{ "slug": "counting-sort", "title": "Counting Sort", "category": "Sorting", "difficulty": "medium", "complexity": { "time": "O(n + k)", "space": "O(n + k)" }, "idea": "For small non-negative integers, count how many of each value there are, turn the counts into end positions with a running sum, then place each input value where it belongs. Stable and comparison-free.", "code": [ "for each value: counts[value]++", "counts[v] += counts[v-1] # end positions", "for i = n-1 down to 0:", " output[--counts[a[i]]] = a[i] # stable", "output is sorted" ], "example": { "array": [ 4, 2, 2, 8, 3, 3, 1 ] }, "invariant": "Each step is the real data structure at that step — the picture is a pure function of it.", "stepCount": 17, "steps": [ { "i": 0, "op": "lanes", "caption": "Counting sort: tally every value, then place them all in order.", "note": null, "codeLine": 1, "stats": null, "lanes": [ { "id": "in", "label": "input", "array": [ 4, 2, 2, 8, 3, 3, 1 ], "focus": [ null, null, null, null, null, null, null ], "markers": [], "regions": [] }, { "id": "count", "label": "counts (index = value)", "array": [ 0, 0, 0, 0, 0, 0, 0, 0, 0 ], "focus": [ null, null, null, null, null, null, null, null, null ], "markers": [], "regions": [] }, { "id": "out", "label": "output", "array": [ 0, 0, 0, 0, 0, 0, 0 ], "focus": [ null, null, null, null, null, null, null ], "placeholders": [ true, true, true, true, true, true, true ], "markers": [], "regions": [] } ] }, { "i": 1, "op": "lanes", "caption": "input[0] = 4 → counts[4] = 1.", "note": null, "codeLine": 1, "stats": null, "lanes": [ { "id": "in", "label": "input", "array": [ 4, 2, 2, 8, 3, 3, 1 ], "focus": [ "read", null, null, null, null, null, null ], "markers": [], "regions": [] }, { "id": "count", "label": "counts (index = value)", "array": [ 0, 0, 0, 0, 1, 0, 0, 0, 0 ], "focus": [ null, null, null, null, "write", null, null, null, null ], "markers": [], "regions": [] }, { "id": "out", "label": "output", "array": [ 0, 0, 0, 0, 0, 0, 0 ], "focus": [ null, null, null, null, null, null, null ], "placeholders": [ true, true, true, true, true, true, true ], "markers": [], "regions": [] } ] }, { "i": 2, "op": "lanes", "caption": "input[1] = 2 → counts[2] = 1.", "note": null, "codeLine": 1, "stats": null, "lanes": [ { "id": "in", "label": "input", "array": [ 4, 2, 2, 8, 3, 3, 1 ], "focus": [ null, "read", null, null, null, null, null ], "markers": [], "regions": [] }, { "id": "count", "label": "counts (index = value)", "array": [ 0, 0, 1, 0, 1, 0, 0, 0, 0 ], "focus": [ null, null, "write", null, null, null, null, null, null ], "markers": [], "regions": [] }, { "id": "out", "label": "output", "array": [ 0, 0, 0, 0, 0, 0, 0 ], "focus": [ null, null, null, null, null, null, null ], "placeholders": [ true, true, true, true, true, true, true ], "markers": [], "regions": [] } ] }, { "i": 3, "op": "lanes", "caption": "input[2] = 2 → counts[2] = 2.", "note": null, "codeLine": 1, "stats": null, "lanes": [ { "id": "in", "label": "input", "array": [ 4, 2, 2, 8, 3, 3, 1 ], "focus": [ null, null, "read", null, null, null, null ], "markers": [], "regions": [] }, { "id": "count", "label": "counts (index = value)", "array": [ 0, 0, 2, 0, 1, 0, 0, 0, 0 ], "focus": [ null, null, "write", null, null, null, null, null, null ], "markers": [], "regions": [] }, { "id": "out", "label": "output", "array": [ 0, 0, 0, 0, 0, 0, 0 ], "focus": [ null, null, null, null, null, null, null ], "placeholders": [ true, true, true, true, true, true, true ], "markers": [], "regions": [] } ] }, { "i": 4, "op": "lanes", "caption": "input[3] = 8 → counts[8] = 1.", "note": null, "codeLine": 1, "stats": null, "lanes": [ { "id": "in", "label": "input", "array": [ 4, 2, 2, 8, 3, 3, 1 ], "focus": [ null, null, null, "read", null, null, null ], "markers": [], "regions": [] }, { "id": "count", "label": "counts (index = value)", "array": [ 0, 0, 2, 0, 1, 0, 0, 0, 1 ], "focus": [ null, null, null, null, null, null, null, null, "write" ], "markers": [], "regions": [] }, { "id": "out", "label": "output", "array": [ 0, 0, 0, 0, 0, 0, 0 ], "focus": [ null, null, null, null, null, null, null ], "placeholders": [ true, true, true, true, true, true, true ], "markers": [], "regions": [] } ] }, { "i": 5, "op": "lanes", "caption": "input[4] = 3 → counts[3] = 1.", "note": null, "codeLine": 1, "stats": null, "lanes": [ { "id": "in", "label": "input", "array": [ 4, 2, 2, 8, 3, 3, 1 ], "focus": [ null, null, null, null, "read", null, null ], "markers": [], "regions": [] }, { "id": "count", "label": "counts (index = value)", "array": [ 0, 0, 2, 1, 1, 0, 0, 0, 1 ], "focus": [ null, null, null, "write", null, null, null, null, null ], "markers": [], "regions": [] }, { "id": "out", "label": "output", "array": [ 0, 0, 0, 0, 0, 0, 0 ], "focus": [ null, null, null, null, null, null, null ], "placeholders": [ true, true, true, true, true, true, true ], "markers": [], "regions": [] } ] }, { "i": 6, "op": "lanes", "caption": "input[5] = 3 → counts[3] = 2.", "note": null, "codeLine": 1, "stats": null, "lanes": [ { "id": "in", "label": "input", "array": [ 4, 2, 2, 8, 3, 3, 1 ], "focus": [ null, null, null, null, null, "read", null ], "markers": [], "regions": [] }, { "id": "count", "label": "counts (index = value)", "array": [ 0, 0, 2, 2, 1, 0, 0, 0, 1 ], "focus": [ null, null, null, "write", null, null, null, null, null ], "markers": [], "regions": [] }, { "id": "out", "label": "output", "array": [ 0, 0, 0, 0, 0, 0, 0 ], "focus": [ null, null, null, null, null, null, null ], "placeholders": [ true, true, true, true, true, true, true ], "markers": [], "regions": [] } ] }, { "i": 7, "op": "lanes", "caption": "input[6] = 1 → counts[1] = 1.", "note": null, "codeLine": 1, "stats": null, "lanes": [ { "id": "in", "label": "input", "array": [ 4, 2, 2, 8, 3, 3, 1 ], "focus": [ null, null, null, null, null, null, "read" ], "markers": [], "regions": [] }, { "id": "count", "label": "counts (index = value)", "array": [ 0, 1, 2, 2, 1, 0, 0, 0, 1 ], "focus": [ null, "write", null, null, null, null, null, null, null ], "markers": [], "regions": [] }, { "id": "out", "label": "output", "array": [ 0, 0, 0, 0, 0, 0, 0 ], "focus": [ null, null, null, null, null, null, null ], "placeholders": [ true, true, true, true, true, true, true ], "markers": [], "regions": [] } ] }, { "i": 8, "op": "lanes", "caption": "Running-sum the counts, so each value knows its end position.", "note": null, "codeLine": 2, "stats": null, "lanes": [ { "id": "in", "label": "input", "array": [ 4, 2, 2, 8, 3, 3, 1 ], "focus": [ null, null, null, null, null, null, null ], "markers": [], "regions": [] }, { "id": "count", "label": "counts (index = value)", "array": [ 0, 1, 3, 5, 6, 6, 6, 6, 7 ], "focus": [ "read", "read", "read", "read", "read", "read", "read", "read", "read" ], "markers": [], "regions": [] }, { "id": "out", "label": "output", "array": [ 0, 0, 0, 0, 0, 0, 0 ], "focus": [ null, null, null, null, null, null, null ], "placeholders": [ true, true, true, true, true, true, true ], "markers": [], "regions": [] } ] }, { "i": 9, "op": "lanes", "caption": "Place 1 at output[0] (right to left keeps it stable).", "note": null, "codeLine": 4, "stats": null, "lanes": [ { "id": "in", "label": "input", "array": [ 4, 2, 2, 8, 3, 3, 1 ], "focus": [ null, null, null, null, null, null, "read" ], "markers": [], "regions": [] }, { "id": "count", "label": "counts (index = value)", "array": [ 0, 0, 3, 5, 6, 6, 6, 6, 7 ], "focus": [ null, "write", null, null, null, null, null, null, null ], "markers": [], "regions": [] }, { "id": "out", "label": "output", "array": [ 1, 0, 0, 0, 0, 0, 0 ], "focus": [ "insert", null, null, null, null, null, null ], "placeholders": [ false, true, true, true, true, true, true ], "markers": [], "regions": [] } ] }, { "i": 10, "op": "lanes", "caption": "Place 3 at output[4] (right to left keeps it stable).", "note": null, "codeLine": 4, "stats": null, "lanes": [ { "id": "in", "label": "input", "array": [ 4, 2, 2, 8, 3, 3, 1 ], "focus": [ null, null, null, null, null, "read", null ], "markers": [], "regions": [] }, { "id": "count", "label": "counts (index = value)", "array": [ 0, 0, 3, 4, 6, 6, 6, 6, 7 ], "focus": [ null, null, null, "write", null, null, null, null, null ], "markers": [], "regions": [] }, { "id": "out", "label": "output", "array": [ 1, 0, 0, 0, 3, 0, 0 ], "focus": [ null, null, null, null, "insert", null, null ], "placeholders": [ false, true, true, true, false, true, true ], "markers": [], "regions": [] } ] }, { "i": 11, "op": "lanes", "caption": "Place 3 at output[3] (right to left keeps it stable).", "note": null, "codeLine": 4, "stats": null, "lanes": [ { "id": "in", "label": "input", "array": [ 4, 2, 2, 8, 3, 3, 1 ], "focus": [ null, null, null, null, "read", null, null ], "markers": [], "regions": [] }, { "id": "count", "label": "counts (index = value)", "array": [ 0, 0, 3, 3, 6, 6, 6, 6, 7 ], "focus": [ null, null, null, "write", null, null, null, null, null ], "markers": [], "regions": [] }, { "id": "out", "label": "output", "array": [ 1, 0, 0, 3, 3, 0, 0 ], "focus": [ null, null, null, "insert", null, null, null ], "placeholders": [ false, true, true, false, false, true, true ], "markers": [], "regions": [] } ] }, { "i": 12, "op": "lanes", "caption": "Place 8 at output[6] (right to left keeps it stable).", "note": null, "codeLine": 4, "stats": null, "lanes": [ { "id": "in", "label": "input", "array": [ 4, 2, 2, 8, 3, 3, 1 ], "focus": [ null, null, null, "read", null, null, null ], "markers": [], "regions": [] }, { "id": "count", "label": "counts (index = value)", "array": [ 0, 0, 3, 3, 6, 6, 6, 6, 6 ], "focus": [ null, null, null, null, null, null, null, null, "write" ], "markers": [], "regions": [] }, { "id": "out", "label": "output", "array": [ 1, 0, 0, 3, 3, 0, 8 ], "focus": [ null, null, null, null, null, null, "insert" ], "placeholders": [ false, true, true, false, false, true, false ], "markers": [], "regions": [] } ] }, { "i": 13, "op": "lanes", "caption": "Place 2 at output[2] (right to left keeps it stable).", "note": null, "codeLine": 4, "stats": null, "lanes": [ { "id": "in", "label": "input", "array": [ 4, 2, 2, 8, 3, 3, 1 ], "focus": [ null, null, "read", null, null, null, null ], "markers": [], "regions": [] }, { "id": "count", "label": "counts (index = value)", "array": [ 0, 0, 2, 3, 6, 6, 6, 6, 6 ], "focus": [ null, null, "write", null, null, null, null, null, null ], "markers": [], "regions": [] }, { "id": "out", "label": "output", "array": [ 1, 0, 2, 3, 3, 0, 8 ], "focus": [ null, null, "insert", null, null, null, null ], "placeholders": [ false, true, false, false, false, true, false ], "markers": [], "regions": [] } ] }, { "i": 14, "op": "lanes", "caption": "Place 2 at output[1] (right to left keeps it stable).", "note": null, "codeLine": 4, "stats": null, "lanes": [ { "id": "in", "label": "input", "array": [ 4, 2, 2, 8, 3, 3, 1 ], "focus": [ null, "read", null, null, null, null, null ], "markers": [], "regions": [] }, { "id": "count", "label": "counts (index = value)", "array": [ 0, 0, 1, 3, 6, 6, 6, 6, 6 ], "focus": [ null, null, "write", null, null, null, null, null, null ], "markers": [], "regions": [] }, { "id": "out", "label": "output", "array": [ 1, 2, 2, 3, 3, 0, 8 ], "focus": [ null, "insert", null, null, null, null, null ], "placeholders": [ false, false, false, false, false, true, false ], "markers": [], "regions": [] } ] }, { "i": 15, "op": "lanes", "caption": "Place 4 at output[5] (right to left keeps it stable).", "note": null, "codeLine": 4, "stats": null, "lanes": [ { "id": "in", "label": "input", "array": [ 4, 2, 2, 8, 3, 3, 1 ], "focus": [ "read", null, null, null, null, null, null ], "markers": [], "regions": [] }, { "id": "count", "label": "counts (index = value)", "array": [ 0, 0, 1, 3, 5, 6, 6, 6, 6 ], "focus": [ null, null, null, null, "write", null, null, null, null ], "markers": [], "regions": [] }, { "id": "out", "label": "output", "array": [ 1, 2, 2, 3, 3, 4, 8 ], "focus": [ null, null, null, null, null, "insert", null ], "markers": [], "regions": [] } ] }, { "i": 16, "op": "lanes", "caption": "Sorted — the output holds every value in order.", "note": null, "codeLine": 5, "stats": null, "lanes": [ { "id": "in", "label": "input", "array": [ 4, 2, 2, 8, 3, 3, 1 ], "focus": [ null, null, null, null, null, null, null ], "markers": [], "regions": [] }, { "id": "count", "label": "counts (index = value)", "array": [ 0, 0, 1, 3, 5, 6, 6, 6, 6 ], "focus": [ null, null, null, null, null, null, null, null, null ], "markers": [], "regions": [] }, { "id": "out", "label": "output", "array": [ 1, 2, 2, 3, 3, 4, 8 ], "focus": [ "settled", "settled", "settled", "settled", "settled", "settled", "settled" ], "markers": [], "regions": [ { "label": "done", "from": 0, "to": 6, "tone": "settled" } ] } ] } ] }