption": "Copy the merged run back into the array [0..3].", "note": null, "codeLine": 5, "stats": null, "lanes": [ { "id": "arr", "label": "array", "array": [ 2, 4, 6, 8, 3, 7, 1, 5 ], "focus": [ "write", "write", "write", "write", null, null, null, null ], "markers": [], "regions": [ { "label": "range", "from": 0, "to": 3, "tone": "window" } ] }, { "id": "buf", "label": "merge buffer", "array": [ 2, 4, 6, 8, 0, 0, 0, 0 ], "focus": [ null, null, null, null, null, null, null, null ], "placeholders": [ false, false, false, false, true, true, true, true ], "markers": [], "regions": [] } ] }, { "i": 15, "op": "lanes", "caption": "Merge sorted halves [4..4] and [5..5].", "note": null, "codeLine": 3, "stats": null, "lanes": [ { "id": "arr", "label": "array", "array": [ 2, 4, 6, 8, 3, 7, 1, 5 ], "focus": [ null, null, null, null, null, null, null, null ], "markers": [], "regions": [ { "label": "range", "from": 4, "to": 5, "tone": "window" } ] }, { "id": "buf", "label": "merge buffer", "array": [ 0, 0, 0, 0, 0, 0, 0, 0 ], "focus": [ null, null, null, null, null, null, null, null ], "placeholders": [ true, true, true, true, true, true, true, true ], "markers": [], "regions": [] } ] }, { "i": 16, "op": "lanes", "caption": "3 vs 7 → take 3 into buffer[4].", "note": null, "codeLine": 4, "stats": null, "lanes": [ { "id": "arr", "label": "array", "array": [ 2, 4, 6, 8, 3, 7, 1, 5 ], "focus": [ null, null, null, null, "read", null, null, null ], "markers": [ { "name": "i", "index": 4, "tone": "compare" }, { "name": "j", "index": 5, "tone": "compare" } ], "regions": [ { "label": "range", "from": 4, "to": 5, "tone": "window" } ] }, { "id": "buf", "label": "merge buffer", "array": [ 0, 0, 0, 0, 3, 0, 0, 0 ], "focus": [ null, null, null, null, "insert", null, null, null ], "placeholders": [ true, true, true, true, false, true, true, true ], "markers": [ { "name": "k", "index": 4, "tone": "pivot" } ], "regions": [] } ] }, { "i": 17, "op": "lanes", "caption": "Copy remaining 7 → buffer[5].", "note": null, "codeLine": 4, "stats": null, "lanes": [ { "id": "arr", "label": "array", "array": [ 2, 4, 6, 8, 3, 7, 1, 5 ], "focus": [ null, null, null, null, null, "read", null, null ], "markers": [ { "name": "i", "index": 4, "tone": "compare" }, { "name": "j", "index": 5, "tone": "compare" } ], "regions": [ { "label": "range", "from": 4, "to": 5, "tone": "window" } ] }, { "id": "buf", "label": "merge buffer", "array": [ 0, 0, 0, 0, 3, 7, 0, 0 ], "focus": [ null, null, null, null, null, "insert", null, null ], "placeholders": [ true, true, true, true, false, false, true, true ], "markers": [ { "name": "k", "index": 5, "tone": "pivot" } ], "regions": [] } ] }, { "i": 18, "op": "lanes", "caption": "Copy the merged run back into the array [4..5].", "note": null, "codeLine": 5, "stats": null, "lanes": [ { "id": "arr", "label": "array", "array": [ 2, 4, 6, 8, 3, 7, 1, 5 ], "focus": [ null, null, null, null, "write", "write", null, null ], "markers": [], "regions": [ { "label": "range", "from": 4, "to": 5, "tone": "window" } ] }, { "id": "buf", "label": "merge buffer", "array": [ 0, 0, 0, 0, 3, 7, 0, 0 ], "focus": [ null, null, null, null, null, null, null, null ], "placeholders": [ true, true, true, true, false, false, true, true ], "markers": [], "regions": [] } ] }, { "i": 19, "op": "lanes", "caption": "Merge sorted halves [6..6] and [7..7].", "note": null, "codeLine": 3, "stats": null, "lanes": [ { "id": "arr", "label": "array", "array": [ 2, 4, 6, 8, 3, 7, 1, 5 ], "focus": [ null, null, null, null, null, null, null, null ], "markers": [], "regions": [ { "label": "range", "from": 6, "to": 7, "tone": "window" } ] }, { "id": "buf", "label": "merge buffer", "array": [ 0, 0, 0, 0, 0, 0, 0, 0 ], "focus": [ null, null, null, null, null, null, null, null ], "placeholders": [ true, true, true, true, true, true, true, true ], "markers": [], "regions": [] } ] }, { "i": 20, "op": "lanes", "caption": "1 vs 5 → take 1 into buffer[6].", "note": null, "codeLine": 4, "stats": null, "lanes": [ { "id": "arr", "label": "array", "array": [ 2, 4, 6, 8, 3, 7, 1, 5 ], "focus": [ null, null, null, null, null, null, "read", null ], "markers": [ { "name": "i", "index": 6, "tone": "compare" }, { "name": "j", "index": 7, "tone": "compare" } ], "regions": [ { "label": "range", "from": 6, "to": 7, "tone": "window" } ] }, { "id": "buf", "label": "merge buffer", "array": [ 0, 0, 0, 0, 0, 0, 1, 0 ], "focus": [ null, null, null, null, null, null, "insert", null ], "placeholders": [ true, true, true, true, true, true, false, true ], "markers": [ { "name": "k", "index": 6, "tone": "pivot" } ], "regions": [] } ] }, { "i": 21, "op": "lanes", "caption": "Copy remaining 5 → buffer[7].", "note": null, "codeLine": 4, "stats": null, "lanes": [ { "id": "arr", "label": "array", "array": [ 2, 4, 6, 8, 3, 7, 1, 5 ], "focus": [ null, null, null, null, null, null, null, "read" ], "markers": [ { "name": "i", "index": 6, "tone": "compare" }, { "name": "j", "index": 7, "tone": "compare" } ], "regions": [ { "label": "range", "from": 6, "to": 7, "tone": "window" } ] }, { "id": "buf", "label": "merge buffer", "array": [ 0, 0, 0, 0, 0, 0, 1, 5 ], "focus": [ null, null, null, null, null, null, null, "insert" ], "placeholders": [ true, true, true, true, true, true, false, false ], "markers": [ { "name": "k", "index": 7, "tone": "pivot" } ], "regions": [] } ] }, { "i": 22, "op": "lanes", "caption": "Copy the merged run back into the array [6..7].", "note": null, "codeLine": 5, "stats": null, "lanes": [ { "id": "arr", "label": "array", "array": [ 2, 4, 6, 8, 3, 7, 1, 5 ], "focus": [ null, null, null, null, null, null, "write", "write" ], "markers": [], "regions": [ { "label": "range", "from": 6, "to": 7, "tone": "window" } ] }, { "id": "buf", "label": "merge buffer", "array": [ 0, 0, 0, 0, 0, 0, 1, 5 ], "focus": [ null, null, null, null, null, null, null, null ], "placeholders": [ true, true, true, true, true, true, false, false ], "markers": [], "regions": [] } ] }, { "i": 23, "op": "lanes", "caption": "Merge sorted halves [4..5] and [6..7].", "note": null, "codeLine": 3, "stats": null, "lanes": [ { "id": "arr", "label": "array", "array": [ 2, 4, 6, 8, 3, 7, 1, 5 ], "focus": [ null, null, null, null, null, null, null, null ], "markers": [], "regions": [ { "label": "range", "from": 4, "to": 7, "tone": "window" } ] }, { "id": "buf", "label": "merge buffer", "array": [ 0, 0, 0, 0, 0, 0, 0, 0 ], "focus": [ null, null, null, null, null, null, null, null ], "placeholders": [ true, true, true, true, true, true, true, true ], "markers": [], "regions": [] } ] }, { "i": 24, "op": "lanes", "caption": "3 vs 1 → take 1 into buffer[4].", "note": null, "codeLine": 4, "stats": null, "lanes": [ { "id": "arr", "label": "array", "array": [ 2, 4, 6, 8, 3, 7, 1, 5 ], "focus": [ null, null, null, null, null, null, "read", null ], "markers": [ { "name": "i", "index": 4, "tone": "compare" }, { "name": "j", "index": 6, "tone": "compare" } ], "regions": [ { "label": "range", "from": 4, "to": 7, "tone": "window" } ] }, { "id": "buf", "label": "merge buffer", "array": [ 0, 0, 0, 0, 1, 0, 0, 0 ], "focus": [ null, null, null, null, "insert", null, null, null ], "placeholders": [ true, true, true, true, false, true, true, true ], "markers": [ { "name": "k", "index": 4, "tone": "pivot" } ], "regions": [] } ] }, { "i": 25, "op": "lanes", "caption": "3 vs 5 → take 3 into buffer[5].", "note": null, "codeLine": 4, "stats": null, "lanes": [ { "id": "arr", "label": "array", "array": [ 2, 4, 6, 8, 3, 7, 1, 5 ], "focus": [ null, null, null, null, "read", null, null, null ], "markers": [ { "name": "i", "index": 4, "tone": "compare" }, { "name": "j", "index": 7, "tone": "compare" } ], "regions": [ { "label": "range", "from": 4, "to": 7, "tone": "window" } ] }, { "id": "buf", "label": "merge buffer", "array": [ 0, 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